Citation
Comparing Poisson, Hurdle, and Zip Model Fit under Varying Degrees of Skew and Zero-Inflation

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Title:
Comparing Poisson, Hurdle, and Zip Model Fit under Varying Degrees of Skew and Zero-Inflation
Creator:
MILLER, JEFFREY MONROE ( Author, Primary )
Copyright Date:
2008

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Subjects / Keywords:
Arithmetic mean ( jstor )
Binomials ( jstor )
Gaussian distributions ( jstor )
Modeling ( jstor )
Parametric models ( jstor )
Proportions ( jstor )
Skewed distribution ( jstor )
Standard deviation ( jstor )
Statistical models ( jstor )
Statistics ( jstor )

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Source Institution:
University of Florida
Holding Location:
University of Florida
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Copyright Jeffrey Monroe Miller. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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7/12/2007
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660162326 ( OCLC )

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Full Text





COMPARING POISSON, HURDLE, AND ZIP MODEL FIT
UNDER VARYING DEGREES OF SKEW AND ZERO-INFLATION























By

JEFFREY MONROE MILLER


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007
































Copyright 2007

by

Jeffrey Monroe Miller



































To the memory of my grandfather, Rev. Harold E. Cato.









ACKNOWLEDGMENTS

Several people helped make this study possible. I would like to thank my stepfather and

mother, Dr. Daniel and Gail Jacobs as well as my father and stepmother, Jerry and Darnelle

Miller for their many years of encouragement. I would also like to thank my supervisory

committee chair, M. David Miller, for his unyielding guidance, patience, and support. I thank Dr.

Jon Morris for the numerous training experiences. Many professors are appreciated for providing

the educational foundations for the dissertation topic including Dr. James Algina and Dr. Alan

Agresti.

The idea to research zero-inflation was inspired by experiences with data while

consulting on proj ects. To this extent, I thank those clients Dr. Courtney Zmach and Dr. Lori

Burkhead. Undergraduate faculty that I would like to acknowledge for their inspiration and

direction include Blaine Peden, Patricia Quinn, and Lee Anna Rasar. Several friends have been a

source of encouragement including Matt Grezik and Rachael Wilkerson. Finally, I thank those

who made it financially possible to complete this dissertation including my consulting clients,

the University of Florida College of Education, the Lastinger Center, and Adsam, LLC.












TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ............ ...... .__. ...............8....


LIST OF FIGURES ........._.___..... .__. ...............11....


AB S TRAC T ............._. .......... ..............._ 12...


CHAPTER


1 INTRODUCTION ................. ...............14.......... ......


Statement of the Problem ................. ...............15................
Rationale for the Study ................... .......... ...............15......
Purpose and Significance of the Study .............. ...............16....
Research Questions............... ...............1


2 REVIEW OF THE LITERATURE ................. ...............18.......... ....


Zero-Inflated Count Data ................. ...............18........... ....
Count Data ................. ...............18.................
Zero-Inflation .................... ...............19.
The Sources ofZero-Inflation .............. ...............20....

Impact of Zero-Inflation on Analyses .............. ...............21....
Simple Solutions to Zero-Inflation ................. ...............22................
Deleting zeros............... ...............22.
Assuming normality .............. ...............22....
Transforming Zeros ................. ...............23.................
Generalized Linear Models .............. .. ...... .. ..............2
The Binomial Distribution and the Logit Link ................. ...............28..............
Evaluating M odel Fit .................. ... .......... .. ................3 1....
The Poisson Distribution and the Log Link .............. ...............35....
Iterative Estimation .............. ...............38....
Interpretation of Coefficients .............. ...............38....
Hypothesis testing .............. ...............39....
O verdispersion ............................ .... ....................3
Poisson and Negative Binomial Models with Zero-Inflation............... ..............4
The Hurdle model .................. ......__ ...............45...
The Negative Binomial Hurdle model .............. ...............48....
The Zero-Inflated Poisson (ZIP) model .............. ..... ...............49..
The Negative Binomial Zero-Inflated Poisson model............_ .. ......._ ........51
Model Comparison Testing for Zero-Inflated Data....................... ..............5
Review of Research Pertaining to and Using Zero-Inflated Count Data. .............. ..... ........._.53
Hurdle M odel ............ ..... .._ ...............53...












Statistical .............. ...............53....

Applications .................. ...............54..
Zero-Inflated Poisson Model ................. ...............55..._._._......
Statistical .............. ...............55....

Applications .............. ...............59..
ZIP and Hurdle Model-Comparisons .............. ...............63....
Statistical .............. ...............64....

Applications .............. ...............64....
Discrepant Findings ....___ ................ .........__..........6


3 METHODOLOGY .............. ...............73....


Research Questions............... ...............7
Monte Carlo Study Design .............. ...............73....
Monte Carlo Sampling............... ...............75
P seudo-P opul ati on ................... ...............75.......... ....
The Prespecified Zero Proportions ........._.. ...._._..... ...............76...
Pre-Specified Skew .............. ...............76....
Random Number Generation............... ...............7

Sam ple Size .............. ...............78....
Simulation Size............... ...............78..
Iteration Size............... ...............79..
Distribution Generation ........._..... ...._... ...............80.....
Monte Carlo Model s.........._...._ ......_.. ...............80.....
Monte Carlo Analysis Procedures .............. ...............82....
Analysis Design .........._..._.. ...............84........ ......

4 RE SULT S .............. ...............94....


Pseudo-Population Results .................. ...............94..
Pseudo-Population Poisson Models .............. ...............95....
Pseudo-Population Hurdle Models ..........._.._.. ...............96...._._. .....
Hurdle vs. Negative Binomial Hurdle ..........._.... ....._._ ....._.._.........9
Poisson vs. Hurdle............... ...... .. .. .. ............9

Negative Binomial Poisson vs. Negative Binomial Hurdle .............. ...................98
Pseudo-Population ZIP Models............... ...............98.
Comparing AIC's For All Models .........._...._. ...............99......... ..
Monte Carlo Simulation Results .........._...._. ...............101.._.... .....
Positively Skewed Distribution .........._...._ ......_. ...._... ............10
Normal Distribution. ..........._.._. ...............111....... ......

Negatively Skewed Distribution.................. ............12
Review of Positively Skewed Distribution Findings ..........._.._.. ..........._.. ...._.._ ...133
Review of Normal Distribution Findings .............. ...............135....
Review of Negatively Skewed Distribution Findings .............. ...............136....


5 DI SCUS SSION ........._..... ...._... ..............._ 176..












The Impact of the Event Stage Distribution ................. ......... ......... ...........7
Positively Skewed Event-Stage Distributions ................. ...............................176
Normal Event-Stage Distributions .............. ...............181....
Negatively Skewed Event-Stage Distributions .............. ...............183....
Summary of Findings .............. ...............185....
Limitations ................. ...............186................
Discrete Conditions .............. ...............186....
Convergence and Optimization ................. ......... ...............186 ....
Under di spersi on ................. ...............187................
Other m odels .............. .. ............... ...........18
Validity of Model-Fitting and Model-Comparisons ................. ..............................188
Suggestions for Future Research .............. ...............190....
Application in Educational Research............... ...............19
Maj or Contribution of Findings ........._.___..... .___ ...............191...

LIST OF REFERENCES ........._.___..... .___ ...............195....


BIOGRAPHICAL SKETCH .............. ...............201....










LIST OF TABLES


Table page

2-1 Five pairs of nested models valid for statistical comparison ................ ......................71

2-2 Summary of literature on zero-inflation ..........._ ..... ..__ ....___ ............._..72

3-1 Proportions of counts as a function of zeros and skew ....._____ ... .....___ ..............87

3-2 Frequencies of counts as a function of zeros and skew .......... ................ ...............87

3-3 Descriptive statistics for each distribution............... ..............8

3-4 Poisson model: pseudo-population parameters ................. ...............88...............

3-5 Negative Binomial Poisson model: pseudo-population parameters ................. ...............89

3-6 Hurdle model (zeros): pseudo-population parameters ................. ......... ................89

3-7 Hurdle model (events): pseudo-population parameters ......... ................ ...............90

3-8 Negative Binomial Hurdle model (zeros): pseudo-population parameters ................... .....90

3-9 Negative Binomial Hurdle model (events): pseudo-population parameters ................... ...91

3-10 ZIP model (zeros): pseudo-population parameters ................. ............... ......... ...91

3-11 ZIP Model (events): pseudo-population parameters ................. ................ ......... .92

3-12 Negative Binomial ZIP model (zeros): pseudo-population parameters.................... .........92

3-13 Negative Binomial ZIP model (events): pseudo-population parameters ................... ........93

4-1 Deviance statistics comparing Poisson and negative binomial Poisson models. ............138

4-2 Deviance statistics comparing Hurdle and negative binomial Hurdle models ................138

4-3 Deviance statistics comparing Poisson and Hurdle models............... ................13

4-4 Deviance statistics comparing NB Poisson and NB Hurdle models................ ...............138

4-5 Deviance statistics comparing ZIP and negative binomial ZIP models ..........................139

4-6 Log-likelihood comparisons for positively skewed distribution with .10 zeros ..............139

4-7 AIC's for positively skewed distribution models with a .10 proportion of zeros............ 140

4-8 Log-likelihood comparisons for positively skewed distribution with .25 zeros .............140










4-9 AIC's for positively skewed distribution models with a .25 proportion of zeros............ 141

4-10 Log-likelihood comparisons for positively skewed distribution with .50 zeros .............141

4-11 AIC's for positively skewed distribution models with a .50 proportion of zeros............1 42

4-12 Log-likelihood comparisons for positively skewed distribution with .75 zeros..........._...142

4-13 AIC's for positively skewed distribution models with a .75 proportion of zeros............ 143

4-14 Log-likelihood comparisons for positively skewed distribution with .90 zeros ..............143

4-15 AIC's for positively skewed distribution models with a .90 proportion of zeros............1 44

4-16 Log-likelihood comparisons for normal distribution with .10 zeros .............. ..... ........._.144

4-17 AIC's for normal distribution models with a .10 proportion of zeros ................... ..........145

4-18 Log-likelihood comparisons for normal distribution with .25 zeros .............. ................145

4-19 AIC's for normal distribution models with a .25 proportion of zeros ................... ..........146

4-20 Log-likelihood comparisons for normal distribution with .50 zeros .............. ................147

4-21 AIC's for normal distribution models with a .50 proportion of zeros ................... ..........147

4-22 Log-likelihood comparisons for normal distribution with .75 zeros .............. ................148

4-23 AIC's for normal distribution models with a .75 proportion of zeros ................... ..........148

4-24 Log-likelihood comparisons for normal distribution with .90 zeros .............. ................149

4-25 AIC's for normal distribution models with a .90 proportion of zeros ................... ..........149

4-26 Log-likelihood comparisons for negatively skewed distribution with .10 zeros.............1 50

4-27 AIC's for negatively skewed models with a .10 proportion of zeros .............. .... ...........150

4-28 Log-likelihood comparisons for negatively skewed distribution with .25 zeros.............1 51

4-29 AIC's for negatively skewed models with a .25 proportion of zeros .............. .... ...........151

4-30 Log-likelihood comparisons for negatively skewed distribution with .50 zeros.............1 52

4-3 1 AIC's for negatively skewed models with a .50 proportion of zeros .............. .... ...........152

4-32 Log-likelihood comparisons for negatively skewed distribution with .75 zeros.............1 53

4-33 AIC's for negatively skewed models with a .75 proportion of zeros .............. .... ...........153










4-34 Log-likelihood comparisons for negatively skewed distribution with .90 zeros.............1 54

4-3 5 AIC's for negatively skewed models with a .90 proportion of zeros .............. .... ...........154

4-36 Positively skewed distribution: percentage of simulations favoring complex model......155

4-37 AIC's: positively skewed distribution (all conditions) ................ .........................155

4-38 Normal distribution: percentage of simulations favoring complex model. .....................155

4-39 AIC's: normal distribution (all conditions)............... ..............15

4-40 Negatively skewed distribution: percentage of simulations favoring complex model....156

4-41 AIC's: negatively skewed distribution (all conditions) .............. ....................15

4-42 Convergence frequencies: positively skewed distribution ................. ............ .........156

4-43 Convergence frequencies: normal distribution ................ ...............156........... ...

4-44 Convergence frequencies: negatively skewed distribution ................. ......................157










LIST OF FIGURES


Figure page

4-1 Boxplot of AIC's for all models for a .10 proportion of zeros ........._._.... ......._. .....158

4-2 Boxplot of AIC's for all models for a .25 proportion of zeros ........._._.... ......._. .....159

4-3 Boxplot of AIC's for all models for a .50 proportion of zeros ........._._.... ......._. .....160

4-4 Boxplot of AIC's for all models for a .75 proportion of zeros ........._._.... ......._. .....161

4-5 Boxplot of AIC's for all models for a .90 proportion of zeros ........._._.... ......._. .....162

4-6 Boxplot of AIC's for all models for a .10 proportion of zeros ........._._.... ......._. .....163

4-7 Boxplot of AIC's for all models for a .25 proportion of zeros ........._._.... ......._. .....164

4-8 Boxplot of AIC's for all models for a .50 proportion of zeros ........._._.... ......._. .....165

4-9 Boxplot of AIC's for all models for a .75 proportion of zeros ........._._.... ......._. .....166

4-10 Boxplot of AIC's for all models for a .90 proportion of zeros ........._._.... ......._. .....167

4-11 Boxplot of AIC's for all models for a .10 proportion of zeros ........._._.... ......._. .....168

4-12 Boxplot of AIC's for all models for a .25 proportion of zeros ........._._..........._........169

4-13 Boxplot of AIC's for all models for a .50 proportion of zeros ........._._..........._........170

4-14 Boxplot of AIC's for all models for a .75 proportion of zeros ........._._..........._........171

4-15 Boxplot of AIC's for all models for a .90 proportion of zeros ........._._..........._........172

4-16 AIC rank order for positively skewed distribution models .............. ....................17

4-17 AIC rank order for normal distribution models............... ...............174

4-18 AIC rank order for negatively skewed distribution models .............. .....................7

5-1 Poisson, NB Poisson, and Hurdle over all proportions of zeros ................. ................. 192

5-2 Hurdle, NB Hurdle, and NB Poisson over all proportions of zeros ................. ...............193

5-3 ZIP, NB ZIP, Hurdle, and NB Hurdle over all proportions of zeros .............. ..... .......... 194









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COMPARING POISSON, HURDLE, AND ZIP MODEL FIT
UNDER VARYING DEGREES OF SKEW AND ZERO-INFLATION

By

Jeffrey Monroe Miller

May 2007

Chair: M. David Miller
Major Department: Educational Psychology

Many datasets are characterized as count data with a preponderance of zeros. Such data

are often analyzed by ignoring the zero-inflation and assuming a Poisson distribution. The

Hurdle model is more sophisticated in that it considers the zeros to be completely separate from

the nonzeros. The zero-inflated Poisson (ZIP) model is similar to the Hurdle model; however, it

permits some of the zeros to be analyzed along with the nonzeros. Both models, as well as the

Poisson, have negative binomial formulations for use when the Poisson assumption of an equal

mean and variance is violated.

The choice between the models should be guided by the researcher' s beliefs about the

source of the zeros. Beyond this substantive concern, the choice should be based on the model

providing the closest fit between the observed and predicted values. Unfortunately, the literature

presents anomalous findings in terms of model superiority.

Datasets with zero-inflation may vary in terms of the proportion of zeros. They may also

vary in terms of the distribution for the nonzeros. Our study used a Monte Carlo design to sample

1,000 cases from positively skewed, normal, and negatively skewed distributions with

proportions of zeros of .10, .25, .50, .75, and .90. The data were analyzed with each model over










2,000 simulations. The deviance statistic and Akaike's Information Criterion (AIC) value were

used to compare the fit between models.

The results suggest that the literature is not entirely anomalous; however, the accuracy of

the findings depends on the proportion of zeros and the distribution for the nonzeros. Although

the Hurdle model tends to be the superior model, there are situations when others, including the

negative binomial Poisson model, are superior. The findings suggest that the researcher should

consider the proportion of zeros and the distribution for the nonzeros when selecting a model to

accommodate zero-inflated data.









CHAPTER 1
INTTRODUCTION

Analyzing data necessitates determination of the type of data being analyzed. The most

basic assumption is that the data follows a normal distribution. However, there are many other

types of distributions. The validity of the results can be affected by the dissimilarity between the

distribution of the data and the distribution assumed in the analysis. As such, it is imperative that

the researcher choose a method for analyzing the data that maintains a distribution similar to that

of the observed data.

Counts are an example of data which does not readily lend itself to the assumption of a

normal distribution. Counts are bounded by their lowest value, which is usually zero. A

regression analysis assuming a normal distribution would permit results below zero. Further,

counts are discrete integers while the normal distribution assumes continuous data. Finally,

counts often display positive skew such that the frequency for low counts is considerably higher

than the frequencies as the count levels increase.

It is not uncommon to find count data analyzed in a more appropriate manner than

assuming a normal distribution. Typically, more appropriate analysis includes specification of a

Poisson distribution with a log link, rather than a normal distribution with a Gaussian link.

However, this does not guarantee accurate and valid results as other features of the data may

warrant an even more sophisticated model.

An example of data requiring a more rigorous treatment of the data is the case of zero-

inflation. In this scenario, there are far more zeros than would be expected using the Poisson

distribution. As such, a number of methods including the zero-inflated Poisson (ZIP) model and

the Hurdle model are available. Further, there are negative binomial variations of these for use

when particular assumptions appear to be violated. The choice between the models depends on









whether the researcher believes the zeros are all a complete lack of the quantity being measured

or that at least some of the zeros are purely random error.

Statement of the Problem

The results from both simulated and actual data sets in the zero-inflation literature are in

much disagreement. Lambert (1992) found the ZIP model to be superior to the negative binomial

Poisson model, which was superior to the Poisson model. Greene (1994) found the negative

binomial Poisson model to be superior to the ZIP model, which was superior to the Poisson

model. Slymen, Ayala, Arredondo, and Elder (2006) found the ZIP and negative binomial ZIP

models to be equal. Welsh, Cunningham, Donnelly, and Lindenmayer found the Hurdle and ZIP

models to be equal while Pardoe and Durham (2003) found the negative binomial ZIP model to

be superior to both the Poisson and Hurdle models.

One striking characteristic of these articles and others is their differences in terms of the

proportion of zeros and the distribution for the nonzeros. Some research (Boihning, Dietz,

Schlattmann, Mendonga, and Kirchner, 1999) analyzed data in which the proportion of zeros was

as low as .216 while others (Zorn, 1996) used proportions as high as .958. Further, the nonzeros

varied in terms of their distributions from highly positively skewed to normal to uniform. It is

possible that different models yield different results depending on the proportion of zeros and the

distribution for the nonzeros.

Rationale for the Study

The best model is the one that appropriately answers the researcher' s question. Beyond

this, a superior model is one that has close proximity between the observed data and that

predicted by the model. In other words, a superior model is one with good fit to the data.

This study compared the fit between the Poisson, ZIP, and Hurdle models as well as their

negative binomial formulations. Each analysis was performed for five different proportions of









zeros and three different amounts of skew for the nonzero distribution. The intended results

would clarify the discrepant Eindings of previous research.

Purpose and Significance of the Study

The primary purpose of this study was to determine superiority of fit for various models

under varying proportions of zero-inflation and varying levels of skew. As such, determination

can be made as to which model has better fit given data with a particular proportion of zeros and

a particular distribution. The secondary purpose was to elucidate the reasons for discrepant

Endings in previous research.

The superior model is the appropriate model given the research question. However, there

are situations in which the appropriate model is unknown or

unclear. Further, there may be situations in which a simpler model such as the Poisson may be

used in lieu of the more sophisticated Hurdle and ZIP models. This research provides results that

aid researchers in determining the appropriate model to use given zero-inflated data.

Research Questions

Model comparisons in this research were based on two measures. One is the deviance

statistic, which is a measure of the difference in log-likelihood between two models, permitting a

probabilistic decision as to whether one model is adequate or whether an alternative model is

superior. This statistic is appropriate when one model is nested within another model. The other

measure is Akaike's Information Criterion (AIC). This statistic penalizes for model complexity

and permits comparison of nonnested models; however, it can only be used descriptively. These

two measures of model fit were used to compare results from data simulations where each

dataset included 2,000 cases and each model was analyzed 1,000 times. Specifically, the

measures of model fit were used to answer the following research questions:










* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated log-likelihood between a) the
Negative binomial Poisson model vs. Poisson model; b) the Hurdle model vs. Poisson
model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the
Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model
vs. ZIP model?

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated AIC between all models?









CHAPTER 2
REVIEW OF THE LITERATURE

Zero-Inflated Count Data

Count Data

As the name implies, count data is data that arises from counting. They are the

"realization of a nonnegative integer-valued random variable" (Cameron & Travedi, 1998, p. 1).

As such, the response values take the form of discrete integers (Zorn, 1996). Although the lower

boundary can feasibly be any integer, it is usually the case that its value is zero. Strictly

speaking, there can be no nonnegative numbers. Hence, the data are constrained by this lower

bound of zero and no upper bound.

Acknowledgment of concerns over zero-inflation, ignoring covariates, likely dates to

Cohen (1954). Cameron and Triverdi (1989, p. 10-11) identified many areas in which special

models have been used to analyze count data including "models of counts of doctor visits and

other types of health care utilization; occupational injuries and illnesses; absenteeism in the

workplace; recreational or shopping trips; automobile insurance rate making; labor mobility;

entry and exits from industry; takeover activity in business; mortgage prepayments and loan

defaults; bank failures; patent registration in connection with industrial research and

development; and frequency of airline accidents .. as well as in many disciplines including

demographic economics, in crime victimology, in marketing, political science and government,

[and] sociology". Surprisingly, there was no mention of research in education. Examples of

variables in educational research that yield count data include a student' s number of days absent,

number of test items scored correct or incorrect, and number of referrals for disciplinary action.









The lower bound constraint of zero presents the biggest obstacle toward analyzing count

data when assuming a normal distribution. It is common for count data to have a skewed

distribution that is truncated at the lower bound.



skew(Y)= "= (2-1)
(N -1l)s,3

Hence, the data are heteroscedastic with variance increasing as the count increases.

Therefore, standard models, such as ordinary least squares regression, are not appropriate since

they assume that the residuals are distributed normally with a mean of zero and a standard

deviation of one (Slymen, Ayala, Arredondo, & Elder, 2006). Cameron and Triverdi (1998)

clarify that the use of standard OLS regression "leads to significant deficiencies unless the mean

of the counts is high, in which case normal approximation and related regression methods may

be satisfactory" (p.2).

An example of a count data variable is the number of household members under the age

of 21 reported by respondents in the Adult Education for Work-Related Reasons (AEWR) survey

administered by National Council for Educational Statistics in 2003 (Hagedorn, Montaquila,

Vaden-Kiernan, Kim & Chapman, 2004). The sample size was 12,725. This variable has a lower

count boundary of zero and an upper count boundary of six. The count distribution is positively

skewed at 1.971. The distribution mean of 0.54 is certainly not an accurate measure of central

tendency; the median and mode are both zero (i.e., the lower-bound itself), and the standard

deviation of 0.999 permits negative values in the lower 68% confidence interval.

Zero-Inflation

It is not uncommon for the outcome variable in a count data distribution to be

characterized by a preponderance of zeros. As Tooze, Grunwald, & Jones (2002, p.341) explain,










Typically, [for count data] the outcome variable measures an amount that must be non-
negative and may in some cases be zero. The positive values are generally skewed, often
extremely so .. Distributions of data of this type follow a common form: there is a spike
of discrete probability mass at zero, followed by a bump or ramp describing positive
values.

The occurrence is primarily in the case of interval/ratio count data and sometimes ordinal

data (Boihning, Dietz, Schlattmann, Mendonga, & Kirchner, 1999). Regarding continuous data,

Hall and Zhang (2004) explain that these distributions "have a null probability of yielding a zero

.. there is little motivation for a model such as [zero-inflated] normal, because all observed

zeros are unambiguous .. (p. 162). If continuous zeros are inflated and those zeros are of

concern, they can be analyzed separately from the nonzeros. The null probability of continuous

zeros is evident in measures such as height and age.

The condition of excessive zeros is known as zero-inflation (Lachenbruch, 2002) or as a

probability mass that clumps at zeros (Tooze, Grunwald, & Jones, 2002). It has been recognized

as an area of research in the mid-60'sl (Lachenbruch, 2002) when Weiler (1964) proposed a

method for mixing discrete and continuous distributions. Min and Agresti (2005) formally define

zero-inflation as "data for which a generalized linear model has lack of fit due to

disproportionately many zeroes" (p. 1). There are simply "a greater number of zero counts than

would be expected under the Poisson or some of its variations" (Zorn, 1996, p. 1).

The Sources of Zero-Inflation

The zeros can be classified as being either true zeros or sampling zeros. True zeros

represent responses of zero that are truly null. Suppose an educational inventory item states

"How many college preparatory workshops have you attended?" Some of the respondents in the

sample may have no intentions to apply for college. Hence, the number of preparatory


SAltemnatively, if a scale is bound, it is reasonable to consider an inflated upper bound. In this case, scale reversal
and subsequent appropriate analysis if justified (Lachenbruch, 2002).









workshops attended may never be greater than zero. Sampling zeros, on the other hand, arise as a

probability. There are a proportion of college-bound students who have not attended a workshop

due to the possibility that the workshop was not (or is not yet) available. Alternatively, some

college-bound students may feel prepared and have no reason to participate in a workshop.

Hence, the mechanism underlying zero-inflation can arise from one or both of 1) a

possibility that no other response is probabilistic, or 2) that the response is within a random

sample of potential count responses. Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-

Choy, Tyre, and Possingham (2005) term the sampling zeros as 'false zeros' and include error as

a source of zeros. They state, "Zero inflation is often the result of a large number of 'true zero'

observations caused by the real .. effect of interest .. However, the term [zero-inflation] can

also be applied to data sets with 'false zero' observations because of sampling or observer errors

in the course of data collection" (p. 1235).

Often, the data contains both types of zeros. This is the result of a dual data generating

process (Cameron & Trivedi, 1998). For example, some adults in the AEWR sample may have

had true-zero household members under the age of 21 because they are unable to bear children or

desire to bear children. Alternatively, they may have random-zero household members under the

age of 21 because these adults do have such children but not as members of the household.

Impact of Zero-Inflation on Analyses

"Much of the interest in count data modeling appears to stem from the recognition that

the use of continuous distributions to model integer outcomes might have unwelcome

consequences including inconsistent parameter estimates" (Mullahy, 1986, p.341). In the typical

count data scenario, the zero left-bound implies heteroscedasticity (Zorn, 1996). An even greater

problem with zero-inflated distributions, beyond this inadequacy of analyzing such a skewed and

heteroscedastic distribution as if it were normal (Tooze, Grunwald, & Jones, 2002) is that they










yield "surprisingly large inefficiencies and nonsensical results" (King, 1989, pl26). Martin,

Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and Possingham (2005) and

McCullagh and Nelder (1989) explain that zero-inflation is a special case of overdispersion in

which the variance is greater than it should be given a particular distributional shape and

measure of central tendency. The impact is biased/inconsistent parameter estimates, inflated

standard errors and invalid inferences (Jang, 2005; Martin, Brendan, Wintle, Rhodes, Kuhnert,

Field, Low-Choy, Tyre, and Possingham, 2005).

Simple Solutions to Zero-Inflation

Deleting zeros

The simplest of solutions is to delete all cases having responses of zero on the variable of

interest. A large proportion of total responses would then be removed from the total dataset. This

would then result in a loss of valuable information impacting statistical conclusion validity

(Tooze, Grunwald, & Jones, 2002). The sample size may also then be too small for analyses of

the non-zero values.

Assuming normality

Another simple solution is to ignore the zero-inflation, assume asymptotic normality, and

analyze the data using standard techniques such as ordinary least squares regression.

hhml = P, + fl,Sexl + PAge, + (2-2)

According to this model, the number of household members under the age of 21 for adult

respondent i is predicted from the overall mean, a coefficient relating the respondent's sex to

hhm, a coefficient relating the respondent' s age to hhm, and error. The model assumes that the

residuals for hhm are distributed normally with a mean of zero and a common variance, cr For

the first equation, y is a vector of responses, X is a design matrix for the explanatory variable










responses, p is a vector of regression coefficients relating y to X, and E is a vector of residuals

measuring the deviation between the observed values of the design matrix and those predicted

from the fitted equation.

Transforming Zeros

Another simple solution is to transform the counts to coerce a more normal distribution

(Slymen, Ayala, Arredondo, & Elder, 2006). Since count distributions often appear to be

positively skewed, one reasonable transformation involves taking the natural logarithm of the

responses to the predictor variables. However, assuming the zeros haven't been deleted, the

transformation will not work since the natural logarithm of zero is undefined (Zhou & Tu, 1999;

King, 1989).

Sometimes natural log transformations for zero are handled by adding a small value, such

as .001, to the zeros. However, this then leads to an inflation of that transformed adjusted value.

If 70% of the scores are zero, the resulting transformed distribution will have a 70% abundance

of the transformed value (Delucchi & Bostrom, 2004). 2 Further, since the transformation is

linear, this technique has been shown to yield biased parameter estimates that differ as a function

of the adjustment quantity (King, 1989). Although the undefined log zero problem has been

handled, the original problems pervade. As Welsh, Cunningham, Donnelly, & Linenmayer

(1996) state, "It is clear for data with many zero values that such an approach will not be valid as

the underlying distributional assumptions (linearity, homoscedasticity and Gaussianity) will

[still] be violated" (p.298). Finally, for any technique, transformations sometimes create a new

problem while solving the old one; "a transform that produces constant variance may not

produce normality .. (Agresti, 1996, p.73)".


2 This implies then that, beyond the dual generating process for zeros, the problem can be generalized from inflated
zeros to inflated lower boundaries for count data.









Generalized Linear Models

Bryk and Raudenbush (1996) state, "There are important cases .. for which the

assumption of linearity and normality are not realistic, and no transformation can make them so"

(p.291). As previously demonstrated, count data is likely to be one such case. Instead of deleting

cases or transforming the data, it is more reasonable to specify a different distribution. As

explained by Hox (2002), although it is nice to be able to transform data, "modeling inherently

nonlinear functions directly is sometimes preferable, because it may reflect some 'true'

developmental process" (pp. 93-94). In order for a model to be 'inherently nonlinear' (Hox,

2002), there must be no transformation that makes it linear.3 These nonlinear models belong to

the class of generalized linear models (GLM).

The following explanation of generalized linear models based on the seminal work of

McCullagh and Nelder (1989) with additional clarification by Lawal (2003) and Agresti (1996).

Lawal (2003) explains that generalized linear models are a subset of the traditional linear models

that permit other possibilities than modeling the mean as a linear function of the covariates. All

GLM possess a random component, a systematic component, and a link function. As explained

by Agresti (1996), the random component requires the specification of the distribution for the

outcome variable. One could specify this distribution to be normal; hence, classical models such

as ordinary least squares regression and analysis of variance models are included within this

broader class of generalized linear models. Other possible random components that could be

specified include the binomial distribution, negative-binomial distribution, gamma distribution,

and Poisson distribution. Specifying the random component depends on the expected population

distribution of the outcome variable. Given both zero-inflation and truncated count data yielding


3 Transforming covariates (e.g., including polynomial terms) may graphically appear to be nonlinear while still be
linear in the parameters (Singer & Willett, 2003).









an odd-shaped skewed distribution, the random component plays an important part in obtaining

valid results.

In order to better understand the formulation of the three components, it is necessary to

clarify the theoretical foundations of distributions. The probability density function for a normal

distribution is

1 (y U)2
f (y; u, a) = exp(- ), (2-3)


which, given random variable X ~ N(pu,o-'), reduces to the standard normal probability density

function

Sy2
f(y)= exp(- ),(2-4)
J2~i2

which when transformed to the cumulative density function yields

1 uZ
~(DUy = FOy) = exp(- )u.(2-5)


A convenient method for obtaining the parameters is to use the distribution's moment

generating function (Rice, 1995). For the normal distribution, this function is

0 t2
M, y(t) = E[exp(tY)] = exp(put + ) (2-6)


The logarithm of the moment generating function yields the cumulant generating function,

which then yields the moments of the distribution. For the normal distribution, the first moment

is the mean (pU), and the second moment is the variance (0 ).

Strictly speaking, a requirement for GLM is that the outcome has a distribution within the

exponential family of models (EFM) (McCullagh and Nelder, 1996). These distributions are









defined primarily by a vector of natural parameters (0) and a scale parameter (#). The

formulation is given by

(yB b(0))
f, (y; 8, #) = exp (( ) + c(y, #)) (2-7)
a(#)

At first glance, it seems odd to include the normal distribution in the EFM; however, first

recall the probability density function


f (y; u, a) = exp(- ) (2-8)


Algebraic manipulation reveals that the normal distribution is indeed an EFM formulation.

( y p U)2 / 2) 1 y
EF~Zw,,u) = f y(8, ~) = exp (( ) o(zc) (2-9)
o 20

Here, the natural (i.e., canonical) parameter is pu, and the scale parameter is a .

These parameters need to be estimated. McCullagh and Nelder (1996) explain the

estimation as follows: "In the case of generalized linear models, estimation proceeds by defining

a measure of goodness of fit between the observed data and the fitted values that minimizes the

goodness-of-fit criterion. We shall be concerned primarily with estimates obtained by

maximizing the likelihood or log likelihood of the parameters for the data observed" (p. 23-24).

This turns out to be the log of the EFM function.

e(0, #; y) =log( f, (y; 0, #)) (2-10)

The natural and scale parameters are estimated by derivations revealing the mean function

E(Y)= pu = b '(0), (2-11)

and the variance function

var(Y) = b "(0)a(#) (2-12)









Note that the mean function depends on only one parameter. However, as McCullagh and

Nelder (1989) explain, .. the variance of Y is the product of two functions; one, b "(0) ,

depends on the [canonical] parameter (and hence on the mean) only and will be called the

variance ju~nction [denoted V(pu)], while the other [a(#) ] is independent of I and depends only

on # .. The function a(#) is commonly of the form a(#) = / w (p.29) and is commonly called

the dispersion parameter. For the normal distribution, the natural parameter is the mean (pu); the

variance function, V( pu); equals 1.0, and the dispersion parameter is cr

The systematic component is simply the model for the predictors established as a linear

combination and is denoted r. The link function, g(-), brings together the random component

and the systematic component hence linking the function for the mean, pu, and the function for the

systematic component, r, as r = g( pu). In other words, it specifies how the population mean of

the outcome variable with a particular distribution is related to the predictors in the model. If

g( pu) redundantly equals pu, then the population mean itself is related to the predictors. This is

termed the identity link and is exactly the function used to link the mean of the normal

distribution to its covariates.

The key advantage of GLM is that they are not restricted to one particular link function.

Many other links are available. For example, one could specify the log link as g( pu) = log( pu) or

the logit link as g( pu) = log[ pu / (1- p)]. However, each random component has one common

'canonical' link function that is best suited to the random component (McCullagh & Nelder.

1996). Alternatively, "Each potential probability distribution for the random component has one

special function of the mean that is called its natural parameter" (Agresti, 1996, p.73). For

example, a normal random component usually corresponds to an identity link, a Poisson









distributed random component usually corresponds to a log link, and a binomial distributed

random component usually corresponds to a logit link. In sum, the canonical link and natural link

are two equivalent terms for specifying the most suitable link connecting a particular distribution

for the outcome variable with its linear systematic covariate function.

The Binomial Distribution and the Logit Link

Suppose we were interested in the differences between households with zero children

under age 21 and households with one or more children over the age of 21. We could feasibly

collapse all nonzero responses in the AEWR data into a value of one. Now, 71.58% of the values

are zeros, and 28.42% of the values are ones. This distribution is obviously not normal. We have

now introduced both a lower bound (zero), an upper bound (one), and an inherently nonnormal

distribution; hence, a different random component and link can be specified to accommodate

these constraints.

Variables that take on only one of two values are known as binary, or Bernoulli,

variables, and the distribution of multiple independent trials for these variables is termed

binomial. Bernoulli responses are modeled in terms of the probability (Pr) that the outcome

variable (Y) is equal to either zero or one. The random component over multiple independent

trials is thus a binomial distribution with parameters n for the number of trials and ai for the

probability that Y= 1.

Y ~ B(n, zi) (2-13)

The binomial distribution assumes that the responses are dichotomous, mutually exclusive,

independent, and randomly selected (Agresti, 1996). Since the responses are discrete, the

probability density function is termed the probability mass function and is defined as


f (k; n, p) = n!" k n-"k]i. (2-14)
k !(n k)!i









This function gives the p probability of k ones (i.e., heads, hits, successes) over n trials. Rice

(1995) clarifies, "Any particular sequence of k successes occurs with probability [pkl n-"k

from the multiplication principle [i.e., independent probabilities of realizations being


multiplicative]. The total number of such sequences is [permutations], since there are
k !(n k)!i


ways to assign k successes to n trials" (p.36). The moment generating function is
k !(n k)!i

{1- xi + xi exp(5)) ", and the cumulant generating function is n logt 1- ai + xi exp(5) } .

The log-likelihood function is virtually the same as the probability mass function. However, now

we are determining the value of p as a function of n and k (rather than determining k as a

function of p and n) while taking the log of this maximum at


e[f(p; n,k)] = n! k1 -k] (2-15)
k !(n k)!i

The estimates are obtained through derivations of the likelihood function as was previously

discussed for the normal distribution. Just as the normal distribution population mean, pu, has the

best maximum likelihood estimates of X the binomial distribution population probability, zi,

has the best maximum likelihood estimate of k divided by n, which is the proportion of ones,

hits, or successes. This greatly reduces calculations when a quick estimate is needed and the

random component is not linked to any predictors.

The binomial distribution eventually converges to a normal distribution. However, the

speed of this convergence is primarily a function of skew,


Men y) =(2-16)


with p = .50 yielding the fastest convergence (McCullagh and Nelder, 1996).









The link function should account for the binomial outcome variable. If linear predictors

are used to predict a probability, then we have predicted values in an infinite range rather than

constrained to be between zero and one. What is needed is a link function that will map a

bounded zero-one probability onto this range of infinite values. The canonical link for the

binomial distribution is the logit link.

r = g(s) = log(ir /(1- zi)) (2-17)

A logit is the natural log of an odds ratio or log[p / (1-p)]. An odds ratio is equal to a

probability divided by one minus that probability. Hence, if the probability is .5, then the odds

are .5 / (1-.5) = 1 meaning that the odds of a one are the same as the odds of a zero [i.e., the odds

of success and failure are identical]. If the probability is .75, then the odds are .75 / (1-.75) = 3

meaning that a response of one is three times more likely than a response of zero. The reciprocal

odds of 1/3 means that a response of one is three times

less likely than a response of zero, which is equivalent to stating that a response of zero is

three times more likely than a response of one. When using the logit link to connect the binomial

random distribution and the systematic component, the generalized linear model is


logity)= log( )= f, + fX, +...X (2-18)

A probability of .50, which is an odds of one, corresponds to a logit of zero. Odds favoring

a response of one yield a positive logit, and odds favoring a response of zero yield a negative

logit. Hence, the mapping is satisfied since the logit can be any real number (Agresti, 1996).

The regression parameters are slope-like in that they determine the relative rate of change

of the curve. The exact rate of change depends on each probability with the best approximation









at that probability being Pfr(1 zi) with the steepest rate of change being at ai = 0.50, which is

where X = -a /p (Agresti & Finlay, 1997).

Since natural logs can be reversed through exponentiation and since odds can be converted

to probabilities by dividing the odds by the sum of the odds and one, the fitted equation can be

used to predict probabilities via

exp(a + ,X, + ---+ PkXk)
a = (2-19)
1+ [exp(a + ,X, + ---+ PkXkl

It is more common to interpret logistic regression coefficients by only exponentiating

them. Then, the coefficient has a slope, rather than slope-like, interpretation; however, the

relationship is multiplicative rather than additive. Specifically, the expected outcome is

multiplied by exp( P) for each one-unit increase in X.

Evaluating Model Fit

The next step in interpreting generalized linear model results is to determine how well the

estimated model fits the observed data, where fit is the degree of discrepancy between the

observed and predicted values. McCullagh and Nelder (1989) explain, "In general the pus will

not equal the y' s exactly, and the question then arises of how discrepant they are, because while a

small discrepancy might be tolerable a large discrepancy is not" (p.33). The goodness of fit

improves as the observed values and predicted values approach equality. For example, if a

scatterplot reveals that all points fall on a straight line, then the predictive power of the

regression equation would be perfect, and the subsequent fit would be perfect.

The comparison is usually performed through some statistical comparison of the observed

outcome values and the predicted (i.e., fitted) outcome values. Rather than compare and

summarize the actual observed and predicted values, it is common to gain summary information










by inspecting the log-likelihood value produced from the estimation procedure. Since the model

parameters are estimated are from the data, perfect fit (i.e., observed-fitted = 0) is rare.4 Hence,

the goodness of the fit is measured to determine whether the difference is small enough to be

tolerated.

There are many measures of model fit. Typically, the model is compared either to a null

model in which the only parameter is the mean or a full model in which the number of

parameters is equal to the sample size. "It is well-known that minus twice the LR statistic has a

limiting chi-square distribution under the null hypothesis" (Vuong, 1989, p.308). McCullagh

and Nelder (1989) equivalently state, "The discrepancy of a fit is proportional to twice the

difference between the maximum log likelihood achievable and that achieved by the model

under investigation" (p.33). This deviance statistic (G2) iS then considered to be asymptotically

distributed chi-square with degrees of freedom equal to the number of parameters subtracted

from the sample size.' A significant p-value indicates that the deviance is greater than what

would be expected under a null hypothesis that the model with less parameters is adequate;

hence, the observed model with an additional parameter or parameters is considered a significant

improvement over the null model.

Another measure of model fit is Pearson's X2; however, unlike G2, it is not additive for

nested models. Yet another measure of model fit is Akaike's Information Criterion (AIC), which

penalizes the deviance for the number of parameters in the model.6 The notion is that increasing



4 Perfect fit is always obtained if the number of parameters and the sample size are identical (McCullagh & Nelder,
1989).
5 The relationship is not always exact since sometimes the deviance is scaled and/or the likelihood is more difficult
to estimate than in the simple logistic regression scenario presented here (McCullagh & Nelder, 1989).

6 Other measures such as the Bayesian Information Criterion (BIC) penalize for both the number of parameters and
the sample size.









the number of parameters will increase the log-likelihood regardless of the model and the data.

Hence, the AIC penalizes the log-likelihood with regard to the number of parameters. There are

two variations that provide further penalties. The Bayesian Information Criterion (BIC) penalizes

for sample size; the Consistent Akaike Information Criterion (CAIC) penalizes even further by

considering sample size and adding a small adjustment (Cameron & Trivedi, 1998). These

indices can be compared to those of competing models; however, this must be done

descriptively, not inferentially. The disadvantage is that the AIC can not be compared to a

statistical distribution resulting in probabilities for significance testing; however, the advantage is

that, as a descriptive statistic, it can be used to compare nonnested models.

The explanation thus far points to the fact that models can be compared to null, full, or

other models. Statistical comparison is valid to the extent that one model is nested within the

other, which is to say that both models share the same parameters, and one model has at least one

parameter that is not included in the other. Alternatively, Clarke (2001) defines the models as

follows: "Two models are nested if one model can be reduced to the other model by imposing a

set of linear restrictions on the parameter vector .. Two models are nonnested, either partially

or strictly, if one model cannot be reduced to the other model by imposing a set of linear

restrictions on the parameter vector" (p.727). The deviance for comparing two models is

calculated as the difference in log likelihood between the two models then multiplied by -2.

This quantity is asymptotically distributed chi-square with degrees of freedom equal to the

difference in parameters between the two models (Agresti, 1996). A significant p-value indicates

that the deviance is greater than what would be expected under a null hypothesis of model

equivalence; hence, the more complex model with an additional parameter or parameters is

considered a significant improvement over the nested model.









The difference in log-likelihood statistics (i.e., deviance) can not be used to statistically

test nonnested models. This is due to the fact that neither of the models can be considered the

simple or more complex models with additional variables leading to a probabilistically higher

log-likelihood. A t-test (or F-test) is a sensible alternative that eliminates concern for nesting.

However, Monte Carlo simulations have demonstrated that, for model comparison tests, the F-

test is lacking in sufficient power and can result in multicollinearity (Clarke, 2001).

The motivation for the AIC statistic is that, all else being equal, "the greater the number of

coefficients, the greater the log-likelihoods" (Clarke, 2001, p.731). Hence, model fit becomes

impacted by the number of variables in the model along with the effects of those variables.

Hence, the AIC penalizes for the number of parameters. The formula is

AIC = -2(LL) + 2K (2-20)

where LL is the log-likelihood estimate and K is number of parameters in the model including the

intercept. Hence, now the log-likelihood is adjusted to accommodate simplicity and parsimony

(Mazerolle, 2004).

In actuality, one could compare log-likelihoods between nonnested models. However,

beyond the lack of parameter penalty, this technique might lead to the statistical hypothesis

testing associated with log-likelihood statistics (i.e., test for the deviance approximated

by X2).The AIC, on the other hand, should not be used in a formal statistical hypothesis test

regardless of whether the model is nested or nonnested (Clarke, 2001). Generally, the researcher

looks at several AIC indices and decides which model fits best based on a lower-is-better

criterion. Mazerolle (2004) states, "The AIC is not a hypothesis test, does not have a p-value, and

does not use notions of significance. Instead, the AIC focuses on the strength of evidence .. and

gives a measure of uncertainty for each model" (p. 181).










Logistic modeling necessitated treating all nonzero numbers of children as a value of one.

Depending on the research question, this may be a loss of valuable information (Slymen, Ayala,

Arredondo, & Elder, 2006). Although sometimes it is necessary to model zero-inflated binomial

data (Hall, 2000), specifying a binary distribution and logit link is not an ideal method for

handling zero-inflated count data. The generalized linear model that specifies a binomial

distribution and a logit link becomes more relevant when discussing the technique of splitting

zero-inflated data into a model for the probability of zero separate from or combined with a

model for the counts.

The Poisson Distribution and the Log Link

McCullagh and Nelder (1989), Lawal (2003), and Rice (1995) are the key references for the

technical underpinnings for this model and distribution. The generalized linear 'Poisson' model

is considered to be the benchmark model for count data (Cameron & Triverdi, 1998).7 This is

primarily attributed to the fact that the Poisson distribution has a nonnegative mean (Agresti,

1996). If y is a nonnegative random variable, the Poisson probability mass function is given by

ef~Z
f (k; A2) = Pr(Y = k) = ,k = 0, 1, 2, .. (2-21)


where Ai is standard Poisson notation for the mean ( p) and k is the range of counts. Derivations

by Rice (1995) show that the expected value of a random Poisson variable is Ai; hence, "the

parameter Ai of the Poisson distribution can thus be interpreted as the average count" (p. 113).

Alternatively, lambda ( Ai) represents "the unobserved expected rate or occurrence of events ...'

(Zorn, 1996, p.1). The moment generating function is



SIt is also commonly used to model event count data, which is "data composed of counts of the number of events
occurring within a specific observation period .. [taking the] form of non-negative integers (Zorn, 1996, p.1)".










E(e") = e;"'exp't)-1). (2-22)

The resulting cumulant generating function is Al(exp(t)-1i), which, with a variance function of

Ai and a dispersion parameter equal to one, leads to mean and variance both being equal to Ai, and

the skew equal to one divided by the square root of Ai This equivalence of the mean and

variance defined by a single parameter (Cameron & Triverdi, 1998; Agresti, 1996) is the result

of a function that yields residuals that sum to zero (Jang, 2005); hence, the systematic portion of

a Poisson GLM has no error term.

The Poisson distribution is a generalization of a sequence of binomial distributions.

Rodriguez (2006) explained that "the Poisson distribution can be derived as a limiting form of

the binomial distribution if you consider the distribution of the number of successes in a very

larger number of Bernoulli trials with a small probability of success in each trial. Specifically, if

Y ~ B(n, zi), then the distribution of Y as n + oo and 71 + 0 with pu = nzi remaining Eixed

approaches a Poisson distribution with mean pu Thus, the Poisson distribution provides an

approximation to the binomial for the analyses of rare events, where ri is small and n is large

(p.3). Rice (1995) clarified, "The Poisson distribution can be derived as the limit of a binomial

distribution as the number of trials, n, approaches infinity and the probability of success on each

trial, p, approaches zero in such a way that np = Ai (p.43). Scheaffer (1995) and Rice (1995)

have derived the generalization.

Further, just as a binomial distribution converges to a normal distribution given sufficient

trials, the Poisson distribution converges to a normal distribution given a large mean.

The log-likelihood function for the Poisson distribution is

(A, y) = C y, log A, -1,, (2-23)









with the maximum likelihood estimate of Ai simply being the sample mean (Rice, 1995) and

with the related deviance function being

Devian~ce(A, y) = 2C { y, log(y, / A) -(y, A,~) }. (2-24)

McCullagh and Nelder (1989) state that the second term is often ignored. "Provided that the

fitted model includes a constant term, or intercept, the sum over the units of the second term is

identically zero, justifying its omission" (McCullagh & Nelder, 1989, p.34).

The systematic portion of the generalized linear model takes the form

Ai, = exp(x, p) = exp(x,l p,) exp(x,l p,)... exp(xk; pk) (2-25)

which is often equivalently expressed as

log(A~) =P' XI (2-26)

with p derived by solving the equation


(ex(y, )- epxu), = 0 (2-27)


by using iterative computations such as the Newton-Raphson. The canonical link for a

generalized linear model with a Poisson random component specification is the log link (Stokes,

Davis, & Koch, 1991).

9= lo(A),Y ~ (A).(2-28)

The Poisson distribution is not limited to count variates. Cameron and Triverdi (1998)

explain that, although counts are usually in the purview of directly observable cardinal numbers,

they may also arise through a latent process. In other words, ordinal rankings such as school

course grades may be discretized as pseudocounts and assumed to have a Poisson distribution.

Hence, the results of an analysis based on a Poisson distribution and the results using an ordinal

analytic technique are often comparable.










As is almost always the case, it is common to identify other variables associated with the

count variable (i.e., misspecification). However, the Poisson model has in interesting feature in

that it assumes that there are no variables excluded from the model that are related to the count

variable. In other words, there is no stochastic variation (i.e., no error term) (Cameron & Trivedi,

1998). Modifications must be made when one wishes to use a Poisson model with stochastic

variation.

Iterative Estimation

Agresti (1996) clarifies the iterative estimation procedure. "The Newton-Raphson

algorithm approximates the log-likelihood function in a neighborhood of the initial guess by a

simpler polynomial function that has shape of a concave (mound-shaped) parabola. It has the

same slope and curvature location of the maximum of this approximating polynomial. That

location comprises the second guess for the ML estimates. One then approximates the log-

likelihood function in a neighborhood of the second guess by another concave parabolic

function, and the third guess is the location of its maximum. The successive approximations

converge rapidly to the ML estimates, often within a few cycles" (p.94). The most common

methods for estimating standard errors include Hessian maximum likelihood (MLH) (i.e., second

partial derivative based) standard errors and maximum likelihood outer products (MLOP) (i.e.,

summed outer product of first derivative) estimation.

Interpretation of Coefficients

Standard ordinary linear squares regression lends an interpretation of P as the predicted

additive change in the response variable per one-unit change in the predictor variable. However,

as was the case with the binomial distribution, the interpretation differs when considering

exponential distributions. For the Poisson distribution, "a one-unit increase in X has a

multiplicative of exp(/7) on the pu The mean of Y at x+1 equals the mean of Y at x multiplied










by exp( P)" (Agresti, 1996, p.81).s Due to the inherent difficulty in interpretation, it is common

to express in one of three alternative ways. First, the direction of the sign of P indicates a

positive or negative 'effect' of the predictor on the count variable. Second, the fitted value can be

calculated at the mean.9 Third, some interpret the coefficient in terms of percent change; hence,

if f =1.64, then as X increases to X+1, the predicted probability increases by 64% (Agresti,

1996).

Hypothesis testing

After conducting the analysis and estimating parameters, hypotheses can be tested in

several ways as explained by Agresti (1996). One could test the hypothesis that P=0 using the

traditional Wald :-test via z = b / seb Some programs provide Wald test results that are actually

Z2; this is the Wald X2 Statistic with one degree of freedom and appropriate only for a two-tailed

test. A third method, the Score test, is "based on the behavior of the log-likelihood function at the

null value for p =0" (Agresti, 1996, p.94) yielding a chi-square distributed statistic with one

degree of freedom.

Overdispersion

In practice, the assumption of an equal mean and variance is the exception rather than the

norm (McCullagh & Nelder, 1989). It is often the case that the sample variance is greater than or

less than the observed sample mean with these two seldom being statistically equivalent

(Cameron & Trivedi, 1998), especially for zero-inflated data (Welsh, Cunningham, Donnelly,




SThis is similar to the interpretation for the binomial distribution with logit link: however. now the multiplicative
effect is directly on p rather than on the odds of p.

9 This can be particularly troublesome since that fitted value will only hold at the mean. It provides no valid
inference for values greater than or less than the mean since the function is a curve with steepness that can vary
drastically between separate values for the predictors.










and Lindenmayer, 1996). This condition is known as overdispersionl0 (underdispersion) and is a

violation of a maj or tenet of the Poisson distribution that the conditional mean and conditional

variance of the dependent variable are equal (i.e., equidispersion, nonstochasticity) (Jang, 2005;

Zorn, 1996). 1 This assumption of equidispersion is the analog of the ordinary least squares

regression assumption of homoscedasticity.

The overdispersion has been explained as heterogeneity that "has not been accounted for

[that is] unobserved (i.e., the population consists of several subpopulations, in this case of

Poisson type, but the subpopulation membership is not observed in the sample" (Boihning, Dietz,

Shlattman, Mendonca, & Kirchner, 1999, p. 195). The impact of violation is one of incorrect

conclusions due to inaccurate t-statistics and standard errors (Cameron & Triverdi, 1998;

Agresti, 1996). "The estimates of the coefficients can still be consistent using Poisson regression,

but the standard errors can be biased and they will be too small" (Jewell & Hubbard, 2006, p.14).

Alternatively, Slymen, Ayala, Arredondo, and Elder (2006) state that "Confidence intervals for

regression estimates may be too narrow and tests of association may yield p-values that are too

small" (p.2). The underlying mechanism for overdispersion is explained as unobserved

heterogeneity in responses.

It is apparent that some modification to the variance to accommodate over-dispersion is

ideal. Typically, maximum likelihood procedures are used to estimate parameters in the model.

"The term pseudo- (or quasi-) nzaxinsun likelihood estimation is used to describe the situation in

which the assumption of correct specification of the density is relaxed. Here the first moment

[i.e., the mean] of the specified linear exponential family density is assumed to be correctly

'o Overdispersion is sometimes referred to as extra-Poisson variation (Bdhning, Dietz, Shlattman, Mendonca, &
Kirchner, 1999).

11 The under- or overdispersion may disappear when predictors are added to the model: however, this is likely not
the case if the variance is more than twice the mean. (Cameron & Trivedi, 1989).










specified, while the second [i.e., the variance] and other moments are permitted to be incorrectly

specified" (Cameron & Triverdi, 1998, p.19). Hence, the Poisson distribution as a baseline

(Ridout, Demetrio, & Hinde, 1998) can be modified to accommodate overdispersion and

underdispersion (Cameron & Triverdi, 1998).

Rice (1995) states that "gamma densities provide a fairly flexible class for modeling

nonnegative random variables" (p. 52). One way to accommodate overdispersion is to consider

the unobserved heterogeneity as a gamma distributed disturbance added to the Poisson

distributed count data (Jang, 2005). In other words, an individual score may be distributed

Poisson with a mean of 2 but then this mean is "regarded as a random variable which we may

suppose in the population to have a gamma distribution with mean pu and index p / # "

(McCullagh & Nelder, 1989, p. 199). This mixture leads to the negative binomial distribution.

Given gamma function, r, and count y, the negative binomial probability mass function is


pr (Y = y; pu, #) = (2-29)


Cameron and Trivedi (1998) provide a formulation where the negative binomial extends from

the Poisson rather than being explained as a mixture distribution. Given a set of predictor

variables, we can define

co, = pu, +apu,P (2-3 0)

where a is scalar parameter to be specified or estimated, and p is a pre-specified power term. If

the scalar parameter is set to zero then the resulting variance is equal to the mean, and the

Poisson distribution holds. Hence, the Poisson model is nested within the negative binomial

model .









The standard formulation for the negative binomial formulation of the function,

sometimes called NB2 (Cameron & Trivedi, 1998), leads to a variance that is quadratic by

setting p to 2

co, = pu, + a p, (2 31)

This is the formulation seen in most textbooks (Rodriguez, 2006; Scheaffer, 1996). Its mean is

the same as that of the Poisson distribution; however, its variance is derived from the gamma

distribution (Cameron & Trivedi, 1998). Just as the Poisson distribution converges to a binomial

distribution, the negative binomial distribution converges to a Poisson distribution (Jewell &

Hubbard, 2006).

Poisson and Negative Binomial Models with Zero-Inflation

As elaborated upon previously, the zero-inflation problem is two-fold. First, the proportion

of zeros is higher than expected given the specified population distribution shape resulting in an

excess zeros problem. This can be descriptively determined by calculating the expected number

of zeros as

E(fq(Y))= fq(Y)*(exp(-Y)) = ne" (2-32)

For example, Zorn' s example had a frequency of 4,052 with a A = 0.11. The expected frequency

of zeros would be

E( fq(Y)) = 4052* (exp(-0. 109)) = 3, 634, (2-33)

which is less than the 3,882 zeros observed in the data. It turns out to be (3 882/3634)* 100 =

107% of the expectation (Zorn, 1996).

Second, the zeros can be a mixture of structural (i.e., true) zeros (Ridout, Demetrio, &

Hinde, 1998) and sampled zeros reflecting the multiple sources of zeros problem. Shankar,

Milton, and Mannering (1997) state that if "a two-state process is modeled as a single process ..










if applying traditional Poisson and NB distributions, the estimated models will be inherently

biased because there will be an over-representation of zero observations in the data, many of

which do not follow the assumed distribution of [the] frequencies" (p.830O)" Shankar, Milton, and

Mannering (1997) note that the negative binomial model "can spuriously indicate overdispersion

when the underlying process actually consists of a zero-altered splitting mechanism" (p.835-

836). In sum, the sources of zeros arise from a dual generating process (i.e., structural and

sampled) leading to two sources of unequal mean/variance dispersion (i.e., that due to

unobserved heterogeneity of responses and that due to zero-inflation).

Most complex methods for analyzing zero-inflated count data model a mixture of two

different distributions. The justification for splitting the distribution into two pieces is well-

reasoned by Delucci and Bostrom (2004). "If it is deemed more reasonable to consider the zeros

as indicators of cases without a problem, a more appropriate approach is to ask two questions: is

there a difference in the proportion of subjects without the problem [i.e., structural true zeros],

and, for those who have a problem [sampled false zeros], is there a difference in severity" (p.

1164).

Zorn (1996) refers to 'dual regime' models "wherein an observation experiences a first

stage in which there is some probability that its count will move from a 'zero-only' state to one

in which it may be something other than zero" (p.2). Typically, the dual-regime is composed of a

ttrttrttrtranstion~rtrt~t stage based on a binomial distribution and an events stage based on some type of

Poisson distribution.

There are many ways to model two-part distributions. For example, Mullahy (1986) and

King (1989) proposed a Hurdle model in which the zeros are analyzed separately from the

nonzeros. Lambert (1992) proposed a zero-inflated Poisson (ZIP) model in which different










proportions of zeros are analyzed separately and along with the nonzeros. Another early

formulation (Heilborn, 1989) was the zero-altered Poisson (ZAP) model. "Arbitrary zeros are

introduced by mixing point mass at 0 with a positive Poisson that assigns no mass to 0 rather

than a standard Poisson (Lambert, 1992, p.1)". Mullahy (1986) presented a variation of the

Hurdle model based on a geometric distributionl2 for use when specifying a Poisson distribution

is not reasonable. Another possibility is to specify a log-gamma distribution for the event stage

(Moulton, Curriero, & Barruso, 2002). Lambert (1989) presented a variation to the ZIP model

known as ZIP(z), which introduced a multiplicative constant to the event stage covariance matrix

in order to account for the relationship between the two models. Gupta, Gupta, and Tripathi

(1996) derived an adjusted generalized Poisson regression model for handling both zero-inflation

and zero-deflation; however, accuracy was suggested to be contingent on the amount of inflation

or deflation. 13

It is also possible to formulate the model with different link functions. Lambert (1989)

mentions the possibility of using the log-log link, complementary log-log link (Ridout, Demetrio,

& Hinde, 1998), and additive log-log link while Lachenbruch (2002 ) mentions the lognormal

and log-gamma distributions. Hall (2000) formulated a two-part model for zero-inflated binomial

data. Gurmu (1997) describes a semi-parametric approach that avoids some distributional

assumptions (Ridout, Demetrio, & Hinde, 1998).

There is some research on the extension of two-part models to accommodate random

effects (Min & Agresti, 2005; Hall & Zhang, 2004; Hall, 2004; Hall, 2002; Olsen, 1999). Hall's

(2000) model for zero-inflated binomial data permits both fixed and random effects. Dobbie and


12 This distribution is an extension of the binomial distribution where the sequence in infinite. It is typically used in
cases where the researcher is concerned with probability up to and including the first success (Rice, 1995).

13 Min (I 'li 1)Stated that the Hurdle model also has this feature.










Walsh (2001) permit correlated count data. Finally, Crepon and Duguet (1997) consider the

cases where the variables are latent and correlated.

The evolution of the research to date has led to an emphasis on the standard Hurdle model

and ZIP models (along with their negative binomial extensions) with a binary distribution for the

transition stage and a Poisson distribution for the events stage and with fixed covariates. For both

models, estimates are obtained from maximum likelihood procedures, although there has been

some research on the use of generalized estimating equations (GEE) (Hall & Zhang, 2004;

Dobbie & Welsh, 2001). The models are primarily distinguished by whether zeros are permitted

in the event stage. In other words, their differences are a reflection of the researcher' s notions

about the potentially multiple sources of zeros in the data and their relationship to excess zeros.

They also differ in terms of the transition stage cumulative probability function. To be

clarified in the sections that follow, Zorn (1996) summarizes the differences as follows: .. the

hurdle model has asymmetric hurdle probability while in the ZIP specification p, is symmetrical.

Also, the hurdle model does not permit zero values to occur once the 'hurdle' has been crossed,

while in the ZIP model zeros may occur in either stage" (p.4). Choosing between the models is a

matter of validity; hence, the choice rests on substantive ground as well as statistical

considerations. As Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and

Possingham (2005) note, "it is imperative that the source of zero observations be considered and

modeled accordingly, or we risk making incorrect inferences .. (p.1243-1244).

The Hurdle model

The Hurdle model was developed separately by Mullahy (1986) in economics and King

(1989) in political science, although the term itself was most likely coined by Cragg (1971). 14


14 Cragg (1971) proposed the basic two-step process in which the probability of occurrence is modeled separately
from frequencies of occurrence.









Welsh, Cunningham, Donnelly, and Lindenmayer (1996) refer to it as a 'conditional Poisson

model'. "The idea underlying the hurdle formulations is that a binomial probability model

governs the binary outcome whether a count variate has a zero or a positive realization [i.e., a

transition stage]. If the realization is positive the 'hurdle' is crossed, and the conditional

distribution of the positives is governed by a truncated-at-zerol5 count data model [i.e., events

stage]" (Mullahy, 1986, p.345) such as a truncated Poisson or truncated negative binomial

distribution (Min & Agresti, 2005). In other words, one distribution addresses the zeros while

another distribution addresses the positive nonzero counts. For example, for grade-retention data,

there would be a model for schools with no dropouts and a model for school with at least one

dropout.

It is a "finite mixture generated by combining the zeros generated by one density with the

zeros and positives generated by a second zero-truncated density separately .. (Cameron &

Trivedi, 1998, p.124). Log-likelihood values are estimated separately for each density. A key

feature of the transition model is asymmetry in that the probability of crossing the hurdle

increases more quickly as the covariates increase than it decreases as the covariates decrease.

The function is then "asymmetrical" (King, 1989) leading to validity concerns supporting or

refuting the substantive theory underlying the model.

The two-part distribution of the dependent variable is given first by the transition stage g;

probability mass function

P(Y I= 0) =g,(0) 16 (2-34)





'5 The lower bound is then one.

16 This is an alternative representation of the aforementioned Pr(Y = 0) = 1 Fr .









modeling whether the response crosses the hurdle of zero. Assuming a Poisson distribution and

log link, Zorn (1996) expands the cumulative distribution function to include covariates as

p,= 1- exp[- exp(Xof 0)] (2-3 5)

The basic model for the event stage is then the probability for a nonzero realization multiplied by

the probability for the counts. 1


P(Y I = j) = (1 -g,(0)) 2 ..j = 1, 2,... (2-3 6)
1- g2 (0)

Greene (1994) notates the models with a binomial distribution and logit link for the transition

stage and a Poisson distribution with a log link for the event stage asls

Transition: Pr(y, = 0) = p, (2-3 7)
1- p e A/2
Event: Pr(y, = k) = ( ( ') ,2..(2-38)
1-e k!

Here, p is the probability of a count of zero while ii is the truncated Poisson mean for the counts

greater than zero. The generalized linear models as a function of covariates is then


Transition Stage: log( ) = xllB, (2-3 9)
1 p,
Event Stage: log(il,) = x22B2 (2-40)

The two vectors of parameters are estimated j ointly.

A, = [n {exp[- exp(XP)]} C {-exp[-ex p(Xp)]}\1 (2-41)
A2 = On Xp(yXp) /({ex ep (X p)]-Y R1}y!)],, (2-42)

An alternative notation provided by Min (2003) is




17 The second term is an alternative representation of the aforementioned f (k; ii) = Pr(Y = k) =

1s Note that p is used to representative the probability of a zero rather than the conventional representation of a
probability of one.










8 (P>)= [logg(y, = 0; P,, x,l)]+C [log(1 I(yI = 0; P,,x,l))]








Since the two models are functionally independent, the likelihood functions can be maximized

separately (Min & Agresti, 2005; Min, 2003, Cameron & Trivedi, 1998; Mullahy, 1986).

A;,,,,vie = log(Az) + log(A,) (2-45)
e(p,, p,) = (p, )+ (p,) (2-46)

This is because "the large sample covariance between the two sets of parameters is zero so

the joint covariance matrices can be obtained from the separate fits" (Welsh, Cunningham,

Donnelly, & Lindermayer, 1996, p.300). Solving the likelihood equations uses either the

Newton-Raphson algorithm or the Fisher scoring algorithm, both giving equivalent results (Min,

2003). The Poisson model is nested within the Hurdle model (Zorn, 1996). Hence, fit of these

two models can be compared statistically. 19

The Negative Binomial Hurdle model

In the case of zero-inflated data, it is possible to have two sources of overdispersion. The

variance can be greater than the mean due to the preponderance of zeros. However, there is now

the possibility of unobserved heterogeneity in the event stage (Mazerolle, 2004; Min & Agresti,

2004). The former scenario has been referred to as zero-driven overdispersion (Zorn, 1996); the

latter is Poisson overdispersion. Just as was the case with the Poisson model, it is possible to nest

the Hurdle model within a more general negative binomial framework. Further, the negative

binomial Poisson model is nested within the negative binomial Hurdle model. Hence, the fit of a)


19 Zhou and Tu (1999) developed likelihood ratio for count data with zeros; however, it was not generalized to any
particular zero-inflation model.









the Poisson model and the Hurdle model, b) the negative binomial Poisson model and the

negative binomial Hurdle model, and c) the Hurdle model and the negative binomial Hurdle

model can be compared using statistical tests.20 Estimation is typically performed by solving the

maximum likelihood equations using the Newton-Raphson algorithm.

The Zero-Inflated Poisson (ZIP) model

The Zero-Inflated Poisson, or ZIP, model is another model that one can use when the

zeros in a dataset are argued to be caused by both chance and systematic factors (Min & Agresti,

2005). The transition stage addresses zero-inflation while the event stage addresses unobserved

heterogeneity of responses including zeros (Jang, 2005). Welsh, Cunningham, Donnelly, and

Lindenmayer (1996) refer to it as a mixture model.

This two-part model, developed by Lambert (1992) permits zeros to occur in both the

transition stage and event stage (Cameron & Trivedi, 1998); "crossing the 'hurdle' in the ZIP

model does not guarantee a positive realization of Y (Zorn, 1996, p.4). Further, the probability

function in the transition stage is now symmetrical (Zorn, 1996). Lachenbruch (2002) explains

that "ZIP regression inflates the number of zeros by mixing point mass at 0 with a Poisson

distribution" (p. 12). Zorn (1996, p.4) clarifies the distinction between the ZIP and Hurdle

models as follows:

As a special case of the general model, the ZIP regression is thus seen to make
substantially different assumptions about the nature of the data generating process than the
hurdle model. Whether parameterized as a logit or a probit, the probability exiting the zero-
only stage is assumed to follow a symmetric cumulative distribution. Likewise, even those
cases which make the transition to the events stage may nevertheless have zero counts;
crossing the "hurdle" in the ZIP model does not guarantee a positive realization of Y ..
The sole difference in assumptions here is that the hurdle model's count distribution is
assumed to be truncated at zero whereas the ZIP specification count data may take on zero



20 It is not the case that the negative binomial Poisson model is nested within the Hurdle model: hence, one can not
statistically compare all four models collectively.









values in the event stage.

Another difference is that, unless the ZIP model is overly parameterized, only the Hurdle

model can handle zero deflation (Min & Agresti, 2005).

Compared to the Hurdle model, the equations for the event stage are very similar. The

exception is that (1-p,) is divided by (1- e ) in the Hurdle model before being multiplied by the

remaining elements of the equation. However, the transition stage equations are strikingly

different. For the Hurdle model, the equation is PrOy, = 0) = p; the ZIP model includes the

addition of the probability of a nonzero multiplied by the exponentiated Poisson mean. This is

the mathematical characteristic that distinguishes the Hurdle model's exclusion of zeros in the

event stage and the ZIP model's potential inclusion of zeros in the event stage. Rather than

model the probability of a zero in the transition stage, the ZIP also models the probability that the

counts have a Poisson distribution hence permitting zeros from both a perfect state and a Poisson

state (Hur, 1999). Given this, 32 parameterizes the mean of this Poisson distribution (Welsh,

Cunningham, Donnelly, & Lindenmayer, 1996). When adding covariates, the Hurdle and ZIP

generalized linear model appear the same.


Transition Stage: log( ) = xllB, (2-47)
1 p,
Event Stage: log(il,) = x22B2 (2-48)

Unlike the Hurdle model, the ZIP model likelihood function can not be maximized

separately for the transition and event stage. Hence, the Hurdle model "has the attractive

advantage of an orthogonal parameterization which makes it simpler to fit and interpret than the

mixture model" (Welsh, Cunningham, Donnelly, & Lindenmayer, 1996) with the disadvantage

of asymmetrical transition to counts. The likelihood function derived by Lambert (1992) is










L=Clog(ewr +exp(-e ))+C(y:BS-e a)-[log(1+ear)-[loggy!) (2-49)
Y;=o y,>0 I=1 y,>0
where PB is the vector of coefficients and matrix of scores for the event stage and 7G is the

vector of coefficients and matrix of scores for the transition stage, and where iterations are based

on the EM or Newton-Raphson algorithms (Min, 2003; Lambert, 1992)21

Strictly speaking, the Poisson model is not nested within the ZIP model; therefore, it would

not be wise to conduct a formal model fit test (Zorn, 1996; Greene, 1994). However, it is

interesting to note that the log-likelihood of 10,607 is slightly lower than that produced by the

negative binomial Hurdle model. This is in line with Greene' s (1994) observation when using the

Vuong statistic as an alternative for testing nonnested models. "For present purposes, the

important question is whether the ZIP models .. provide any improvement over the basic

negative binomial .. The log-likelihood functions are uniformly higher, but as noted earlier,

since the models are not nested, these are not directly comparable. The Vuong statistic, however,

is consistent with the observation" (Greene, 1994, p.26).

The Negative Binomial Zero-Inflated Poisson model

The ZIP model can be extended to the negative binomial model just as the Poisson was

extended to the negative binomial and as the Hurdle was extended to the Hurdle negative

binomial. This may be necessary as Min (2003) explains that "Sometimes such simple models

for overdispersion are themselves inadequate. For instance, the data might be bimodal, with a

clump at zero and a separate hump around some considerably higher value. This might happen

for variables for which a certain fraction follows some distribution have positive probability of a

zero outcome" (p. 13). He further explains, "The equality of the mean and variance assumed by



21 Alternatively, Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and Possingham (2005),
compared relative means and credible intervals estimate from a bootstrap procedure.









the ZIP model .. is often not realistic. Zero-inflated negative binomial models would likely

often be more appropriate than ZIP models" (p.15). Unfortunately, current iterative techniques

lead to a greater risk of nonconvergence than when using the other two-part models (Fanoye &

Singh, 2006; Lambert, 1992).

The key assumption in accepting these results over those produced from the Hurdle models

is that there are some zeros that belong with the counts representing no household members

under the age of 21 for reasons other than their never having such household members at all.

Although it is not valid to statistically compare the fit of the ZIP model with the Hurdle and

Poisson models, it is reasonable to test the fit of the ZIP model within the negative binomial ZIP

model .

In sum, the choice between Hurdle models and ZIP models is ultimately guided by the

assumptions one makes about the data generating process. Min (2004) states, "The zero-inflated

models are more natural when it is reasonable to think of the population as a mixture, with one

set of subj ects that necessarily has a 0 response. However, they are more complex to fit, as the

model components must be fitted simultaneously. By contrast, one can separately fit the two

components in the hurdle model. The hurdle model is also suitable for modeling data with fewer

zeros than would be expected under standard distributional assumptions" (p.20).

Model Comparison Testing for Zero-Inflated Data

The Poisson model is nested within the negative binomial Poisson differing only by the

dispersion parameter. In fact, the two models are equivalent when one uses the negative binomial

model and restricts the dispersion parameter to 1.0 achieving Poisson equidispersion (Cameron

& Trivedi, 1998). Likewise, the Hurdle model is nested within the negative binomial Hurdle

model, and the ZIP model is nested within the negative binomial ZIP model. However, since

each involves estimation of a transition stage and event stage model, the nesting rule applies to









both equations. In other words, if the event stage contains 3 parameters and the transition stage 4

parameters for the nested model, then the more complex model must contain at least these same

3 parameters in the event stage and at least the same 4 parameters in the transition stage.

According to Zorn (1996), the Poisson model is nested within the Hurdle model, and the

negative binomial Poisson model is nested within the negative binomial Hurdle model. This is

reasonable given that the Hurdle models are estimating the Poisson models in the event stage and

that these likelihood statistics are independent of those produced in the transition stage. The ZIP

models, on the other hand, are not estimated in this manner; it is not reasonable to assume that

the Poisson models are nested within the ZIP Models (Greene, 1994). This leads to the hierarchy

of permissible model-testing displayed in Table 2-1. Other models can be compared

descriptively using the aforementioned Akaike's Information Criterion (AIC). The remainder of

the dissertation will use the deviance statistic for model comparison inference and the AIC for

model comparison description.

Review of Research Pertaining to and Using Zero-Inflated Count Data

Hurdle Model

Statistical

As previously discussed, Mullahy (1986) presented the underlying statistical foundations

for the Hurdle model. He specified it in terms of extending from both the Poisson and geometric

distributions. He also presented an extension to the Hurdle model that he termed the "with-zeros"

(WZ) model. The WZ model adjusts for zero-inflation by augmenting or reducing the probability

by an additive constant (rather than having them specified by the parent distribution). Subsequent

research has focused on the Hurdle model rather than the WZ model; this is most likely due to

the fact that the WZ turns out to be a special case that collapses into a Hurdle model in some

specifications, and estimates are often similar between the models (Mullahy, 1986). Finally,









Mullahy presented tests for specifications including a technically complex information matrix

test as well as the Score test typically provided in software output. King's (1989) key

contribution to the Hurdle model was a Monte Carlo study confirming that the counts can be

viewed as arising from a Poisson process (Civetinni & Hines, 2005).

Min and Agresti (2004) conducted a simulation to compare a zero-inflation and a zero-

deflation condition. They found that the estimates were reasonable for the ZIP model under zero-

inflation. However, the coefficient and standard error for the event stage were both very large

under zero-deflation. They explain that logit models simply can not accommodate too few zeros.

"The zero-inflated model is only suitable for zero-inflation problems. However, the hurdle model

is also suitable for modeling data with fewer zeros than would be expected under standard

distributional assumptions. In fact, when a data set is zero-deflated at some levels of the

covariates, the zero-inflation model may fail" (Min & Agresti, 2004, p.5).

In contrast, Mullahy (1986) stated that, "a particularly interesting feature of the modified

count data specifications considered here [i.e., Hurdle model] is that they provide a natural

means for modeling overdispersion or underdispersion of the data. Specifically, overdispersion

and underdispersion are viewed as arising from a misspecification of the maintained parent [data

generating process] in which the relative probabilities of zero and non-zero (positive) realizations

implied by the parent distribution are not supported by the data" (Mullahy, 1986, p.342).

Applications

Mullahy (1986) researched daily consumption of coffee, tea, and milk where each is a

count variable. Covariates included age, years of completed schooling, family income, sex, race,

and marital status. For the Hurdle model, the test statistics [p-values] for the coffee-model were

substantially smaller than those for tea-model and milk-model. Mullahy argued that this is not

surprising given the ratio of estimates and standard errors. However, closer inspection of the data









reveals that it is the coffee variable that has the lowest proportion of zeros at only 26.26%. It is

possible that the Hurdle model was unable to adequately account for the additional

overdispersion witnessed in the other two models (61.63% and 40.37% zeros). In other words, a

two-part model may be a necessary but not sufficient condition for handling overdispersion in

zero-inflated count models, and negative binomial formulations may be increasingly necessary as

the proportion of zeros increases.

King (1989) applied the Hurdle model to data for the relationship between the number of

nations entering war in a period of time as a function of those in formal international alliances.

The hurdle model was formulated based on the premises of Mullahy (1986) and justified due to

there being some countries who will not go to war and others who will not at first but will later

be 'dragged in' by alliances. Hence, this is a classic example of the justification for true zeros

and event-driven zeros. This was demonstrated statistically by comparing Hurdle and Poisson

results. The Poisson alliance coefficient of .007 was significant, the Hurdle model event-stage

coefficient of .007 was significant, and the Hurdle model transition stage coefficient of .001 was

not significant. Hence, the Poisson interpretation would be that increased alliances lead to

increased war frequency. However, the Hurdle results clarify that this is only true after the onset

of war (i.e., the hurdle has been crossed). Further statistical evidence supported the Hurdle model

based on the likelihood-ratio model comparison test.

Zero-Inflated Poisson Model

Statistical

As previously discussed, Lambert (1989) presented the original formulation for the ZIP

model and its ZIP(z) formulation. She also presented the models' extension from the Poisson and

negative binomial as well as the derivation of the maximum likelihood (EM) estimates. She ran

several simulations to test the adequacy of the model. The first simulation varied sample size,









with one covariate taking on a fixed coefficient and a fixed variance in both parts of the model.

The result was an average of 50% zeros in the transition stage and 23% zeros in the event stage.

The results suggest that the ZIP model consistently converges at n=25 when using EM and at

n=100 when using the Newton-Raphson algorithm. An examination of confidence intervals

revealed that the normal-theory intervals are not reliable at n=100; however, almost all simulated

likelihood-ratio confidence intervals contained the true mean even at n=25. "To summarize,

these simulations with one covariate for both 32 and p are encouraging. The ZIP and ZIP(z)

regressions were not difficult to compute, and as long as inference was applied only when the

observed information matrix was nonsingular, estimated coefficients, standard errors based on

observed information, and estimated properties of Y could be trusted" (Lambert, 1992, p.7).

Warton (2005) compared 20 datasets of varying sample sizes, proportions of zeros, and

factors/levels. The ordinary least squares version included the addition of one to all counts before

taking the logarithm. The other models not accommodating zero-inflations were the Poisson and

four formulations of the negative binomial Poisson (including the aforementioned quasi-Poisson

where the variance is set to
negative binomial ZIP model. The Akaike Information Criterion (AIC) values were calculated

for a total of 1,672 variables averaged over datasets and rescaled to a minimum AIC of zero for

each dataset.

As expected, when overdispersion was present, the negative binomial formulations

outperformed the models without these formulations. However, when overdispersion was not

present, the reverse was true for 53% of the variables. This suggests that the level of skew in the

model interacts with zero-inflation when measuring model adequacy.









When the proportion of zeros is very small, the distribution looks more like a Poisson

distribution truncated at zero. In other words, it shares features modeled by the event stage of a

Hurdle model. This led to estimation problems in which the negative binomial model ZIP model

rarely converged. When it did converge, its fit was better than that of the ZIP model for only

1 1% of the datasets. Possibly, as previously discussed, the zero-inflated Hurdle would converge

more often since it can handle both zero-inflation and zero-deflation (Min & Agresti, 2005; Min

& Agresti, 2004).

A very interesting finding pertained to the transformed OLS fit indices. "Although

transformed least squares was not the best fitting model for data, it fitted the data reasonably

well. Surprisingly, transformed least squares appeared to fit data about as well as the zero-

inflated negative binomial model .. The AIC for transformed least squares was not as small as

for the negative binomial model overall, although it was smaller for 20 per cent of the variables

considered here" (Warton, 2005, p.283).

Averaging over all datasets, the AIC was lowest for all negative binomial models

followed by a close tie between the transformed OLS model and the negative binomial ZIP

model, which was followed by the ZIP model. All models were a drastic improvement over the

standard Poisson model. The implications are that, although the Poisson is rarely adequate when

the data is not equidispersed and/or is inflated or deflated, an intuitive climb up the ladder of

models may not be reasonable. There were features of these datasets including varying degrees

of zero-inflation and overall distributions that warrant further investigation toward appropriate

model selection. "If one were to fit a zero-inflated model, it would be advisable to present

quantitative evidence that the zero-inflation term was required. Based on the present results, it is

likely that a term for extra zeros is not needed, and a simpler model will usually suffice ..










special techniques are not generally necessary to account for the high frequency of zeros. The

negative binomial was found to be a good model for the number of zeros in counted abundance

datasets, suggesting that a good approach to analyzing such data will often be to use negative

binomial log-linear models" (Warton, 2005, p.287-288).

In regard to their problems with zero-inflated negative binomial convergence, Fanoye and

Singh (2006) developed an extension that improves convergence termed the zero-inflated

generalized Poisson regression (ZIGP) model. Their recent research revealed convergence in less

than 20 iterations for all trials. However parameter estimates and standard errors were often very

different than those produced by the ZIP model. They conclude, "Even though the ZIGP

regression model is a good competitor of ZINB regression model, we do not know under what

conditions, if any, which one will be better. The only observation we have in this regard at this

time is that in all of the datasets fitted to both models, we successfully fitted the ZIGP regression

model to all datasets. However, in a few cases, the iterative technique to estimate the parameters

of ZINB regression model did not converge" (p.128).

Greene (1994) used the Vuong statistic when comparing the Poisson, negative binomial

Poisson, and ZIP models. It was noted that the rank order for the Vuong statistics and the log-

likelihood estimates were in alignment. The conclusion suggested future research using the

Vuong statistic. "The use of Vuong' s statistic to test the specification seems not to have appeared

in the recent literature .. We conjecture that the Vuong testing procedure offers some real

potential for testing the distributional assumption in the discrete data context. In the cases

examined, it appears to perform well and in line with expectations" (Greene, 1994, p.30).

Shankar, Milton, and Mannering (1997) used the Vuong statistic to decide between the

negative binomial, ZIP, and negative binomial ZIP model for traffic accident data. They clarify









the interpretation of the statistic stating, "A value >1.96 (the 95% confidence level of the t-test)

for V favors the ZINB while a value < -1.96 favors the parent-NB (values in between 1.96 an -

1.96 mean that the test is indecisive) .. This test can also be applied for the ZIP(z) and ZIP

cases" (p.831).

Civettini and Hines (2005) explored the effects of misspecification on negative binomial

ZIP models. This included misspecification by leaving a variable out of the event stage that was

present in the event stage and misspecification by shifting a variable from the transition stage to

the event stage.

Applications

Lambert (1992), in formulating the ZIP model, applied it to the analysis of defects in

manufacturing. In terms of improperly soldered leads, 81% of circuit boards had zero defects

relative to the 71% to be expected under a Poisson distribution linked to a model with a three-

way interaction. This most complicated model had a log-likelihood of -638.20. This

dropped to -511.2 for the ZIP model. Although comparing to a different combination of

covariates, the negative binomial Poisson model fit better than the Poisson model but not as well

as the ZIP model.

Greene (1994) used credit-reporting data to investigate differences between the Poisson,

negative binomial Poisson, ZIP, negative binomial ZIP, as well as some of their aforementioned

variants and the specification of a probit link rather than the logit link. The data consisted of

1,023 people who had been approved for credit cards. The count variable of concern was the

number of maj or derogatory reports (MDR), which is the number of payment delinquencies in

the past 60 days.

For this sample, 89.4% had zero MDR. Given a mean of 0. 13, this frequency of 804 is

nearly double the 418 we might expect in a Poisson distribution. The skew of 4.018 is reduced to









2.77 when ignoring the zeros while the mean increases to 1.22. As expected, the negative

binomial Poisson resulted in improved fit (based on the Vuong test statistic), increased standard

errors and different parameter estimates. The ZIP model resulted in slightly worse fit than the

negative binomial Poisson while remaining much better compared to the Poisson model. If all of

the overdispersion was due to unobserved response heterogeneity then the results should be

similar for the negative binomial ZIP model. However, this model produced the best fit of all.

It is interesting to note that, again, the standard errors increase while the parameter

estimates are different relative to the ZIP model. In fact, of the 6 parameters, 4 estimates

decreased, 2 increased, and 1 switched in sign. Hence, there are two implications. First, the

negative binomial ZIP model was necessary to accommodate two sources of overdispersion to

adjust standard errors. Second, ignoring the negative binomial formulations would have led to

nonsensical parameter estimates driven by a sample mean of 0. 13.

Boihning, Dietz, Schlattmann, Mendonga, and Kirchner (1999) compared pre- and post-

intervention scores on the decayed, missing, and filled teeth index (DIVFT) for 797 children in

one of six randomly assigned treatment conditions. The results were not exhaustive; however,

the log-likelihood did decrease from -1473.20 to -1410.27 when going from the Poisson model to

the ZIP model. This study was somewhat unique in that all the covariates (sex, ethnicity, and

condition) were categorical, and that the conditions were dummy-coded represented as five

parameters. Also, this data had features that might suggest that zero-inflated models weren't

necessary. For pre-intervention, the proportion of zeros was 21.58%, which increased to only

28.99% at post-intervention. The means, with the zeros in the data, were 3.24 and 1.85,

respectively. Ignoring the zeros changed these means to 4.13 and 2.61, respectively. The skew,

with the zeros in the data, was 0.20 and 0.65, respectively. Ignoring zeros changed the skew to









0.08 and 0.63, respectively. In other words, many features of the data were consistent with what

would be expected of a Poisson, and possibly normal, distribution. Nonetheless, with these

means and frequencies, the Poisson distribution suggests overdispersion with 31 permissible

zeros for the pre-intervention and 125 permissible for the post-intervention whereas the data

revealed 173 zeros and 232 zeros, respectively. It then becomes a matter of whether the

overdispersion was due to the proportion of zeros in each condition or unobserved heterogeneity

in the event stage. The negative binomial ZIP model was not used to analyze this data.

Xie, He, and Goh (2001) analyzed the number of computer hard disk read-write errors.

Approximately 87% of the 208 cases were zeros. Given that the Poisson mean was 8.64, the

authors noted that the ZIP model is to be preferred over the Poisson model. However, this mean

is due to several values between 1 and 5, a few between 6 and 15, and 2 values of 75. These

latter two values appear to be so far from the others that they should have been treated as outliers

and addressed in some other manner.

Jang (2005) analyzed the number of non-home based trips per day from 4,416 households

in Jeonju City, Korea. The provided bar graph suggested that approximately 45% of the cases

were zeros. The Vuong statistic (Vuong, 1989) was used for model selection given that the

Poisson is not nested within the ZIP or negative binomial ZIP models.

The purpose of the article by Delucchi and Bostrom (2004) was to provide a brief

introduction to many possible methods for handling zero-inflation including standard t-tests,

bootstrapping,22 and nonparameteric methods. In doing so, they provided results from a study

involving 179 patients with opioid dependence assigned to either a methadone-maintenance or

methadone-assi sted-detoxification treatment. Five out of seven ways to segment the sample


22 See Jung, Jhun, and Lee (2005) for bootstrap procedures and simulation results for Type I and Type II errors.









resulted in zero-inflation ranging from 17% to approximately 66% zeros. The only two-part

model to be used was the ZIP model. The table of results revealed that, in terms of p-values, the

ZIP model performs either very similarly or very differently from the Pearson X2 test for the

proportion of zero values, the Mann-Whitney-Wilcoxon test of nonzero values, and/or the Mann-

Whitney-Wilcoxon test of difference in mean scores between treatment groups. It is possible that

these tests become more similar as the proportion of zeros declines but such conclusions are

based purely on the table of p-values.

Desouhant, Debouzie, and Menu (1998) researched the frequency of immature weevils in

chestnuts. One tree was measured over 16 years, another was measured over 11 years, and three

trees were measured on 1 year. The means ranged from .06 to .63. "None of the 30 distributions

fits a Poisson, X2 ValUeS being always very significant .. The ZIP distribution fits 25 out of 3 1

cases .. The NB distribution fits 20 out of the 31" (Desouhant, Debouzie, & Menu, 1998,

p.3 84). This led to the conclusion that researchers should consider both true zeros and

overdispersion (i.e., trees noted as 'contagious' and trees varying in random oviposition

behavior).

Shankar, Milton, and Mannering (1997) analyzed a 2-year summary of traffic accident

frequencies. For principal arterials, they chose a negative binomial model with data ranging from

0 to 84 (M~= 0.294, SD = 1.09). For minor arterials, they chose the negative binomial ZIP model

for data ranging from 0 to 7 (M~= 0.09, SD = 0.346). For collector arterials, they chose the ZIP

model for data ranging from 0 to 6 (M~= 0.61, SD = 0.279). Model selection was based on the

Vuong statistic. For example, they state, "As suspected previously, inherent overdispersion in the

data is due to the parent NB process and this was validated when the [negative binomial ZIP]










specification failed to provide a statistically better fit (the Vuong statistic <1.96, which

corresponds to the 95% confidence limit of the t-test" (p.833).

Slymen, Ayala, Arredondo, and Elder (2006) analyzed percent calories from fat and

number of days of vigorous physical activity from 357 females participating in a baseline

condition and one of three treatment conditions. Covariates included employment status,

education, martial status, cigarette smoking, and self-reported health. The zero-inflation was 294

out of 357 (82.4%). They compared models using likelihood ratio tests between the Poisson and

negative binomial Poisson and likewise between the ZIP and negative binomial ZIP. The AIC's

were inspected to compare the Poisson and ZIP models.

Not surprisingly, the negative binomial model fit better than the Poisson model. However,

the ZIP model did not fit better or worse than the negative binomial ZIP, and the parameter

estimates and standard errors were nearly identical. This suggests almost no overdispersion in the

data. Indeed, the nonzero percentages were as follows: 1 = 2.8%, 2 = 3.4%, 3 = 4.8%, 5 = 2.0%,

6 = 0.0%, and 7 = 2.0%. This suggests strong equidispersion leaning toward a uniform nonzero

distribution. The AIC's for both models were also nearly equal although both being considerably

smaller than the AIC for the Poisson model and somewhat smaller than the AIC for the negative

binomial Poisson model. Based on a 'smaller-is-better' heuristic, the authors favored the ZIP

model with an AIC of 562.5 over the zero-inflated ZIP model with an AIC of 565.

ZIP and Hurdle Model-Comparisons

The purpose of this section is to present a breadth of literature in which both the Hurdle

and ZIP models were either compared statistically and/or used to analyze real data. This also

includes extensions such as the negative binomial and tau formulations (e.g., ZIP(r)). Some

authors presented alternatives that seem to depart from the ZIP and Hurdle models too drastically

to be within the scope of this dissertation. For example, Lachenbruch (2001) used a two-part










model; however, the splitting formulation was not consistent with the literature. Further, the

model was compared to atypical formulations such as the Wilcoxon, Kolmogorov-Smirnov, and

z tests. As such, these types of articles are not included in the subsequent review. One exception

is Xie, He, and Goh (2001) who included a likelihood-ratio test for comparing the Poisson and

ZIP models.

Statistical

Greene (1994) proposed several 'zero-altered count models' for comparison. First, he took

Mullahy' s with-zero' s (WZ) adaptation of the Hurdle model and included a scalar estimate for

ease on computational burdens. Greene also presented an adaptation of Lambert' s ZIP known as

ZIP(r) and modified it for the negative binomial formulations terming them ZINB and

ZINB(r ). The intention was to identify a "procedure which will enable us to test the zero

inflated model against the simple Poisson model or against the negative binomial model. The

latter will allow us to make a statement as to whether the excess zeros are the consequence of the

splitting mechanism or are a symptom of unobserved heterogeneity" (Greene, 1994, p.10).

Greene developed a method for comparing the models; however, he noted that there was no a

priori reason to think that the Vuong statistic would be inferior.

Applications

Zorn (1996) examined the counts of actions taken by Congress addressing Supreme Court

decisions between 1953 and 1987. The zeros were seen to arise from two sources since many

cases will not be addressed unless there are lobbyists to pressure redress. Covariates included the

year of the decision, the political orientation of the decision, the presence of lower court

disagreement, the presence of precedence alteration, declaration of unconstitutionality, and

unanimous vote. The number of actions ranged from 0 to 11 (M~= 0. 11, SD = .64); however,

3,882 (95.8%) of the 4,052 counts were zeros. This contributed to an exceptionally high skew of









7.97. When ignoring the zeros, the skew was reduced to 1.86 (M~= 2.59, SD = 1.53). The

observed zeros were 107% of that which would be Poisson-expected.

Regardless, Poisson model results were in line with theory-driven expectations. However,

the test of overdispersion was significant when comparing the Poisson and negative binomial

Poisson resulting in fewer significant predictors than if ignoring overdispersion. The author also

fitted a generalized negative binomial model in which "the variance parameter is allowed to vary

as an exponential function of the same independent variables included in the model of the count"

(Zorn, 1996, p.9), which led to even better model Sit. However, due to zero-inflation, no model

provided reasonable estimate sizes given the low mean count.

Their analyses using the ZIP and Hurdle models yielded several findings. First, the

probability of remaining a zero in the transition stage was considerably lower for the Hurdle

model than for the ZIP model at lower levels of a predictor. This is a reflection of the asymmetry

of the Hurdle model. Second, parameter estimates and standard errors were similar between the

two models. They concluded that "at least in some circumstances the performance of ZIP and

hurdle Poisson models will be quite similar. This suggests that, as a practical matter and barring

any strong theoretical considerations favoring one over the other, the choice between them may

be made largely on the basis of convenience of estimation" (Zorn, 1996, p. 11).

Pardoe and Durham (2003) compared the Poisson, Hurdle, and ZIP models as well as

their negative binomial formulations using wine sales data. Of the 1,425 counts, 1,000 (70.2%)

were zeros. The authors noted that this is greater than the 67.8% that a Poisson distribution is

capable of predicting. Based on the AIC, the zero-inflated negative binomial ZIP performed best.

Surprisingly, the Hurdle model fit more poorly than did the Poisson model. It is possible, given

the unrealistically high AIC relative to the other models, that the value wasn't calculated









correctly. Alternatively, the distribution may not have been correctly specified since their

analysis included Bayesian estimates of the prior distributions. No discussion pertaining to the

Hurdle model was included. However, they did provide a novel procedure for comparing the

range of fit statistics across the zero-inflation models. This 'parallel coordinate plot for goodness

of fit measures' consists of an x-axis labeled Min on the left and Max on the right. The y-axis is

a series of horizontal lines each pertaining to a fit statistic (e.g., AIC, BIC). The ceiling x-axis

contains the labels for the models being compared. Then, points for each model are plotted on

the lines for the fit statistics at their relative location between Min and Max. A similar procedure

restricted to the AIC was used by Warton (2005). This technique was adapted to display the

coverage for simulated fit statistics.

Welsh, Cunningham, Donnelly, and Lindenmayer (1996) used zero-inflated rare species

count data to compare the Poisson, negative binomial Poisson, Hurdle, negative binomial Hurdle,

and ZIP models. Approximately 66% of the observations were zeros. They found little difference

between the Hurdle, negative binomial Hurdle, and ZIP model results. Since there was no

overdispersion, the authors recommended using the Poisson model. This is in line with Warton's

(2005) assertion that the more complex zero-inflation models may not always be necessary; at

least, this appears to be the case with 66% zero-inflation and equidispersion.

Discrepant Findings

What is exactly meant by zero-inflation? Min and Agresti (2005) define zero-inflated

count data as "data for which a generalized linear model has lack of fit due to disproportionately

many zero" (p. 1). This raises the question, "At what point does the frequency of zeros become

disproportionate to the frequency of non-zeros?" One statistical definition states that the

proportion of zeros is greater than that to be expected given the posited distribution (Zorn, 1996).

For example, for count data, the proportion of zeros should not be greater than that expected by a









Poisson distribution. However, there are three problems with this. First, there may be many

different proportions of zeros greater than that expected by a particular Poisson distribution.

Second, the definition assumes that the full model is distributed Poisson. Third, it ignores the two

potential sources of overdispersion for Poisson zero-inflated data.

The aforementioned AEWR example displayed a zero proportion of .7158 with a mean of

.54, a standard deviation of 1, and a skew of 1.971. Ignoring the zeros, although this event stage

distribution remains negative skewed, the mean increased to 1.91, and the level of skew dropped

to 0.96. The distribution for the categorical sex variable was binomial with approximately 43%

males and 47% females. The distribution for the age variable was roughly normal with a mean of

48.86, a standard deviation of 17.41, and a skew 0.87. Hence, with 71% zeros, a heavily skewed

distribution of 1.971, a moderately skewed nonzero distribution of 0.96, a normally distributed

continuous predictor, and a two-level categorical predictor led to the following findings:

1) the Hurdle model fit better than Poisson model; 2) the negative binomial Hurdle fit better than

negative binomial Poisson model; 3) the negative binomial Hurdle fit better than the Hurdle

model; 4) the negative binomial ZIP fit better than ZIP model; 5) the negative binomial ZIP

model descriptively fit better than all others; and, 6) the Hurdle and negative binomial Hurdle

model yielded nearly identical estimates and p-values. Hence, findings between the zero-

inflation models differed in terms of both fit and the significance of parameter estimates.

Although not all research presented sufficient information (e.g., data necessary to calculate

skew), there is clearly enough variation in results to warrant further research. Mullahy's (1986)

Hurdle model analyses were impacted by zero-inflation of .263 and not by zero-inflation of .616

or .404; however, this is in disagreement with findings that the Hurdle model adequately handles

zero-deflation (Min, 2003). Lambert' s ZIP analysis with 71.8% zeros favored the ZIP over the










negative binomial ZIP. Greene's (1994) ZIP analyses resulted in nonsensical results under .894

zeros and heavy skew (4.02); the negative binomial ZIP corrected this. Slymen, Ayala,

Arredondo, and Elder' s (2006) ZIP and negative binomial ZIP results were virtually identical;

however, their event stage distribution was uniform. This was confirmed by Warton's (2005)

finding that the negative binomial fits better than the ZIP only when zero-inflation and

overdispersion both are indeed present. Extending from this is Boihning, Dietz, Schlattmann,

Mendonga, and Kirchner' s (1999) findings that the ZIP actually fit better than the Poisson given

.216 and .289 zero-deflation. This is again in contrast to the suggestion that the Hurdle, and not

the ZIP, is appropriate for zero-deflated data. However, it could be argued that a normal

distribution should have been assumed given that the event stage distribution was relatively

normal .

When comparing the Hurdle model to the ZIP model, Zorn (1996) found similar results

given .958 zero-inflation, skew of 7.97, and a reduction of skew to 1.86 for the event stage.

These findings are in contrast to Zorn' s (1996) findings of greater zero-inflation and,

subsequently, greater skew. Welsh, Cunningham, Donnelly, and Lindenmayer (1997) also found

little difference between the Hurdle and ZIP models. Table 2-2 summarizes the findings from the

zero-inflation literature.

There are three factors that may have caused the anomalous findings. The first possibility

pertains to the effect of different types, quantities, and values for predictors. The second

possibility is the proportion of zeros for the outcome variable. The third possibility is the degree

of skew in the event stage for the outcome variable.

It has already been suggested that the Hurdle and ZIP models should be chosen given a

priori research about the source and nature of the zeros. Further, it has been established that the










negative binomial formulations are meant to handle additional overdispersion in the event stage.

However, the previous findings suggest that there are additional considerations such as the

proportion of zeros and the nature of the event stage distribution. The proportion of zeros in this

research ranged from as low as .20 (Delucchi & Bostrom, 1994) to .958 (Zorn, 1996).

Distributions for the event stage included those that were heavily positively skewed (Greene,

1994), moderately positively skewed (AEWR example), distributed normally (Boihning, Dietz,

Schlattmann, Mendonga, & Kirchner, 1999), and distributed uniformly (Slymen, Ayala,

Arredondo, & Elder, 2006).

The first possibility, pertaining to the covariates, can only be tested by varying an

incredibly large set of conditions ranging from small to large quantities of predictors as well as

their types (e.g., nominal, ordinal), and distributions. However, given a particular set of

covariates and corresponding values, the other two possibilities pertaining to zero-inflation and

skew can be explored. It is possible to vary the proportion of zeros for the outcome variable and

to simultaneously vary the degree of skew of the nonzeros for this outcome variable. Hence, the

following research questions are presented:

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated log-likelihood between a) the
Negative binomial Poisson model vs. Poisson model; b) the Hurdle model vs. Poisson
model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the
Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model
vs. ZIP model?

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated AIC between all models?

These questions were answered by establishing several levels of skew and zero-inflation.

The covariates and their values were Eixed as one continuous variable from a standard normal

distribution and one binary variable. Data for the outcome variable, for all levels of skew and









zero-inflation, were simulated to estimate log-likelihood values, AIC indices, covariate

coefficients, standard errors, and p-values. The details are delineated in the following chapter.

The obj ective is consistent with Zorn' s (1996) advice: "First and foremost, work should be

undertaken to better ascertain the statistical properties of the various estimators outlined here. It

is important that we determine the robustness of these techniques to skewness in the dependent

variable, model misspecification, and the host of other problems that all too frequently plague

political science researchers .. perhaps using Monte Carlo methods to assess under what

circumstances the results of the two may diverge" (Zorn, 1996, p. 12).









Table 2-1. Five pairs of nested models valid for statistical comparison
Valid Comparisons (Nested Models)
1 2 3 4 5
Poisson Simple Simple
NB Poisson Complex Simple
Hurdle Simple Complex
NB Hurdle Complex Complex
ZIP Simple
NB ZIP Complex















































NB ZIP over
OLS/NB ZIP
over Poisson
Equal


Zeros
Simulation

Simulation

.26, .62, .41
.718


.894


.824


.87


Superior Model Comments


Models Compared
Hurdle vs. ZIP

Hurdle vs. ZIP

Hurdle
ZIP vs. NB Poisson
vs. Poisson

ZIP vs. NB ZIP vs.
NB Poisson

Poisson vs. NB
Poisson; ZIP vs. NB
ZIP
ZIP vs. Poisson

Poisson vs. ZIP





Hurdle vs. ZIP
Poisson vs. Hurdle
vs. ZIP

ZIP vs. NB ZIP

ZIP vs. NB ZIP

NB ZIP vs. OLS vs.
NB ZIP vs. Poisson

Hurdle vs. ZIP


Table 2-2. Summary of literature on zero-inflation


Researchers)
Min & Agresti
(2004)
Min & Agresti
(2004)
Mullahy (1986)
Lambert (1992)


Greene (1994)


Slymen, Ayala,
Arredondo, and
Elder (2006)
Xie, He, and Goh
(2001)
Boihning, Dietz,
Schlattmann,
Mendonga, and
Kirchner' s
(1999)
Zorn (1996)
Pardoe and
Durham (2003)

Warton (2005)

Warton (2005)

Warton (2005)


Wel sh,
Cunningham,
Donnelly, and
Lindenmayer
(1997)


Hurdle


Zero-deflation

Zero-inflation


Equal


Hurdle .26
ZIP over NB
Poisson over
Poisson
NB ZIP over ZIP;
NB Poisson over
ZIP over Poisson
NB Poisson;
Equal

ZIP


Heavy skew;
Probit link

Uniform event
stage; Overall,
AIC's favor ZIP
Outliers

Zero-deflation;
normal event stage




Heavy skew
Based on AIC's


Overdi sp ersi on
favored NB ZIP
Rare convergence
for NB ZIP
Based on AIC's


No overdispersion


.216, .289 ZIP


Equal
NB ZIP over
Poisson over
Hurdle
ZIP or NB ZIP


.958
.702


Various


Very low ZIP


Various


.66









CHAPTER 3
METHODOLOGY

The previous chapter fulfilled three obj ectives. First, it described the statistical models and

methods for analyzing count data including that which is zero-inflated. Second, it presented

research, both technical and applied, pertaining to three models (Poisson, Hurdle, and ZIP) as

well as their negative binomial formulations. Third, it was concluded that there is considerable

divergence in findings between models and that such differences should be explored by

examining different levels of zero-inflation and skew for the count outcome variable. This led to

the following research questions:

Research Questions

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated log-likelihood between a) the
Negative binomial Poisson model vs. Poisson model; b) the Hurdle model vs. Poisson
model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the
Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model
vs. ZIP model?

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated AIC between all models?

As recommended by Zorn (1996), these questions were answered using a Monte Carlo

study in which the proportion of zeros and the skew for the distribution of the event stage counts

varied between simulations.

Monte Carlo Study Design

Monte Carlo studies begin in the same manner as other research methods. First, a problem

is identified as a research question. Second, the problem is made concrete in the form of a

hypothesis or set of hypotheses. Third, the hypotheses are tested using rigorous methods. Fourth,

conclusions are drawn from these results. Fifth, the implications and limitations are elucidated

for future researchers and practitioners.









For this study, the problem was zero-inflation for count outcomes. The problem was then

clarified in the form of the research questions stated above. It is at this point that Monte Carlo

studies differ from most other methods. For the Monte Carlo study, no real data is gathered.

Rather, a set of samples are generated based on given parameter specifications resulting in a

sampling distribution that is considered to be equivalent to that which would have been obtained

had this many real participants been available. This is a clear advantage over the true experiment

where statistics are used to infer from one sample to an entire population (Mooney, 1997). The

Monte Carlo study is not limited to a finite number of participants and is subsequently not prone

to violations of asymptotic theory (Paxton, Curran, Bollen, Kirby, & Chen, 2001). Paxton,

Curran, Bollen, Kirby, and Chen (2001, p.287) provide a succinct explanation as follows:

The researcher begins by creating a model with known population parameters (i.e., the
values are set by the researcher). The analyst then draws repeated samples of size Nfrom
that population and, for each sample, estimates the parameters of interest. Next, a sampling
distribution is estimated for each population parameter by collecting the parameter
estimates from all the samples. The properties of that sampling distribution, such as its
mean or variance, come from this estimated sampling distribution.

Similarly, Mooney (1997, p.2) explains,

Monte Carlo simulation offers an alternative to analytical mathematics for understanding a
statistic's sampling distribution and evaluating its behavior in random samples. Monte
Carlo simulation does this empirically using random samples fiom Imown populations of
simulated data to track a statistic' s behavior. The basic concept is straightforward: If a
statistic' s sampling distribution is the density function of the values it could take on in a
given population, then its estimate is the relative frequency distribution of the values of
that statistic that were actually observed in many samples drawn from that population.

The Monte Carlo simulations were performed using the R programming language (R

Development Core Team, 2006). R is an open-source language based on the commercially










available S-Plus program (Insightful, 2005) and is just one of many programs that can be used to

model zero-inflated data.23

Monte Carlo Sampling

Pseudo-Population

Mooney (1997) explains that defining the population parameters requires defining the

pseudo-population. In the pseudo-population, the values for a categorical factor, X1, with two

levels were constant at either 0 or 1; this is representative of a typical two-level categorical

factor such as sex. In the dataset, these values alternated. The Excel 2003 random number

generator was used to draw a random selection of 1,000 normally distributed values to represent

the continuous variable, X2~- N(0,1). This resulted in a pseudo-population of N = 1,000 where

the values for categorical X1 and continuous X2 were known. The values for X2 ranged from -

2.945 to 3.28 with a mean of 0, a median of 0.05, a standard deviation of 0.986, and skew of -

0. 127. These two sets of covariates and their distributions were chosen as a parsimonious

generalization to the basic ANCOVA general linear model that extends from analyses with either

quantitative or qualitative predictors.24 The simulations varied in terms of a) the amount of zero-

inflation present in the outcome variable scores; b) the amount of skew present in the event stage

outcome variable scores, and c) the generalized linear model.

The outcome variable, Y, was established as a deterministic variable in that it varied

systematically as a function of the specified distributions. As clarified by Mooney (1997),

"Deterministic variables are vectors of numbers that take on a range of values in a prespecified,

nonrandom manner" (p.6). The regression coefficients for X1 and X2 are random variables that

23 Others include Stata, LIMDEP, COUNT, MATLAB, and SAS. Preliminary analyses compared results between
SAS-code incorporated in research (Min & Agresti, 2004; Min, 2003) to publicly available R-code written by Simon
Jackman of the Stanford Political Science Computing Laboratory to verify the comparability of results.

24 The model is similar to that of the AEWR examples.










take on their realizations as a result of the relationship between deterministic Y and the two

random X covari ate s.

The Prespecified Zero Proportions

Justification for generating values with prespecified proportions of event counts with

frequencies also determined by prespecified proportions of zeros is justified by Mooney (1997).

"If we know (or are willing to make assumptions about) the components that make up a statistic,

then we can simulate these components, calculate the statistic, and explore the behavior of the

resulting estimates" (Mooney, 1997, p.67). In his case, the concern was bias determined by

calculating statistics and inspecting graphs as a result of Monte Carlo simulations. For this study,

the concern was goodness-of-fit for six models by comparing log-likelihoods and AIC's and

inspecting graphs as a result of Monte Carlo simulations over three levels of skew and five levels

of zero proportions.

Previous research displayed zero-inflation ranging from .20 (Mullahy, 1986) to .96 (Zorn,

1996). To reflect a range including both zero-deflation and zero-inflation, six pseudo-populations

were established differing in the proportion of zeros present in the count outcome variable. The

pseudo-populations contained either 0. 10, 0.25, 0.50, 0.75, or 0.90 proportions of zeros. 25

Pre-Specified Skew

To manipulate skew, the event stage distributions for the count outcome variable were

varied over three conditions. For each condition, proportions were specified and values were

drawn randomly from a multinomial distribution such that the frequencies of the values added to

the frequency for those already drawn to represent zero-inflation summed to 1,000. In other



25 Originally, a 0.00 proportion of zeros was included as a control condition. However, for the case of negative
skew, this is simply a count model truncated at one. And, for a normal distribution event-stage, this is semi-
continuous data often tested with different methods from those for zero-inflation (Min, 2002).










words, if .50 (or 500) observations were assigned a value of zero then the remaining .50 (or 500)

observations had values drawn from the prespecified multinomial distribution.

Event stage values ranging from one to five were sampled in order to represent a range

small enough to distinguish the distribution from one that might be analyzed as continuous given

a particular shape. The prespecified probabilities for each of the five counts were determined

primarily to achieve a particular level of skew and secondarily in order to achieve a convergent

and admissible solution. Hence, the proportions were not always exactly equal to .10, .25, .50,

.75, and .90; the approximates were selected to achieve convergence leading to more trust in

convergence for the Monte Carlo simulations. Table 3-1 displays the proportions of each count

as a function of the three levels of skew and five levels of zeros. Table 3-2 displays this same

information in terms of frequencies instead of proportions. Table 3-3 displays the descriptive

statistics for each distribution.

Random Number Generation

By definition, a random number is one in which there is no way possible to a priori

determine its value. Most statistical analysis software packages include random number

generators. However, these generated random numbers are not truly random. Usually, one

specifies a seed; when replications are performed using the same seed, the generated numbers are

identical to the first. Hence, the values are pseudo-random (Bonate, 2001). However, this

limitation is actually an advantage in that the researcher can check for errors in model

programming and run the analysis again with the same generated sample (Mooney, 1997).26





26 Technically, this is only true for most random number generators. The R programming language, by default, bases
its generation on the Wichman-Hill algorithm and the system clock resulting in a 626-integer seed (R Development
Core Team, 2006).









Another feature of Monte Carlo random sampling pertains to the desired distributions.

Typically, the random numbers are drawn from a uniform distribution, which is then followed by

a transformation to the desired distribution. As Mooney (1997) explains, "In its standard form,

U(0, 1), the uniform distribution is the building block of all Monte Carlo simulation work in that

from it, in one way or another, variables with all other distribution functions are derived. This is

because the U(0, 1) distribution with its 0 < x < 1 range, can be used to simulate a set of random

probabilities, which are used to generate other distribution functions through the inverse

transformation and acceptance-rejection methods" (p.10).

The random number generation was performed using R 2.3.1 (R Development Core Team,

2006). The procedure requires the generic sample command in which the following were

specified: 1) a range of counts, which in this study, was from one to five (not zero to five since

proportions of zeros were already drawn from the pseudo-population), 2) the number of values to

draw, which in this study was one minus the prespecified proportion of zeros, 3) the proportions

for each value in the range, which in this case was one of three possibilities determining skew,

and 4) the specification to sample with replacement. The seed was arbitrarily set at 6770.

Sample Size

Determining the appropriate sample size for each simulate is an important concern. This

could range from zero to infinity. However, if the sample size is too small then it is not safe to

assume that estimates are asymptotically normal. On the other hand, computer time and burden

increases as sample size increases. The sample size was based on the highest found in the

literature pertaining to zero-inflated count data, which was n = 1,000 (Civentti & Hines, 2005).

Simulation Size

Determining the number of simulations is also an important concern since too few

replications may result in inaccurate estimates and too many replications may unnecessarily









overburden computer time and performance (Bonate, 2001). Hur' s (1999) research pertaining to

the ZIP model with random effects set the number of simulations at 200. Min and Agresti (2005)

were able to sufficiently compare the goodness of fit for several competing models using 1,000

simulations. Likewise, Civettini and Hines (2005) selected 1,000 simulations when researching

misspecification in negative binomial ZIP models. Lambert (1989) set the number of simulations

at 2,000 when researching the asymptotic properties of the ZIP model.

Mooney (1997) states that "The best practical advice on how many trials are needed for a

given experiment is "lots"!i Most simulations published recently report upward from 1,000 trials,

and simulations of 10,000 and 25,000 trials are common" (p.58). Given the previously noted

problems with convergence for the negative binomial ZIP model, it seems prudent to minimize

the number of simulations as much as possible. However, it is also important to simulate under

conditions already found to produce asymptotic results. Hence, similar to Lambert' s (1989)

seminal study and equal to the maximum found in the literature, the number of simulations was

set at 2,000 for each condition (S = 2,000).

Iteration Size

Iteration size is not much of a concern for the basic Poisson and negative binomial

Poisson model. However, obtaining valid estimates for the Hurdle model, ZIP model, and their

negative binomial counterparts requires selecting an appropriate maximum number of iterations.

Too few iterations can lead to incorrect estimates or, even worse, premature declaration of

nonconvergence. Too many iterations results in unnecessary computer time and burden. Various

procedures for analyzing these models in R have maximum iterations of 500, 5,000, and 50,000.

It was important to determine an iteration size that would be equal across analyses and large

enough given some models' potential for nonconvergence. These concerns were deemed more










important than excessive computational burden. Hence, the procedure with the largest iteration

size was selected for all procedures leading to 50,000 iterations per analysis.

Distribution Generation

The following describes the procedure for generating the distributions for each

simulation.

* The total proportion of counts out of 1,000 to be sampled was reduced by the prespecified
proportion of zeros. Hence, if the proportion of zeros was .50 then the proportion of event
stage counts was 1.00-0.50 = .50. Translated into frequencies, this is 1,000-(0.50 1,000) =
500

* The generic R 'sample' procedure was used to sample with replacement from the event stage
counts according the specified proportions depending on the skew condition. The seed was
set at 6770.

* The values were sampled over N = 1,000.

* Each sample was simulated S= 2,000 times.

* The data over all S = 2,000 at N= 1,000 were then stored in separate files as they were
created. The filenames conformed to the labeling format where the model was replaced by an
underscore (e.g., _25Pos was the filename for the S = 1,000 datasets at N=2,000 where the
proportion of zeros was .25 and the skew for the event stage was positive).

Monte Carlo Models

As previously discussed, generalized linear models include a random component, a

systematic component, and a link function. The X1 and X2 COnstants form the systematic

component in the pseudo-population's generalized linear model. The random component

specification for the distribution of the outcome mean varied from pseudo-population to pseudo-

population. The base level generalized linear model assuming a normally distributed outcome is

given by

Y = Po + fl,(X,l )+ A(X23> + (3-1)









Subsequent models extended from this base model to form the six distributions for

deterministic Y The first model, which was the Poisson generalized linear model with a log

link, is given by

log(il) = o, + (XI)+ 2,(X27) (3 -2)

Table 3-4 displays the parameters for this model over all conditions of skew and zero-

inflation. For example, the analysis with a .10 proportion of zeros and a positively skewed event

stage distribution yielded

log(il,) = .450 +.004(X,) -.037(X2) (3-3)

For both predictors, the coefficient near zero transforms to an exponentiation near one. The

coefficient for X1 is lowest (.001) for the negatively distributed data with a .25 proportion of

zeros and highest (.007) for the positively distributed data with a .50 proportion of zeros. The

coefficient for X2 is lowest (-.007) for the negatively distributed data with a .10 proportion of

zeros and highest (-. 165) for the normally distributed data with a .90 proportion of zeros. Hence,

for the two-level categorical X1 variable, changing from zero to one multiplies the mean of the

outcome variable by approximately exp(0.00), which equals one. For the simulated data, the test

for the coefficient estimates corresponding to these pseudo-population parameter values is

approximately Ho: P = 0.

The second model, which was the negative binomial formulation of the Poisson model, is

the same as the Poisson model with the addition of a dispersion parameter. Table 3-5 displays the

parameters for this model over all conditions of skew and zero-inflation. Like the Poisson model,

the test for the simulated data is Ho: p = 0. The dispersion parameter is also included in Table 3-

5. For the simulated data, the test that this parameter equals zero has equivalent results to the

model comparison tests conducted.









The third and fifth models were the general formulations for the Hurdle and ZIP models

given by


logit (p, )= log( )= P, + 7,(Z,I)+ P(Z,I) (3 -4)
1 p,
log(l,) = P,+ P(X,I)+ P(X,I) (3-5)

while the fourth and six models were their negative binomial formulations. Tables 3-6 through 3-

13 display the pseudo-population coefficients, standard errors, log-likelihood values, and AIC

values over all conditions of proportions of zero and skew. Although there are four models (i.e.,

Hurdle, negative binomial Hurdle, ZIP, and negative binomial ZIP), there are eight tables since,

for each model, there are separate results for the transition (zeros) stage and events (nonzero

counts) stage.

Monte Carlo Analysis Procedures

A generic looping procedure was written in the R programming language to retrieve each

of the 15 sets of simulated data. Each dataset was analyzed with each of the six models. The

Poisson model used the generic glm procedure in R, which requires specification of the model,

Poisson distribution, and log link. The negative binomial Poisson model used the glm.nb

procedure in R from the generic M4rSS library. This procedure requires only the specification of

the model. The R Core Development Team (2006) describes the procedure as follows: "An

alternating iteration process is used. For given 'theta' [dispersion] the GLM is fitted using the

same process as used by 'glm()'. For fixed means the 'theta' parameter is estimated using score

and information iterations. The two are alternated until convergence of both is achieved" (R

Core Development Team, 2006).

The Hurdle model used the hurdle procedure in the pscl library authored by Simon

Jackman, PhD of the Political Science Computing Laboratory at Stanford University. The










procedure requires specification of the count response variable (Y), the Poisson distribution for

the event stage, the logit link function for the transition stage, and the models for both the

transition stage and event stage, which for this study were X1 and X2. The negative binomial

Hurdle model also used the hurdle procedure but specified a negative binomial rather than a

Poisson distribution for the event stage. This hurdle model procedure maximizes the log-

likelihood using either the Broyden-Fletcher-Goldfarb-Shanno (BFGS) or Nelder-Mead

methods. The Nelder-Mead (default) was selected for solution optimization.

The ZIP model used the zicounts procedure in the zicounts library authored by Samuel

Mwalili, doctoral student in biostatistics at Katholieke Universeteit Leuven (Netherlands).

Similar to the hurdle procedure, zicounts requires specification of the count outcome variable and

models for both the transition stage and event stage. The distribution is specified as "ZIP".

Optimization procedures include the BFGS, Nelder-Mead, and conj oint gradient (CG) methods;

for consistency, the Nelder-Mead was chosen. The negative binomial ZIP used the same

procedure with the specification of the "ZINB" distribution. The specific procedure for analyzing

the data with the models was as follows:

* Three separate loops were established for the negatively skewed, normal, and positively
skewed distributions.

* Arrays were created to store the log-likelihood, AIC, coefficient estimates, standard errors,
and p-values.

* The outermost loop for each distribution pertained to the five conditions of varying zero
proportions; at each loop, the data corresponding to this zero-condition and distribution was
loaded.

* Within these loops, another looping procedure pertained to the six models that analyzed the
data.

* Within these loops another looping procedure was defined for the number of simulations in
the data set.










* It was at this point that the data corresponding to a particular distribution, proportion of
zeros, model, and simulation were analyzed with the calculated AIC, log-likelihood,
coefficient estimates, standard errors, and p-values transferred to the appropriate array.
Models that failed to converge were automatically coded as NA. 27

* The results for each statistic over all simulations for a particular distribution, model, and
proportion of zeros were then exported in comma-separated format for subsequent analyses.
Hence, the three distributions by fiye proportion of zeros conditions by six models yielded 90
data Hiles each containing columns pertaining to a particular statistic or set of statistics and
2,000 rows of simulated results.
Analysis Design

A series of tests were conducted using the simulated data. The results and graphical

output were created using R 2.01 (R Development Core Team, 2006) and SPSS 14.0. The design

for a particular analysis depended on the research question. These questions were as follows:

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated log-likelihood between a) the
Negative binomial Poisson model vs. Poisson model; b) the Hurdle model vs. Poisson
model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the
Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model
vs. ZIP model?

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated AIC between all models?

The first question was answered by calculating log-likelihood values for the six models

over the six zero proportion conditions and the three skew conditions. The deviance statistic was

then calculated as -2(LLs-LLc) where LLs is the model with less parameters (i.e., the simple

model) than the LLc (i.e., the more complex model). Since there were 5 conditions for the

proportions of zeros and 3 conditions for skew, this led to a) a total of 15 separate analyses

comparing the fit of the negative binomial Poisson model and the Poisson model; b) 15 separate

analyses comparing the fit of the Poisson model and the Hurdle model; c) 15 separate analyses

comparing the fit of the negative binomial Poisson model and the negative binomial Hurdle

27 This number was chosen to reduce the probability of obtaining a valid result that would be identified as a
nonconvergant solution










model; and d) 15 separate analyses comparing the fit of ZIP model and the negative binomial

ZIP model.

Test statistics for the deviance were assumed asymptotically chi-square with degrees of

freedom equal to the difference in the number of parameters between the two models. Some

models differed only by the dispersion parameter; these were a) the comparison of the Poisson

model and negative binomial Poisson model; b) the comparison of the Hurdle model and

negative binomial Hurdle model; and c) the comparison of the ZIP model and negative binomial

ZIP model. The log-likelihood statistic for the Hurdle model is based on the log-likelihood

statistics from each of its two parts. Given three parameters in the Poisson model (i.e., the

intercept and the two predictor coefficients), there are six parameters in the Hurdle model.

Hence, the degrees of freedom for the model comparison test are the difference, which are three.

The same is true for comparing the negative binomial Poisson model to the negative binomial

Hurdle model where including the dispersion parameter leads to subtracting 4 parameters from 7

parameters .

Each of the 2,000 goodness-of-fit statistics for the simpler model was subtracted from each

of the 2,000 goodness-of-fit statistics for the more complex model. These results were then

multiplied by -2. Each of these values was then compared to the chi-square distribution with the

appropriate degrees of freedom. This yielded a p-value representing the probability of obtaining

a statistic this high or higher given that the simpler model adequately fits the data. Values

exceeding this critical chi-square statistic based on a Type I error rate of a = .05 were coded "1"

with these results suggesting that the more complex model fits the data better than the Poisson

model. Values failing to exceed the critical value were coded "O" with these results suggesting

adequate fit for the simpler model.









The results were thus the proportion of simulated datasets favoring the more complex

model over the simpler model for a particular proportion of zeros and level of skew. Output

included 1.) descriptive statistics for the goodness-of-fit for each model and 2.) boxplots for the

difference in goodness-of-fit between the two models.

Answering the second question was done in a similar fashion to answering the first

question. This is due to the fact that the AIC is a linear transformation of the log-likelihood

statistic with a result that is positive in sign and is interpreted in a lower-is-better fashion.

However, these analyses did not involve comparisons to the chi-square distribution. As

previously explained, the AIC should not be used in this manner. However, the advantage is that

the AIC can be used to descriptively compare all models regardless of whether one is nested or

not within another. Boxplots were created to display the range of AIC's produced over

simulations for each analysis.










Table 3-1. Proportions of counts as a function of zeros and skew
Study Proportion Proportion of Remaining Nonzero Values
ofZeros
Ones Twos Threes Fours Fives
Posl0 0.10 0.504 0.227 0.088 0.054 0.027
Norml10 0.099 0.198 0.306 0.198 0.099
Negl10 0.027 0.054 0.091 0.227 0.501
Pos25 0.25 0.418 0.190 0.075 0.046 0.021
Norm25 0.083 0.166 0.254 0.166 0.081
Neg25 0.022 0.045 0.075 0.188 0.420
Pos50 .50 0.280 0.125 0.050 0.030 0.015
Norm50 0.053 0.107 0.175 0.110 0.055
Neg50 0.015 0.030 0.050 0.125 0.280
Pos75 0.75 0.140 0.062 0.025 0.015 0.008
Norm75 0.028 0.052 0.089 0.053 0.028
Neg75 0.008 0.015 0.025 0.062 0.140
Pos90 0.90 0.056 0.025 0.010 0.006 0.003
Norm90 0.011 0.021 0.035 0.024 0.009
Neg90 0.004 0.005 0.010 0.025 0.056

Table 3-2. Frequencies of counts as a function of zeros and skew
Study Frequency Frequency of Individual Nonzero Values
of Zeros
Ones Twos Threes Fours Fives
Posl0 100 504 227 88 54 27
Norml0 99 198 306 198 99
Negl0 27 54 91 227 501
Pos25 250 418 190 75 46 21
Norm25 83 166 254 166 81
Neg25 22 45 75 188 420
Pos50 500 280 125 50 30 15
Norm50 53 107 175 110 55
Neg50 15 30 50 125 280
Pos75 750 140 62 25 15 8
Norm75 28 52 89 53 28
Neg75 8 15 25 62 140
Pos90 900 56 25 10 6 3
Norm90 11 21 35 24 9
Neg90 4 5 10 25 56











Table 3-3. Descriptive statistics for each distribution
Range = 0 to 5


Skew


Mean


Range = 1 to 5
Mean Std.Dev.


Std.Dev.
1.127
1.414
1.619
1.181
1.635
2.054
1.149
1.710
2.253
0.928
1.422
1.914
0.622
0.965
1.318


Skew
1.45
0.08
-1.43
1.42
0.08
-1.86
1.44
-0.01
-1.44
1.44
-0.00
-1.44
1.46
-0.06
-1.51


Posl0
Norm10
Negl0
Pos25
Norm25
Neg25
Pos50
Norm50
Neg50
Pos75
Norm75
Neg75
Pos90
Norm90
Neg90


1.570
2.700
3.820
1.310
2.250
3.190
0.880
1.510
2.130
0.440
0.750
1.060
0.180
0.300
0.420


1.155
-0.310
-1.356
1.092
-0.069
-0.673
1.514
0.587
0.236
2.577
1.660
1.349
4.490
3.268
2.908


1.750
3.000
4.250
1.750
2.990
4.840
1.750
3.010
4.250
1.760
3.000
4.240
1.750
2.990
4.240


1.051
1.150
1.053
1.046
1.149
0.367
1.053
1.140
1.053
1.064
1.149
1.064
1.058
1.124
1.084


Table 3-4. Poisson model:


pseudo-population parameters


Pos10
Nonn10
Neg10
Pos25
Nonn25
Neg25
Pos50
Nonn50
Neg50
Pos75
Nonn75
Neg75
Pos90
Nonn90
Neg90


.450
.992
1.340
0.270
.807
1.159
-138
.407
.752
-829
-.291


-1475.3
-1806.7
-2019.2
-1468.1
-1913.9
-2265.35
-1334.3
-1888.5
-2370.5


2956.6
3619.3
4044.5
2942.3
3833.8
4534.7
2674.5
3783.0
4746.9
1951.7
2929.6
3845.3
1099.0
1711.1
2310.2


.068 .005 .095 -.077
.052 .003 .073 -.079


.048 -972.9
.037 -1461.8
.031 -1919.7


.057 .043 .002 .061 -.044


-546.5
-852.6
-1152.1














Table 3-5. Negative Binomial Poisson model: pseudo-population parameters

Po sw P s, p, .s LL


Pos10

Norml0

Neg10

Pos25

Norm25

Neg25

Pos50

Norm50

Neg50

Pos75

Norm75

Neg75

Pos90

Norm90

Neg90


-1594.5

-1998.9

-2277.9

-1509.4

-1931.5

-2235.5

-1290.1

-1723.0

-2041.3

-910.4

-1275.6

-1567.4

-511.6

-751.1

-957.3


(zeros): pseudo-population parameters

sP, A1 saq A


AIC


Pos10

Norml0

Neg10

Pos25

Norm25

Neg25

Pos50

Norm50

Neg50

Pos75

Norm75

Neg75

Pos90

Norm90

Neg90


-2.198

-2.198

-2.198

-1.099

-1.098

-1.099

.000

-.005

.000

1.099

1.100

1.099

2.205

2.206

2.204


-1409.5

-1797.1

-1928.3

-1465.5

-1787.6

-1897.9

-1296.6

-1510.1

-1583.9

-865.2

-970.3

-1008.1

-444.9

-486.4

-503.0


2830.9

3606.1

3868.7

2943.0

3587.2

3807.9

2605.1

3023.2

3179.7

1742.4

1952.6

2028.3

901.8

984.8

1018.1


Table 3-6. Hurdle model

A























































Pos10

Norml0

~Neg10

Pos25

Norm25

~Neg25

Pos50

Norm50

~Neg50

Pos75

Norm75

~Neg75

Pos90

Norm90

Neg90


-2.197

-2.197

-2.205

-1.099

-1.099

-1.099

.000

.000

.001

1.099

1.099

1.099

2.206

2.244

2.205


-1395.7

-1800.2

-1936.1

-1455.3

-1790.2

-1904.4

-1288.9

-1511.9

-1588.1

-861.1

-971.2

-1010.3

-443.3

-487.3

-503.9


Table 3-7. Hurdle model (events): pseudo-population parameters


SP, 1


Sa P2


AIC


Pos10

Norml0

Neg10

Pos25

Norm25

Neg25

Pos50

Norm50

Neg50

Pos75

Norm75

Neg75

Pos90

Norm90

Neg90


.212

1.035

1.430

.214

1.033

1.431

.208

1.035

1.430

.213

1.032

1.428

.179

1.106

1.420


-1409.5

-1797.1

-1928.3

-1465.5

-1787.6

-1897.9

-1296.6

-1510.1

-1583.9

-865.2

-970.3

-1008.2

-444.9

-486.4

-503.0


2830.9

3606.1

3868.7

2943

3587.2

3807.9

2605.1

3023.2

3179.7

1742.4

1952.6

2028.3

901.8

984.8

1018.1


Table 3-8. Negative Binomial Hurdle

0o Sp, 1


model (zeros): pseudo-population

S4 P2 S ,


parameters
LT,


AIC


2803.4

3612.3

3884.1

2922.5

3592.3

3820.8

2589.8

3035.8

3188.2

1734.3

1954.4

2032.5

898.7

986.6

1019.8














Table 3-9. Negative Binomial Hurdle model (events): pseudo-population parameters

P S~G S P .S Theta LL AIC


Pos10

Nonn10

Neg10

Pos25

Nonn25

Neg25

Pos50

Nonn50

Neg50

Pos75

Nonn75

Neg75

Pos90

Nonn90

Neg90


-.067 .108

1.033 .030

1.429 .024

-.042 .114

1.030 .033

1.431 .026

-.072 .146

1.037 .041

1.429 .032

-.075 .208

1.031 .058

1.427 .046

-.109 .332

1.051 .091

1.421 .072


.009 .091

.004 .043

.002 .034

.009 .098

.005 .047

.001 .037

.020 .122

.006 .057

.004 .046

.012 .173

.005 .081

.002 .064

.078 .273

-051 .130

.007 .102


-.071 .046

-012 .022

-002 .017

-.067 .051

-027 .025

-015 .019

-.065 .062

-025 .029

-012 .023

-.097 .088

-054 .041

-011 .032

-.081 .140

-048 .066

-028 .052


2.039

165.718

164.970

2.234

166.348

166.006

2.030

166.574

166.526

1.950

166.445

166.377

2.047

91.979

166.606


-1395.7

-1800.2

-1936.1

-1455.3

-1790.2

-1904.4

-1288.9

-1511.9

-1588.1

-861.1

-971.2

-1010.3

-443.3

-487.3

-503.9


2803.4

3612.3

3884.1

2922.5

3592.3

3820.8

2589.8

3035.8

3188.2

1734.3

1954.4

2032.5

898.7

986.6

1019.8


Table 3-10. ZIP model (zeros): pseudo-population parameters

Ao sp, A1 sa4 A


AIC


Pos10

Nonn10

Neg10

Pos25

Nonn25

Neg25

Pos50

Nonn50

Neg50

Pos75

Nonn75

Neg75

Pos90

Nonn90

Neg90


-27.010

-3.114

-2.365

-15.460

-1.379

-1.162

-.890

-.126

-031

.610

1.032

1.079

1.797

2.133

2.188


4434.000

.038

.175

3834.000

.132

.109

.198

.097

.091

.142

.057

.110

.183

.152

.071


11.490

.002

.002

-3.418

.008

.001

.051

.002

.000

.010

.005

-001

.032

.002

.007


4435.000 -.363

.527 .050

.246 .049

.002 1.213

.186 -.056

.154 -.025

.270 -.050

.137 .030

.129 .035

.198 -.013

.081 -.054

.147 .044

.255 .104

.214 .128

.101 -.028


150.700

.253

.125

.001

.098

.078

.141

.070

.065

.010

.040

.075

.129

.108

.051


-1478.3

-1800.1

-1931.3

-1471.1

-1790.6

-1900.9

-1299.5

-1513.1

-1586.9

-868.2

-973.3

-1011.2

-447.9

-489.4

-506.0


2968.6

3612.1

3874.6

2954.3

3593.1

3813.9

2611.1

3038.3

3185.7

1748.5

1958.5

2034.3

907.7

990.8

1024.1
















sP 4


sa P


AIC


Pos10

Nonn10

Neg10

Pos25

Nonn25

Neg25

Pos50

Nonn50

Neg50

Pos75

Nonn75

Neg75

Pos90

Nonn90

Neg90


.450

1.035

1.430

.270

1.032

1.431

.205

1.039

1.429

.213

1.015

1.428

.180

1.017

1.422


-1478.3

-1800.1

-1931.3

-1471.1

-1790.6

-1900.9

-1299.5

-1513.1

-1586.9

-868.2

-973.3

-1011.2

-447.9

-489.4

-506.0


2968.6

3612.1

3874.6

2954.3

3593.1

3813.9

2611.1

3038.3

3185.7

1748.5

1958.5

2034.3

907.7

990.8

1024.1


Table 3-12.


Negative



-15.280

-3.110

-2.370

-17.090

-1.380

-1.162

-4.895

-.131

-031

.130

1.015

1.080

1.462

2.133

2.187


Binomial ZIP model (zeros):

sp P1 s,


pseudo-population parameters

2 Sp LL


AIC


Pos10

Nonn10

Neg10

Pos25

Nonn25

Neg25

Pos50

Nonn50

Neg50

Pos75

Nonn75

Neg75

Pos90

Nonn90

Neg90


142.300

.375

.175

501.400

.132

.109

3.373

.097

.091

.364

.107

.104

.386

.152


-3.880

.0165

-001

2.390

-.002

.000

1.352

.008

.000

.015

.001

-002

.047

.002


830.400

.527

.247

473.600

.186

.037

2.211

.014

.129

.271

.151

.147

.298

.214


.719

.0510

.047

1.700

-.059

-016

-1.086

.031

.035

-.040

.033

.044

.098

.128


102.100

.253

.125

133.500

.983

.019

1.304

.069

.065

.136

.076

.075

.151

.108


-1479.3

-1801.1

-1932.3

-1471.4

-1791.6

-1901.9

-1292.4

-1514.1

-1587.9

-865.1

-974.3

-1012.2

-447.3

-490.4


2972.6

3616.1

3878.7

2956.7

3597.1

3817.9

2598.8

3042.3

3189.7

1744.3

1962.6

2038.3

908.7

994.8


.150 -.001


.212 .136


.107 -507.0 1028.1


Table 3-11. ZIP Model (events): pseudo-population parameters














Table 3-13. Negative Binomial ZIP model (events): pseudo-population parameters

o S, P1 Sr P Sp Theta LL AIC


Pos10

Nonn10

Neg10

Pos25

Nonn25

Neg25

Pos50

Nonn50

Neg50

Pos75

Nonn75

Neg75

Pos90

Nonn90

Neg90


.036 .004

.030 .004

.024 .001

.040 .003

.033 .005

.026 .001

.067 .045

.040 .007

.032 .004

.204 .010

.057 .005

.045 .000

.327 .072

.091 .017

.071 .006


-.037 .026

-013 .021

-003 .017

-.024 .029

-027 .025

-016 .019

-.080 .052

-026 .029

-012 .023

-.099 .087

-054 .040

-010 .032

-.080 .140

-052 .065

-028 .051


14.69

12.64

13.02

3.16

10.88

15.94

.57

11.70

14.19

.69

12.02

12.99

.73.

12.76

14.73


-1479.3

-1801.1

-1932.3

-1471.4

-1791.6

-1901.9

-1292.4

-1514.1

-1587.9

-865.1

-974.3

-1012.2

-447.3

-490.4

-507.0


2972.6

3616.1

3878.7

2956.7

3597.1

3817.9

2598.8

3042.3

3189.7

1744.3

1962.6

2038.3

908.7

994.8

1028.1









CHAPTER 4
RESULTS

This chapter presents the results based on the previously discussed methods and

procedures for analyzing data with varying proportions of zeros and varying event stage

distributions. First, the results are presented using the data for the pseudo-population in which

the proportions for each count level are exactly that which was rando mly sampled from in the

simulations. Tables and figures are included to support interpretation. Second, the results are

presented outlined by the skew level (i.e., positive, normal, and negative) and by the proportion

of zeros within that skew level (i.e., .10, .25, .50, .75, and .90). For each combination of

conditions, the results are presented for the five model comparisons. Third, the results are

summarized separately for the negative, normal, and positive event count distributions. Tables

and figures are included to support interpretation. The primary purpose of the results was to

assist in answering the research questions, which were as follows:

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated log-likelihood between a) the
Negative binomial Poisson model vs. Poisson model; b) the Hurdle model vs. Poisson
model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the
Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model
vs. ZIP model?

* Given one two-level categorical covariate with known values and one continuous covariate
with known values, what is the difference in the estimated AIC between all models?

Pseudo-Population Results

Before addressing the results of the Monte Carlo simulations, it is necessary to discuss

the results when each model was analyzed with the pseudo-population data. The prespecified

proportions for each count were displayed in Table 3-1. Table 3-2 displayed this same

information in terms of frequencies instead of proportions. Table 3-3 displayed the descriptive

statistics for each distribution. Tables 3-4 through 3-13 displayed the coefficients, standard









errors, log-likelihood values, and AIC values with each table pertaining to either a specific model

or one of the two stages for a specific model.

The following sections provide the results when comparing models using the pseudo-

population data. First, the Poisson model results were compared to the negative binomial Poisson

model results, both descriptively via AIC's and inferentially via the deviance model comparison

test. Second, the results are presented in a likewise manner for the three Hurdle model

comparisons. These comparisons were a) the Hurdle model vs. the negative binomial Hurdle

model, b) the Poisson model vs. the Hurdle model, and c) the negative binomial Poisson model

vs. the negative binomial Hurdle model. Third, the results are presented for the comparison of

the ZIP model and the negative binomial ZIP model. Fourth, descriptive results are presented

comparing all models for each of the five proportions of zeros.

Pseudo-Population Poisson Models

Based on the AIC, for all proportions of zeros the data fit the Poisson model better when

the distribution was positively skewed than when normally distributed. In addition, this data fit

the Poisson model better when the distribution was normally distributed than when it was

negatively skewed. For example, for the negatively skewed distribution with a .10 proportion of

zeros, the AIC was 4,044.5. As the curve shifted left to a normal distribution, the AIC dropped to

3,619.3, and as the curve shifted further left to a positively skewed distribution, the AIC dropped

to 2,956.3.

The same pattern emerged for the negative binomial Poisson models. For example, for the

negatively skewed distribution with a .10 proportion of zeros, the AIC was 4,563.8. As the curve

shifted left to a normal distribution, the AIC dropped to 4,005.9, and as the curve shifted further

left to a positively skewed distribution, the AIC dropped to 3,196.9.










For the .10 proportion of zeros condition, the Poisson models had a lower AIC than that

calculated for the negative binomial Poisson model. For the .25 proportion of zeros condition,

the AIC's were approximately equal between the two models. For the .50, .75, and .90

proportions of zeros conditions, the AIC was lower for the negative binomial Poisson model than

for the Poisson model.

The deviance statistic was calculated comparing the Poisson log-likelihood and negative

binomial Poisson log-likelihood for each skew condition and zero proportion condition. These

statistics are displayed in Table 4-1. For all analyses, the Poisson model is nested within its

negative binomial Poisson formulation differing by one degree of freedom (i.e., the dispersion

parameter in the negative binomial Poisson model). Hence, at the .05 Type I error rate and

assuming deviance statistics asymptotically distributed chi-square, a deviance exceeding 3.84

suggests better fit for the more complex negative binomial Poisson model. This was the result for

all analyses in which the proportion of zeros was .50 or greater. However, for all analyses with

.10 and .25 proportions of zeros, the statistic was significant in favor of the Poisson model.

Pseudo-Population Hurdle Models

Hurdle vs. Negative Binomial Hurdle

For the Hurdle models, regardless of the proportion of zeros, the positively skewed

distributions had a lower AIC than did the normal distributions. These, in turn, had a lower AIC

than the negatively skewed distribution. The same was true for the negative binomial Hurdle

models. However, for both models, the difference between AIC's for the three skew conditions

decreased as the proportion of zeros decreased (i.e., the models became more similar in fit).

When comparing the Hurdle models and negative binomial Hurdle models, the AIC's appear to

be similar regardless of the proportion of zeros and regardless of skew.










The deviance statistic was calculated comparing the Hurdle log-likelihood and negative

binomial Hurdle log-likelihood for each skew condition and zero proportion condition. These

statistics are displayed in Table 4-2. For all analyses, the Hurdle model is nested within its

negative binomial Hurdle formulation differing by one degree of freedom (i.e., the dispersion

parameter in the negative binomial Hurdle model). Hence, at the .05 Type I error rate and

assuming deviance statistics asymptotically distributed chi-square, a deviance exceeding 3.84

suggests better fit for the more complex negative binomial Hurdle model.

For the positively skewed distributions, the negative binomial Hurdle model fit

significantly better than the Hurdle model, except when the proportion of zeros was .90. In this

case the deviance statistic did not exceed 3.84. For the normal distributions, the Hurdle model fit

significantly better than the negative binomial Hurdle model when the proportion of zeros was

.10 or .25. When the proportion of zeros was .50, .75, or .90, the deviance did not exceed 3.84.

For the negatively skewed distributions, the Hurdle model fit significantly better than the

negative binomial Hurdle model when the proportion of zeros was .10, .25, .50, or .75. When the

proportion of zeros of .90, the deviance statistic did not exceed 3.84.

Poisson vs. Hurdle

Deviance statistics were also calculated to compare the Poisson log-likelihood and Hurdle

log-likelihood for each skew condition and zero proportion condition. These statistics are

displayed in Table 4-3. For all analyses, the Poisson model is nested within the Hurdle model

differing by three degrees of freedom. Hence, at the .05 Type I error rate and assuming deviance

statistics asymptotically distributed chi-square, a deviance exceeding 7.82 suggests better fit for

the more complex Hurdle model.

The deviances were large supporting the Hurdle model fit over the Poisson model fit. In

fact, several of the deviances were over 1,000. There was only one analysis that did not favor the









Hurdle model fit over the Poisson model fit. This was the deviance calculated for the positively

skewed distribution with a .25 proportion of zeros.

Negative Binomial Poisson vs. Negative Binomial Hurdle

Deviance statistics were also calculated to compare the negative binomial Poisson log-

likelihood and the negative binomial Hurdle log-likelihood for each skew condition and zero

proportion condition. These statistics are displayed in Table 4-4. For all analyses, the negative

binomial Poisson model is nested within the negative binomial Hurdle model differing by three

degrees of freedom (i.e., the duplication of parameters in the negative binomial Hurdle model to

represent both a transition stage and an event stage).28 Hence, at the .05 Type I error rate and

assuming deviance statistics asymptotically distributed chi-square, a deviance exceeding 7.82

suggests better fit for the more complex negative binomial Hurdle model.

As was the case when comparing the Poisson model and the Hurdle model, the deviances

were large. Likewise, there was one deviance that did not exceed the critical value. However, it

was for the positively skewed distribution with .50 zeros rather than the positively skewed

distribution with .25 zeros.

Pseudo-Population ZIP Models

The pseudo-population results for the ZIP models were rather similar to those obtained for

the Hurdle models. Graphically, regardless of skew condition and proportions of zeros, the AIC's

for the ZIP models and negative binomial ZIP models were very similar. Additionally, the AIC's

appeared to be equal between the .10 and .25 proportions of zeros conditions.

The deviance statistic was calculated comparing the ZIP log-likelihood and negative

binomial ZIP log-likelihood for each skew condition and zero proportion condition. These



28The presence of a dispersion parameter is now redundant between models.









statistics are displayed in Table 4-5. For all analyses, the ZIP model is nested within its negative

binomial ZIP formulation differing by one degree of freedom (i.e., the dispersion parameter in

the negative binomial ZIP model). Hence, at the .05 Type I error rate and assuming deviance

statistics asymptotically distributed chi-square, a deviance exceeding 3.84 suggests better fit for

the more complex negative binomial ZIP model.

The deviance was significant for only two comparisons. When the distribution was

positively skewed, the fit was significantly better for the negative binomial ZIP model than for

the ZIP model when the proportion of zeros was either .50 or .75. All other results suggested that

the ZIP model fit was adequate.

Comparing AIC's For All Models

For the .10 proportion of zeros condition with negative skew, the AIC's were

approximately equal for all models except for the Poisson models. The AIC for the Poisson

model was higher than for the other models, and the AIC for the negative binomial Poisson

model was highest of all. When the distribution was normal, the only model to have a noticeably

different AIC was the negative binomial Hurdle, which was again highest of all. For the

positively skewed distribution, there was some separation between the ZIP and Hurdle models,

with the Hurdle models having the lower AIC. The differences between these models and their

negative binomial formulations appeared to be trivial. The Poisson model appeared to have the

same AIC as those displayed for the ZIP models. The AIC for the negative binomial was again

highest of all. Between the three distributions, the AIC declined from the negatively skewed

distribution to the normal distribution to the positively skewed distribution.

For the .25 proportion of zeros, there was little distinction between the Poisson models

and negative binomial Poisson models for all distributions. Further, there was no distribution

displaying a nontrivial distinction between the Hurdle models and the ZIP models. For the









positively skewed distribution, all six distributions appeared approximately equal with a slightly

higher AIC apparent for the negative binomial Poisson model. Between the three distributions,

the AIC's appeared equal for the negatively skewed distribution and the normal distribution;

However, the AIC's for the normal distribution were considerably smaller than the AIC's for the

other two distributions.

For the .50 proportion of zeros condition with negative skew, the results for the Poisson

model and the negative binomial Poisson model reversed. For the negatively skewed distribution,

the AIC for the negative binomial Poisson was higher than those for the ZIP and Hurdle models,

while the AIC for the Poisson model was higher yet. As in the .50 proportion of zeros condition,

the AIC appeared equal for the ZIP and Hurdle models. Also, as in the .50 proportion of zeros

condition, there appeared to be no difference between any of the models for the positively

skewed distribution. Between the three distributions, the AIC's declined from the negatively

skewed distribution to the normal distribution to the positively skewed distribution with declines

being most rapid for the Poisson and negative binomial Poisson models.

Beyond the fact that the overall AIC was lower for the .90 proportion of zeros condition

than for the .75 condition, these last two conditions displayed similar results. For each

distribution, the Hurdle and ZIP model AIC's were approximately equal. They declined slightly

from the negatively skewed distribution to the normal distribution and declined a bit more from

this distribution to the positively skewed distribution. For the negatively skewed distribution, the

negative binomial Poisson AIC was considerably higher than the AIC's for the ZIP and Hurdle

models; the AIC for the Poisson model was highest of all. For the normal distribution, both the

Poisson and negative binomial Poisson model AIC's declined by approximately 50% of their

value for the negatively skewed distribution. For the positively skewed distribution, they




Full Text

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COMPARING POISSON, HURDLE, AND ZIP MODEL FIT UNDER VARYING DEGREES OF SKEW AND ZERO-INFLATION By JEFFREY MONROE MILLER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 1

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Copyright 2007 by Jeffrey Monroe Miller 2

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To the memory of my grandfather, Rev. Harold E. Cato. 3

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ACKNOWLEDGMENTS Several people helped make this study possibl e. I would like to thank my stepfather and mother, Dr. Daniel and Gail Jacobs as well as my father and stepmother, Jerry and Darnelle Miller for their many years of encouragement. I would also like to thank my supervisory committee chair, M. David Miller, for his unyieldi ng guidance, patience, and support. I thank Dr. Jon Morris for the numerous training experiences Many professors are appreciated for providing the educational foundations for the dissertation topic including Dr. James Algina and Dr. Alan Agresti. The idea to research zero-inflation was in spired by experiences with data while consulting on projects. To this extent, I thank those clients Dr. Courtney Zmach and Dr. Lori Burkhead. Undergraduate faculty that I would like to acknowle dge for their inspiration and direction include Blaine Peden, Patricia Quinn, and Lee Anna Ra sar. Several friends have been a source of encouragement including Matt Grezik and Rachael Wilkerson. Finally, I thank those who made it financially possible to complete this dissertation including my consulting clients, the University of Florida College of Educa tion, the Lastinger Cent er, and Adsam, LLC. 4

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TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.......................................................................................................................11 ABSTRACT...................................................................................................................................12 CHAPTER 1 INTRODUCTION..................................................................................................................14 Statement of the Problem....................................................................................................... .15 Rationale for the Study........................................................................................................ ...15 Purpose and Significance of the Study...................................................................................16 Research Questions............................................................................................................. ....16 2 REVIEW OF THE LITERATURE........................................................................................18 Zero-Inflated Count Data........................................................................................................18 Count Data.......................................................................................................................18 Zero-Inflation..................................................................................................................19 The Sources of Zero-Inflation.........................................................................................20 Impact of Zero-Inflation on Analyses.............................................................................21 Simple Solutions to Zero-Inflation..................................................................................22 Deleting zeros...........................................................................................................22 Assuming normality.................................................................................................22 Transforming Zeros..................................................................................................23 Generalized Linear Models.............................................................................................24 The Binomial Distribution and the Logit Link................................................................28 Evaluating Model Fit.......................................................................................................31 The Poisson Distribution and the Log Link....................................................................35 Iterative Estimation.........................................................................................................38 Interpretation of Coefficients..........................................................................................38 Hypothesis testing....................................................................................................39 Overdispersion.........................................................................................................39 Poisson and Negative Binomial Models with Zero-Inflation..........................................42 The Hurdle model.....................................................................................................45 The Negative Binomial Hurdle model.....................................................................48 The Zero-Inflated Poisson (ZIP) model...................................................................49 The Negative Binomial Zero-Inflated Poisson model..............................................51 Model Comparison Testing for Zero-Inflated Data.........................................................52 Review of Research Pertaining to and Using Zero-Inflated Count Data................................53 Hurdle Model...................................................................................................................53 5

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Statistical..................................................................................................................53 Applications.............................................................................................................54 Zero-Inflated Poisson Model...........................................................................................55 Statistical..................................................................................................................55 Applications.............................................................................................................59 ZIP and Hurdle Model-Comparisons..............................................................................63 Statistical..................................................................................................................64 Applications.............................................................................................................64 Discrepant Findings............................................................................................................ ....66 3 METHODOLOGY.................................................................................................................73 Research Questions............................................................................................................. ....73 Monte Carlo Study Design.....................................................................................................73 Monte Carlo Sampling............................................................................................................75 Pseudo-Population...........................................................................................................75 The Prespecified Zero Proportions..................................................................................76 Pre-Specified Skew.........................................................................................................76 Random Number Generation...........................................................................................77 Sample Size.....................................................................................................................78 Simulation Size................................................................................................................ 78 Iteration Size....................................................................................................................79 Distribution Generation...................................................................................................80 Monte Carlo Models...............................................................................................................80 Monte Carlo Analysis Procedures..........................................................................................82 Analysis Design......................................................................................................................84 4 RESULTS...............................................................................................................................94 Pseudo-Population Results.....................................................................................................94 Pseudo-Population Poisson Models................................................................................95 Pseudo-Population Hurdle Models..................................................................................96 Hurdle vs. Negative Binomial Hurdle......................................................................96 Poisson vs. Hurdle....................................................................................................97 Negative Binomial Poisson vs. Negative Binomial Hurdle.....................................98 Pseudo-Population ZIP Models.......................................................................................98 Comparing AICs For All Models...................................................................................99 Monte Carlo Simulation Results...........................................................................................101 Positively Skewed Distribution.....................................................................................101 Normal Distribution.......................................................................................................111 Negatively Skewed Distribution....................................................................................122 Review of Positively Skewed Distribution Findings............................................................133 Review of Normal Distribution Findings.............................................................................135 Review of Negatively Skew ed Distribution Findings..........................................................136 5 DISCUSSION.......................................................................................................................176 6

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The Impact of the Event Stage Distribution.........................................................................176 Positively Skewed Event-Stage Distributions...............................................................176 Normal Event-Stage Distributions................................................................................181 Negatively Skewed Event-Stage Distributions.............................................................183 Summary of Findings....................................................................................................185 Limitations.................................................................................................................... ........186 Discrete Conditions.......................................................................................................186 Convergence and Optimization.....................................................................................186 Underdispersion.............................................................................................................187 Other models.................................................................................................................18 8 Validity of Model-Fitting and Model-Comparisons.............................................................188 Suggestions for Future Research..........................................................................................190 Application in Educational Research....................................................................................190 Major Contribution of Findings............................................................................................191 LIST OF REFERENCES.............................................................................................................195 BIOGRAPHICAL SKETCH.......................................................................................................201 7

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LIST OF TABLES Table page 2-1 Five pairs of nested models valid for statistical comparison.............................................71 2-2 Summary of literature on zero-inflation............................................................................72 3-1 Proportions of counts as a function of zeros and skew......................................................87 3-2 Frequencies of counts as a function of zeros and skew.....................................................87 3-3 Descriptive statisti cs for each distribution.........................................................................88 3-4 Poisson model: pseudo-population parameters..................................................................88 3-5 Negative Binomial Poisson model: pseudo-population parameters..................................89 3-6 Hurdle model (zeros): pseudo-population parameters.......................................................89 3-7 Hurdle model (events) : pseudo-population parameters.....................................................90 3-8 Negative Binomial Hurdle model (zeros): pseudo-population parameters........................90 3-9 Negative Binomial Hurdle model (events): pseudo-population parameters......................91 3-10 ZIP model (zeros): pseudo-population parameters............................................................91 3-11 ZIP Model (events): pseudo-population parameters..........................................................92 3-12 Negative Binomial ZIP model ( zeros): pseudo-population parameters.............................92 3-13 Negative Binomial ZIP model (events): pseudo-population parameters...........................93 4-1 Deviance statistics comparing Poiss on and negative binomial Poisson models..............138 4-2 Deviance statistics comparing Hurdle and negative binomial Hurdle models................138 4-3 Deviance statistics comparing Poisson and Hurdle models.............................................138 4-4 Deviance statistics comparing NB Poisson and NB Hurdle models................................138 4-5 Deviance statistics comparing ZIP and negative binomial ZIP models..........................139 4-6 Log-likelihood comparisons for posit ively skewed distribution with .10 zeros..............139 4-7 AICs for positively skewed distribu tion models with a .10 proportion of zeros............140 4-8 Log-likelihood comparisons for posit ively skewed distribution with .25 zeros..............140 8

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4-9 AICs for positively skewed distribu tion models with a .25 proportion of zeros............141 4-10 Log-likelihood comparisons for posit ively skewed distribution with .50 zeros..............141 4-11 AICs for positively skewed distribu tion models with a .50 proportion of zeros............142 4-12 Log-likelihood comparisons for posit ively skewed distribution with .75 zeros..............142 4-13 AICs for positively skewed distribu tion models with a .75 proportion of zeros............143 4-14 Log-likelihood comparisons for posit ively skewed distribution with .90 zeros..............143 4-15 AICs for positively skewed distribu tion models with a .90 proportion of zeros............144 4-16 Log-likelihood comparisons fo r normal distribution with .10 zeros...............................144 4-17 AICs for normal distribution models with a .10 proportion of zeros.............................145 4-18 Log-likelihood comparisons fo r normal distribution with .25 zeros...............................145 4-19 AICs for normal distribution models with a .25 proportion of zeros.............................146 4-20 Log-likelihood comparisons fo r normal distribution with .50 zeros...............................147 4-21 AICs for normal distribution m odels with a .50 proportion of zeros.............................147 4-22 Log-likelihood comparisons fo r normal distribution with .75 zeros...............................148 4-23 AICs for normal distribution models with a .75 proportion of zeros.............................148 4-24 Log-likelihood comparisons fo r normal distribution with .90 zeros...............................149 4-25 AICs for normal distribution models with a .90 proportion of zeros.............................149 4-26 Log-likelihood comparisons for negatively skewed distribution with .10 zeros.............150 4-27 AICs for negatively skewed models with a .10 proportion of zeros..............................150 4-28 Log-likelihood comparisons for negatively skewed distribution with .25 zeros.............151 4-29 AICs for negatively skewed models with a .25 proportion of zeros..............................151 4-30 Log-likelihood comparisons for negatively skewed distribution with .50 zeros.............152 4-31 AICs for negatively skewed models with a .50 proportion of zeros..............................152 4-32 Log-likelihood comparisons for negatively skewed distribution with .75 zeros.............153 4-33 AICs for negatively skewed models with a .75 proportion of zeros..............................153 9

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4-34 Log-likelihood comparisons for negatively skewed distribution with .90 zeros.............154 4-35 AICs for negatively skewed models with a .90 proportion of zeros..............................154 4-36 Positively skewed distribution: percentage of simulations favoring complex model......155 4-37 AICs: positively skewed distribution (all conditions)....................................................155 4-38 Normal distribution: percentage of simulations favoring complex model......................155 4-39 AICs: normal dist ribution (all conditions)......................................................................155 4-40 Negatively skewed distribution: percen tage of simulations favoring complex model....156 4-41 AICs: negatively skewed distribution (all conditions)...................................................156 4-42 Convergence frequencies: positively skewed distribution...............................................156 4-43 Convergence frequencies: normal distribution................................................................156 4-44 Convergence frequencies: negatively skewed distribution..............................................157 10

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LIST OF FIGURES Figure page 4-1 Boxplot of AICs for all models for a .10 proportion of zeros........................................158 4-2 Boxplot of AICs for all models for a .25 proportion of zeros........................................159 4-3 Boxplot of AICs for all models for a .50 proportion of zeros........................................160 4-4 Boxplot of AICs for all models for a .75 proportion of zeros........................................161 4-5 Boxplot of AICs for all models for a .90 proportion of zeros........................................162 4-6 Boxplot of AICs for all models for a .10 proportion of zeros........................................163 4-7 Boxplot of AICs for all models for a .25 proportion of zeros........................................164 4-8 Boxplot of AICs for all models for a .50 proportion of zeros........................................165 4-9 Boxplot of AICs for all models for a .75 proportion of zeros........................................166 4-10 Boxplot of AICs for all models for a .90 proportion of zeros........................................167 4-11 Boxplot of AICs for all models for a .10 proportion of zeros........................................168 4-12 Boxplot of AICs for a ll models for a .25 proportion of zeros........................................169 4-13 Boxplot of AICs for a ll models for a .50 proportion of zeros........................................170 4-14 Boxplot of AICs for a ll models for a .75 proportion of zeros........................................171 4-15 Boxplot of AICs for a ll models for a .90 proportion of zeros........................................172 4-16 AIC rank order for positively skewed distribution models.............................................173 4-17 AIC rank order for normal distribution models...............................................................174 4-18 AIC rank order for negatively skewed distribution models............................................175 5-1 Poisson, NB Poisson, and Hurdle over all proportions of zeros......................................192 5-2 Hurdle, NB Hurdle, and NB Poiss on over all proportions of zeros.................................193 5-3 ZIP, NB ZIP, Hurdle, and NB Hu rdle over all proportions of zeros...............................194 11

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Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPARING POISSON, HURDLE, AND ZIP MODEL FIT UNDER VARYING DEGREES OF SKEW AND ZERO-INFLATION By Jeffrey Monroe Miller May 2007 Chair: M. David Miller Major Department: Educational Psychology Many datasets are characterized as count data with a prepond erance of zeros. Such data are often analyzed by ignoring the zero-inflation and assuming a Poisson distribution. The Hurdle model is more sophisticated in that it cons iders the zeros to be completely separate from the nonzeros. The zero-inflated Poisson (ZIP) mode l is similar to the Hurdle model; however, it permits some of the zeros to be analyzed along with the nonzeros. Both models, as well as the Poisson, have negative binomial formulations for use when the Poisson assumption of an equal mean and variance is violated. The choice between the models should be guided by the researchers beliefs about the source of the zeros. Beyond this substantive co ncern, the choice should be based on the model providing the closest fit between the observed and predicted va lues. Unfortunately, the literature presents anomalous findings in terms of model superiority. Datasets with zero-inflation may vary in terms of the proportion of zeros. They may also vary in terms of the distribution for the nonzeros Our study used a Monte Carlo design to sample 1,000 cases from positively skewed, normal, a nd negatively skewed distributions with proportions of zeros of .10, .25, .50, .75, and .90. The da ta were analyzed with each model over 12

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2,000 simulations. The deviance statistic and Akai kes Information Criterion (AIC) value were used to compare the fit between models. The results suggest that the literature is not entirely anomalous; however, the accuracy of the findings depends on the proportion of zeros and the distribution for the nonzeros. Although the Hurdle model tends to be the superior model, there are situations when others, including the negative binomial Poisson model, are superior. The findings sugge st that the researcher should consider the proportion of zeros and the distribution for the nonzer os when selecting a model to accommodate zero-inflated data. 13

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CHAPTER 1 INTRODUCTION Analyzing data necessitates determination of the type of data being analyzed. The most basic assumption is that the data follows a nor mal distribution. However, there are many other types of distributions. The validity of the results can be affected by the dissimilarity between the distribution of the data and the di stribution assumed in the analysis. As such, it is imperative that the researcher choose a me thod for analyzing the data that maintains a distribution similar to that of the observed data. Counts are an example of data which does not readily lend itself to the assumption of a normal distribution. Counts are bounded by their lowest value, which is usually zero. A regression analysis assuming a normal distribution would permit results below zero. Further, counts are discrete integers while the normal distribution a ssumes continuous data. Finally, counts often display positive skew such that the frequency for low counts is considerably higher than the frequencies as the count levels increase. It is not uncommon to find count data an alyzed in a more appropriate manner than assuming a normal distribution. Typically, more a ppropriate analysis includes specification of a Poisson distribution with a log link, rather than a normal dist ribution with a Gaussian link. However, this does not guarantee accurate and valid results as other features of the data may warrant an even more sophisticated model. An example of data requiring a more rigorous treatment of the data is the case of zeroinflation. In this scenario, ther e are far more zeros than would be expected using the Poisson distribution. As such, a number of methods including the zero-i nflated Poisson (ZIP) model and the Hurdle model are available. Further, there ar e negative binomial variations of these for use when particular assumptions appear to be violated. The choice between the models depends on 14

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whether the researcher believes the zeros are all a complete lack of the quantity being measured or that at least some of the zeros are purely random error. Statement of the Problem The results from both simulated and actual data sets in the zero-inflation literature are in much disagreement. Lambert (1992) found the ZIP m odel to be superior to the negative binomial Poisson model, which was superior to the Poisson model. Greene (1994) found the negative binomial Poisson model to be superior to the ZIP model, which was superior to the Poisson model. Slymen, Ayala, Arredondo, and Elder (2006) found the ZIP and negative binomial ZIP models to be equal. Welsh, Cunningham, Donne lly, and Lindenmayer found the Hurdle and ZIP models to be equal while Pardoe and Durham (2 003) found the negative binomial ZIP model to be superior to both the Poisson and Hurdle models. One striking characteristic of these articles and others is their diffe rences in terms of the proportion of zeros and the distribution for the nonzeros. Some research (Bhning, Dietz, Schlattmann, Mendona, and Kirchner, 1999) analy zed data in which the proportion of zeros was as low as .216 while others (Zorn, 1996) used proportions as high as .958. Further, the nonzeros varied in terms of their distributions from highly positively skewed to normal to uniform. It is possible that different models yi eld different results depending on the proportion of zeros and the distribution for the nonzeros. Rationale for the Study The best model is the one that appropriat ely answers the resear chers question. Beyond this, a superior model is one that has close proximity between the observed data and that predicted by the model. In other words, a supe rior model is one with good fit to the data. This study compared the fit between the Poiss on, ZIP, and Hurdle models as well as their negative binomial formulations. Each analysis wa s performed for five different proportions of 15

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zeros and three different amounts of skew for the nonzero distribution. The intended results would clarify the discrepant fi ndings of previous research. Purpose and Significance of the Study The primary purpose of this study was to determine superiority of fit for various models under varying proportions of zero-in flation and varying levels of skew. As such, determination can be made as to which model has better fit give n data with a particular proportion of zeros and a particular distribution. The s econdary purpose was to elucidat e the reasons for discrepant findings in previous research. The superior model is the appropriate mode l given the research question. However, there are situations in which the appropriate model is unknown or unclear. Further, there may be situations in wh ich a simpler model such as the Poisson may be used in lieu of the more sophisticated Hurdle an d ZIP models. This resear ch provides results that aid researchers in determining the appropriate model to us e given zero-inflated data. Research Questions Model comparisons in this research were based on two measures. One is the deviance statistic, which is a measure of the difference in log-likelihood between two models, permitting a probabilistic decision as to whether one model is adequate or whether an alternative model is superior. This statistic is appropr iate when one model is nested within another model. The other measure is Akaikes Information Criterion (AIC). This statistic penalizes for model complexity and permits comparison of nonnested models; howe ver, it can only be used descriptively. These two measures of model fit were used to comp are results from data simulations where each dataset included 2,000 cases a nd each model was analyzed 1,000 times. Specifically, the measures of model fit were used to an swer the following research questions: 16

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Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated log-likelihood between a) the Negative binomial Poisson model vs. Poiss on model; b) the Hurdle model vs. Poisson model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model vs. ZIP model? Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated AIC between all models? 17

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CHAPTER 2 REVIEW OF THE LITERATURE Zero-Inflated Count Data Count Data As the name implies, count data is data that arises from counting. They are the "realization of a nonnegative integer-valued ra ndom variable (Camer on & Travedi, 1998, p.1). As such, the response values take the form of discrete integers (Zorn, 1996). Although the lower boundary can feasibly be any integer, it is usua lly the case that its value is zero. Strictly speaking, there can be no nonnegative numbers. Hence, the data are constrained by this lower bound of zero and no upper bound. Acknowledgment of concerns over zero-inflatio n, ignoring covariat es, likely dates to Cohen (1954). Cameron and Triverdi (1989, p.1011) identified many areas in which special models have been used to analyze count data including models of counts of doctor visits and other types of health care utilization; occupational injuries and illnesses; absenteeism in the workplace; recreational or shopping trips; automobile insurance rate making; labor mobility; entry and exits from industry; takeover activity in business; mortgage prepayments and loan defaults; bank failures; patent registration in connection wi th industrial research and development; and frequency of airline accidents . as well as in many disciplines including demographic economics, in crime victimology, in marketing, political science and government, [and] sociology. Surprisingly, there was no menti on of research in education. Examples of variables in educational research that yield count data include a students numb er of days absent, number of test items scored correct or incorrect, and number of referrals for disciplinary action. 18

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The lower bound constraint of zero presents the biggest obstacle to ward analyzing count data when assuming a normal distribution. It is common for count data to have a skewed distribution that is tr uncated at the lower bound. 3 1 3() () (1)N i i YYY skewY Ns (2-1) Hence, the data are heteroscedastic with variance increasing as the count increases. Therefore, standard models, such as ordinary le ast squares regression, are not appropriate since they assume that the residuals are distribute d normally with a mean of zero and a standard deviation of one (Slymen, Ayala, Arredondo, & Elder, 2006). Cameron and Triverdi (1998) clarify that the use of standard OLS regression l eads to significant deficiencies unless the mean of the counts is high, in which case normal a pproximation and related regression methods may be satisfactory (p.2). An example of a count data variable is the number of household members under the age of 21 reported by respondents in the Adult Education for Work-Rel ated Reasons (AEWR) survey administered by National Council for Educationa l Statistics in 2003 (Hagedorn, Montaquila, Vaden-Kiernan, Kim & Chapman, 2004). The sample size was 12,725. This variable has a lower count boundary of zero and an upper count boundary of six. The count distribution is positively skewed at 1.971. The distribution mean of 0.54 is certainly not an accurate measure of central tendency; the median and mode are both zero (i.e., the lower-bound itself), and the standard deviation of 0.999 permits negative values in the lower 68% confidence interval. Zero-Inflation It is not uncommon for the outcome variable in a count data distribution to be characterized by a preponderance of zeros. As Tooze, Grunwald, & Jones (2002, p.341) explain, 19

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Typically, [for count data] the outcome variable measures an amount that must be nonnegative and may in some cases be zero. The positive values are generally skewed, often extremely so . Distributions of data of this type follow a common form: there is a spike of discrete probability mass at zero, followed by a bump or ramp describing positive values. The occurrence is primarily in the case of interva l/ratio count data and sometimes ordinal data (Bhning, Dietz, Schlattmann, Mendona, & Kirchner, 1999). Regarding continuous data, Hall and Zhang (2004) explain that these distributions have a null probability of yielding a zero . there is little motivation for a model such as [zero-inflated] normal, because all observed zeros are unambiguous . (p.162). If continuo us zeros are inflated and those zeros are of concern, they can be analyzed separately from the nonzeros. The null probability of continuous zeros is evident in measures such as height and age. The condition of excessive zeros is known as zero-inflation (Lachenbruch, 2002) or as a probability mass that clumps at zeros (Tooze, Gr unwald, & Jones, 2002). It has been recognized as an area of research in the mid-60s1 (Lachenbruch, 2002) when Weiler (1964) proposed a method for mixing discrete and continuous distribu tions. Min and Agresti (2005) formally define zero-inflation as data for which a generali zed linear model has lack of fit due to disproportionately many zeroes (p.1). There are simply a greater number of zero counts than would be expected under the Poisson or some of its variations (Zorn, 1996, p.1). The Sources of Zero-Inflation The zeros can be classified as being either true zeros or sampling zeros. True zeros represent responses of zero that are truly null. Suppose an educational inventory item states How many college preparatory wo rkshops have you attended? Some of the respondents in the sample may have no intentions to apply for college. Hence, the number of preparatory 1 Alternatively, if a scale is bound, it is reasonable to consider an inflated u pper bound. In this case, scale reversal and subsequent appropriate analysis if justified (Lachenbruch, 2002). 20

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workshops attended may never be greater than zer o. Sampling zeros, on the other hand, arise as a probability. There are a proportion of college-bo und students who have not attended a workshop due to the possibility that the workshop was not (o r is not yet) available. Alternatively, some college-bound students may feel prepared and have no reason to participate in a workshop. Hence, the mechanism underlying zero-infla tion can arise from one or both of 1) a possibility that no other respons e is probabilistic, or 2) that the response is within a random sample of potential count res ponses. Martin, Brendan, Wintle Rhodes, Kuhnert, Field, LowChoy, Tyre, and Possingham (2005) term the sampling zeros as false zeros and include error as a source of zeros. They state, Zero inflation is often the result of a large number of true zero observations caused by the real . effect of intere st . However, the term [zero-inflation] can also be applied to data sets with false zero observations because of sa mpling or observer errors in the course of data collection (p.1235). Often, the data contains both types of zeros. This is the result of a dual data generating process (Cameron & Trivedi, 1998). For example, some adults in the AEWR sample may have had true-zero household members under the age of 21 because they are unable to bear children or desire to bear children. Alternatively, they may have random-zero household members under the age of 21 because these adults do have such ch ildren but not as members of the household. Impact of Zero-Inflation on Analyses Much of the interest in count data modeli ng appears to stem from the recognition that the use of continuous distri butions to model integer out comes might have unwelcome consequences including inconsistent parameter estimates (Mullahy, 1986, p.341). In the typical count data scenario, the zero left-bound implies heteroscedasticity (Zorn, 1996). An even greater problem with zero-inflated distri butions, beyond this inadequacy of analyzing such a skewed and heteroscedastic distribution as if it were normal (Tooze, Grunwald, & Jones, 2002) is that they 21

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yield surprisingly large ineffi ciencies and nonsensical results (King, 1989, p126). Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and Possingham (2005) and McCullagh and Nelder (1989) explai n that zero-inflation is a speci al case of overdispersion in which the variance is greater than it should be given a particular distributional shape and measure of central tendency. The impact is bi ased/inconsistent parameter estimates, inflated standard errors and invalid inferences (Jang, 2005; Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and Possingham, 2005). Simple Solutions to Zero-Inflation Deleting zeros The simplest of solutions is to delete all ca ses having responses of zero on the variable of interest. A large proportion of total responses would then be remove d from the total dataset. This would then result in a loss of valuable inform ation impacting statistical conclusion validity (Tooze, Grunwald, & Jones, 2002). The sample size may also then be too small for analyses of the non-zero values. Assuming normality Another simple solution is to ignore the zer o-inflation, assume asymptotic normality, and analyze the data using standard techniques such as ordinary least squares regression. 012 iihhmSexAgei i (2-2) According to this model, the number of hous ehold members under the age of 21 for adult respondent i is predicted from the overall mean, a co efficient relating the respondents sex to hhm, a coefficient relating the respondents age to hhm, and error. The model assumes that the residuals for hhm are distributed normally with a m ean of zero and a common variance,2 For the first equation, y is a vector of responses, X is a design matrix for the explanatory variable 22

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responses, is a vector of regression coefficients relating y to X, and is a vector of residuals measuring the deviation between the observed values of the design matrix and those predicted from the fitted equation. Transforming Zeros Another simple solution is to transform the counts to coerce a mo re normal distribution (Slymen, Ayala, Arredondo, & Elder, 2006). Sinc e count distributions often appear to be positively skewed, one reasonable transformation involves taking the natu ral logarithm of the responses to the predic tor variables. However, assuming the zeros havent been deleted, the transformation will not work since the natural logarithm of zero is undefined (Zhou & Tu, 1999; King, 1989). Sometimes natural log transformations for zer o are handled by adding a small value, such as .001, to the zeros. However, this then leads to an inflation of that tran sformed adjusted value. If 70% of the scores are zero, the resulting transformed distri bution will have a 70% abundance of the transformed value (Delucchi & Bostrom, 2004).2 Further, since the transformation is linear, this technique has been shown to yield biased parameter estimates that differ as a function of the adjustment quantity (King, 1989). Although the undefined log zero problem has been handled, the original problems pervade. As Welsh, Cunningham, Donnelly, & Linenmayer (1996) state, It is clear for data with many zero va lues that such an approa ch will not be valid as the underlying distributional assumptions (linear ity, homoscedasticity and Gaussianity) will [still] be violated (p.298). Finally, for any technique, transformations sometimes create a new problem while solving the old one; a transform that produces constant variance may not produce normality . (Agresti, 1996, p.73). 2 This implies then that, beyond the dual generating proce ss for zeros, the problem can be generalized from inflated zeros to inflated lower boundaries for count data. 23

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Generalized Linear Models Bryk and Raudenbush (1996) state, There ar e important cases . for which the assumption of linearity and normality are not realistic, and no transformation can make them so (p.291). As previously demonstrated, count data is likely to be one such case. Instead of deleting cases or transforming the data, it is more reas onable to specify a diffe rent distribution. As explained by Hox (2002), although it is nice to be able to transform data, modeling inherently nonlinear functions directly is sometimes pref erable, because it may reflect some true developmental process (pp. 93-94). In order for a model to be inhe rently nonlinear (Hox, 2002), there must be no transfor mation that makes it linear.3 These nonlinear models belong to the class of generalized linear models (GLM). The following explanation of generalized lin ear models based on the seminal work of McCullagh and Nelder (1989) with additional clarif ication by Lawal (2003) and Agresti (1996). Lawal (2003) explains that generalized linear models are a subset of the traditional linear models that permit other possibilities th an modeling the mean as a linear function of the covariates. All GLM possess a random component, a systematic co mponent, and a link function. As explained by Agresti (1996), the random compone nt requires the specificati on of the distri bution for the outcome variable. One could specif y this distribution to be norma l; hence, classical models such as ordinary least squares regres sion and analysis of variance m odels are included within this broader class of generalized lin ear models. Other possible random components that could be specified include the binomial distribution, negative-binomial di stribution, gamma distribution, and Poisson distribution. Speci fying the random component depends on the expected population distribution of the outcome vari able. Given both zero-inflation a nd truncated count data yielding 3 Transforming covariates (e.g., including polynomial terms) may graphically appear to be nonlinear while still be linear in the parameters (Singer & Willett, 2003). 24

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an odd-shaped skewed distribution, the random co mponent plays an importa nt part in obtaining valid results. In order to better understand the formulation of the three components, it is necessary to clarify the theoretical foundations of distributions. The probabili ty density function for a normal distribution is 2 21( (;,)exp() 2 2 y fy ) (2-3) which, given random variable 2~(,) XN reduces to the standard normal probability density function 21 ()exp() 2 2 y fy (2-4) which when transformed to the cumulative density function yields 21 ()()exp() 2 2yu yFy du. (2-5) A convenient method for obtaining the paramete rs is to use the distributions moment generating function (Rice, 1995). For the normal distribution, this function is 22()[exp()]exp() 2yt MtEtYt (2-6) The logarithm of the moment generating function yields the cumulant generating function, which then yields the moments of the distribution. For the normal distribution, the first moment is the mean ( ), and the second moment is the variance (2 ). Strictly speaking, a requirement for GLM is that the outcome has a distribution within the exponential family of models (EFM) (McCullagh and Nelder, 1996). Th ese distributions are 25

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defined primarily by a vector of natural parameters ( ) and a scale parameter ( ). The formulation is given by (()) (;,)exp{()(,)} ()Yyb fy cy a (2-7) At first glance, it seems odd to include the nor mal distribution in the EFM; however, first recall the probability density function 2 21( (;,)exp() 2 2 y fy ) (2-8) Algebraic manipulation reveals th at the normal distribution is indeed an EFM formulation. 22 2 (0,1) 22()/2)1 (,)exp{( )(log(2)} 2NYyy EFMfy (2-9) Here, the natural (i.e., canonical) parameter is and the scale parameter is 2 These parameters need to be estimated McCullagh and Nelder (1996) explain the estimation as follows: In the case of generalized linear models, estimation proceeds by defining a measure of goodness of fit between the observed data and the fitted values that minimizes the goodness-of-fit criterion. We shall be concerned primarily with estimates obtained by maximizing the likelihood or log likelihood of the parameters for the data observed (p. 23-24). This turns out to be the log of the EFM function. (,;)log((;,))Yyfy (2-10) The natural and scale parameters are estimated by derivations revealing the mean function ()'() EYb (2-11) and the variance function var()''()() Yba (2-12) 26

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Note that the mean function depends on onl y one parameter. However, as McCullagh and Nelder (1989) explain, . the variance of Y is the product of two functions; one, ''() b depends on the [canonical] parameter (and hen ce on the mean) only and will be called the variance function [denoted V( )], while the other [() a ] is independent of and depends only on . The function () a is commonly of the form()/ aw (p.29) and is commonly c the dispersion parameter. For the normal distribution, the natural parameter is the mean ( alled ); th variance function, V ( e ); equals 1.0, and the dispersion parameter is 2 The systematic component is simply the mode l for the predictors established as a linear combination and is denoted The link function, g ( ), brings together the random component and the systematic component hence linking the function for the mean, and the function for the systematic component, as = g ( ). In other words, it specif ies how the population mean of the outcome variable with a particular distribution is related to the predictors in the model. If g ( ) redundantly equals then the population mean itself is related to the predictors. This is termed the identity link and is exactly the f unction used to link the mean of the normal distribution to its covariates. The key advantage of GLM is that they are no t restricted to one pa rticular link function. Many other links are available. For example, one could specify the log link as g ( ) = log( ) or the logit link as g( ) = log[ / (1)]. However, each random component has one common canonical link function that is best suited to the random component (McCullagh & Nelder. 1996). Alternatively, Each potentia l probability distri bution for the random component has one special function of the mean that is called its natural parameter (A gresti, 1996, p.73). For example, a normal random component usually co rresponds to an ident ity link, a Poisson 27

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distributed random component usua lly corresponds to a log link, and a binomial distributed random component usually corresponds to a logit link. In sum, th e canonical link and natural link are two equivalent terms for specifying the most suitable link connecting a particular distribution covariate function. The B l a different random compone nt and link can be specified to accommodate these e di stribution with parameters n for the number of trials and for the outcome variable with its lin ear systematicinomial Distribution and the Logit Link Suppose we were interested in the differ ences between households with zero children under age 21 and households with one or more children over the age of 21. We could feasibly collapse all nonzero responses in the AEWR data in to a value of one. Now, 71.58% of the values are zeros, and 28.42% of the values are ones. This distribution is obviously not normal. We have now introduced both a lower bound (zero), an up per bound (one), and an inherently nonnorma distribution; hence constraints. Variables that take on only one of tw o values are known as binary, or Bernoulli, variables, and the distribution of multiple i ndependent trials for th ese variables is termed binomial. Bernoulli responses are modeled in term s of the probability (Pr) that the outcome variable ( Y ) is equal to either zero or one. The random component over multiple indpendent trials is thus a binomial for the probability that Y = 1. ~(,) YBn (2-13) The binomial distribution assumes that the res ponses are dichotomous, mutually exc independent, and randomly selected (Agresti, 1996). Since the responses are discrete, lusive, the probability density function is termed the probability mass function and is defined as (;,)[(1)] !()!knn fknp pp knk k. (2-14) 28

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This function gives the p probability of k ones (i.e., heads, hits, successes) over n trials. Rice (1995) clarifies, Any particular sequence of k successes occurs with probability [( from the multiplication principle [i.e., inde pendent probabilities of realizations being multiplicative]. The total number of such sequences is 1)]knppk !()! n knk [permutations], since there are !()! n knk ways to assign k successes to n trials (p.36). The mome nt generating function is {1exp()}n and the cumulant generating function is log{1exp()} n The log-likelihood function is virt ually the same as the probability mass function. However, now we are determining the value of p as a function of n and k (rather than determining k as a function of p and n ) while taking the log of this maximum at [(;,)][(1)] !()!knn fpnk pp knk k. (2-15) The estimates are obtained through derivations of the likelihood func tion as was previously discussed for the normal distribution. Just as the normal distribution population mean, has the best maximum likelihood estimates of X the binomial distribu tion population probability, has the best maximum likelihood estimate of k divided by n which is the proportion of ones, hits, or successes. This greatly reduces calculations when a quick estimate is needed and the random component is not lin ked to any predictors. The binomial distribution even tually converges to a normal distribution. However, the speed of this convergence is primarily a function of skew, (12) () (1) Skewy n (2-16) with p = .50 yielding the fastest conver gence (McCullagh and Nelder, 1996). 29

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The link function should account for the binomial outcome vari able. If linear predictors are used to predict a probability, then we have predicted values in an infinite range rather than constrained to be between zero and one. What is needed is a link function that will map a bounded zero-one probability onto this range of infinite values. The canonical link for the binomial distribution is the logit link. ()log(/(1)) g (2-17) A logit is the natural log of an odds ratio or log[ p / (1-p )]. An odds ratio is equal to a probability divided by one minus that probability. He nce, if the probability is .5, then the odds are .5 / (1-.5) = 1 meaning that the odds of a one are the same as the odds of a zero [i.e., the odds of success and failure are identical] If the probability is .75, th en the odds are .75 / (1-.75) = 3 meaning that a response of one is three times more likely than a response of zero. The reciprocal odds of 1/3 means that a r esponse of one is three times less likely than a response of zero, which is e quivalent to stating that a response of zero is three times more likely than a response of one. Wh en using the logit link to connect the binomial random distribution and the systematic co mponent, the generalized linear model is logit( ) = 01log ... 1pXX (2-18) A probability of .50, which is an odds of one corresponds to a logit of zero. Odds favoring a response of one yield a positive logit, and odds favoring a response of zero yield a negative logit. Hence, the mapping is satisfied since th e logit can be any real number (Agresti, 1996). The regression parameters are slope -like in that they determine the relative rate of change of the curve. The exact rate of change depends on each probability with the best approximation 30

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at that probability being (1) with the steepest rate of change being at = 0.50, which is where X = / (Agresti & Finlay, 1997). Since natural logs can be re versed through exponen tiation and since odds can be converted to probabilities by dividing the odds by the sum of the odds and one, the fitted equation can be used to predict probabilities via 11 11exp( ) 1[exp( )]kk kkXX XX (2-19) It is more common to interpret logistic regression coefficients by only exponentiating them. Then, the coefficient has a slope, rather than slope-like, interpretation; however, the relationship is multiplicative rather than addi tive. Specifically, the expected outcome is multiplied by exp( ) for each one-unit increase in X. Evaluating Model Fit The next step in interpreting generalized linear model results is to determine how well the estimated model fits the observed data, where fit is the degree of di screpancy between the observed and predicted values. McCullagh and Nelder (1989) explain, In general the s will not equal the y s exactly, and the question then arises of how discrepant they are, because while a small discrepancy might be tolerable a large di screpancy is not (p. 33). The goodness of fit improves as the observed values and predicted va lues approach equality. For example, if a scatterplot reveals that all point s fall on a straight line, then the predictive power of the regression equation would be perfect, a nd the subsequent f it would be perfect. The comparison is usually performed through so me statistical comparison of the observed outcome values and the predicted (i.e., fitted) outcome values. Rather than compare and summarize the actual observed and predicted values, it is common to gain summary information 31

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by inspecting the log-likelihood va lue produced from the estimation procedure. Since the model parameters are estimated are from the data, pe rfect fit (i.e., observed-fitted = 0) is rare.4 Hence, the goodness of the fit is measure d to determine whether the difference is small enough to be tolerated. There are many measures of model fit. Typically, the model is compared either to a null model in which the only parameter is the mean or a full model in which the number of parameters is equal to the sample size. It is well-known that minus twice the LR statistic has a limiting chi-square distribution under the null hypothesis (Vuong, 1989, p.308). McCullagh and Nelder (1989) equivalently st ate, The discrepancy of a f it is proportiona l to twice the difference between the maximum log likelihood achievable and that achieved by the model under investigation (p.33). This deviance statistic (G2) is then considered to be asymptotically distributed chi-square with degrees of freedom equal to the number of parameters subtracted from the sample size.5 A significant p -value indicates that the devi ance is greater than what would be expected under a null hypothesis that th e model with less parameters is adequate; hence, the observed model with an additional parame ter or parameters is c onsidered a significant improvement over the null model. Another measure of model fit is Pearsons X2; however, unlike G2, it is not additive for nested models. Yet another measure of model fit is Akaikes Information Criterion (AIC), which penalizes the deviance for the number of parameters in the model.6 The notion is that increasing 4 Perfect fit is always obtained if th e number of parameters and the sample size are identical (McCullagh & Nelder, 1989). 5 The relationship is not always exact since sometimes the deviance is scaled and/or the likelihood is more difficult to estimate than in the simple logistic regression scenario presented here (McCullagh & Nelder, 1989). 6 Other measures such as the Bayesian Information Criteri on (BIC) penalize for both the number of parameters and the sample size. 32

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the number of parameters will increase the log-lik elihood regardless of the model and the data. Hence, the AIC penalizes the loglikelihood with regard to the number of parameters. There are two variations that provide furt her penalties. The Bayesian In formation Criterion (BIC) penalizes for sample size; the Consistent Akaike Inform ation Criterion (CAIC) penalizes even further by considering sample size and adding a small adjustment (Cameron & Trivedi, 1998). These indices can be compared to those of comp eting models; however, this must be done descriptively, not inferentially. The disadvantage is that the AIC can not be compared to a statistical distribution resulting in probabilities for significance testin g; however, the advantage is that, as a descriptive statistic, it can be used to compare nonnested models. The explanation thus far points to the fact that models can be compared to null, full, or other models. Statistical comparis on is valid to the extent that one model is nested within the other, which is to say that both models share the same parameters, and one model has at least one parameter that is not included in the other. Alternatively, Clar ke (2001) defines the models as follows: Two models are nested if one model can be reduced to the other model by imposing a set of linear restrictions on the parameter vector . Two models are nonnested, either partially or strictly, if one model cannot be reduced to the other model by imposing a set of linear restrictions on the parameter vector (p.727). The deviance for comparing two models is calculated as the difference in log likelihoods between the two models then multiplied by -2. This quantity is asymptotically distributed chisquare with degrees of freedom equal to the difference in parameters between the tw o models (Agresti, 1996). A significant p -value indicates that the deviance is greater th an what would be expected und er a null hypothesis of model equivalence; hence, the more complex model wi th an additional parameter or parameters is considered a significant improvement over the nested model. 33

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The difference in log-likelihood statistics (i.e., deviance) can not be used to statistically test nonnested models. This is due to the fact th at neither of the models can be considered the simple or more complex models with additional variables leading to a probabilistically higher log-likelihood. A t -test (or F -test) is a sensible al ternative that eliminat es concern for nesting. However, Monte Carlo simulations have demons trated that, for model comparison tests, the F test is lacking in sufficient power and can result in multicollinearity (Clarke, 2001). The motivation for the AIC statistic is that, al l else being equal, the greater the number of coefficients, the greater the log-likelihoods (C larke, 2001, p.731). Hence, model fit becomes impacted by the number of variables in the mode l along with the effects of those variables. Hence, the AIC penalizes for the number of parameters. The formula is 2()2 A ICLLK (2-20) where LL is the log-likel ihood estimate and K is number of parameters in the model including the intercept. Hence, now the l og-likelihood is adjusted to acco mmodate simplicity and parsimony (Mazerolle, 2004). In actuality, one could compare log-like lihoods between nonnested models. However, beyond the lack of parameter penalty, this tech nique might lead to th e statistical hypothesis testing associated with log-likelihood statistics (i.e., test for the deviance approximated by2 ).The AIC, on the other hand, should not be us ed in a formal statis tical hypothesis test regardless of whether the model is nested or nonnested (Clarke, 2001). Generally, the researcher looks at several AIC indices and decides which model fits best based on a lower-is-better criterion. Mazerolle (2004) states, The AIC is not a hypothesis test, does not have a p -value, and does not use notions of significan ce. Instead, the AIC focuses on the strength of evidence . and gives a measure of uncertainty for each model (p.181). 34

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Logistic modeling necessitated treating all nonzero numbers of children as a value of one. Depending on the research question, this may be a loss of valuable information (Slymen, Ayala, Arredondo, & Elder, 2006). Although sometimes it is necessary to model zero-inflated binomial data (Hall, 2000), specifying a binary distribut ion and logit link is not an ideal method for handling zero-inflated count data. The genera lized linear model that specifies a binomial distribution and a logit link beco mes more relevant when discu ssing the technique of splitting zero-inflated data into a model for the probability of zero separate from or combined with a model for the counts. The Poisson Distribution and the Log Link McCullagh and Nelder (1989), La wal (2003), and Rice (1995) are the key references for the technical underpinnings for this model and dist ribution. The generalized linear Poisson model is considered to be the benchmark mode l for count data (Cameron & Triverdi, 1998).7 This is primarily attributed to the fact that the Poisson distribution has a nonnegative mean (Agresti, 1996). If y is a nonnegative random variable, the Poi sson probability mass function is given by (;)Pr() y e fkYk k k = 0, 1, 2, . (2-21) where is standard Poisson notation for the mean ( ) and k is the range of counts. Derivations by Rice (1995) show that th e expected value of a random Poisson variable is ; hence, the parameter of the Poisson distribution can thus be interpreted as the av erage count (p.113). Alternatively, lambda ( ) represents the unobserved expected ra te or occurrence of events . (Zorn, 1996, p.1). The moment generating function is 7 It is also commonly used to model event count data, which is data composed of counts of the number of events occurring within a specific observation pe riod . [taking the] form of non-negative integers (Zorn, 1996, p.1). 35

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(exp()1)()tX tEee. (2-22) The resulting cumulant generating function is(exp()1) t which, with a variance function of and a dispersion parameter equal to one, lead s to mean and variance both being equal to and the skew equal to one divided by the square root of This equivalence of the mean and variance defined by a single parameter (Cameron & Triverdi, 1998; Agresti, 1996) is the result of a function that yields residuals that sum to zero (Jang, 2005); hence, the systematic portion of a Poisson GLM has no error term. The Poisson distribution is a generalization of a sequence of binomial distributions. Rodrguez (2006) explained that the Poisson distribution can be derived as a limiting form of the binomial distribution if you consider the di stribution of the number of successes in a very larger number of Bernoulli trials with a small proba bility of success in each trial. Specifically, if ~(,) YBn then the distribution of Y as n and 0 with n remaining fixed approaches a Poisson distribution with mean Thus, the Poisson distribution provides an approximation to the binomial for the analyses of rare events, where is small and n is large (p.3). Rice (1995) clarified, The Poisson dist ribution can be derived as the limit of a binomial distribution as the number of trials, n approaches infinity and th e probability of success on each trial, p approaches zero in such a way that np (p.43). Scheaffer (1995) and Rice (1995) have derived the generalization. Further, just as a binomial distribution conve rges to a normal distri bution given sufficient trials, the Poisson distributi on converges to a normal distribution given a large mean. The log-likelihood function for the Poisson distribution is (,)logii iyy (2-23) 36

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with the maximum likelihood estimate of simply being the sample mean (Rice, 1995) and with the related devi ance function being (,)2{log(/)()}iiiiiDevianceyyyy k. (2-24) McCullagh and Nelder (1989) state that the second term is often ignored. Provided that the fitted model includes a constant term, or intercept, the sum over the units of the second term is identically zero, justifying its omi ssion (McCullagh & Nelder, 1989, p.34). The systematic portion of the genera lized linear model takes the form 1122exp()exp()exp()...exp()iiiik ixxx xi (2-25) which is often equivalently expressed as log()'iX (2-26) with derived by solving the equation 1(exp())n iii iyxx0 (2-27) by using iterative computations such as the Newton-Raphson. The canonical link for a generalized linear model with a Poisson random component specifica tion is the log link (Stokes, Davis, & Koch, 1991). log() ~() YP (2-28) The Poisson distribution is not limited to count variates. Cameron and Triverdi (1998) explain that, although count s are usually in the purview of di rectly observable cardinal numbers, they may also arise through a latent process. In other words, ordinal rankings such as school course grades may be discretized as pseudocounts and assumed to have a Poisson distribution. Hence, the results of an analysis based on a Pois son distribution and the r esults using an ordinal analytic technique are often comparable. 37

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As is almost always the case, it is common to identify other variables associated with the count variable (i.e., misspecification). However, th e Poisson model has in interesting feature in that it assumes that there are no variables excluded from the model that are related to the count variable. In other words, there is no stochastic variation (i.e., no error term) (Cameron & Trivedi, 1998). Modifications must be made when one wi shes to use a Poisson model with stochastic variation. Iterative Estimation Agresti (1996) clarifies the iterative estimation procedure. The Newton-Raphson algorithm approximates the log-likelihood function in a neighborhood of the initial guess by a simpler polynomial function that has shape of a concave (mound-shaped) parabola. It has the same slope and curvature location of the maxi mum of this approximating polynomial. That location comprises the second guess for the ML estimates. One then approximates the loglikelihood function in a neighborhood of th e second guess by another concave parabolic function, and the third guess is the location of its maximum. The successive approximations converge rapidly to the ML estimates, often within a few cycles (p.94). The most common methods for estimating standard errors includ e Hessian maximum likelihood (MLH) (i.e., second partial derivative based) standard errors and maximum likelihood outer products (MLOP) (i.e., summed outer product of first derivative) estimation. Interpretation of Coefficients Standard ordinary linear squares re gression lends an in terpretation of as the predicted additive change in the response variable per one-unit change in the predictor variable. However, as was the case with the binomial distribution, the interpre tation differs when considering exponential distributions. For th e Poisson distribution, a on e-unit increase in X has a multiplicative of exp( ) on the The mean of Y at x +1 equals the mean of Y at x multiplied 38

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by exp( ) (Agresti, 1996, p.81).8 Due to the inherent difficulty in interpretation, it is common to express in one of three alternative wa ys. First, the direction of the sign of indicates a positive or negative effect of the predictor on the count variable. Second, the fitted value can be calculated at the mean.9 Third, some interpret the coefficien t in terms of percent change; hence, if =1.64, then as X increases to X+1, the predicted probability incr eases by 64% (Agresti, 1996). Hypothesis testing After conducting the analysis and estimati ng parameters, hypotheses can be tested in several ways as explained by Agresti (1996). One could test the hypothesis that =0 using the traditional Wald z -test via/b z bse Some programs provide Wald test results that are actually z2; this is the Wald 2 statistic with one degree of freedom and appropriate only for a two-tailed test. A third method, the Score test, is based on the behavior of the loglikelihood function at the null value for =0 (Agresti, 1996, p.94) yielding a chi-s quare distributed st atistic with one degree of freedom. Overdispersion In practice, the assumption of an equal mean a nd variance is the exception rather than the norm (McCullagh & Nelder, 1989). It is often the case that the sample variance is greater than or less than the observed sample mean with thes e two seldom being statistically equivalent (Cameron & Trivedi, 1998), especially for zero-inflated data (Welsh, Cunningham, Donnelly, 8 This is similar to the interpretation for the binomial distribution with logit link; however, now the multiplicative effect is directly on rather than on the odds of 9 This can be particularly troublesome since that fitted va lue will only hold at the mean. It provides no valid inference for values greater than or less than the mean since the function is a curve with steepness that can vary drastically between separate values for the predictors. 39

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and Lindenmayer, 1996). This condition is known as overdispersion10 (underdispersion) and is a violation of a major tenet of the Poisson dist ribution that the conditional mean and conditional variance of the dependent variab le are equal (i.e., equidispersion, nonstochastic ity) (Jang, 2005; Zorn, 1996).11 This assumption of equidispersion is th e analog of the ordinary least squares regression assumption of homoscedasticity. The overdispersion has been explained as heterogeneity that has not been accounted for [that is] unobserved (i.e., the population consis ts of several subpopulatio ns, in this case of Poisson type, but the sub population membership is not observed in the sample (Bhning, Dietz, Shlattman, Mendonca, & Kirchner, 1999, p.195). The impact of violation is one of incorrect conclusions due to inaccurate t -statistics and standard erro rs (Cameron & Triverdi, 1998; Agresti, 1996). The estimates of the coefficients can still be consistent using Poisson regression, but the standard errors can be biased and they will be too small (Jewell & Hubbard, 2006, p.14). Alternatively, Slymen, Ayala, Arredondo, and Elder (2006) state that Confidence intervals for regression estimates may be too narrow and tests of association may yield p -values that are too small (p.2). The underlying mechanism for overdispersion is explained as unobserved heterogeneity in responses. It is apparent that some modification to the variance to accommodate over-dispersion is ideal. Typically, maximum likelihood procedures are used to estimate parameters in the model. The term pseudo(or quasi) maximum likelihood estimation is used to describe the situation in which the assumption of correct sp ecification of the density is re laxed. Here the first moment [i.e., the mean] of the specified linear exponent ial family density is assumed to be correctly 10 Overdispersion is sometimes referred to as extra-Poisson variation (Bhning, Dietz, Shlattman, Mendonca, & Kirchner, 1999). 11 The underor overdispersion may disappear when predictors are added to the model; however, this is likely not the case if the variance is more than tw ice the mean. (Cameron & Trivedi, 1989). 40

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specified, while the second [i.e., the variance] and other moments are permitted to be incorrectly specified (Cameron & Triverdi, 1998, p.19). He nce, the Poisson distribution as a baseline (Ridout, Demtrio, & Hinde, 1998) can be m odified to accommodate overdispersion and underdispersion (Cameron & Triverdi, 1998). Rice (1995) states that ga mma densities provide a fairly flexible class for modeling nonnegative random variables (p. 52). One way to accommodate overdispersion is to consider the unobserved heterogeneity as a gamma distri buted disturbance added to the Poisson distributed count data (Jang, 2005). In other words, an individu al score may be distributed Poisson with a mean of but then this mean is regarded as a random variable which we may suppose in the population to have a gamma distribution with mean and index / (McCullagh & Nelder, 1989, p.199). This mixture l eads to the negative bi nomial distribution. Given gamma function, and count y the negative binomial probability mass function is () (;,) !()(1)yy prYy y (2-29) Cameron and Trivedi (1998) provide a formulati on where the negative binomial extends from the Poisson rather than being explained as a mi xture distribution. Gi ven a set of predictor variables, we can define p ii i (2-30) where is scalar parameter to be specified or estimated, and p is a pre-specified power term. If the scalar parameter is set to zero then the r esulting variance is equal to the mean, and the Poisson distribution holds. Hence, the Poisson model is nested within the negative binomial model. 41

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The standard formulation for the negativ e binomial formulation of the function, sometimes called NB2 (Cameron & Trivedi, 1998), leads to a variance that is quadratic by setting p to 2 2 ii i (2-31) This is the formulation seen in most textbooks (Rodrguez, 2006; Schea ffer, 1996). Its mean is the same as that of the Poisson distribution; how ever, its variance is derived from the gamma distribution (Cameron & Trivedi, 1998). Just as the Poisson distribution converges to a binomial distribution, the negative binom ial distribution converges to a Poisson distribution (Jewell & Hubbard, 2006). Poisson and Negative Binomial Models with Zero-Inflation As elaborated upon previously, th e zero-inflation problem is tw o-fold. First, the proportion of zeros is higher than expected given the spec ified population distribution shape resulting in an excess zeros problem. This can be descriptively determined by calculating the expected number of zeros as (())()*(exp()) EfqYfqYY = (2-32) e nFor example, Zorns example had a frequency of 4,052 with a = 0.11. The expected frequency of zeros would be (())4052*(exp(0.109))3,634 EfqY (2-33) which is less than the 3,882 zeros observed in the data. It turns out to be (3882/3634)*100 = 107% of the expectation (Zorn, 1996). Second, the zeros can be a mixture of stru ctural (i.e., true) zero s (Ridout, Demtrio, & Hinde, 1998) and sampled zeros reflecting the mu ltiple sources of zeros problem. Shankar, Milton, and Mannering (1997) state that if a two-state process is modeled as a single process . 42

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if applying traditional Poisson and NB distributions, the estimated models will be inherently biased because there will be an over-representation of zero observations in the data, many of which do not follow the assumed distribution of [the ] frequencies (p.830) Shankar, Milton, and Mannering (1997) note that the ne gative binomial model can spuriously indicate overdispersion when the underlying process actually consists of a zero-altered splitting mechanism (p.835836). In sum, the sources of zeros arise from a dual generating pro cess (i.e., structural and sampled) leading to two sources of unequal m ean/variance dispersion (i.e., that due to unobserved heterogeneity of responses and that due to zero-inflation). Most complex methods for analyzing zero-infl ated count data mode l a mixture of two different distributions. The justification for splitting the distribution into two pieces is wellreasoned by Delucci and Bostrom (2 004). If it is deemed more r easonable to consider the zeros as indicators of cases without a proble m, a more appropriate approach is to ask two questions: is there a difference in the proportion of subjects without the problem [i.e., structural true zeros], and, for those who have a problem [sampled fal se zeros], is there a difference in severity (p. 1164). Zorn (1996) refers to dual regime models wherein an observation experiences a first stage in which there is some probability that its count will move from a zero-only state to one in which it may be something othe r than zero (p.2). T ypically, the dual-regime is composed of a transition stage based on a binomial distribution and an events stage based on some type of Poisson distribution. There are many ways to model two-part di stributions. For example, Mullahy (1986) and King (1989) proposed a Hurdle model in which th e zeros are analyzed separately from the nonzeros. Lambert (1992) proposed a zero-inflated Poisson (ZIP) model in which different 43

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proportions of zeros are analy zed separately and along with the nonzeros. Another early formulation (Heilborn, 1989) w as the zero-altered Poisson (ZAP ) model. Arbitrary zeros are introduced by mixing point mass at 0 with a posit ive Poisson that assigns no mass to 0 rather than a standard Poisson (Lambert, 1992, p.1). Mullahy (1986) presented a variation of the Hurdle model based on a geometric distribution12 for use when specifying a Poisson distribution is not reasonable. Another possib ility is to specify a log-gamma distribution for the event stage (Moulton, Curriero, & Barruso, 2002). Lambert ( 1989) presented a variat ion to the ZIP model known as ZIP( ), which introduced a multiplicative constant to the event stage covariance matrix in order to account for the re lationship between the two models Gupta, Gupta, and Tripathi (1996) derived an adjusted generalized Poisson regression model for hand ling both zero-inflation and zero-deflation; however, accuracy was suggested to be contingent on the amount of inflation or deflation.13 It is also possible to form ulate the model with differen t link functions. Lambert (1989) mentions the possibility of usi ng the log-log link, complementary log-log link (Ridout, Demtrio, & Hinde, 1998), and additive log-log link whil e Lachenbruch (2002 ) mentions the lognormal and log-gamma distributions. Hall (2000) formulated a two-part model for zero-inflated binomial data. Gurmu (1997) describes a semi-parametric approach that avoids some distributional assumptions (Ridout, Demtrio, & Hinde, 1998). There is some research on the extension of two-part models to accommodate random effects (Min & Agresti, 2005; Ha ll & Zhang, 2004; Hall, 2004; Hall, 2002; Olsen, 1999). Halls (2000) model for zero-inflated binomial data pe rmits both fixed and random effects. Dobbie and 12 This distribution is an extension of the binomial distribution where the sequence in infin ite. It is typically used in cases where the researcher is concer ned with probability up to and including the first success (Rice, 1995). 13 Min (2004) stated that the Hurdle model also has this feature. 44

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Walsh (2001) permit correlated count data. Fi nally, Crepon and Duguet (1997) consider the cases where the variables are latent and correlated. The evolution of the research to date has led to an emphasis on the standard Hurdle model and ZIP models (along with their negative binomial extensions) with a binary distribution for the transition stage and a Poisson dist ribution for the events stage and with fixed covariates. For both models, estimates are obtained from maximum likelihood procedures, a lthough there has been some research on the use of generalized estim ating equations (GEE) (Hall & Zhang, 2004; Dobbie & Welsh, 2001). The models are primarily distinguished by whether zeros are permitted in the event stage. In other words, their diffe rences are a reflection of the researchers notions about the potentially multiple sources of zeros in the data and their relationship to excess zeros. They also differ in terms of the transition stage cumulative probability function. To be clarified in the sections that follow, Zorn ( 1996) summarizes the differences as follows: . the hurdle model has asymmetric hurdle proba bility while in the ZIP specification pi is symmetrical. Also, the hurdle model does not permit zero values to occur once the hurdle has been crossed, while in the ZIP model zeros may occur in either stage (p.4). Choosing be tween the models is a matter of validity; hence, the choice rests on substantive ground as well as statistical considerations. As Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and Possingham (2005) note, it is impe rative that the source of zero observations be considered and modeled accordingly, or we risk making incorrect inferences . (p.1243-1244). The Hurdle model The Hurdle model was developed separate ly by Mullahy (1986) in economics and King (1989) in political science, although the term itself was most likely coined by Cragg (1971). 14 14 Cragg (1971) proposed the basic two-step process in wh ich the probability of occurrence is modeled separately from frequencies of occurrence. 45

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Welsh, Cunningham, Donnelly, and Lindenmayer (1996) refer to it as a conditional Poisson model. The idea underlying the hurdle formula tions is that a binomial probability model governs the binary outcome whethe r a count variate has a zero or a positive realization [i.e., a transition stage]. If the reali zation is positive the hurdle is crossed, and the conditional distribution of the positives is governed by a truncated-at-zero15 count data model [i.e., events stage] (Mullahy, 1986, p.345) su ch as a truncated Poisson or truncated negative binomial distribution (Min & Agresti, 2005). In other wo rds, one distribution addresses the zeros while another distribution addresses the positive nonzero counts. For example, for grade-retention data, there would be a model for schools with no dropo uts and a model for school with at least one dropout. It is a finite mixture genera ted by combining the zeros generated by one density with the zeros and positives generated by a second zero-truncated density separately . (Cameron & Trivedi, 1998, p.124). Log-likelihood values are estimated separately for each density. A key feature of the transition model is asymmetry in that the probability of crossing the hurdle increases more quickly as the co variates increase than it d ecreases as the covariates decrease. The function is then as ymmetrical (King, 1989) leading to validity concerns supporting or refuting the substantive theo ry underlying the model. The two-part distributio n of the dependent variable is given first by the transition stage g1 probability mass function 1(0)(iPYg 0) 16 (2-34) 15 The lower bound is then one. 16 This is an alternative representation of the aforementionedPr(0)1 Y 46

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modeling whether the response crosses the hurdle of zero. Assuming a Poisson distribution and log link, Zorn (1996) expands the cumulative distribution function to include covariates as 001exp[exp()]ipX (2-35) The basic model for the event stage is then th e probability for a nonzero realization multiplied by the probability for the counts. 17 2 1 2() ()(1(0))...1,2,... 1(0)igj PYjg j g (2-36) Greene (1994) notates the models with a binomia l distribution and logit link for the transition stage and a Poisson distribution with a log link for the event stage as18 Transition: Pr(0)i y p (2-37) Event: 1 Pr()()(),1,2,... 1!i ik i ie p yk k ek (2-38) Here, p is the probability of a count of zero while is the truncated Poisson mean for the counts greater than zero. The generalized linear m odels as a function of covariates is then Transition Stage: 11log() 1i i ip p x (2-39) Event Stage: (2-40) 22log()ii xThe two vectors of parameters are estimated jointly. 1[{exp[exp()]}{1exp[exp()]} XX (2-41) 2[exp()/({exp[exp()]1}!)] yX Xy (2-42) An alternative notation provided by Min (2003) is 17 The second term is an alternativ e representation of the aforementioned (;)Pr() !ye fkYk k 18 Note that p is used to representative the probability of a zero rather than the conventional representation of a probability of one. 47

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111 11 00()[log(0;,)][log(1(0;,))]iiii i yyPyx Pyx1 1 ie2 (2-43) 1111 01,log(1)i in x i yix' 22 22' 22 2 00()[ log(1)]log(!)x i i i ix e ii i yyyxee y (2-44) Since the two models are func tionally independent, the likelih ood functions can be maximized separately (Min & Agresti, 2005; Min, 2003, Cameron & Trivedi, 1998; Mullahy, 1986). 1log()log()hurdle (2-45) 1212(,)()() (2-46) This is because the large sample covariance be tween the two sets of parameters is zero so the joint covariance matrices can be obtained from the separate fits (Welsh, Cunningham, Donnelly, & Lindermayer, 1996, p.300). Solving the likelihood equations uses either the Newton-Raphson algorithm or the Fisher scoring al gorithm, both giving equi valent results (Min, 2003). The Poisson model is nested within the Hu rdle model (Zorn, 1996). Hence, fit of these two models can be compared statistically.19 The Negative Binomial Hurdle model In the case of zero-inflated data, it is possibl e to have two sources of overdispersion. The variance can be greater th an the mean due to the preponderance of zeros. However, there is now the possibility of unobserved heterogeneity in th e event stage (Mazerolle, 2004; Min & Agresti, 2004). The former scenario has been referred to as zero-driven overdispersion (Zorn, 1996); the latter is Poisson overdispersion. Just as was the c ase with the Poisson model, it is possible to nest the Hurdle model within a more general nega tive binomial framework. Further, the negative binomial Poisson model is nested within the negative binomial Hurdle model. Hence, the fit of a) 19 Zhou and Tu (1999) developed likelihood ratio for count da ta with zeros; however, it was not generalized to any particular zero-inflation model. 48

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the Poisson model and the Hurdle model, b) the negative binomial Poisson model and the negative binomial Hurdle model, and c) the Hurdle model and the negative binomial Hurdle model can be compared using statistical tests.20 Estimation is typically performed by solving the maximum likelihood equations usin g the Newton-Raphson algorithm. The Zero-Inflated Poisson (ZIP) model The Zero-Inflated Poisson, or ZIP, model is another model that one can use when the zeros in a dataset are argued to be caused by both chance and syst ematic factors (Min & Agresti, 2005). The transition stage addresses zero-infla tion while the event stage addresses unobserved heterogeneity of responses including zeros ( Jang, 2005). Welsh, Cunningham, Donnelly, and Lindenmayer (1996) refer to it as a mixture model. This two-part model, developed by Lambert (1992) permits zeros to occur in both the transition stage and event stage (Cameron & Triv edi, 1998); crossing the hurdle in the ZIP model does not guarantee a positive realization of Y (Zorn, 1996, p.4). Further, the probability function in the transition stage is now symmetrical (Zorn, 1996) Lachenbruch (2002) explains that ZIP regression inflates th e number of zeros by mixing point mass at 0 with a Poisson distribution (p. 12). Zorn (1996, p.4) clarifies the distinction between the ZIP and Hurdle models as follows: As a special case of the general model, the ZIP regression is thus seen to make substantially different assumptions about the na ture of the data gene rating process than the hurdle model. Whether parameterized as a logit or a probit, the probabi lity exiting the zeroonly stage is assumed to follow a symmetric cumulative distribution. Likewise, even those cases which make the transition to the events stage may nevertheless have zero counts; crossing the hurdle in the ZIP model does not guarantee a positive realization of Y . The sole difference in assumptions here is that the hurdle models count distribution is assumed to be truncated at zero whereas the ZIP specification count data may take on zero 20 It is not the case that the negative binomial Poisson model is nested within the Hurdle model; hence, one can not statistically compare all fo ur models collectively. 49

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values in the event stage. Another difference is that, unl ess the ZIP model is overly pa rameterized, only the Hurdle model can handle zero defla tion (Min & Agresti, 2005). Compared to the Hurdle model, the equations for the event stage are very similar. The exception is that (1pi) is divided by (1ei ) in the Hurdle model before being multiplied by the remaining elements of the equation. However, the transition stage equations are strikingly different. For the Hurdle m odel, the equation is Pr(yi = 0) = p ; the ZIP model includes the addition of the probability of a nonzero multiplied by the exponentiated Poisson mean. This is the mathematical characteristic that distinguishe s the Hurdle models excl usion of zeros in the event stage and the ZIP models potential inclusion of zeros in the event stage. Rather than model the probability of a zero in the transition stage, the ZIP also models the probability that the counts have a Poisson distribution hence permitting zeros from both a perfect state and a Poisson state (Hur, 1999). Given this, parameterizes the mean of this Poisson distribution (Welsh, Cunningham, Donnelly, & Lindenmayer, 1996). When adding covariates, the Hurdle and ZIP generalized linear model appear the same. Transition Stage: 11log() 1i i ip p x (2-47) Event Stage: (2-48) 22log()ii xUnlike the Hurdle model, the ZIP model likelihood function can not be maximized separately for the transition and event stage. Hence, the Hurdle model has the attractive advantage of an orthogonal parameterization which makes it simpler to fit and interpret than the mixture model (Welsh, Cunningham, Donnelly, & Lindenmayer, 1996) with the disadvantage of asymmetrical transiti on to counts. The likelihood func tion derived by Lambert (1992) is 50

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0010log(exp())()log(1)log(!)ii ii ii in GBBG ii i yy iLeeyBeeyy (2-49) where B is the vector of coefficients and matrix of scores for the event stage and G is the vector of coefficients and matrix of scores for th e transition stage, and where iterations are based on the EM or Newton-Raphson algo rithms (Min, 2003; Lambert, 1992)21. Strictly speaking, the Poisson m odel is not nested within the ZIP model; therefore, it would not be wise to conduct a formal model fit test (Zorn, 1996; Greene, 1994). However, it is interesting to note that the log-likelihood of 10,607 is slightly lower than that produced by the negative binomial Hurdle model. This is in lin e with Greenes (1994) observation when using the Vuong statistic as an alternat ive for testing nonnested models. For present purposes, the important question is whether the ZIP models . provide any improvement over the basic negative binomial . The log-likelihood functio ns are uniformly higher, but as noted earlier, since the models are not nested these are not directly compar able. The Vuong statistic, however, is consistent with the observa tion (Greene, 1994, p.26). The Negative Binomial Zero-Inflated Poisson model The ZIP model can be extended to the negati ve binomial model just as the Poisson was extended to the negative binomial and as the Hu rdle was extended to the Hurdle negative binomial. This may be necessary as Min (2003) expl ains that Sometimes such simple models for overdispersion are themselves inadequate. For instance, the data might be bimodal, with a clump at zero and a separate hump around some c onsiderably higher value. This might happen for variables for which a certain fraction follows some distribution have positive probability of a zero outcome (p.13). He further explains, The equality of the mean and variance assumed by 21 Alternatively, Martin, Brendan, Wintle, Rhodes, Kuhnert, Field, Low-Choy, Tyre, and Possingham (2005), compared relative means and credible intervals estimate from a bootstrap procedure. 51

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the ZIP model . is often not realistic. Zero -inflated negative binomia l models would likely often be more appropriate than ZIP models (p .15). Unfortunately, current iterative techniques lead to a greater risk of nonc onvergence than when using the ot her two-part models (Fanoye & Singh, 2006; Lambert, 1992). The key assumption in accepting these results ov er those produced from the Hurdle models is that there are some zeros that belong w ith the counts representing no household members under the age of 21 for reasons other than thei r never having such household members at all. Although it is not valid to statistically compare the fit of the ZIP model with the Hurdle and Poisson models, it is reasonable to test the fit of the ZIP model within the negative binomial ZIP model. In sum, the choice between Hurdle models and ZIP models is ultimately guided by the assumptions one makes about the data generating process. Min (2004) states, The zero-inflated models are more natural when it is reasonable to think of the population as a mixture, with one set of subjects that necessarily has a 0 response. However, they are more complex to fit, as the model components must be fitted simultaneously. By contrast, one can separately fit the two components in the hurdle model. The hurdle model is also suitable for modeling data with fewer zeros than would be expected under standard distributi onal assumptions (p.20). Model Comparison Testing for Zero-Inflated Data The Poisson model is nested within th e negative binomial Poisson differing only by the dispersion parameter. In fact, the two models ar e equivalent when one uses the negative binomial model and restricts the dispersion parameter to 1.0 achieving Poisson equidispersion (Cameron & Trivedi, 1998). Likewise, the Hurdle model is nested within the negative binomial Hurdle model, and the ZIP model is nested within the negative binomial ZIP model. However, since each involves estimation of a transition stage and event stage model, the n esting rule applies to 52

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both equations. In other words, if the event stag e contains 3 parameters and the transition stage 4 parameters for the nested model, then the more complex model must contain at least these same 3 parameters in the event stage and at least th e same 4 parameters in the transition stage. According to Zorn (1996), th e Poisson model is nested within the Hurdle model, and the negative binomial Poisson model is nested within the negative bi nomial Hurdle model. This is reasonable given that the Hurdle models are estim ating the Poisson models in the event stage and that these likelihood statistics are independent of those produced in the transition stage. The ZIP models, on the other hand, are not estimated in th is manner; it is not reasonable to assume that the Poisson models are nested wi thin the ZIP Models (Greene, 1994) This leads to the hierarchy of permissible model-testing displayed in Table 2-1. Other models can be compared descriptively using the aforementioned Akaikes In formation Criterion (AIC). The remainder of the dissertation will use the deviance statistic for model comparison inference and the AIC for model comparison description. Review of Research Pertaining to and Using Zero-Inflated Count Data Hurdle Model Statistical As previously discussed, Mullahy (1986) pr esented the underlying statistical foundations for the Hurdle model. He specified it in term s of extending from both th e Poisson and geometric distributions. He also presented an extension to the Hurdle model that he termed the with-zeros (WZ) model. The WZ model adjusts for zero-in flation by augmenting or reducing the probability by an additive constant (rather than having them specified by the parent di stribution). Subsequent research has focused on the Hurdle model rather th an the WZ model; this is most likely due to the fact that the WZ turns out to be a special case that collapses into a Hurdle model in some specifications, and estimates are often similar between the models (Mullahy, 1986). Finally, 53

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Mullahy presented tests for specifi cations including a technically complex information matrix test as well as the Score test typically provided in soft ware output. Kings (1989) key contribution to the Hurdle model was a Monte Carlo study confirming that the counts can be viewed as arising from a Poisson process (Civetinni & Hines, 2005). Min and Agresti (2004) conducted a simulati on to compare a zero-inflation and a zerodeflation condition. They found th at the estimates were reasonable for the ZIP model under zeroinflation. However, the coefficient and standard error for the event stage were both very large under zero-deflation. They explain that logit mo dels simply can not accommodate too few zeros. The zero-inflated model is only suitable for zer o-inflation problems. However, the hurdle model is also suitable for modeling data with fewe r zeros than would be expected under standard distributional assumptions. In fact, when a data set is zero-deflated at some levels of the covariates, the zero-inflation model ma y fail (Min & Agresti, 2004, p.5). In contrast, Mullahy (1986) stat ed that, a particularly intere sting feature of the modified count data specifications consider ed here [i.e., Hurdle model] is that they provide a natural means for modeling overdispersion or underdispersion of the data. Specifically, overdispersion and underdispersion are viewed as arising from a misspecification of the maintained parent [data generating process] in which the re lative probabilities of zero and nonzero (positive) realizations implied by the parent distribution are not supported by the data (Mullahy, 1986, p.342). Applications Mullahy (1986) researched daily consumption of coffee, tea, and milk where each is a count variable. Covariates included age, years of completed schooling, family income, sex, race, and marital status. For the Hurdle model, the test statistics [ p -values] for the coffee-model were substantially smaller than those for tea-model a nd milk-model. Mullahy argued that this is not surprising given the ratio of estimates and standard errors. However, closer inspection of the data 54

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reveals that it is the coffee vari able that has the low est proportion of zeros at only 26.26%. It is possible that the Hurdle model was unable to adequately account for the additional overdispersion witnessed in the ot her two models (61.63% and 40.37% zeros). In other words, a two-part model may be a necessary but not suffi cient condition for handling overdispersion in zero-inflated count models, and negative binom ial formulations may be increasingly necessary as the proportion of zeros increases. King (1989) applied the Hurdle model to data for the relationship between the number of nations entering war in a period of time as a function of those in formal interna tional alliances. The hurdle model was formulated based on the premises of Mullahy (1986) and justified due to there being some countries who will not go to war and others who will not at first but will later be dragged in by alliances. Hence, this is a classic example of the justification for true zeros and event-driven zeros. This w as demonstrated statistically by comparing Hurdle and Poisson results. The Poisson alliance co efficient of .007 was significant, the Hurdle model event-stage coefficient of .007 was significant, and the Hurdle model transition stage coefficient of .001 was not significant. Hence, the Poisson interpretati on would be that increas ed alliances lead to increased war frequency. However, the Hurdle results clarify that this is on ly true after the onset of war (i.e., the hurdle has been crossed). Furthe r statistical evidence supported the Hurdle model based on the likelihood-ratio model comparison test. Zero-Inflated Poisson Model Statistical As previously discussed, Lambert (1989) presen ted the original formulation for the ZIP model and its ZIP( ) formulation. She also presented the models extension from the Poisson and negative binomial as well as the derivation of the maximum likelihood (EM) estimates. She ran several simulations to test the adequacy of the model. The first simulation varied sample size, 55

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with one covariate taking on a fixed coefficient and a fixed variance in both parts of the model. The result was an average of 50% zeros in the tr ansition stage and 23% zeros in the event stage. The results suggest that the ZIP model consistently converges at n =25 when using EM and at n =100 when using the Newton-Raphson algorithm. An examination of confidence intervals revealed that the normal-theory intervals are not reliable at n =100; however, almost all simulated likelihood-ratio confidence intervals contained the true mean even at n =25. To summarize, these simulations with one covariate for both and p are encouraging. The ZIP and ZIP( ) regressions were not difficult to compute, and as long as infere nce was applied only when the observed information matrix was nonsingular, estimated coefficien ts, standard errors based on observed information, and estimated properties of Y could be trusted (Lambert, 1992, p.7). Warton (2005) compared 20 datasets of va rying sample sizes, pr oportions of zeros, and factors/levels. The ordinary least squares version included the addi tion of one to all counts before taking the logarithm. The other models not acco mmodating zero-inflations were the Poisson and four formulations of the nega tive binomial Poisson (including the aforementioned quasi-Poisson where the variance is set to ). The zero-inflated models included the ZIP model and the negative binomial ZIP model. The Akaike Info rmation Criterion (AIC) values were calculated for a total of 1,672 variables averaged over data sets and rescaled to a minimum AIC of zero for each dataset. As expected, when overdispersion was pr esent, the negative binomial formulations outperformed the models without these formula tions. However, when overdispersion was not present, the reverse was true for 53% of the variables. This suggests that the level of skew in the model interacts with zero-inflation when measuring model adequacy. 56

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When the proportion of zeros is very small, the distribution looks more like a Poisson distribution truncated at zero. In other words, it shares f eatures modeled by the event stage of a Hurdle model. This led to estimation problems in which the negative bi nomial model ZIP model rarely converged. When it did conve rge, its fit was bett er than that of th e ZIP model for only 11% of the datasets. Possibly, as previously discussed, the zero-inflated Hurdle would converge more often since it can handle both zero-inflatio n and zero-deflation (Min & Agresti, 2005; Min & Agresti, 2004). A very interesting finding pertained to the transformed OLS fit indices. Although transformed least squares was not the best fitti ng model for data, it fitted the data reasonably well. Surprisingly, transformed least squares appeared to fit data about as well as the zeroinflated negative binomial model . The AIC for transformed least squares was not as small as for the negative binomial model overall, although it was smaller for 20 per cent of the variables considered here (Warton, 2005, p.283). Averaging over all datasets, the AIC was lowest for all negative binomial models followed by a close tie between the transfor med OLS model and the negative binomial ZIP model, which was followed by the ZIP model. A ll models were a drastic improvement over the standard Poisson model. The implications are th at, although the Poisson is rarely adequate when the data is not equidispersed and/or is inflated or deflated, an intui tive climb up the ladder of models may not be reasonable. There were feat ures of these datasets including varying degrees of zero-inflation and overall dist ributions that warrant further i nvestigation toward appropriate model selection. If one were to fit a zero-infl ated model, it would be advisable to present quantitative evidence that the zero-inflation term was required. Based on th e present results, it is likely that a term for extra zeros is not needed and a simpler model will usually suffice . 57

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special techniques are not gene rally necessary to account for th e high frequency of zeros. The negative binomial was found to be a good model for the number of zeros in counted abundance datasets, suggesting that a good approach to analyz ing such data will often be to use negative binomial log-linear models (Warton, 2005, p.287-288). In regard to their problems w ith zero-inflated negative bino mial convergence, Fanoye and Singh (2006) developed an extension that im proves convergence termed the zero-inflated generalized Poisson regression (ZIGP) model. Thei r recent research revealed convergence in less than 20 iterations for all trials. However paramete r estimates and standard errors were often very different than those produced by the ZIP m odel. They conclude, Even though the ZIGP regression model is a good competitor of ZINB regression model, we do not know under what conditions, if any, which one will be better. The on ly observation we have in this regard at this time is that in all of the datasets fitted to bot h models, we successfully fitted the ZIGP regression model to all datasets. However, in a few cases, th e iterative technique to estimate the parameters of ZINB regression model did not converge (p.128). Greene (1994) used the Vuong statistic when comparing the Poisson, negative binomial Poisson, and ZIP models. It w as noted that the rank order for the Vuong statistics and the loglikelihood estimates were in alignment. The c onclusion suggested futu re research using the Vuong statistic. The use of Vuongs statistic to test the specifica tion seems not to have appeared in the recent literature . We conjecture that the Vuong testi ng procedure offers some real potential for testing th e distributional assumption in the di screte data contex t. In the cases examined, it appears to perform well and in li ne with expectations (Greene, 1994, p.30). Shankar, Milton, and Mannering (1997) used the Vuong statistic to decide between the negative binomial, ZIP, and nega tive binomial ZIP model for traffic accident data. They clarify 58

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the interpretation of the statistic stating, A value >1.96 (the 95% confidence level of the t -test) for V favors the ZINB while a value < -1.96 favors the parent-NB (values in between 1.96 an 1.96 mean that the test is indecisive) . This test can also be applied for the ZIP( ) and ZIP cases (p.831). Civettini and Hines (2005) explored the effects of missp ecification on negative binomial ZIP models. This included misspecification by leavi ng a variable out of the event stage that was present in the event stage and misspecification by sh ifting a variable from the transition stage to the event stage. Applications Lambert (1992), in formulating the ZIP model, applied it to the anal ysis of defects in manufacturing. In terms of improperly soldered leads, 81% of circu it boards had zero defects relative to the 71% to be expect ed under a Poisson distribution linked to a model with a threeway interaction. This most complicated model had a log-likelihood of -638.20. This dropped to -511.2 for the ZIP model. Although comparing to a differe nt combination of covariates, the negative binomial Poisson model fit better than th e Poisson model but not as well as the ZIP model. Greene (1994) used credit-reporting data to investigate differences between the Poisson, negative binomial Poisson, ZIP, negative binomia l ZIP, as well as some of their aforementioned variants and the specification of a probit link rather than the logit link. The data consisted of 1,023 people who had been approved for credit car ds. The count variable of concern was the number of major derogatory reports (MDR), which is the numb er of payment delinquencies in the past 60 days. For this sample, 89.4% had zero MDR. Given a mean of 0.13, this frequency of 804 is nearly double the 418 we might expe ct in a Poisson distribution. Th e skew of 4.018 is reduced to 59

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2.77 when ignoring the zeros while the mean increases to 1.22. As expected, the negative binomial Poisson resulted in improved fit (based on the Vuong test statisti c), increased standard errors and different parameter estimates. The ZIP model resulted in sligh tly worse fit than the negative binomial Poisson while re maining much better compared to the Poisson model. If all of the overdispersion was due to unobserved response heterogeneity then the results should be similar for the negative binomial ZIP model. However, this model produced the best fit of all. It is interesting to note that, again, the standard errors increase while the parameter estimates are different relative to the ZIP mode l. In fact, of the 6 parameters, 4 estimates decreased, 2 increased, and 1 switc hed in sign. Hence, there are two implications. First, the negative binomial ZIP model was necessary to ac commodate two sources of overdispersion to adjust standard errors. Second, ignoring the negative binomial formulations would have led to nonsensical parameter estimates driv en by a sample mean of 0.13. Bhning, Dietz, Schlattmann, Mendona, and Kirchner (1999) compared preand postintervention scores on the decayed, missing, and filled teeth index (DMFT) for 797 children in one of six randomly assigned treatment conditions The results were not exhaustive; however, the log-likelihood did decrease fr om -1473.20 to -1410.27 when going from the Poisson model to the ZIP model. This study was somewhat unique in that all the covariates (sex, ethnicity, and condition) were categorical, and that the c onditions were dummy-coded represented as five parameters. Also, this data had features that mi ght suggest that zero-in flated models werent necessary. For pre-intervention, the proportion of zeros was 21.58%, which increased to only 28.99% at post-intervention. The means, with the zeros in the data, were 3.24 and 1.85, respectively. Ignoring the zeros changed these m eans to 4.13 and 2.61, respectively. The skew, with the zeros in the data, was 0.20 and 0.65, re spectively. Ignoring zeros changed the skew to 60

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0.08 and 0.63, respectively. In other words, many features of the data were consistent with what would be expected of a Poisson, and possibl y normal, distribution. Nonetheless, with these means and frequencies, the Poisson distributi on suggests overdispersion with 31 permissible zeros for the pre-intervention and 125 permissi ble for the post-intervention whereas the data revealed 173 zeros and 232 zeros, respectively. It then becomes a matter of whether the overdispersion was due to the proportion of zeros in each condition or unobserved heterogeneity in the event stage. The negative binomial ZI P model was not used to analyze this data. Xie, He, and Goh (2001) analyzed the number of computer hard disk read-write errors. Approximately 87% of the 208 cases were zeros. Given that the Poisson mean was 8.64, the authors noted that the ZIP model is to be preferred over the Poisson model. However, this mean is due to several values between 1 and 5, a few between 6 and 15, and 2 values of 75. These latter two values appear to be so far from the ot hers that they should have been treated as outliers and addressed in some other manner. Jang (2005) analyzed the numb er of non-home based trips per day from 4,416 households in Jeonju City, Korea. The provided bar graph suggested that approximately 45% of the cases were zeros. The Vuong statistic (Vuong, 1989) w as used for model selection given that the Poisson is not nested within the ZI P or negative binom ial ZIP models. The purpose of the article by Delucchi a nd Bostrom (2004) was to provide a brief introduction to many possible methods for ha ndling zero-inflation including standard t -tests, bootstrapping,22 and nonparameteric methods. In doing so they provided results from a study involving 179 patients with opioid dependence as signed to either a methadone-maintenance or methadone-assisted-detoxification treatment. Five out of seven ways to segment the sample 22 See Jung, Jhun, and Lee (2005) for bootstrap procedures and simulation results for Type I and Type II errors. 61

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resulted in zero-inflation ra nging from 17% to approximately 66% zeros. The only two-part model to be used was the ZIP model. The ta ble of results revealed that, in terms of p -values, the ZIP model performs either very similarl y or very differently from the Pearson2 test for the proportion of zero values, the Mann-Whitney-Wilc oxon test of nonzero values, and/or the MannWhitney-Wilcoxon test of difference in mean scores between treatment groups It is possible that these tests become more similar as the proportion of zeros declines but such conclusions are based purely on the table of p -values. Desouhant, Debouzie, and Menu (1 998) researched the frequenc y of immature weevils in chestnuts. One tree was measured over 16 years, another was measured over 11 years, and three trees were measured on 1 year. The means ranged fr om .06 to .63. None of the 30 distributions fits a Poisson, 2 values being always very significant . The ZIP distribution fits 25 out of 31 cases . The NB distribution fits 20 out of the 31 (Desouhant, Debouzie, & Menu, 1998, p.384). This led to the conclu sion that researchers should c onsider both true zeros and overdispersion (i.e., trees not ed as contagious and trees va rying in random oviposition behavior). Shankar, Milton, and Mannering (1997) anal yzed a 2-year summary of traffic accident frequencies. For principal arterials, they chose a negative binomial model with data ranging from 0 to 84 ( M = 0.294, SD = 1.09). For minor arterials, they chose the negative binomial ZIP model for data ranging from 0 to 7 ( M = 0.09, SD = 0.346). For collector arterials, they chose the ZIP model for data ranging from 0 to 6 ( M = 0.61, SD = 0.279). Model selection was based on the Vuong statistic. For example, they state, As suspected previous ly, inherent overdispersion in the data is due to the parent NB process and this was validated when the [negative binomial ZIP] 62

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specification failed to provide a statistically better fit (the Vuong statistic <1.96, which corresponds to the 95% c onfidence limit of the t -test (p.833). Slymen, Ayala, Arredondo, and Elder (2006) analyzed pe rcent calories from fat and number of days of vigorous physical activity from 357 females partic ipating in a baseline condition and one of three treatment conditi ons. Covariates included employment status, education, martial status, cigarette smoking, and self-reported health. The zero-inflation was 294 out of 357 (82.4%). They compared models usin g likelihood ratio tests between the Poisson and negative binomial Poisson and likewise between the ZIP and negative binomial ZIP. The AICs were inspected to compare the Poisson and ZIP models. Not surprisingly, the negative bi nomial model fit better than the Poisson model. However, the ZIP model did not fit better or worse than the negative binomial ZIP, and the parameter estimates and standard errors were nearly identical. This suggests almost no overdispersion in the data. Indeed, the nonzero percentages were as follows: 1 = 2. 8%, 2 = 3.4%, 3 = 4.8%, 5 = 2.0%, 6 = 0.0%, and 7 = 2.0%. This suggests strong eq uidispersion leaning to ward a uniform nonzero distribution. The AICs for both models were also nearly equal although both being considerably smaller than the AIC for the Poisson model and somewhat smaller than the AIC for the negative binomial Poisson model. Based on a smaller-is-b etter heuristic, the authors favored the ZIP model with an AIC of 562.5 over the zeroinflated ZIP model with an AIC of 565. ZIP and Hurdle Model-Comparisons The purpose of this section is to present a breadth of literature in which both the Hurdle and ZIP models were either compared statisticall y and/or used to analyze real data. This also includes extensions such as the negative binomial and tau form ulations (e.g., ZIP( )). Some authors presented alternatives that seem to depart from the ZIP and Hurdle models too drastically to be within the scope of this dissertation. For example, Lachenbruch (2001) used a two-part 63

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model; however, the splitting form ulation was not consistent with the literature. Further, the model was compared to atypica l formulations such as the Wilcoxon, Kolmogorov-Smirnov, and z tests. As such, these types of articles are not included in the subsequent review. One exception is Xie, He, and Goh (2001) who included a like lihood-ratio test for comparing the Poisson and ZIP models. Statistical Greene (1994) proposed several zero-altered count models fo r comparison. First, he took Mullahys with-zeros (WZ) adaptation of the Hu rdle model and included a scalar estimate for ease on computational burdens. Greene also pres ented an adaptation of Lamberts ZIP known as ZIP( ) and modified it for the negative binomia l formulations terming them ZINB and ZINB( ). The intention was to identify a procedure which will enable us to test the zero inflated model against the simple Poisson model or against the negative binomial model. The latter will allow us to make a statement as to wh ether the excess zeros are the consequence of the splitting mechanism or are a symptom of unobs erved heterogeneity (Greene, 1994, p.10). Greene developed a method for co mparing the models; however, he noted that there was no a priori reason to think that the Vuong statistic would be inferior. Applications Zorn (1996) examined the counts of actions taken by Congress addr essing Supreme Court decisions between 1953 and 1987. The zeros were seen to arise from two sources since many cases will not be addressed unless there are lobbyist s to pressure redress. Covariates included the year of the decision, the polit ical orientation of the decisi on, the presence of lower court disagreement, the presence of precedence altera tion, declaration of unconstitutionality, and unanimous vote. The number of actions ranged from 0 to 11 ( M = 0.11, SD = .64); however, 3,882 (95.8%) of the 4,052 counts were zeros. This contributed to an exceptionally high skew of 64

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7.97. When ignoring the zeros, th e skew was reduced to 1.86 ( M = 2.59, SD = 1.53). The observed zeros were 107% of that which would be Poisson-expected. Regardless, Poisson model results were in lin e with theory-driven expectations. However, the test of overdispersion w as significant when comparing the Poisson and negative binomial Poisson resulting in fewer significan t predictors than if ignoring overdispersion. The author also fitted a generalized negative binomial model in which the variance parameter is allowed to vary as an exponential function of the same independent variables included in the model of the count (Zorn, 1996, p.9), which led to ev en better model fit. However, due to zero-inflation, no model provided reasonable estimate sizes given the low mean count. Their analyses using the ZIP and Hurdle m odels yielded several findings. First, the probability of remaining a zero in the transition stage was considerably lower for the Hurdle model than for the ZIP model at lower levels of a predictor. This is a reflection of the asymmetry of the Hurdle model. Second, parameter estimates and standard errors we re similar between the two models. They concluded that at least in some circumstan ces the performance of ZIP and hurdle Poisson models will be quite similar. Th is suggests that, as a practical matter and barring any strong theoretical considera tions favoring one over the other, the choice betw een them may be made largely on the basis of conven ience of estimation (Zorn, 1996, p.11). Pardoe and Durham (2003) compared the Poisson, Hurdle, and ZIP models as well as their negative binomial formula tions using wine sal es data. Of the 1,425 counts, 1,000 (70.2%) were zeros. The authors noted that this is greate r than the 67.8% that a Poisson distribution is capable of predicting. Based on the AIC, the zeroinflated negative binomial ZIP performed best. Surprisingly, the Hurdle model fit more poorly than did the Poisson model. It is possible, given the unrealistically high AIC relative to the other models, that the value wasnt calculated 65

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correctly. Alternatively, the di stribution may not have been co rrectly specified since their analysis included Bayesian estimat es of the prior distributions. No discussion pertaining to the Hurdle model was included. However, they did provide a novel procedure for comparing the range of fit statistics across the zero-inflation models. This par allel coordinate plot for goodness of fit measures consists of an x-axis labeled Min on the left and Max on the right. The y-axis is a series of horizontal lines each pe rtaining to a fit sta tistic (e.g., AIC, BIC). The ceiling x-axis contains the labels for the models being comp ared. Then, points for each model are plotted on the lines for the fit statistics at their relative location between Min and Max. A similar procedure restricted to the AIC was used by Warton (2005). This technique was adapted to display the coverage for simulated fit statistics. Welsh, Cunningham, Donnelly, and Lindenmayer (1996) used zero-inflated rare species count data to compare the Poiss on, negative binomial Poisson, Hu rdle, negative binomial Hurdle, and ZIP models. Approximately 66% of the observa tions were zeros. They found little difference between the Hurdle, negative binomial Hurdle, and ZIP model results. Since there was no overdispersion, the authors recommended using the Po isson model. This is in line with Wartons (2005) assertion that the more complex zero-inflation models may not always be necessary; at least, this appears to be the case with 66% zero-inflation and equidispersion. Discrepant Findings What is exactly meant by zero-inflation? Min and Agresti (2005) define zero-inflated count data as data for which a generalized linear model has lack of fit due to di sproportionately many zero (p.1). This raises the question, At what point does the frequency of zeros become disproportionate to the frequenc y of non-zeros? One statistical definition states that the proportion of zeros is greater than that to be expected given th e posited distribution (Zorn, 1996). For example, for count data, the proportion of zeros should not be greater than that expected by a 66

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Poisson distribution. However, there are three problems with this. First, there may be many different proportions of zeros greater than that expected by a pa rticular Poisson distribution. Second, the definition assumes that the full model is distributed Poisson. Th ird, it ignores the two potential sources of overdispersion for Poisson zero-inflated data. The aforementioned AEWR example displayed a zero proportion of .7158 with a mean of .54, a standard deviation of 1, and a skew of 1.971. Ignoring th e zeros, although this event stage distribution remains negative skewed, the mean in creased to 1.91, and th e level of skew dropped to 0.96. The distribution for th e categorical sex variable was bi nomial with approximately 43% males and 47% females. The distribution for the age variable was roughly normal with a mean of 48.86, a standard deviation of 17. 41, and a skew 0.87. Hence, with 71% zeros, a heavily skewed distribution of 1.971, a moderate ly skewed nonzero distribution of 0.96, a normally distributed continuous predictor, and a tw o-level categorical predictor led to the following findings: 1) the Hurdle model fit better than Poisson model; 2) the negative binomial Hurdle fit better than negative binomial Poisson model; 3) the negative binomial Hurdle fit better than the Hurdle model; 4) the negative binomial ZIP fit better than ZIP model; 5) the negative binomial ZIP model descriptively fit better than all others; and, 6) the Hurdle and negative binomial Hurdle model yielded nearly identical estimates and p -values. Hence, findings between the zeroinflation models differed in terms of both fit and the significance of parameter estimates. Although not all research presented sufficient in formation (e.g., data necessary to calculate skew), there is clearly enough variation in results to warrant fu rther research. Mullahys (1986) Hurdle model analyses were impacted by zero-infl ation of .263 and not by zero-inflation of .616 or .404; however, this is in disagreement with findings that the Hurdle model adequately handles zero-deflation (Min, 2003). Lamberts ZIP analys is with 71.8% zeros favored the ZIP over the 67

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negative binomial ZIP. Greene s (1994) ZIP analyses resulted in nonsensical results under .894 zeros and heavy skew (4.02); the negative binomial ZIP corrected this. Slymen, Ayala, Arredondo, and Elders ( 2006) ZIP and negative binomial ZIP r esults were virtually identical; however, their event stage distribution was uniform. This w as confirmed by Wartons (2005) finding that the negative binomia l fits better than the ZIP only when zero-inflation and overdispersion both are indeed present. Extending from this is Bhning, Dietz, Schlattmann, Mendona, and Kirchners (1999) findi ngs that the ZIP actually fit better than the Poisson given .216 and .289 zero-deflation. This is again in contrast to the s uggestion that the Hurdle, and not the ZIP, is appropriate for zer o-deflated data. However, it c ould be argued that a normal distribution should have been assu med given that the event stag e distribution was relatively normal. When comparing the Hurdle model to the ZI P model, Zorn (1996) found similar results given .958 zero-inflation, skew of 7.97, and a reduction of skew to 1.86 for the event stage. These findings are in contrast to Zorns (1996) findings of greater zero-inflation and, subsequently, greater skew. Welsh, Cunningham, Donnelly, and Lindenmayer (1997) also found little difference between the Hurdle and ZIP mode ls. Table 2-2 summarizes the findings from the zero-inflation literature. There are three factors that may have caused the anomalous findings. The first possibility pertains to the effect of di fferent types, quantities, and values for predictors. The second possibility is the proportion of zer os for the outcome variable. The third possibility is the degree of skew in the event stage for the outcome variable. It has already been suggested that the Hurdle and ZIP mode ls should be chosen given a priori research about the source a nd nature of the zeros. Further, it has been established that the 68

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negative binomial formulations are meant to handl e additional overdispersion in the event stage. However, the previous findings suggest that there are additional considerations such as the proportion of zeros and the nature of the event st age distribution. The propor tion of zeros in this research ranged from as low as .20 (Delucch i & Bostrom, 1994) to .958 (Zorn, 1996). Distributions for the event stage included those that were heavily positively skewed (Greene, 1994), moderately positively skewed (AEWR exam ple), distributed norm ally (Bhning, Dietz, Schlattmann, Mendona, & Kirchner, 1999), and distributed uniformly (Slymen, Ayala, Arredondo, & Elder, 2006). The first possibility, pertaining to the cova riates, can only be tested by varying an incredibly large set of conditions ranging from small to large quantities of predictors as well as their types (e.g., nominal, ordinal), and distributions. However, given a particular set of covariates and corresponding values the other two possibilities pertaining to zero-inflation and skew can be explored. It is po ssible to vary the proportion of zer os for the outcome variable and to simultaneously vary the degree of skew of the nonzeros for this outcome variable. Hence, the following research questions are presented: Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated log-lik elihood between a) the Negative binomial Poisson model vs. Poiss on model; b) the Hurdle model vs. Poisson model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model vs. ZIP model? Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated AIC between all models? These questions were answered by establishing several levels of skew and zero-inflation. The covariates and their values were fixed as one continuous variable fr om a standard normal distribution and one binary variab le. Data for the outcome variable, for all levels of skew and 69

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zero-inflation, were simulate d to estimate log-likelihood va lues, AIC indices, covariate coefficients, standard errors, and p -values. The details are delineat ed in the following chapter. The objective is consistent with Zorns (1996) advice: First and fore most, work should be undertaken to better ascertain the st atistical properties of the various estimators outlined here. It is important that we determine the robustness of these techniques to skewness in the dependent variable, model misspecification, and the host of ot her problems that all too frequently plague political science researchers . perhaps using Monte Carlo methods to assess under what circumstances the results of the tw o may diverge (Zorn, 1996, p.12). 70

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Table 2-1. Five pairs of nested m odels valid for statistical comparison Valid Comparisons (Nested Models) 1 2 3 4 5 Poisson Simple Simple NB Poisson Complex Simple Hurdle Simple Complex NB Hurdle Complex Complex ZIP Simple NB ZIP Complex 71

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Table 2-2. Summary of lit erature on zero-inflation Researcher(s) Models Compared Zeros Superior Model Comments Min & Agresti (2004) Hurdle vs. ZIP Simulation Hurdle Zero-deflation Min & Agresti (2004) Hurdle vs. ZIP Simulation Equal Zero-inflation Mullahy (1986) Hurdle .26, .62, .41 Hurdle .26 Lambert (1992) ZIP vs. NB Poisson vs. Poisson .718 ZIP over NB Poisson over Poisson Greene (1994) ZIP vs. NB ZIP vs. NB Poisson .894 NB ZIP over ZIP; NB Poisson over ZIP over Poisson Heavy skew; Probit link Slymen, Ayala, Arredondo, and Elder (2006) Poisson vs. NB Poisson; ZIP vs. NB ZIP .824 NB Poisson; Equal Uniform event stage; Overall, AICs favor ZIP Xie, He, and Goh (2001) ZIP vs. Poisson .87 ZIP Outliers Bhning, Dietz, Schlattmann, Mendona, and Kirchners (1999) Poisson vs. ZIP .216, .289 ZIP Zero-deflation; normal event stage Zorn (1996) Hurdle vs. ZIP .958 Equal Heavy skew Pardoe and Durham (2003) Poisson vs. Hurdle vs. ZIP .702 NB ZIP over Poisson over Hurdle Based on AICs Warton (2005) ZIP vs. NB ZIP Vari ous ZIP or NB ZIP Overdispersion favored NB ZIP Warton (2005) ZIP vs. NB ZIP Very low ZIP Rare convergence for NB ZIP Warton (2005) NB ZIP vs. OLS vs. NB ZIP vs. Poisson Various NB ZIP over OLS/NB ZIP over Poisson Based on AICs Welsh, Cunningham, Donnelly, and Lindenmayer (1997) Hurdle vs. ZIP .66 Equal No overdispersion 72

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CHAPTER 3 METHODOLOGY The previous chapter fulfilled three objectives. First, it described the statistical models and methods for analyzing count data including that which is zeroinflated. Second, it presented research, both technical and app lied, pertaining to three models (Poisson, Hurdle, and ZIP) as well as their negative binomial form ulations. Third, it was concluded that there is considerable divergence in findings between models and th at such differences s hould be explored by examining different levels of zero-inflation and sk ew for the count outcome variable. This led to the following research questions: Research Questions Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated log-lik elihood between a) the Negative binomial Poisson model vs. Poiss on model; b) the Hurdle model vs. Poisson model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model vs. ZIP model? Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated AIC between all models? As recommended by Zorn (1996), these questio ns were answered using a Monte Carlo study in which the proportion of ze ros and the skew for the distribution of the event stage counts varied between simulations. Monte Carlo Study Design Monte Carlo studies begin in the same manner as other research methods. First, a problem is identified as a re search question. Second, the problem is made concrete in the form of a hypothesis or set of hypotheses. Third, the hypotheses are tested using rigorous methods. Fourth, conclusions are drawn from these results. Fifth, the implications and limitations are elucidated for future researchers and practitioners. 73

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For this study, the problem was zero-inflation for count out comes. The problem was then clarified in the form of the rese arch questions stated above. It is at this point that Monte Carlo studies differ from most other me thods. For the Monte Carlo study, no real data is gathered. Rather, a set of samples are generated based on given parameter specifications resulting in a sampling distribution that is consid ered to be equivalent to that which would have been obtained had this many real participants been available. Th is is a clear advantage ov er the true experiment where statistics are used to in fer from one sample to an en tire population (Mooney, 1997). The Monte Carlo study is not limited to a finite number of participants and is subsequently not prone to violations of asymptotic th eory (Paxton, Curran, Bollen, Kirby, & Chen, 2001). Paxton, Curran, Bollen, Kirby, and Chen (2001, p.287) provide a succinct expl anation as follows: The researcher begins by cr eating a model with known population parameters (i.e., the values are set by the research er). The analyst then draws repeated samples of size N from that population and, for each sample, estimates th e parameters of interest. Next, a sampling distribution is estimated for each populat ion parameter by collecting the parameter estimates from all the samples. The properties of that sampling distribution, such as its mean or variance, come from this estimated sampling distribution. Similarly, Mooney (1997, p.2) explains, Monte Carlo simulation offers an alternative to analytical mathematics for understanding a statistics sampling distributi on and evaluating its behavior in random samples. Monte Carlo simulation does this empirically using random samples from known populations of simulated data to track a statistics be havior. The basic concept is straightforward: If a statistics sampling distribution is the densit y function of the values it could take on in a given population, then its estimate is the rela tive frequency distribution of the values of that statistic that were actually observed in many samples drawn from that population. The Monte Carlo simulations were performed using the R programming language (R Development Core Team, 2006). R is an open-source language based on the commercially 74

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available S-Plus program (Insightful, 2005) and is just one of many programs that can be used to model zero-inflated data.23 Monte Carlo Sampling Pseudo-Population Mooney (1997) explains that defining th e population parameters requires defining the pseudo-population. In the pseudo-population, the values for a cat egorical factor, X1, with two levels were constant at either 0 or 1; this is representative of a typical two-level categorical factor such as sex. In the dataset, these values alternat ed. The Excel 2003 random number generator was used to draw a random selection of 1,000 normally di stributed values to represent the continuous variable, X2 ~ N (0,1). This resulted in a pseudo-population of N = 1,000 where the values for categorical X1 and continuous X2 were known. The values for X2 ranged from 2.945 to 3.28 with a mean of 0, a median of 0.05, a standard deviation of 0.986, and skew of 0.127. These two sets of covariates and their distributions were chosen as a parsimonious generalization to the basic ANCOVA general linear model that exte nds from analyses with either quantitative or qual itative predictors.24 The simulations varied in terms of a) the amount of zeroinflation present in the outcome variable scores; b) the amount of skew present in the event stage outcome variable scores, and c) the generalized linear model. The outcome variable, Y was established as a deterministic variable in that it varied systematically as a function of the specified distributions. As clar ified by Mooney (1997), Deterministic variables are vect ors of numbers that take on a ra nge of values in a prespecified, nonrandom manner (p.6). The regression coefficients for X1 and X2 are random variables that 23 Others include Stata, LIMDEP, COUNT, MATLAB, and SAS. Preliminary analyses compared results between SAS-code incorporated in research (Min & Agresti, 2004; Min, 2003) to publicly available R-code written by Simon Jackman of the Stanford Political Science Computing Laboratory to verify the comparability of results. 24 The model is similar to that of the AEWR examples. 75

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take on their realizations as a result of the relationship between deterministic Y and the two random X covariates. The Prespecified Zero Proportions Justification for generating values with prespecified proportions of event counts with frequencies also determined by prespecified proportions of zero s is justified by Mooney (1997). If we know (or are willing to make assumptions about) the components that make up a statistic, then we can simulate these components, calculate the statistic, and explore the behavior of the resulting estimates (Mooney, 1997, p.67). In his case, the concern was bias determined by calculating statistics and inspecti ng graphs as a result of Monte Ca rlo simulations. For this study, the concern was goodness-of-fit for six models by comparing log-likeli hoods and AICs and inspecting graphs as a result of M onte Carlo simulations over three le vels of skew and five levels of zero proportions. Previous research displayed zero-inflation ra nging from .20 (Mullahy, 1986) to .96 (Zorn, 1996). To reflect a range including both zero-deflation and zero-i nflation, six pseudo-populations were established differing in the proportion of zeros present in the count outcome variable. The pseudo-populations contained either 0.10, 0.25, 0.50, 0.75, or 0.90 proportions of zeros.25 Pre-Specified Skew To manipulate skew, the even t stage distributions for the count outcome variable were varied over three conditions. For each condition, proportions were specif ied and values were drawn randomly from a multinomial di stribution such that the freque ncies of the values added to the frequency for those already drawn to represe nt zero-inflation summed to 1,000. In other 25 Originally, a 0.00 proportion of zeros was included as a control condition. However, for the case of negative skew, this is simply a count model truncated at one. And, for a normal distribution event-stage, this is semicontinuous data often tested with different methods from those for zero-inflation (Min, 2002). 76

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words, if .50 (or 500) observations were assigned a value of zero then the remaining .50 (or 500) observations had values drawn from the prespecified multinomial distribution. Event stage values ranging from one to five were sampled in order to represent a range small enough to distinguish the distribution from one that might be analyzed as continuous given a particular shape. The prespecif ied probabilities for each of the five counts were determined primarily to achieve a particular level of skew and secondarily in order to achieve a convergent and admissible solution. Hence, th e proportions were not always exactly equal to .10, .25, .50, .75, and .90; the approximates were selected to achieve convergence leading to more trust in convergence for the Monte Carlo simulations. Table 3-1 displays the proportions of each count as a function of the three levels of skew and five levels of zer os. Table 3-2 displays this same information in terms of frequenc ies instead of proportions. Table 3-3 displays the descriptive statistics for each distribution. Random Number Generation By definition, a random number is one in which there is no way possible to a priori determine its value. Most statistical analys is software packages include random number generators. However, these generated random numbers are not truly random. Usually, one specifies a seed; when re plications are performed using the same seed, the generated numbers are identical to the first. Hence, the values are pseudo-random (Bonate, 2001). However, this limitation is actually an advantage in that th e researcher can check for errors in model programming and run the anal ysis again with the same generated sample (Mooney, 1997).26 26 Technically, this is only true for most random number generators. The R programming language, by default, bases its generation on the Wichman-Hill algorithm and the system clock resulting in a 626-integer seed (R Development Core Team, 2006). 77

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Another feature of Monte Carl o random sampling pertains to the desired distributions. Typically, the random numbers are drawn from a uniform distributi on, which is then followed by a transformation to the desired di stribution. As Mooney (1997) e xplains, In its standard form, U(0, 1), the uniform distribution is the building block of all Monte Carlo simulation work in that from it, in one way or another, variables with all other distributi on functions are derived. This is because the U(0, 1) distribution with its 0 x 1 range, can be used to simulate a set of random probabilities, which are used to generate ot her distribution functi ons through the inverse transformation and acceptance-re jection methods (p.10). The random number generation was performed using R 2.3.1 (R Development Core Team, 2006). The procedure requires the generic sample command in which the following were specified: 1) a range of counts, which in this study, was from one to five (not zero to five since proportions of zeros were alrea dy drawn from the pseudo-populati on), 2) the number of values to draw, which in this study was one minus the pres pecified proportion of ze ros, 3) the proportions for each value in the range, which in this case was one of three possibi lities determining skew, and 4) the specification to sample with replacem ent. The seed was arbitrarily set at 6770. Sample Size Determining the appropriate sample size for each simulate is an important concern. This could range from zero to infinity. However, if the sample size is too small then it is not safe to assume that estimates are asymptotically normal. On the other hand, computer time and burden increases as sample size increases. The sample size was based on the highest found in the literature pertaining to zero-i nflated count data, which was n = 1,000 (Civentti & Hines, 2005). Simulation Size Determining the number of simulations is also an important concern since too few replications may result in inaccurate estimat es and too many replications may unnecessarily 78

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overburden computer time and performance (Bonate 2001). Hurs (1999) r esearch pertaining to the ZIP model with random eff ects set the number of simulations at 200. Min and Agresti (2005) were able to sufficiently compare the goodness of fit for several competing models using 1,000 simulations. Likewise, Civettini and Hines ( 2005) selected 1,000 simulations when researching misspecification in negative binom ial ZIP models. Lambert (1989) set the number of simulations at 2,000 when researching the asympto tic properties of the ZIP model. Mooney (1997) states that The best practical advice on how many trials are needed for a given experiment is lots! Mo st simulations published recently report upward from 1,000 trials, and simulations of 10,000 and 25,000 trials are common (p.58). Given the previously noted problems with convergence for the negative binomi al ZIP model, it seem s prudent to minimize the number of simulations as much as possible. Howe ver, it is also important to simulate under conditions already found to produce asymptotic results. Hence, similar to Lamberts (1989) seminal study and equal to the maximum found in the literature, the number of simulations was set at 2,000 for each condition ( S = 2,000). Iteration Size Iteration size is not much of a concern for the basic Poisson and negative binomial Poisson model. However, obtaining valid estimates for the Hurdle model, ZIP model, and their negative binomial counterparts re quires selecting an appropriate maximum number of iterations. Too few iterations can lead to incorrect estimat es or, even worse, premature declaration of nonconvergence. Too many iterations results in u nnecessary computer time and burden. Various procedures for analyzing these models in R have maximum iterations of 500, 5,000, and 50,000. It was important to determine an iteration size that would be equal ac ross analyses and large enough given some models potential for nonconve rgence. These concerns were deemed more 79

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important than excessive computational burden. He nce, the procedure with the largest iteration size was selected for all procedures leading to 50,000 iterations per analysis. Distribution Generation The following describes the procedure fo r generating the distributions for each simulation. The total proportion of counts out of 1,000 to be sampled was reduced by the prespecified proportion of zeros. Hence, if the proportion of zeros was .50 then the proportion of event stage counts was 1.00-0.50 = .50. Translated into frequencies, this is 1,000-(0.50 1,000) = 500 The generic R sample procedure was used to sample with replacement from the event stage counts according the specified proportions de pending on the skew condition. The seed was set at 6770. The values were sampled over N = 1,000. Each sample was simulated S = 2,000 times. The data over all S = 2,000 at N = 1,000 were then stored in separate files as they were created. The filenames conformed to the labelin g format where the model was replaced by an underscore (e.g., _25Pos w as the filename for the S = 1,000 datasets at N =2,000 where the proportion of zeros was .25 and the sk ew for the event stage was positive). Monte Carlo Models As previously discussed, generalized lin ear models include a random component, a systematic component, and a link function. The X1 and X2 constants form the systematic component in the pseudo-populations genera lized linear model. The random component specification for the distribution of the outcome mean varied from pseudo-population to pseudopopulation. The base level genera lized linear model assuming a normally distributed outcome is given by 01122()()iiYXXi i (3-1) 80

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Subsequent models extended from this base model to form the six distributions for deterministic Y The first model, which was the Poisson generalized linear model with a log link, is given by 01122log()()()iiXXi (3-2) Table 3-4 displays the parameters for this model over all conditions of skew and zeroinflation. For example, the analysis with a .10 proportion of zeros and a positively skewed event stage distribution yielded 1log().450.004().037()iX2X (3-3) For both predictors, the coeffici ent near zero transforms to an exponentiation near one. The coefficient for X1 is lowest (.001) for the negatively distributed data with a .25 proportion of zeros and highest (.007) for the pos itively distributed data with a .50 proportion of zeros. The coefficient for X2 is lowest (-.007) for the negatively di stributed data with a .10 proportion of zeros and highest (-.165) for the normally distribute d data with a .90 proportion of zeros. Hence, for the two-level categorical X1 variable, changing from zero to one multiplies the mean of the outcome variable by approximately exp(0.00), whic h equals one. For the simulated data, the test for the coefficient estimates co rresponding to these pseudo-p opulation parameter values is approximately H0: = 0. The second model, which was the negative binomial formulation of the Poisson model, is the same as the Poisson model with the addition of a dispersion parameter. Table 3-5 displays the parameters for this model over all conditions of sk ew and zero-inflation. Like the Poisson model, the test for the simulated data is H0: = 0. The dispersion parameter is also included in Table 35. For the simulated data, the test that this pa rameter equals zero has equivalent results to the model comparison tests conducted. 81

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The third and fifth models were the general formulations fo r the Hurdle and ZIP models given by logit 01122()log()()() 1i i ipi i p ZZ p (3-4) 45162log()()()iiXXi (3-5) while the fourth and six models were their negative binomial fo rmulations. Tables 3-6 through 313 display the pseudo-population coefficients, standard errors log-likelihood values, and AIC values over all conditions of proportions of zero and skew. A lthough there are four models (i.e., Hurdle, negative binomial Hurdle, ZIP, and negative binomial ZIP), there are eight tables since, for each model, there are separat e results for the transition (zer os) stage and events (nonzero counts) stage. Monte Carlo Analysis Procedures A generic looping procedure was written in the R programming language to retrieve each of the 15 sets of simulated data. Each dataset was analyzed with each of the six models. The Poisson model used the generic glm procedure in R, which requires specification of the model, Poisson distribution, and l og link. The negative binomial Poisson model used the glm.nb procedure in R from the generic MASS library. This procedure requires only the specification of the model. The R Core Development Team (2006) describes the procedure as follows: An alternating iteration process is used For given 'theta' [dispersion] the GLM is fitted using the same process as used by 'glm()'. For fixed means th e 'theta' parameter is estimated using score and information iterations. The two are alternat ed until convergence of both is achieved (R Core Development Team, 2006). The Hurdle model used the hurdle procedure in the pscl library authored by Simon Jackman, PhD of the Political Science Computi ng Laboratory at Stanford University. The 82

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procedure requires specification of the count response variable (Y), the Poisson distribution for the event stage, the logit link function for the transition stage, and the models for both the transition stage and event stage, which for this study were X1 and X2. The negative binomial Hurdle model also used the hurdle procedure bu t specified a negative binomial rather than a Poisson distribution for the event stage. Th is hurdle model procedure maximizes the loglikelihood using either the Br oyden-Fletcher-Goldfarb-Shanno (BFGS) or Nelder-Mead methods. The Nelder-Mead (default) w as selected for solution optimization. The ZIP model used the zicounts procedure in the zicounts library authored by Samuel Mwalili, doctoral student in biostatistics at Katholieke Universeteit Leuven (Netherlands). Similar to the hurdle procedure, zicounts requires specification of th e count outcome variable and models for both the transition st age and event stage. The distri bution is specified as ZIP. Optimization procedures include the BFGS, Nelder-Mead, and conj oint gradient (CG) methods; for consistency, the Nelder-Mead was chosen. The negative binomial ZIP used the same procedure with the specification of the ZINB distribution. The specific procedure for analyzing the data with the models was as follows: Three separate loops were established for the negatively skewed, normal, and positively skewed distributions. Arrays were created to store the log-likelihood, AIC, coeffici ent estimates, standard errors, and p -values. The outermost loop for each dist ribution pertained to the five conditions of varying zero proportions; at each loop, the data correspondi ng to this zero-conditi on and distribution was loaded. Within these loops, another loop ing procedure pertained to the six models that analyzed the data. Within these loops another looping procedure was defined for the number of simulations in the data set. 83

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It was at this point that th e data corresponding to a particular distribution, proportion of zeros, model, and simulation were analyzed with the calculated AIC, log-likelihood, coefficient estimates, standard errors, and p -values transferred to the appropriate array. Models that failed to converge were automatically coded as NA.27 The results for each statistic over all simulati ons for a particular distribution, model, and proportion of zeros were then exported in comma -separated format for subsequent analyses. Hence, the three distributions by five proportion of zeros conditions by six models yielded 90 data files each containing column s pertaining to a particular stat istic or set of statistics and 2,000 rows of simulated results. Analysis Design A series of tests were conduc ted using the simulated data. The results and graphical output were created using R 2. 01 (R Development Core Team, 2006) and SPSS 14.0. The design for a particular analysis depended on the res earch question. These questions were as follows: Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated log-lik elihood between a) the Negative binomial Poisson model vs. Poiss on model; b) the Hurdle model vs. Poisson model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model vs. ZIP model? Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated AIC between all models? The first question was answered by calculating log-likelihood values for the six models over the six zero proportion conditions and the thr ee skew conditions. The deviance statistic was then calculated as -2(LLS-LLC) where LLS is the model with less parameters (i.e., the simple model) than the LLC (i.e., the more complex model). Since there were 5 conditions for the proportions of zeros and 3 conditi ons for skew, this led to a) a total of 15 separate analyses comparing the fit of the negative binomial Pois son model and the Poisso n model; b) 15 separate analyses comparing the fit of th e Poisson model and the Hurdle model; c) 15 separate analyses comparing the fit of the negative binomial Po isson model and the negative binomial Hurdle 27 This number was chosen to reduce the probability of obtaining a valid result that would be identified as a nonconvergant solution 84

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model; and d) 15 separate analyses comparing the fit of ZIP model and the negative binomial ZIP model. Test statistics for the deviance were assume d asymptotically chi-square with degrees of freedom equal to the difference in the number of parameters between the two models. Some models differed only by the dispersion parameter; these were a) the comparison of the Poisson model and negative binomial Po isson model; b) the comparison of the Hurdle model and negative binomial Hurdle model; and c) the comp arison of the ZIP model and negative binomial ZIP model. The log-likelihood statistic for th e Hurdle model is based on the log-likelihood statistics from each of its two parts. Given thre e parameters in the Poisson model (i.e., the intercept and the two predictor coefficients), th ere are six parameters in the Hurdle model. Hence, the degrees of freedom for the model compar ison test are the difference, which are three. The same is true for comparing the negative bino mial Poisson model to the negative binomial Hurdle model where including the dispersion parame ter leads to subtracting 4 parameters from 7 parameters. Each of the 2,000 goodness-of-fit statistics for the simpler mode l was subtracted from each of the 2,000 goodness-of-fit statisti cs for the more complex model. These results were then multiplied by -2. Each of these values was then co mpared to the chi-square distribution with the appropriate degrees of fr eedom. This yielded a p -value representing the probability of obtaining a statistic this high or higher given that the simpler model adequately fits the data. Values exceeding this critical chisquare statistic based on a Type I error rate of = .05 were coded with these results suggesting that the more complex model fits th e data better than the Poisson model. Values failing to exceed the critical valu e were coded with these results suggesting adequate fit for the simpler model. 85

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The results were thus the pr oportion of simulated datasets favoring the more complex model over the simpler model for a particular proportion of zeros and le vel of skew. Output included 1.) descriptive statisti cs for the goodness-of-fit for each model and 2.) boxplots for the difference in goodness-of-fit between the two models. Answering the second question was done in a similar fashion to answering the first question. This is due to the fact that the AI C is a linear transforma tion of the log-likelihood statistic with a result that is positive in sign and is interpreted in a lower-is-better fashion. However, these analyses did not involve compar isons to the chi-square distribution. As previously explained, th e AIC should not be used in this manner. However, the advantage is that the AIC can be used to descriptively compare all models regardless of whether one is nested or not within another. Boxplots were created to display the range of AICs produced over simulations for each analysis. 86

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Table 3-1. Proportions of counts as a function of zeros and skew Study Proportion of Zeros Proportion of Remaining Nonzero Values Ones Twos Threes Fours Fives _Pos10 0.10 0.504 0.227 0.088 0.054 0.027 _Norm10 0.099 0.198 0.306 0.198 0.099 _Neg10 0.027 0.054 0.091 0.227 0.501 _Pos25 0.25 0.418 0.190 0.075 0.046 0.021 _Norm25 0.083 0.166 0.254 0.166 0.081 _Neg25 0.022 0.045 0.075 0.188 0.420 _Pos50 .50 0.280 0.125 0.050 0.030 0.015 _Norm50 0.053 0.107 0.175 0.110 0.055 _Neg50 0.015 0.030 0.050 0.125 0.280 _Pos75 0.75 0.140 0.062 0.025 0.015 0.008 _Norm75 0.028 0.052 0.089 0.053 0.028 _Neg75 0.008 0.015 0.025 0.062 0.140 _Pos90 0.90 0.056 0.025 0.010 0.006 0.003 _Norm90 0.011 0.021 0.035 0.024 0.009 _Neg90 0.004 0.005 0.010 0.025 0.056 Table 3-2. Frequencies of count s as a function of zeros and skew Study Frequency of Zeros Frequency of Individual Nonzero Values Ones Twos Threes Fours Fives _Pos10 100 504 227 88 54 27 _Norm10 99 198 306 198 99 _Neg10 27 54 91 227 501 _Pos25 250 418 190 75 46 21 _Norm25 83 166 254 166 81 _Neg25 22 45 75 188 420 _Pos50 500 280 125 50 30 15 _Norm50 53 107 175 110 55 _Neg50 15 30 50 125 280 _Pos75 750 140 62 25 15 8 _Norm75 28 52 89 53 28 _Neg75 8 15 25 62 140 _Pos90 900 56 25 10 6 3 _Norm90 11 21 35 24 9 _Neg90 4 5 10 25 56 87

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Table 3-3. Descriptive statistics for each distribution Range = 0 to 5 Range = 1 to 5 Mean Std.Dev. Skew Mean Std.Dev. Skew _Pos10 1.570 1.127 1.155 1.750 1.051 1.45 _Norm10 2.700 1.414 -0.310 3.000 1.150 0.08 _Neg10 3.820 1.619 -1.356 4.250 1.053 -1.43 _Pos25 1.310 1.181 1.092 1.750 1.046 1.42 _Norm25 2.250 1.635 -0.069 2.990 1.149 0.08 _Neg25 3.190 2.054 -0.673 4.840 0.367 -1.86 _Pos50 0.880 1.149 1.514 1.750 1.053 1.44 _Norm50 1.510 1.710 0.587 3.010 1.140 -0.01 _Neg50 2.130 2.253 0.236 4.250 1.053 -1.44 _Pos75 0.440 0.928 2.577 1.760 1.064 1.44 _Norm75 0.750 1.422 1.660 3.000 1.149 -0.00 _Neg75 1.060 1.914 1.349 4.240 1.064 -1.44 _Pos90 0.180 0.622 4.490 1.750 1.058 1.46 _Norm90 0.300 0.965 3.268 2.990 1.124 -0.06 _Neg90 0.420 1.318 2.908 4.240 1.084 -1.51 Table 3-4. Poisson model: pseudo-population parameters 0 0s 1 1s 2 2s LL AIC _Pos10 .450 .036 .004 .050 -.037 .026 -1475.3 2956.6 _Norm10 .992 .027 .003 .038 -.015 .020 -1806.7 3619.3 _Neg10 1.340 .023 .002 .032 -.007 .016 -2019.2 4044.5 _Pos25 0.270 .039 .003 .055 -.024 .028 -1468.1 2942.3 _Norm25 .807 .030 .004 .042 -.016 .021 -1913.9 3833.8 _Neg25 1.159 .025 .001 .035 -.009 .018 -2265.35 4534.7 _Pos50 -.138 .048 .007 .068 -.047 .034 -1334.3 2674.5 _Norm50 .407 .036 .004 .052 -.038 .026 -1888.5 3783.0 _Neg50 .752 .031 .003 .043 -.030 .022 -2370.5 4746.9 _Pos75 -.829 .068 .005 .095 -.077 .048 -972.9 1951.7 _Norm75 -.291 .052 .003 .073 -.079 .037 -1461.8 2929.6 _Neg75 .057 .043 .002 .061 -.044 .031 -1919.7 3845.3 _Pos90 _Norm90 -1.77 -1.23 .109 .083 .036 022 .151 116 -.158 -.165 .076 058 -546.5 -852.6 1099.0 1711.1 _Neg90 -.875 .069 .011 .097 -.149 .049 -1152.1 2310.2 88

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Table 3-5. Negative Binomial Poisson model: pseudo-population parameters 0 0s 1 1s 2 2s LL AIC _Pos10 .450 .032 .004 .045 -.037 .023 .45 -1594.5 _Norm10 .991 .023 .003 .033 -.015 .017 .32 -1998.9 _Neg10 1.340 .019 .002 .027 -.007 .014 .24 -2277.9 _Pos25 .269 .040 .004 .057 -.025 .029 .64 -1509.4 _Norm25 .807 .033 .004 .046 -.016 .023 .56 -1931.5 _Neg25 1.159 .029 .001 .041 -.009 .021 .51 -2235.5 _Pos50 -.139 .059 .008 .083 -.048 .042 1.05 -1290.1 _Norm50 .041 .051 .005 .072 -.039 .036 1.11 -1723.0 _Neg50 .751 .048 .004 .067 -.030 .034 1.16 -2041.3 _Pos75 -.829 .095 .005 .134 -.078 .068 1.61 -910.4 _Norm75 -.292 .085 .004 .120 -.078 .061 1.95 -1275.6 _Neg75 .057 .081 .003 .114 -.044 .058 2.26 -1567.4 _Pos90 orm90 -1.77 -1.23 .161 .146 .037 020 .225 204 -.159 -.165 .113 103 2.04 2.70 -511.6 -751.1 _N _Neg90 -.874 .140 .008 .197 -.149 .010 3.38 -957.3 6. Hurdlezer eudpulaaras Table 3model ( os): ps o-po tion p meter 0 0s 1 1s 2 2s LL AIC _Pos10 -2.198 .149 .003 .211 .044 .107 -1409.5 2830.9 _Norm10 -2.198 .149 .001 .210 .044 .107 -1797.1 3606.1 _Neg10 -2.198 .149 .001 .211 .044 .107 -1928.3 3868.7 _Pos25 -1.099 .103 .000 .146 -.019 .074 -1465.5 2943.0 _Norm25 -1.098 .103 -.001 .146 -.020 .074 -1787.6 3587.2 _Neg25 -1.099 .103 .000 .146 -.020 .074 -1897.9 3807.9 _Pos50 .000 .089 .000 .127 .037 .064 -1296.6 2605.1 _Norm50 -.005 .089 .005 .127 .035 .064 -1510.1 3023.2 _Neg50 .000 .089 .000 .127 .036 .064 -1583.9 3179.7 _Pos75 1.099 .103 .002 .146 .046 .074 -865.2 1742.4 _Norm75 1.100 .103 .000 .146 .047 .074 -970.3 1952.6 _Neg75 1.099 .103 .000 .146 .045 .074 -1008.1 2028.3 _Pos90 2.205 .150 -.001 .211 .137 .107 -444.9 901.8 _Norm90 2.206 .150 -.001 .211 .137 .107 -486.4 984.8 _Neg90 2.204 .150 -.004 .211 .138 .106 -503.0 1018.1 89

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Table 3-7. Hurdle model (event s): pseudo-population parameters 0 0s 1 1s 2 2s LL AIC _Pos10 .212 .051 .008 .072 -.063 .036 -1409.5 2830.9 _Norm10 1.035 .030 .003 .042 -.012 .021 -1797.1 3606.1 _Neg10 1.430 .024 .002 .033 -.003 .017 -1928.3 3868.7 _Pos25 .214 .056 .008 .078 -.062 .041 -1465.5 2943 _Norm25 1.033 .033 .005 .047 -.027 .024 -1787.6 3587.2 _Neg25 1.431 .026 .001 .037 -.016 .019 -1897.9 3807.9 _Pos50 .208 .068 .018 .096 -.060 .049 -1296.6 2605.1 _Norm50 1.035 .040 .011 .057 -.023 .029 -1510.1 3023.2 _Neg50 1.430 .032 .004 .045 -.012 .023 -1583.9 3179.7 _Pos75 .213 .096 .010 .135 -.087 .067 -865.2 1742.4 _Norm75 1.032 .057 .005 .081 -.054 .040 -970.3 1952.6 _Neg75 1.428 .045 .002 .064 -.011 .032 -1008.2 2028.3 _Pos90 .179 .156 .060 .215 -.070 .107 -444.9 901.8 _N _Neg90 1.420 .071 .009 .101 -.028 .051 -503.0 orm90 1.106 .091 .018 .128 -.052 .064 -486.4 984.8 1018.1 Table 3-8. Neinomrdle l (zerseudoulatiom gative B ial Hu m ode os): p -pop n para eters 0 0s 1 1s 2 2s LL AIC _Pos10 -2.197 .149 -.001 .211 .044 .107 -1395.7 2803.4 _Norm10 -2.197 .149 .000 .211 .044 .107 -1800.2 3612.3 _Neg10 -2.205 .150 -.001 .212 .047 .108 -1936.1 3884.1 _Pos25 -1.099 .103 .001 .146 -.019 .074 -1455.3 2922.5 _Norm25 -1.099 .103 .000 .146 -.019 .074 -1790.2 3592.3 _Neg25 -1.099 .103 .000 .146 -.019 .074 -1904.4 3820.8 _Pos50 .000 .089 .000 .127 .036 .064 -1288.9 2589.8 _Norm50 .000 .089 -.001 .127 .037 .064 -1511.9 3035.8 _Neg50 .001 .089 -.001 .127 .036 .064 -1588.1 3188.2 _Pos75 1.099 .103 .000 .146 .045 .074 -861.1 1734.3 _Norm75 _Neg75 1.099 1.099 .103 .103 .000 .000 .146 .146 .045 .045 .074 .074 -971.2 -1010.3 1954.4 2032.5 _Pos90 2.206 .150 -.001 .211 .137 .107 -443.3 898.7 _Norm90 2.244 .152 -.029 .213 .181 .107 -487.3 986.6 _Neg90 2.205 .150 -001 .211 .138 .107 -503.9 1019.8 90

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Table 3-9. Negative Binomial Hurdle m odel (events): pseudo-population parameters 0s1s2s0 1 2 Theta LL AIC _Pos10 -.067 .108 .009 .091 -.071 .046 2.039 -1395.7 2803.4 _Norm10 1.033 .030 .004 .043 -.012 .022 165.718 -1800.2 3612.3 _Neg10 1.429 .024 .002 .034 -.002 .017 164.970 -1936.1 3884.1 _Pos25 -.042 .114 .009 .098 -.067 .051 2.234 -1455.3 2922.5 _Norm25 1.030 .033 .005 .047 -.027 .025 166.348 -1790.2 3592.3 _Neg25 1.431 .026 .001 .037 -.015 .019 166.006 -1904.4 3820.8 _Pos50 -.072 .146 .020 .122 -.065 .062 2.030 -1288.9 2589.8 _Norm50 1.037 .041 .006 .057 -.025 .029 166.574 -1511.9 3035.8 _Neg50 1.429 .032 .004 .046 -.012 .023 166.526 -1588.1 3188.2 _Pos75 -.075 .208 .012 .173 -.097 .088 1.950 -861.1 1734.3 _Norm75 1.031 .058 .005 .081 -.054 .041 166.445 -971.2 1954.4 _Neg75 1.427 .046 .002 .064 -.011 .032 166.377 -1010.3 2032.5 _Pos90 orm90 -.109 1.051 .332 091 .078 -.051 .273 130 -.081 -.048 .140 .066 2.047 91.979 -443.3 -487.3 898.7 986.6 _N _Neg90 1.421 .072 .007 .102 -.028 .052 166.606 -503.9 1019.8 Table 3-10. Zl ulaet IP mode (zeros): pse do-popu tion param ers 0 0s1s2s 1 2 LL AIC _Pos10 -27.010 4434.000 11.490 44 1 35.000 -.363 50.700 -1478.3 2968.6 _Norm10 -3.114 .038 .002 .527 .050 .253 -1800.1 3612.1 _Neg10 -2.365 .175 .002 .246 .049 .125 -1931.3 3874.6 _Pos25 -15.460 38 _Pos90 1.797 .183 .032 .255 .104 .129 -447.9 907.7 _Norm90 2.133 .152 .002 .214 .128 .108 -489.4 990.8 _Neg90 2.188 .071 .007 .101 -.028 .051 -506.0 1024.1 34.000 -3.418 .002 1.213 .001 -1471.1 2954.3 _Norm25 -1.379 .132 .008 .186 -.056 .098 -1790.6 3593.1 _Neg25 -1.162 .109 .001 .154 -.025 .078 -1900.9 3813.9 _Pos50 -.890 .198 .051 .270 -.050 .141 -1299.5 2611.1 _Norm50 -.126 .097 .002 .137 .030 .070 -1513.1 3038.3 _Neg50 -.031 .091 .000 .129 .035 .065 -1586.9 3185.7 _Pos75 .610 .142 .010 .198 -.013 .010 -868.2 1748.5 _Norm75 1.032 .057 .005 .081 -.054 .040 -973.3 1958.5 _Neg75 1.079 .110 -.001 .147 .044 .075 -1011.2 2034.3 91

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Table 3-11. ZIP Model (event s): pseudo-population parameters 0s 1s 2s0 1 2 LL AIC _Pos10 .450 .004 .004 .050 -.037 .003 -1478.3 2968.6 _Norm10 1.035 .003 .004 .043 -.013 .002 -1800.1 3612.1 _Neg10 1.430 .024 .002 .033 -.003 .002 -1931.3 3874.6 _Pos25 .270 .040 .003 .055 -.024 .028 -1471.1 2954.3 _Norm25 1.032 .033 .005 .047 -.028 .025 -1790.6 3593.1 _Neg25 1.431 .026 .001 .037 -.016 .019 -1900.9 3813.9 _Pos50 6.0 1024.1 .205 .069 .024 .096 -.063 .050 -1299.5 2611.1 _Norm50 1.039 .040 .007 .057 -.025 .029 -1513.1 3038.3 _Neg50 1.429 .032 .004 .045 -.012 .023 -1586.9 3185.7 _Pos75 .213 .096 .001 .135 -.085 .067 -868.2 1748.5 _Norm75 1.015 .107 .001 .151 .033 .076 -973.3 1958.5 _Neg75 1.428 .045 .002 .064 -.011 .032 -1011.2 2034.3 _Pos90 orm90 .180 1.017 .155 091 .057 .017 .216 128 -.069 -.052 .108 065 -447.9 -489.4 907.7 990.8 _N _Neg90 1.422 .071 .007 .101 -.028 .051 -50 Table 3-12. Ni P ze do-atet egative B nomi al ZI model ( ros): p seu popul ion param ers LL AIC 0s 1s 2s 0 1 2 _Pos10 -15.280 142.300 -3.880 830.400 .719 102.100 -1479.3 2972.6 _Norm10 -3.110 .375 .0165 .527 .0510 .253 -1801.1 3616.1 _Neg10 -2.370 .175 -.001 .247 .047 .125 -1932.3 3878.7 _Pos25 -17.090 501.400 2.390 473.600 1.700 133.500 -1471.4 2956.7 _Norm25 -1.380 .132 -.002 .186 -.059 .983 -1791.6 3597.1 _Neg25 -1.162 .109 .000 .037 -.016 .019 -1901.9 3817.9 _Pos50 -4.895 3.373 1.352 2.211 -1.086 1.304 -1292.4 2598.8 _Norm50 -.131 .097 .008 .014 .031 .069 -1514.1 3042.3 _Neg50 -.031 .091 .000 .129 .035 .065 -1587.9 3189.7 _Pos75 .130 .364 .015 .271 -.040 .136 -865.1 1744.3 _Norm75 1.015 .107 .001 .151 .033 .076 -974.3 1962.6 _Neg75 1.080 .104 -.002 .147 .044 .075 -1012.2 2038.3 _Pos90 1.462 .386 .047 .298 .098 .151 -447.3 908.7 _Norm90 2.133 .152 .002 .214 .128 .108 -490.4 994.8 _Neg90 2.187 .150 -.001 .212 .136 .107 -507.0 1028.1 92

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Table 3-13. Negative Bino mial ZIP model (events): pseudo-population parameters Theta LL AIC 0s1s2s0 1 2 _Pos10 .450 .036 .004 .050 -.037 .026 14.69 -1479.3 2972.6 _Norm10 1.040 .030 .004 .043 -.013 .021 12.64 -1801.1 3616.1 _Neg10 1.430 .024 .001 .033 -.003 .017 13.02 -1932.3 3878.7 _Pos25 .270 .040 .003 .057 -.024 .029 3.16 -1471.4 2956.7 _Norm25 1.030 .033 .005 .047 -.027 .025 10.88 -1791.6 3597.1 _Neg25 1.431 .026 .001 .037 -.016 .019 15.94 -1901.9 3817.9 _Pos50 -.127 .067 .045 .108 -.080 .052 .57 -1292.4 2598.8 _Norm50 1.039 .040 .007 .057 -.026 .029 11.70 -1514.1 3042.3 _Neg50 1.429 .032 .004 .045 -.012 .023 14.19 -1587.9 3189.7 _Pos75 -.066 .204 .010 .172 -.099 .087 .69 -865.1 1744.3 _Norm75 1.033 .057 .005 .081 -.054 .040 12.02 -974.3 1962.6 _Neg75 1.430 .045 .000 .064 -.010 .032 12.99 -1012.2 2038.3 _Pos90 _Norm90 -.102 1.016 .327 091 .072 017 .273 128 -.080 -.052 .140 065 .73. 12.76 -447.3 -490.4 908.7 994.8 _Neg90 1.422 .071 .006 .101 -.028 .051 14.73 -507.0 1028.1 93

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CHAPTER 4 RESULTS This chapter presents the results based on the previously discussed methods and procedures for analyzing data with varying proportions of zeros a nd varying event stage distributions. First, the results are presented using the data for the pseudo-population in which the proportions for each count level are exactly that which was rando mly sampled from in the simulations. Tables and figures ar e included to support interpretation. Second, the results are presented outlined by the skew level (i.e., positiv e, normal, and negative) and by the proportion of zeros within that skew level (i.e., .10, 25, .50, .75, and .90). For each combination of conditions, the results are presented for the five model comparisons. Third, the results are summarized separately for the negative, normal, and positive event coun t distributions. Tables and figures are included to suppor t interpretation. The primary purpose of the r esults was to assist in answering the research que stions, which were as follows: Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated log-lik elihood between a) the Negative binomial Poisson model vs. Poiss on model; b) the Hurdle model vs. Poisson model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model vs. ZIP model? Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated AIC between all models? Pseudo-Population Results Before addressing the results of the Monte Ca rlo simulations, it is necessary to discuss the results when each model was analyzed with the pseudo-population data. The prespecified proportions for each count were displayed in Table 3-1. Table 3-2 displayed this same information in terms of frequenc ies instead of proportions. Table 3-3 displayed the descriptive statistics for each distribution. Tables 3-4 through 3-13 displaye d the coefficients, standard 94

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errors, log-likelihood values, and AIC values with each table pertaining to either a specific model or one of the two stages for a specific model. The following sections provide the results when comparing models using the pseudopopulation data. First, the Poisson model results were compared to the negative binomial Poisson model results, both descriptively via AICs and inferentially vi a the deviance model comparison test. Second, the results are presented in a likewise manne r for the three Hurdle model comparisons. These comparisons were a) the Hurdle model vs. the negative binomial Hurdle model, b) the Poisson model vs. the Hurdle model, and c) the ne gative binomial Poisson model vs. the negative binomial Hurdle model. Third, the results are presented for the comparison of the ZIP model and the negative binomial ZIP mo del. Fourth, descriptive results are presented comparing all models for each of the five proportions of zeros. Pseudo-Population Poisson Models Based on the AIC, for all proportions of zeros the data fit the Poisson model better when the distribution was positively skewed than when nor mally distributed. In addition, this data fit the Poisson model better when the distribution was normally di stributed than when it was negatively skewed. For example, for the negativel y skewed distribution wi th a .10 proportion of zeros, the AIC was 4,044.5. As the curve shifted le ft to a normal distribution, the AIC dropped to 3,619.3, and as the curve shifted further left to a positively skewed distribution, the AIC dropped to 2,956.3. The same pattern emerged for the negative binom ial Poisson models. For example, for the negatively skewed distribution w ith a .10 proportion of zeros, the AIC was 4,563.8. As the curve shifted left to a normal distribut ion, the AIC dropped to 4,005.9, and as the curve shifted further left to a positively skewed distri bution, the AIC dropped to 3,196.9. 95

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For the .10 proportion of zeros condition, the Poisson models had a lower AIC than that calculated for the negative binomial Poisson model. For the .25 proportion of zeros condition, the AICs were approximately equal between the two models. For the .50, .75, and .90 proportions of zeros conditions, th e AIC was lower for the negative binomial Poisson model than for the Poisson model. The deviance statistic was cal culated comparing the Poisson log-likelihood and negative binomial Poisson log-likelihood for each skew condition and zero proportion condition. These statistics are displayed in Table 4-1. For all analyses, the Poiss on model is nested within its negative binomial Poisson formulation differing by one degree of freedom (i.e., the dispersion parameter in the negative binomial Poisson model) Hence, at the .05 T ype I error rate and assuming deviance statistics asymp totically distributed chi-square, a deviance exceeding 3.84 suggests better fit for the more complex negative binomial Poisson model. This was the result for all analyses in which the proporti on of zeros was .50 or greater. However, for all analyses with .10 and .25 proportions of zeros, the statistic was significant in favor of the Poisson model. Pseudo-Population Hurdle Models Hurdle vs. Negative Binomial Hurdle For the Hurdle models, regardless of the proportion of zeros, the positively skewed distributions had a lower AIC than did the normal distributions. These, in turn, had a lower AIC than the negatively skewed distribution. The sam e was true for the negative binomial Hurdle models. However, for both models, the differenc e between AICs for the three skew conditions decreased as the proportion of zer os decreased (i.e., the models became more similar in fit). When comparing the Hurdle models and negative binomial Hurdle models, the AICs appear to be similar regardless of the proportion of zeros and regardless of skew. 96

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The deviance statistic was cal culated comparing the Hurdle log-likelihood and negative binomial Hurdle log-likelihood for each skew condition and zero proportion condition. These statistics are displayed in Table 4-2. For all analyses, the Hurdle model is nested within its negative binomial Hurdle formulation differing by one degree of freedom (i.e., the dispersion parameter in the negative binomial Hurdle model) Hence, at the .05 T ype I error rate and assuming deviance statistics asymp totically distributed chi-square, a deviance exceeding 3.84 suggests better fit for the more comp lex negative binomial Hurdle model. For the positively skewed distributions, the negative binomial Hurdle model fit significantly better than th e Hurdle model, except when the proportion of zeros was .90. In this case the deviance statistic did not exceed 3.84. For the normal distributions, the Hurdle model fit significantly better than the negative binomial Hurdle mode l when the proportion of zeros was .10 or .25. When the proportion of zeros was .50, .75, or .90, the deviance did not exceed 3.84. For the negatively skewed distributions, the Hu rdle model fit significantly better than the negative binomial Hurdle model when the proportion of zeros was .10, .25, .50, or .75. When the proportion of zeros of .90, the devi ance statistic did not exceed 3.84. Poisson vs. Hurdle Deviance statistics were also calculated to compare the Po isson log-likelihood and Hurdle log-likelihood for each skew condition and zer o proportion condition. These statistics are displayed in Table 4-3. For all analyses, the Po isson model is nested within the Hurdle model differing by three degrees of freedom. Hence, at the .05 Type I error rate and assuming deviance statistics asymptotically distribut ed chi-square, a deviance exceedi ng 7.82 suggests better fit for the more complex Hurdle model. The deviances were large supporting the Hurd le model fit over the Po isson model fit. In fact, several of the deviances were over 1,000. There was only one analysis that did not favor the 97

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Hurdle model fit over the Poisson model fit. Th is was the deviance calculated for the positively skewed distribution with a .25 proportion of zeros. Negative Binomial Poisson vs. Negative Binomial Hurdle Deviance statistics were also calculated to compare the negative binomial Poisson loglikelihood and the negative binomial Hurdle log-likelihood for each skew condition and zero proportion condition. These statistics are displayed in Table 4-4. For all analyses, the negative binomial Poisson model is nested within the ne gative binomial Hurdle model differing by three degrees of freedom (i.e., the duplication of paramete rs in the negative binomial Hurdle model to represent both a transition stage and an event stage).28 Hence, at the .05 Type I error rate and assuming deviance statistics asymp totically distributed chi-square, a deviance exceeding 7.82 suggests better fit for the more comp lex negative binomial Hurdle model. As was the case when comparing the Poisson model and the Hurdle model, the deviances were large. Likewise, there was one deviance that did not exceed the critical value. However, it was for the positively skewed distribution with .50 zeros rather than the positively skewed distribution with .25 zeros. Pseudo-Population ZIP Models The pseudo-population results for the ZIP models were rather similar to those obtained for the Hurdle models. Graphically, regardless of skew condition and proportions of zeros, the AICs for the ZIP models and negative bi nomial ZIP models were very similar. Additionally, the AICs appeared to be equal between the .10 a nd .25 proportions of zeros conditions. The deviance statistic was calculated comparing the ZI P log-likelihood and negative binomial ZIP log-likelihood for each skew c ondition and zero propor tion condition. These 28 The presence of a dispersion parameter is now redundant between models. 98

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statistics are displayed in Table 45. For all analyses, the ZIP model is nested within its negative binomial ZIP formulation differing by one degree of freedom (i.e., the di spersion parameter in the negative binomial ZIP model). Hence, at th e .05 Type I error rate and assuming deviance statistics asymptotically distribut ed chi-square, a deviance exceedi ng 3.84 suggests better fit for the more complex negative binomial ZIP model. The deviance was significant for only two comparisons. When the distribution was positively skewed, the fit was significantly better for the negative binomial ZIP model than for the ZIP model when the proportion of zeros was either .50 or .75. All other results suggested that the ZIP model fit was adequate. Comparing AICs For All Models For the .10 proportion of zeros conditi on with negative skew, the AICs were approximately equal for all models except for the Poisson models. The AIC for the Poisson model was higher than for the other models, and the AIC for the negative binomial Poisson model was highest of all. When the distribution was normal, the only model to have a noticeably different AIC was the negative binomial Hurdle which was again highest of all. For the positively skewed distribution, there was some separation between the ZIP and Hurdle models, with the Hurdle models having the lower AIC. The differences between these models and their negative binomial formulations appeared to be tr ivial. The Poisson model appeared to have the same AIC as those displayed for the ZIP models The AIC for the negative binomial was again highest of all. Between the th ree distributions, the AIC declined from the negatively skewed distribution to the normal distribution to the positivel y skewed distribution. For the .25 proportion of zeros, there was lit tle distinction between the Poisson models and negative binomial Poisson models for all di stributions. Further, th ere was no distribution displaying a nontrivial distinc tion between the Hurdle models and the ZIP models. For the 99

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positively skewed distribution, all six distributions appeared approximately equal with a slightly higher AIC apparent for the negative binomial Po isson model. Between the three distributions, the AICs appeared equal for the negatively skew ed distribution and the normal distribution; However, the AICs for the normal distribution were considerably smaller than the AICs for the other two distributions. For the .50 proportion of zeros condition with negative skew, the r esults for the Poisson model and the negative binomial Poisson model re versed. For the negativel y skewed distribution, the AIC for the negative binomial Poisson was highe r than those for the ZIP and Hurdle models, while the AIC for the Poisson model was higher ye t. As in the .50 propor tion of zeros condition, the AIC appeared equal for the ZIP and Hurdle models. Also, as in the .50 proportion of zeros condition, there appeared to be no difference between any of the models for the positively skewed distribution. Between th e three distributions, the AICs declined from the negatively skewed distribution to the normal distribution to the positively sk ewed distribution with declines being most rapid for the Poisson and negative binomial Poisson models. Beyond the fact that the overa ll AIC was lower for the .90 proportion of zeros condition than for the .75 condition, these last two c onditions displayed similar results. For each distribution, the Hurdle and ZIP model AICs were approximately equal. They declined slightly from the negatively skewed distribution to the no rmal distribution and declined a bit more from this distribution to the positively skewed distribution. For the negatively skewed distribution, the negative binomial Poisson AIC w as considerably higher than the AICs for the ZIP and Hurdle models; the AIC for the Poisson model was highest of all. For the normal distribution, both the Poisson and negative binomial Poisson model AICs declined by approximately 50% of their value for the negatively skewed distribution. For the positively skewed distribution, they 100

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declined further yet revealing a Poisson model AIC slightly hi gher than that of the negative binomial AIC model, which was slightly higher than that of the nearly equal Hurdle and ZIP model AICs. Monte Carlo Simulation Results Positively Skewed Distribution These results pertain to the data in which 100 of the 1,000 observations in each of the 2,000 simulated datasets were constant at zero. The remaining 900 values were sampled from the positively skewed pseudo-population distribution with the count proportions and frequencies displayed in Table 31 and Table 3-2. For the .10 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 271.5 to -201.98 with a mean of -238.24 and a standard deviation of 10.43. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 2,000 model comparisons. The average AIC for the Poisson model was 2957.06, while the average AIC for the negative binomial Poisson model was 3193.3, which descriptively supports the inferential findings of better fit for the Poisson model. For the .10 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 5.14 to 69.8 with a m ean of 27.93 and a standard deviation of 9.0. Given a Type I error rate of .05, the deviance st atistic was significant for all of the 2,000 model comparisons. The average AIC for the Hurdle model was 2838.81 while the average AIC for the negative binomial Hurdle model was 2804.88, which descriptively suppo rts the inferential findings of better fit for the negative binomial Hurdle model. For the .10 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 86. 36 to 187.64 with a mean of 132.25 and a standard deviation of 14.58. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 101

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valid model comparisons. The average AIC for the Poisson model was 2957.06 while the average AIC for the Hurdle model was 2838.81, which descriptively supports the inferential findings of better fit for the Hurdle model. For the .10 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 332.81 to 470.81 w ith a mean of 398.42 and a standard deviation of 19.86. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 2,000 model comparisons. The average AIC for th e negative binomial Poisson model was 3193.3 while the average AIC for the negative binomial Hurdle model was 2804.88, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .10 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -2. 02 to -2.00 with a mean of -2 .00 and a standard deviation of .001 Given a Type I error rate of .05, the de viance statistic was signifi cant for none of the 112 valid model comparisons. The average AIC for the ZIP model was 2967.85 the average AIC for the negative binomial ZIP model was 2984.79, which descriptively suggests be tter fit for the ZIP model. Table 4-6 summarizes the log-likelihood statistics for the positively skewed distribution model comparisons with a .25 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. 102

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A comparison of AICs over all models revealed a minimum of 2622.53 and a maximum of 3096.30 across all 9,678 valid model comparis ons. Between the six m odels, the minimum was 2804.88 (i.e., the negative binomial Hurdle mo del), and the maximum was 3193.30 (i.e., the negative binomial Poisson model). The rank or der from lowest AIC to highest AIC was as follows: negative binomial Hurdle, Hurdle, Poisson, ZIP, negative binomial ZIP, negative binomial Poisson. These rankings are in line with the rankings for th e pseudo-population AICs. Table 4-7 displays the descriptive statistics for these two models. Figure 4-1 is a boxplot illustrating these results. The following set of results pertains to the data in which 250 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 750 values were sampled from a positively skewed pse udo-population distribution with the count proportions and frequencies displayed in Table 3-1 and Table 3-2. For the .25 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -126. 64 to -31.15 with a mean of -83. 2 and a standard deviation of 12.3. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 model comparisons. The average AIC for the Po isson model was 2939.65 while the average AIC for the negative binomial Poisson model w as 3020.85, which descriptively supports the inferential findings of better fit for the Poisson model. For the .25 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from .87 to 22.71 with a mean of 5.76 and a standard deviation of 3.47. Given a Type I error rate of .05, the deviance st atistic was significant for 1,994 of the 2,000 valid model comparisons. The average AIC for the Hu rdle model was 2947.9 while the average AIC 103

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for the negative binomial Hurdle model was 2921, which descriptively supports the 99.7% inferential findings of better fit for th e negative binomial Hurdle model. For the .25 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from .07 to 22.7 with a mean of 5.76 a nd a standard deviation of 3.47. Given a Type I error rate of .05, the deviance st atistic was significant for 503 of the 2,000 valid model comparisons. The average AIC for the Po isson model was 2939.65 while the average AIC for the Hurdle model was 2947.9, which descriptivel y supports better fit for the Poisson model. For the .25 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 75.94 to 156.11 with a mean of 109.85 and a standard deviation of 12.33. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for th e negative binomial Poisson model was 3020.9, while the average AIC for the negative binom ial Hurdle model was 2921, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .25 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 10.55 to 17.14 with a mean of 54 and a standard deviation of 3.86. Given a Type I error rate of .05, the de viance statistic was signifi cant for 344 of the 1,850 valid model comparisons. The average AIC fo r the ZIP model was 2951.12 while the average AIC for the negative binomial ZIP model was 2956.15 which descriptively supports better fit for the ZIP model. Table 4-8 summarizes the log-likelihood statistics for the positively skewed distribution model comparisons with a .25 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the 104

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simulated dataset sample, and LLs LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sa mple size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 2754.05 and a maximum of 3127.33 across all valid model comparisons Between the six models, the minimum was 2921.00 (i.e., the negative binomial Hurdle mo del), and the maximum was 3020.85 (i.e., the negative binomial Poisson model). The rank or der from lowest AIC to highest AIC was as follows: negative binomial Hurdle, Poisson, Hurdle, ZIP, negative binomial ZIP, negative binomial Poisson. These rankings are in line with the rankings for th e pseudo-population AICs. Table 4-9 displays the descriptive statistics for these two models. Figure 4-2 is a boxplot illustrating these results. The following set of results pertains to the data in which 500 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 500 values were sampled from a positively skewed pse udo-population distribution with the count proportions and frequencies displayed in Ta ble 3-1 and Table 3-2. For the .50 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 31.23 to 142.04 with a mean of 87.9 and a standard deviation of 16.37. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 2672.88 while the average AIC for the negative binomial Poisson model was 2582.98, which descriptively supports the inferential findings of better fit for the negative binomial Poisson model. For the .50 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from .84 to 43.88 with a mean of 15.69 and a standard deviation of 6.8. Given a Type I error rate of .05, the deviance st atistic was significant for 1,961 of the 2,000 valid 105

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model comparisons. The average AIC for the Hu rdle model was 2601.93 while the average AIC for the negative binomial Hurdle model was 2588. 24, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .50 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 35.5 to 142.74 with a mean of 76.95 and a standard deviation of 17.08. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 2674.5 while the average AIC for the Hurdle model was 2601.93, which descr iptively supports the inferential findings of better fit for the Hurdle model. For the .50 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from .12 to 18.73 with a mean of 4.74 and a standard deviation of 2.87. Given a Type I error rate of .05, the deviance st atistic was significant for 268 of the 2,000 valid model comparisons. The average AIC for the negative binomial Poisson model was 2582.98 while the average AIC for the negative binomial Hurdle model was 2588.24, which descriptively supports better fit for the negative binomial Poisson model. For the .50 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -1.16 to 38.97 with a mean of 13.61 and a standard deviation of 6.51. Given a Type I error rate of .05, the devi ance statistic was significa nt for 1,894 of the 2,000 valid model comparisons. The average AIC for the ZIP model was 2607.84 while the average AIC for the negative binomial ZIP model was 2596.23 which descriptively supports the 94.7% inferential findings of better fit for the negative binomial ZIP model. Table 4-10 summarizes the log-likelihood statis tics for the positively skewed distribution model comparisons with a .50 proportion of zeros. In this table, LL represents the mean log106

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L likelihood for the pseudo-population, LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 2451.80 and a maximum of 2862.94 across all 10,000 valid model comparis ons. Between the six models, the minimum was 2582.09 (i.e., the negative binomial Poiss on model), and the maximum was 2672.88 (i.e., the Poisson model). The rank order from lowest AIC to highest AIC was as follows: negative binomial Poisson, negative binomial Hurdle, negative binomial ZIP, Hurdle, ZIP, Poisson. These rankings are in line with the ra nkings for the pseudopopulation AICs. Table 4-11 displays the descriptive statistics for these two models. Figure 4-3 is a boxplot illustrating these results. The following set of results pertains to the data in which 750 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 250 values were sampled from a positively skewed pse udo-population distribution with the count proportions and frequencies displayed in Ta ble 3-1 and Table 3-2. For the .75 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 68.65 to 180.23 with a mean of 125.85 and a standard deviation of 16.52. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 1955.22, while the average AIC for the negative binomial Poisson model was 1827.36, which descriptively supports the inferential findings of better fit for the negative binomial Poisson model. For the .75 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -1.45 to 32.0 with a mean of 8.3 and a standard deviation of 4.77. 107

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Given a Type I error rate of .05, the deviance st atistic was significant for 1,963 of the 2,000 valid model comparisons. The average AIC for the Hu rdle model was 1749.71 while the average AIC for the negative binomial Hurdle model was 1735.4, which descriptiv ely supports the 98.1% inferential findings of better fit for th e negative binomial Hurdle model. For the .75 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 95.23 to 363.75 with a mean of 219.51 and a standard deviation of 35.63. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 1955.22 while the average AIC for the Hurdle model was 1749.7, which descriptively s upports the in ferential findings of better fit for the Hurdle model. For the .75 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 35.6 to 186.3 with a mean of 101.96 and a standard deviation of 19.64. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for the negative binomial Poisson model was 1827.36 while the average AIC for the negative binomial Hurdle model was 1735.4, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .75 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -2.0 to 29.66 with a mean of 6.27 and a standard deviation of 4.73. Given a Type I error rate of .05, the deviance st atistic was significant for 1,304 of the 2,000 valid model comparisons. The average AIC for the ZI P model was 1747.72, while the average AIC for the negative binomial ZIP model was 1743.44, which descriptively supports the 65.2% inferential findings of better fit for the negative binomial ZIP model. 108

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Table 4-12 summarizes the log-likelihood statis tics for the positively skewed distribution model comparisons with a .75 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 1617.27 and a maximum of 2188.38 across all valid model comparisons Between the six models, the minimum was 1735.40 (i.e., the negative binomial Hurdle mo del), and the maximum was 1827.36(i.e., the negative binomial Poisson model). The rank or der from lowest AIC to highest AIC was as follows: negative binomial Hurdle, negative binomial ZIP, ZIP, Hurdle, negative binomial Poisson, Poisson. These rankings are not in line with the rank ings for the pseudo-population AICs; in the pseudo-population, the Hurdle mo del has a lower AIC than the negative binomial ZIP model. Table 4-13 displays the descriptive statistics for these two mo dels. Figure 4-4 is a boxplot illustrating these results. The following set of results pertains to the data in which 900 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 100 values were sampled from a positively skewed pse udo-population distribution with the count proportions and frequencies displayed in Ta ble 3-1 and Table 3-2. For the .90 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 29.24 to 188.86 with a mean of 69.82 and a standard deviation of 12.5. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 1098.79 while the 109

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average AIC for the negative binomial Poisson model was 1026.98, which descriptively supports the inferential findings of better fit for the negative binomial Poisson model. For the .90 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -4.1 to 17.15 with a mean of 2.96 and a standard deviation of 2.69. Given a Type I error rate of .05, the deviance st atistic was significant for 611 of the 1,992 valid model comparisons. The average AIC for the Hu rdle model was 906.36 while the average AIC for the negative binomial Hurdle model was 897. 5, which descriptively supports the 30.7% inferential findings of better fit for th e negative binomial Hurdle model. For the .90 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 86.47 to 367.79 with a mean of 206.43 and a standard deviation of 41.18. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 1098.79 while the average AIC for the Hurdle model was 906.36, which descriptively s upports the in ferential findings of better fit for the Hurdle model. For the .90 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from 55.21 to 251.73 with a mean of 139.66 and a standard deviation of 28.92. Given a Type I error rate of .05, the devi ance statistic was significa nt for all of the 1,992 valid model comparisons. The average AIC for the negative binomial Poisson model was 1026.98 while the average AIC for the negative binomial Hurdle model was 897.5, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .90 proportion of zeros condition with a positively skewed distribution, the deviance statistic ranged from -2.11 to 14.85 with a mean of .96 and a standard deviation of 2.66. 110

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Given a Type I error rate of .05, the deviance st atistic was significant for 265 of the 1,992 valid model comparisons. The average AIC for the ZI P model was 904.34 while the average AIC for the negative binomial ZIP model was 905.38, which descriptively supports better fit for the ZIP model. Table 4-14 summarizes the log-likelihood statis tics for the positively skewed distribution model comparisons with a .90 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models reveal ed a minimum of 834.28 and a maximum of 1313.41 across all valid model comparisons. Be tween the six models, the minimum was 897.50 (i.e., the negative binomial Hurdle model), and the maximum was 1098.79 (i.e., the Poisson model). The rank order from lowest AIC to highest AIC was as follows: negative binomial Hurdle, ZIP, negative binomial ZIP, Hurdle, ne gative binomial Poisson, Poisson. These rankings are in line with the rankings for the pseudo-popul ation AICs. Table 4-15 displays the descriptive statistics for these two models. Figure 4-5 is a boxplot illustrating these results. Normal Distribution The following set of results pertains to the data in which 100 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 900 values were sampled from a normal pseudo-population distribution with the count proportions and frequencies displayed in Table 31 and Table 3-2. 111

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For the .10 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -425.53 to -347.98 with a mean of -3 85.1 and a standard deviation of 11.5. Given a Type I error rate of .05, the deviance statistic was significant for none of the 2,000 valid model comparisons. The average AIC for the Poison model was 3618.76 while the average AIC for the negative binomial Poisson model was 4005.86, which descriptively supports the inferential findings of better fit fo r the Poisson model. For the .10 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -40.51 to -4.94 with a mean of -6.61 and a standard deviation of 2.75. Given a Type I error rate of .05, the devi ance statistic was significant for none of the 1,503 valid model comparisons. The average AIC for the Hurdle model was 3605.03 while the average AIC for the negative binomial Hurdle model was 3613.55, which descriptively suppo rts the inferential findings of better fit for the Hurdle model. For the .10 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 9.87 to 31.88 with a mean of 19.73 and a standard deviation of 2.94. Given a Type I error rate of .05, the devi ance statistic was significant fo r all of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 3618.76 while the average AIC for the Hurdle model was 3605.03, which descriptively suppor ts the inferential findi ngs of better fit for the Hurdle model. For the .10 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 359.25 to 439.66 with a mean of 397. 78 and a standard deviation of 12.5. Given a Type I error rate of .05, the deviance statistic was significant for all of the 1,503 valid model comparisons. The average AIC for the negative binomial Poisson model was 4005.86 while the 112

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average AIC for the negative binomial Hurdle model was 3613.55, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .10 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -16.62 to -1.83 with a mean of -2.04 and a standard de viation of .7 Given a Type I error rate of .05, the devian ce statistic was significant (ZIP) for 7 of the 1,982 valid model comparisons. The average AIC for the ZIP m odel was 3611.05 while the average AIC for the negative binomial ZIP model was 3615 which descriptively supports the 99.6% inferential findings of better fit for the ZIP model. Table 4-16 summarizes the log-likelihood statistics for the normal distribution model comparisons with a .10 proportion of zeros. In this table, LL represents the mean log-likelihood for the pseudo-population, L LminLLmax LLsrepresents the mean log-likeli hood for the simulated dataset samples, and LL represent the loglikelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviation and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples A comparison of AICs over all models revealed a minimum of 3530.45 and a maximum of 4071.95 across all valid mode l comparisons. Between the six models, the minimum was 3611.05 (i.e., the Hurdle model), and the ma ximum was 4005.86 (i.e., th e negative binomial Poisson model). The rank order from lowest AI C to highest AIC was as follows: Hurdle, ZIP, negative binomial Hurdle, negative binomial ZIP, Poisson, negative binomial Poisson. These rankings are in line with the ra nkings for the pseudopopulation AICs. Table 4-17 displays the descriptive statistics for these two models. Figur e 4-19 is a boxplot illustrating these results. 113

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The following set of results pertains to the data in which 250 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 750 values were sampled from a normal pseudo-population distribution with the count proportions and frequencies displayed in Table 31 and Table 3-2. For the .25 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -69.34 to -5.87 with a mean of -35.75 and a standa rd deviation of 9.23. Given a Type I error rate of .05, the deviance statistic was significant for none of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 3832.24 while the average AIC for the negative binomial Poisson model was 3869.99, wh ich descriptively supports the inferential findings of better fit fo r the Poisson model. For the .25 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -19.83 to -3.93 with a mean of -5.2 and a standard de viation of .82. Given a Type I error rate of .05, the devi ance statistic was significant fo r all of the 1,634 valid model comparisons. The average AIC for the Hurdle model was 3586.47 while the average AIC for the negative binomial Hurdle model was 3593.63 which descriptively suppo rts the inferential findings of better fit for the Hurdle model. For the .25 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 202.52 to 312.73 with a mean of 251. 77 and a standard deviation of 14.35. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 3832.24 while the average AIC for the Hurdle model was 3586.47, which descriptively suppor ts the inferential findi ngs of better fit for the Hurdle model. 114

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For the .25 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 228.78 to 330.2 with a mean of 282. 08 and a standard deviation of 14.95. Given a Type I error rate of .05, the deviance statistic was significant for all of the 1,634 valid model comparisons. The average AIC for the negative binomial Poisson model was 3869.99 while the average AIC for the negative binomial Hurdle model was 3593.63, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .25 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -3.09 to -1.94 with a mean of -2.01 and a standard de viation of .05. Given a Type I error rate of .05, the devi ance statistic was significant fo r none of the 2,000 valid model comparisons. The average AIC for the ZIP m odel was 3592.44 while the average AIC for the negative binomial ZIP model was 3596.45, whic h descriptively suggests better fit for the ZIP model. Table 4-18 summarizes the log-likelihood statistics for the normal distribution model comparisons with a .25 proportion of zeros. In this table, LL represents the mean log-likelihood for the pseudo-population, L LminLLmax LLsrepresents the mean log-likeli hood for the simulated dataset samples, and LL represent the log-likelihood minimu m and maximum for the simulated dataset sample, and LLs represent the standard deviation a nd standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 3529.93 and a maximum of 2966.65 across all 11,634 valid model comparis ons. Between the six models, the minimum was 3586.47 (i.e., the Hurdle model), and the ma ximum was 3869.99 (i.e., the negative binomial Poisson model). The rank order from lowest AI C to highest AIC was as follows: Hurdle, ZIP, negative binomial Hurdle, negative binomial ZIP, Poisson, negative binomial Poisson. These 115

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rankings are not in line with the rankings for the pseudo-pop ulation AICs; in the pseudopopulation, the negative binomial Hurdle model h as a lower AIC (i.e., better fit) than the ZIP model Table 4-19 displays the descriptive st atistics for these two models. Figure 4-20 is a boxplot illustrating these results. The following set of results pertains to the data in which 500 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 500 values were sampled from a normal pseudo-population distribution with the count proportions and frequencies displayed in Table 31 and Table 3-2. For the .50 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 286.84 to 381.11 with a mean of 331. 24 and a standard deviation of 13.74. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 3642.82 while the average AIC for the negative binomial Poisson model was 3356.12, wh ich descriptively supports the inferential findings of better fit for the nega tive binomial Poisson model. For the .50 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -41.05 to 3.15 with a mean of -3.86 and a standard de viation of 2.19 Given a Type I error rate of .05, the devi ance statistic was significant for none of the 1,602 valid model comparisons. The average AIC for the Hurdle model was 2978.57 while the average AIC for the negative binomial Hurdle model was 2994.57, which descriptively suppo rts the inferential findings of better fit for the Hurdle model. For the .50 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 655.17 to 857.57 with a mean of 758. 23 and a standard deviation of 31.92. Given a Type I error rate of .05, the deviance statistic was significan t all of the 2,000 valid model 116

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comparisons. The average AIC for the Poisson m odel was 3642.82 while the average AIC for the Hurdle model was 3023.52, which descriptively suppor ts the inferential findi ngs of better fit for the Hurdle model. For the .50 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 356.28 to 489.60 with a mean of 423. 12 and a standard deviation of 21.81. Given a Type I error rate of .05, the deviance statistic was significant for all of the 1,601 valid model comparisons. The average AIC for the mode l was 3356.12 while the average AIC for the negative binomial Poisson model was 2994.57, wh ich descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .50 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -2.57 to -1.93 with a mean of -2.01 and a standard deviation of .029 Given a Type I error rate of .05, the devi ance statistic was significant for none of the 2,000 valid model comparisons. The average AIC for the ZIP m odel was 2992.55 while the average AIC for the negative binomial ZIP model was 2996.55, which descriptively suggests better fit for the ZIP model. Table 4-20 summarizes the log-likelihood statistics for the normal distribution model comparisons with a .50 proportion of zeros. In this table, LL represents the mean log-likelihood for the pseudo-population, L LminLLmax LLsrepresents the mean log-likeli hood for the simulated dataset samples, and LL represent the log-likelihood minimu m and maximum for the simulated dataset sample, and LLs represent the standard deviation a nd standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 2992.55 and a maximum of 3926.85 across all 12,000 valid model comparis ons. Between the six models, the minimum 117

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was 3023.52 (i.e., the Hurdle model), and the ma ximum was 3783.75 (i.e., the Poisson model). The rank order from lowest AIC to highest AI C was as follows: Hurdle, ZIP, negative binomial Hurdle, negative binomial ZIP, negative binomial Poisson, Poisson. Thes e rankings are not in line with the rankings for the pseudo-population AI Cs; in the population, the negative binomial Hurdle model has a lower AIC (i.e., better fit) than the ZIP model. Ta ble 4-21 displays the descriptive statistics for these two models. Figur e 4-21 is a boxplot illustrating these results. The following set of results pertains to the data in which 750 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 250 values were sampled from a normal pseudo-population distribution with the count proportions and frequencies displayed in Table 31 and Table 3-2. For the .75 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 313.34 to 435.26 with a mean of 372. 68 and a standard deviation of 17.63. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 2930.85 while the average AIC for the negative binomial Poisson model was 2561.17, wh ich descriptively supports the inferential findings of better fit for the nega tive binomial Poisson model. For the .75 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -36.75 to .10 with a mean of -2.29 a nd a standard deviation of .24. Given a Type I error rate of .05, the devi ance statistic was significant fo r none of the 1,491 valid model comparisons. The average AIC for the Hurdle model was 1952.86 while the average AIC for the negative binomial Hurdle model was 1957.07, which descriptively suppo rts the inferential findings of better fit for the Hurdle model. 118

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For the .75 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 827.03 to 1132.06 with a mean of 983. 00 and a standard deviation of 47.54. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 2930.85 while the average AIC for the Hurdle model was 1952.86, which descriptively suppor ts the inferential findi ngs of better fit for the Hurdle model. For the .75 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 499.96 to 701.73 with a mean of 608. 46 and a standard deviation of 30.96. Given a Type I error rate of .05, the deviance statistic was significant for all of the 1,490 valid model comparisons. The average AIC for the negative binomial Poisson model was 2560.17 while the average AIC for the negative binomial Hurdle model was 1957.07, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .75 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -5.86 to -.64 with a mean of -2.01 and a standard deviation of .10. Given a Type I error rate of .05, the devi ance statistic was significant fo r none of the 2,000 valid model comparisons. The average AIC for the ZIP m odel was 1958.85 while the average AIC for the negative binomial ZIP model was 1962.87, which suggests adequate fit for the ZIP model. Table 4-22 summarizes the log-likelihood statistics for the normal distribution model comparisons with a .75 proportion of zeros. In this table, LL represents the mean log-likelihood for the pseudo-population, L LminLLmax LLsrepresents the mean log-likeli hood for the simulated dataset samples, and LL represent the log-likelihood minimu m and maximum for the simulated dataset sample, and LLs represent the standard deviation a nd standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. 119

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A comparison of AICs over all models revealed a minimum of 1918.91 and a maximum of 3103.14 across all valid model comparisons Between the six models, the minimum was 1952.86 (i.e., the Hurdle model), and the maximum was 2930.85 (i.e., the Poisson model). The rank order from lowest AIC to highest AIC w as as follows: Hurdle, negative binomial Hurdle, ZIP, negative binomial ZIP, negative binomial Po isson, Poisson. These rankings are in line with the rankings for the pseudo-popula tion AICs. Table 4-23 displays the descriptive statistics for these two models. Figure 4-22 is a b oxplot illustrating these results. The following set of results pertains to the data in which 900 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 100 values were sampled from a normal pseudo-population distribution with the count proportions and frequencies displayed in Table 31 and Table 3-2. For the .90 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 157.17 to 249.62 with a mean of 204. 06 and a standard deviation of 14.38. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson m odel was 1716.21 while the average AIC for the negative binomial Poisson model was 1514.15, wh ich descriptively supports the inferential findings of better fit for the nega tive binomial Poisson model. For the .90 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -20.63 to 13.74 with a mean of -1.11 and a standa rd deviation of 1.62 Given a Type I error rate of .05, the deviance statistic was significant for none of the 1,537 valid model comparisons. The average AIC for the Hurdle model was 984.24 while the average AIC for the negative binomial Hurdle model was 987.32, which descriptively supports the inferential findings of better fit for the Hurdle model. 120

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For the .90 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 583.77 to 893.39 with a mean of 737. 97 and a standard deviation of 49.17. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 1716.2 while the average AIC for the Hurdle model was 1514.15, which descriptively suppor ts the inferential findi ngs of better fit for the negative binomial Hurdle model. For the .90 proportion of zeros condition with a normal distribution, the deviance statistic ranged from 425.97 to 642.72 with a mean of 532. 39 and a standard deviation of 34.87. Given a Type I error rate of .05, the deviance statistic was significant for all of the 1,537 valid model comparisons. The average AIC for the model negative binomial Poisson was 1514.15 while the average AIC for the negative binomial Hurdle model was 984.24, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .90 proportion of zeros condition with a normal distribution, the deviance statistic ranged from -2.53 to -1.97 with a mean of -2.01 and a standard de viation of .02. Given a Type I error rate of .05, the devi ance statistic was significant fo r none of the 2,000 valid model comparisons. The average AIC for the ZIP m odel was 990.22 while the average AIC for the negative binomial ZIP model was 994.23 which descriptively suggests better fit for the ZIP model. Table 4-24 summarizes the log-likelihood statistics for the normal distribution model comparisons with a .90 proportion of zeros. In this table, LL represents the mean log-likelihood for the pseudo-population, L LminLLmaxrepresents the mean log-likeli hood for the simulated dataset samples, and LL represent the log-likelihood minimu m and maximum for the simulated 121

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dataset sample, and LLs LLs represent the standard deviation a nd standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 962.46 and a maximum of 1881.02 across all 11,536 valid model comparisons Between the six models, the minimum was 984.24 (i.e., the Hurdle model), and the maximu m was 1716.20 (i.e., the Poisson model). The rank order from lowest AIC to highest AIC w as as follows: Hurdle, negative binomial Hurdle, ZIP, negative binomial ZIP, negative binomial Po isson, Poisson. These rankings are in line with the rankings for the pseudo-popula tion AICs. Table 4-25 displays the descriptive statistics for these two models. Figure 4-23 is a b oxplot illustrating these results. Negatively Skewed Distribution The following set of results pertains to the data in which 100 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 900 values were sampled from a negatively skewed p seudo-population distribution wi th the count proportions and frequencies displayed in Ta ble 3-1 and Table 3-2. For the .10 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -572.46 to -470.93 with a mean of -517.56 and a standard deviation of 13.66. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 model comparisons. The average AIC for the Poisson model was 4044.50 while the average AIC for the negative binomial Poisson model was 4563.58, which descriptively supports the inferential findings of better fit for the Poisson model. For the .10 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 13.4 to 61.02 with a mean of 15.75 and a standard deviation of .28. Given a Type I error rate of .05, the devi ance statistic was signifi cant for all of the 1,626 122

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valid model comparisons. The average AIC for the Hurdle model was 3868.06 while the average AIC for the negative binomial Hurdle model was 3886.31, which descri ptively support the inferential findings of better fit for the Hurdle model. For the .10 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 162.77 to 202.59 w ith a mean of 181.96 and a standard deviation of 5.72. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 model comparisons. The average AIC for the Po isson model was 4044.02 while the average AIC for the Hurdle model was 3868.06, which descriptiv ely support the inferen tial findings of better fit for the Hurdle model. For the .10 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 609.07 to 754.25 w ith a mean of 682.91 and a standard deviation of 17.72. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 1,626 valid model comparisons. The average AIC for the negative binomial Poisson model was 4563.58 while the average AIC for the negative binomial Hurdle model was 3886.31, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .10 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -2.49 to -1.92 with a mean of -2.01 and a standard deviation of 0.03. Given a Type I error rate of .05, the devi ance statistic was significa nt for none of the 2,000 valid model comparisons. The average AIC fo r the ZIP model was 3874.07 while the average AIC for the negative binomial Hurdle model was 3878.08, which descriptivel y better fit for the ZIP model. 123

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Table 4-26 summarizes the log-likelihood statistics for the negatively skewed distribution model comparisons with a .10 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 3819.85 and a maximum of 4598.21 across all 9,626 valid model comparis ons. Between the six m odels, the minimum was 3868.06 (i.e., the Hurdle model), and the ma ximum was 4563.58 (i.e., th e negative binomial Poisson model). The rank order fr om lowest AIC to highest AIC was as follows: Hurdle model, ZIP model, negative binomial ZI P model, negative binomial Hurd le model, Poisson model, and negative binomial Poisson model. These rankings are in line with the rankings for the pseudopopulation AICs. Table 427 displays the descriptive statisti cs for these two models. Figure 4-24 is a boxplot illustrating these results. The following set of results pertains to the data in which 250 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 750 values were sampled from a negatively skewed p seudo-population distribution wi th the count proportions and frequencies displayed in Ta ble 3-1 and Table 3-2. For the .25 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 30.72 to 86.61 with a mean of 57.02 and a standard deviation of .99. Given a Type I error rate of .05, the devi ance statistic was signifi cant for of the all 2,000 model comparisons. The average AIC for the Po isson model was 4534.49 while the average AIC 124

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for the negative binomial Poisson model w as 4479.47, which descriptively supports the inferential findings of better fit for th e negative binomial Poisson model. For the .25 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -48. 44 to -10.95 with a mean of -13. 01 and a standard deviation of 1.47. Given a Type I error rate of .05, the devi ance statistic was significa nt for of the all 1,517 valid model comparisons. The average AIC for the Hurdle model was 3807.17 while the average AIC for the negative binomial Hurdle model was 3822.26 which descriptively the inferential findings of better fit for the Hurdle model. For the .25 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 674.99 to 784.98 w ith a mean of 733.31 and a standard deviation of 15.50. Given a Type I error rate of .05, th e deviance statistic was significant for of the all 2,000 valid model comparisons. The average AIC for the Poisson model was 4534.49, while the average AIC for the Hurdle model was 3807.17 which descriptively supports the inferential findings of better fit for the Hurdle model. For the .25 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 602.99 to 723.43 w ith a mean of 663.28 and a standard deviation of 19.81. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 1,517 valid model comparisons. The average AIC for the negative binomial Poisson model was 4479.47 while the average AIC for the negative binomial Hurdle model was 3822.26, which descriptively the inferential findings of better f it for the negative binomial Hurdle model. For the .25 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -4.51 to -2.01 with a mean of -1.95 and a standard deviation of .06. Given a Type I error rate of .05, the devi ance statistic was signifi cant for none of the 2,000 125

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valid model comparisons. The average AIC fo r the ZIP model was 3813.16 while the average AIC for the negative binomial ZIP model was 38 17.17, which descriptively supports better fit for the ZIP model. Table 4-28 summarizes the log-likelihood statistics for the negatively skewed distribution model comparisons with a .25 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 3765.15 and a maximum of 4576.05 across all 11,517 valid model comparis ons. Between the six models, the minimum was 3,807.17 (i.e., the Hurdle model), and the maximum was 4534.49 (i.e., the Poisson model). The rank order from lowest AIC to highest AI C was as follows: Hurdle model, ZIP model, negative binomial ZIP model, negative binom ial Hurdle model, negative binomial Poisson model, and Poisson model These rankings ar e in line with the rankings for the pseudopopulation AICs. Table 429 displays the descriptive statisti cs for these two models. Figure 4-25 is a boxplot illustrating these results. The following set of results pertains to the data in which 500 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 500 values were sampled from a negatively skewed p seudo-population distribution wi th the count proportions and frequencies displayed in Ta ble 3-1 and Table 3-2. For the .50 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 618.37 to 694.3 with a mean of 658.45 and a standard deviation of 126

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9.97. Given a Type I error rate of .05, the de viance statistic was signifi cant for all 2,000 valid model comparisons. The average AIC for th e model Poisson model was 4747.63 while the average AIC for the negative binomial Poisson model was 4091.18, which descriptively supports the inferential findings of better fit for the negative binomial Poisson model. For the .50 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -4.36 to 57.05 with a mean of 8.65 and a standard deviation of 2.17. Given a Type I error rate of .05, the devi ance statistic was significa nt for 1,665 of the 1,672 valid model comparisons. The average AIC for the Hurdle model was 3179.05 while the average AIC for the negative binomial Hurdle mode l was 3189.83 which descriptively the 99.58% inferential findings of better fit for the Hurdle model. For the .50 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 1472.82 to 1677. 02 with a mean of 1574.58 and a standard deviation of 31.48. Given a Type I error rate of .05, the deviance statistic was significant for of all 2,000 valid model comparisons. The average AIC for the Poisson model was 4747.63 while the average AIC for the Hurdle model was 3179. 05 which descriptively supports the inferential findings of better fit for the Hurdle model. For the .50 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 823. 6 to 984.79 with a mean of 907.16 and a standard deviation of 24.35. Given a Type I error rate of .05, the de viance statistic was signifi cant for all 1,672 valid model comparisons. The average AIC for the negative binomial Poisson model was 4091.18 while the average AIC for the negative binomia l Hurdle model was 3189.83, which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. 127

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For the .50 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -2.36 to -1.87 with a mean of -2.00 and a standard deviation of .02. Given a Type I error rate of .05, the devi ance statistic was signifi cant for none of the 1,998 valid model comparisons. The average AIC for the ZIP model was 3184.98 while the average AIC for the negative binomial ZIP model was 31 80.98, which descriptively supports better fit for the negative binomial ZIP model. Table 4-30 summarizes the log-likelihood statistics for the negatively skewed distribution model comparisons with a .50 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 3145.34 and a maximum of 4850.32 across all valid model comparisons Between the six models, the minimum was 3179.05 (i.e., the Hurdle model), and the maximum was 4747.63 (i.e., the Poisson model). The rank order from lowest AIC to highest AIC was as follows: Hurdle model, negative binomial ZIP model, ZIP model, negative binomial Hurdle model, negative binomial Poisson model, and Poisson model. These rankings are in line with the rankings for the pseudo-population AICs. Table 4-31 displays the descriptive statistics for these two models. Figure 4-26 is a boxplot illustrating these results. The following set of results pertains to the data in which 750 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 250 values were sampled 128

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from a negatively skewed p seudo-population with the count proportions a nd frequencies displayed in Table 31 and Table 3-2. For the .75 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 647.7 to 760.47 with a mean of 704.52 and a standard deviation of 16.27. Given a Type I error rate of .05, the deviance statistic was significant for all 2,000 valid model comparisons. The average AIC for the Po isson model was 3845.4 while the average AIC for the negative binomial Poisson model w as 3142.87, which descriptively supports the inferential findings of better fit for th e negative binomial Poisson model. For the .75 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -171. 64 to 9.79 with a mean of -4.89 and a standard deviation of 5.61. Given a Type I error rate of .05, the de viance statistic was signi ficant for 1418 of the 1563 valid model comparisons. The average AIC for the Hurdle model was 2027.47 while the average AIC for the negative binomial Hurdle model was 2034.39, which descri ptively supports the 90.72% inferential findings of better fit for the Hurdle model. For the .75 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 1472.82 to 1677. 82 with a mean of 1574.58 and a standard deviation of 31.48. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The averag e AIC for the Poisson model was 3845.4 while the average AIC for the Hurdle model was 2027. 47 which descriptively supports the inferential findings of better fit for the Hurdle model. For the .75 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 976.32 to 1218.4 w ith a mean of 1114.55 and a standard deviation of 31.27. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 129

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1,562 valid model comparisons. The average AIC for the negative binomial Poisson model was 3142.87 while the average AIC for the negative binomial Hurdle model was 2034.39 which descriptively supports the findings of better fit for the negative binomial Hurdle model. For the .75 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -8.54 to -1.59 with a mean of -2.01 and a standard deviation of .15. Given a Type I error rate of .05, the devi ance statistic was signifi cant for 1of the 1,999 valid model comparisons. The average AIC for the ZI P model was 2033.45 while the average AIC for the negative binomial ZIP model was 2037.46, whic h descriptively supports the one inferential finding of better fit for the ZIP model. Table 4-32 summarizes the log-likelihood statistics for the negatively skewed distribution model comparisons with a .75 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 2000.2 and a maximum of 3994.94 across all 11,561 valid model comparisons Between the six models, the minimum was 2027.47 (i.e., the Hurdle model), and the maxi mum was 3845.4 (i.e., the Poisson model ). The rank order from lowest AIC to highest AIC was as follows: Hurdle model, ZIP model, negative binomial Hurdle model, negative binomial ZIP model, negative binomial Poisson model, and Poisson model. These rankings are, for the most part, in line with the rankings for the pseudopopulation AICs. However, in the pseudo-population, the AICs for the Hurdle model and the 130

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negative binomial Hurdle model were tied at 2020.3. Table 4-33 displays the descriptive statistics for these two models Figure 4-27 is a boxplot i llustrating these results. The following set of results pertains to the data in which 900 of the 1,000 observations in each of the 2,000 simulated datase ts were fixed at zero. The re maining 100 values were sampled from a negatively skewed p seudo-population distribution wi th the count proportions and frequencies displayed in Table 3-1 and Table 3-2. For the .90 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 334.64 to 438.18 w ith a mean of 389.67 and a standard deviation of 14.71 Given a Type I error rate of .05, th e deviance statistic was significant for all of the 2,000 valid model comparisons. The average AIC for the Poisson model was 2311.08, while the average AIC for the negative binomial Poiss on model was 1923.4, which descriptively the inferential findings of better fit for th e negative binomial Poisson model. For the .90 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -40. 33 to -1.06 with a mean of -2.03 and a standard deviation of 2.11 Given a Type I error rate of .05, the de viance statistic was signifi cant for 57 of the 1,478 valid model comparisons. The average AIC for the Hurdle model was 1017.47, while the average AIC for the negative binomial Hurdle model was 1021.39 which descript ively supports findings of better fit for the Hurdle model; inferentially, better fit was found for the negative binomial Hurdle model for 3.9% of the simulations. For the .90 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 1119.3 to 1443.11 with a mean of 1299.61 and a standard deviation of 48.19. Given a Type I error rate of .05, the deviance statistic was significant for all of the 2,000 valid model comparisons. The av erage AIC for the Poisson model was 2311.08 131

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while the average AIC for the Hurdle model was 1017.47, which descri ptively supports the inferential findings of better fit for the Hurdle model. For the .90 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from 783.55 to 997.99 w ith a mean of 907.75 and a standard deviation of 33.26. Given a Type I error rate of .05, th e deviance statistic was significant for all of the 1,478 valid model comparisons. The average AIC for the negative binomial Poisson model was 1923.40 while the average AIC for the negative binomial Hurdle model was 1021.39 which descriptively supports the inferential findings of better fit for the negative binomial Hurdle model. For the .90 proportion of zeros condition w ith a negatively skewed distribution, the deviance statistic ranged from -3.33 to -1.97 with a mean of -2.01 and a standard deviation of .04. Given a Type I error rate of .05, the devi ance statistic was signifi cant for none of the 2,000 valid model comparisons. The average AIC fo r the ZIP model was 1023.46 while the average AIC for the negative binomial ZIP model was 10 27.47, which descriptively supports better fit for the ZIP model. Table 4-34 summarizes the log-likelihood statistics for the negatively skewed distribution model comparisons with a .90 proportion of zeros. In this table, LL represents the mean loglikelihood for the pseudo-population, L LminLLmaxLLLLsrepresents the mean log-likelihood for the simulated dataset samples, and represent the log-likelihood minimum and maximum for the simulated dataset sample, and LLs represent the standard deviat ion and standard error for the simulated dataset samples, and n represents the sample size for the simulated dataset samples. A comparison of AICs over all models revealed a minimum of 997.99 and a maximum of 2454.31 across all 11,478 valid model comparisons Between the six models, the minimum was 132

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1017.47 (i.e., the Hurdle model), and the maxi mum was 2311.08 (i.e., the Poisson model ). The rank order from lowest AIC to highest AIC w as as follows: Hurdle model, negative binomial Hurdle model, negative binomial Poisson, ZIP m odel, negative binomial ZIP model, and Poisson model. These rankings are in line with the rank ings for the pseudo-popula tion AICs. Table 4-35 displays the descriptive statistics for these two models. Figure 4-28 is a boxplot illustrating these results. Review of Positively Skewed Distribution Findings For the .50, .75, and .90 proportions of zer os conditions, the negative binomial Poisson model displayed significantly better fit than the Poisson model; however, for the .10 and .25 proportions of zeros conditions, the Poisson model fit significantly better than the negative binomial Poisson model. For all conditions, the Hurdle model displayed significantly better fit than the Poisson model except at the .25 propor tion of zeros condition where the Hurdle model displayed better fit for 25.2% of the simulations. The negative binomial Hurdle model displayed significantly better fit than the negative binomial Poi sson model for all proportions zeros conditions and all simulations except for the .50 proportion of zeros condition in which the negative binomial Hurdle disp layed significantly be tter fit for .133 of the simulations. There was considerable variability in result s comparing the Hurdle model and negative binomial Hurdle model as well as results compar ing the ZIP model and negative binomial ZIP model. The negative binomial Hurdle displayed significantly better fit for all simulations when the proportion of zeros was .10 and .50. The proportion of simulations favoring the negative binomial Hurdle model dropped to .831 when the proportion of zeros of .75. This proportion significant dropped further to .307 for the .90 proportion of zeros conditions and dropped further yet to .252 for the .25 proportion of zeros conditions. 133

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There were no conditions in which the negative binomial ZIP model displayed significantly better fit than th e ZIP model. The proportion of such cases was .947 at the .50 proportion of zeros condition. The proportion dro pped to .652 at the .75 proportion of zeros. There were only .133 and .186 simu lations favoring the negative binomial ZIP model at .25 and .90 zeros, respectively. Finally, there were no simulations favoring the negative binomial ZIP model when the proportion of zeros was .10. However, as displayed in Table 4-42, there were convergence problems when the proportion of zero s was .10 and .25. Table 4-36 displays the percentage of simulations favoring the more complex model. In terms of AIC, the results were similarl y inconsistent. The negative binomial Hurdle model typically displayed the best fit; however the negative binomial Poisson model, which displayed the worst fit for the .10 and .25 proporti on of zeros conditions then displayed best fit for the .50 proportion of zeros condition. The Hu rdle model displayed the second best fit when the proportion of zeros was .10. However, its ra nk decreased over subsequent simulations from third to fourth best-fitting model. The Poisson model also displayed inconsistent findings. For the .10 proportion of zeros condition, this mode l was the third best-fi tting model. It then improved in fit at the .25 proportion of zeros condition but then became the worst fitting model for the remaining proportion of zeros conditions. The ZIP model displayed fourth best fit when the proportion of zeros was .10 and .25; this rank was superseded by the negativ e binomial ZIP model. However, the results reversed for the .50 and .75 proportion of zeros conditions and re versed again at the .90 proportion of zeros conditions. It is important to note that these resu lts are based on AICs; inspection of previous AIC figures suggests that the ra nk order is purely for compar ison since the actual AIC values may be nearly equal between models and/or pr oportion of zeros conditio ns. For example, the 134

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superiority of the Hurdle model over the ZIP mo del .25 zeros is based on an AIC difference of 3.22 while the superiority of the Hurdle model over the ZIP model at .10 zeros is based on an AID difference of 129.04. Table 4-37 presents th e AICs for the positively skewed distribution over all models and proportions of zeros. Figur e 4-29 illustrates the rank-ordering of these AICs. Review of Normal Distribution Findings The results for the normally distributed event stage were consistent for all proportion of zeros. For all proportions, the Hurdle model displa yed the best fit. Ther e were no significant findings of support for the negative binomial Hu rdle model. The two models did become more similar in terms of AICs at the .75 and .90 pr oportions of zeros. This same pattern was found for the ZIP models. However, it is important to not e that at the .25 and higher proportion of zeros conditions, Hurdle model comparisons converged for approximately 75% of the simulations while the ZIP model comparisons converged for 100% of the model comparisons. The convergence frequencies are displa yed in Table 4-43. Both infere ntially and descriptively, the Poisson model was superior to the negative binomial Poisson model when the proportion of zeros was .10 or .25; the reverse was true when the proportion of zeros was .50, .75, or .90. Table 4-38 displays the percentage of simulations significantly favoring the more complex model. It is important to note that these results ar e based on AICs; inspect ion of previous AIC figures suggests that the rank orde r is purely for comparison since the actual AIC values may be nearly equal between models and/or proporti on of zeros conditions. For example, for all proportions of zeros, the difference in AICs between the ne gative binomial Hurdle, ZIP, and ZIP models are quite small. Table 4-39 presents the AICs for the normal distribution over all models and proportions of zeros Figure 4-30 illustrates the rank-ordering over conditions. 135

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Review of Negatively Skew ed Distribution Findings For all conditions except for the .10 proporti on of zeros, the negative binomial Poisson model displayed significantly be tter fit than the Po isson model. Between all zero proportion conditions, the Hurdle model consistently displaye d significantly better fit when compared to the Poisson model while the negative binomial Hurd le model consistently displayed significantly better fit when compared to the negative binomial Poisson model. In general, the Hurdle model displayed significantly better fit than the negative binomial Hurdle model; however, the results should be inte rpreted cautiously since for all conditions, the negative binomial Hurdle had convergence problems of roughly 25%. This is displayed in Table 4-44. The negative binomial ZIP model never di splayed significantly better fit than the ZIP model. Table 4-40 displays the percentage of simulations significantly favoring the more complex model. Descriptively, the Hurdle model displayed the best fit regardless of the proportion of zeros. The Poisson model displayed the worst fit excep t for the .10 proportion of zeros condition in which the negative binomial Poisson model displaye d the worst fit. Consequently, the next worst fitting model was this negative binomial Poisson m odel except for the .10 proportion of zeros. In between the best fitting and tw o worst fitting models, for the .10 and .25 proportions of zeros conditions, the ZIP model fit better than the negative binomial ZIP model, which fit better than the negative binomial Hurdle model. For the .50 pr oportion of zeros conditi on, the rank order for the ZIP and negative binomial ZIP was reversed with the negative binomial ZIP model displaying better fit. However, for the .75 propo rtion of zeros condition the ZIP model returned to being the second best-fitting model. Finally, for the .90 proportion of zeros condition the negative binomial Hurdle model gained in rank again to become the second best-fitting model. 136

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The negative binomial Poisson also rose in rank to be the third best-fitting model. This was followed by the ZIP and negativ e binomial ZIP models. It is important to note that these results ar e based on AICs; inspect ion of previous AIC figures suggests that the rank orde r is purely for comparison since the actual AIC values may be nearly equal between models and/or proportion of zeros conditions. For example, although the AIC is lower for the Hurdle models than for the negative binomial Hurdle models, the difference between the AICs declines sharply as the propor tion of zeros increases. Table 4-41 presents the AICs for the negatively skewed distribution over all models and proportions of zeros. Figure 431 illustrates the rank-ordering over conditions. 137

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Table 4-1. Deviance statistics comparing Po isson and negative binomial Poisson models Distribution Zeros Positive Normal Negative .10 -238.40 -384.40 -517.40 .25 -82.60 -35.20 -1534.80 .50 88.40 331.00 658.40 .75 125.00 372.40 704.60 .90 69.80 203.00 389.60 Table 4-2. Deviance statistics comparing Hu rdle and negative binomial Hurdle models Distribution Zeros Positive Normal Negative .10 27.60 -6.20 -15.60 .25 20.40 -5.20 -13.00 .50 15.40 -3.60 -8.40 .75 8.20 -1.80 -4.40 .90 3.20 -1.80 -1.80 Table 4-3. Deviance statistics comp aring Poisson and Hurdle models Distribution Zeros Positive Normal Negative .10 131.60 19.20 181.80 .25 5.20 252.60 859.60 .50 75.40 756.80 1573.20 .75 215.40 983.00 1823.20 .90 203.20 732.40 1298.20 Table 4-4. Deviance statistics compari ng NB Poisson and NB Hurdle models Distribution Zeros Positive Normal Negative .10 397.60 397.40 683.60 .25 108.20 282.60 662.20 .50 2.40 422.20 906.40 .75 98.60 608.80 1114.20 .90 136.60 527.60 906.80 138

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Table 4-5. Deviance statistics compari ng ZIP and negative binomial ZIP models Distribution Zeros Positive Normal Negative .10 -2.00 -2.00 -2.00 .25 -0.60 -2.00 -2.00 .50 14.20 -2.00 -2.00 .75 6.20 -2.00 -2.00 .90 1.20 -2.00 -2.00 Table 4-6. Log-likelihood comparisons for positively skewed distri bution with .10 zeros Model LL L LminLLmaxLLLLs LLs n Poisson -1475.3 -1475.53 -1539.15 -1409.23 19.27 0.430995 2000 1 NB Poisson -1594.5 -1594.65 -1650.38 -1539.67 15.79 0.353131 2000 Hurdle -1409.5 -1409.40 -1495.97 -1315.41 26.27 0.587599 2000 2 NB Hurdle -1395.7 -1395.44 -1483.98 -1304.26 25.68 0.574254 2000 Poisson -1475.3 -1475.53 -1539.15 -1409.23 19.27 0.430995 2000 3 Hurdle -1409.5 NB -1409.40 -1495.97 -1315.41 26.27 0.587599 2000 Poiss on -1594.5-1594.65 -1650.38-1539.6715.790.3531312000 4 le 1 NB Hurd -1395.7 -1395.44 -1483.98 -1304.26 25.68 0.574254 2000 ZIP -1478.3 -1477.92 -1542.15 -1412.24 19.8380 0.505194 542 5 NB ZIP -1479.3-1485.34 -1531.90-1439.61 18.7300 0 1.570000136 139

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Table 4-7. AICs for positively skewed distri bution models with a .10 proportion of zeros Model AIC A ICmin A ICmax A ICAICs A ICs n Poisson 2956.6 2957.06 2824.47 3084.30 38.55 0.86 2000 NB Poisson le 2 IP 2962.6 2967.852836.473096.3039.68 1.01 1542 136 3194.9 3193.30 3083.34 3304.77 31.58 0.71 2000 Hurdle 2822.9 2838.81 2650.83 3011.94 52.56 1.18 2000 NB Hurd 2975.4 2804.88 2622.53 2981.96 51.36 1.15 000 Z NB ZIP 2964.6 2984.793077.792984.6936.55 3.13 Table 4ics vd d ti zeros Model LL-8. Log-l kelihood omparison for positi ely skewe istri bu on with .25 L L min max LLs LL LL LLn s Poisson -1468.1 -1466.83-1542.31-1388.7722.760.509050 2000 1 NB Poisson -1509.4 -1508.43-1561.66-1448.0817.540.392323 2000 Hurdle -1465.5 -1463.95-1542.28-1377.4124.290.543358 2000 2 NB Hurdle -1455.3 -1453.50-1523.70-1370.0323.670.529329 2000 Poisson -1468.1 -1466.83-1542.31-1388.7722.760.509050 2000 3 Hurdle -1465.5 -1463.95-1542.28-1377.4124.290.543358 2000 NB Poisson -1509.4 -1508.43-1561.66-1448.0817.540.392323 2000 4 NB Hurdle -1455.3 -1453.50-1523.70-1370.0323.670.529329 2000 ZIP -1471.1 -1469.56-1544.96-1391.7722.710.511613 1972 5 NB ZIP -1471.4 -1471.07-1537.28-1407.3620.360.470331 1874 140

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Table 4-9. AICs for positively skewed distri bution models with a .25 proportion of zeros Model AIC A ICmin A ICmax A ICAICs A ICs n Poisson 2942.3 2939.654 2783.538 3090.621 45.53082 1.018100 2000 NB Poisson 3026.8 3020.851 2900.160 3127.329 35.09046 0.784647 2000 Hurdle 2943.0 2947.899 2774.825 3104.551 48.59943 1.086716 2000 NB Hurdle 2922.5 2921.001 2754.053 3061.392 47.34459 1.058657 2000 ZIP 2954.3 2951.124 2795.538 3101.917 45.43863 1.023227 1972 NB ZIP 2956.7 2956.147 2828.716 3088.553 40.72097 0.940661 1874 Table 4-10. Log-likelihood comparisons for pos itively skeed distrLLw i bution with .50 zeros Model L LminLLmaxLLLLs LLs n Poisson -1334.3 -1333.44 -1428.47 -1239.52 29.11 0.650 2000 1 -1290.1 -1289.49-1361.70-1223.9021.340.477 2000 Hurdle -1296.6 -1294.96-1357.10-1221.9020.800.465 2000 NB Poisson 2 NB Hurdle -1288.9 -1287.12-1352.33-1220.4220.260.906 2000 Poisson -1334.3 -1333.44-1428.47-1239.5229.110.650 2000 3 Hurdle -1296.6 -1294.96-1357.10-1221.9020.800.465 2000 NBPoisson -1290.1 -1289.49-1361.70-1223.9021.340.477 2000 4 NB Hurdle -1288.9 -1287.12-1352.33-1220.4220.260.906 2000 ZIP -1299.5 -1297.92-1360.09-1224.6320.800.465 2000 5 NB ZIP -1292.4 -1291.12-1356.33-1224.0020.170.451 2000 141

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Table 4-11. AICs for positively skewed distribution models with a .50 proportion of zeros Model AIC A ICmin A ICmax A ICAICs A ICs n Poisson 2674.5 2672.876 2485.036 2862.935 58.22 1.301853 2000 NB Poisson e IP 2611.1 2607.8402461.2622732.18941.610.930524 2000 2000 2588.3 2582.975 2451.801 2727.394 42.68 0.954555 2000 Hurdle 2605.1 2601.929 2455.800 2726.197 41.60 0.930422 2000 NBHurdl 2589.8 2588.240 2454.830 2718.660 40.52 0.905000 2000 Z NB ZIP 2598.8 2596.2322462.0042726.65440.350.902392 Table 4-12. Log-likelihood comor p ewed distri bution with .75 zeros Model LLparisons f os itively sk L L minLL max LLs LL LLsn Poisson -972.9 -974.608-1088.690-853.76131.925 0.7132000 1 on -910.4 -911.681-999.070-819.43823.872 0.5332000 Hurdle -865.2 -864.852-907.808-806.14614.387 0.3212000 NB Poiss 2 NB on Hurdle -861.1 -860.701 -905.917 -801.635 14.148 0.316 2000 Poiss -972.9 -974.608 1088.690 -853.761 31.925 0.713 2000 3 Hurdle on -865.2 -864.852 -907.808 -806.146 14.387 0.321 2000 NB Poiss -910.4 -911.681 -999.070 -819.438 23.872 0.533 2000 4 NB Hurd le ZIP -868.2 -867.858-910.807-809.10514.382 0.3212000 -861.1 -860.701 -905.917 -801.635 14.148 0.316 2000 5 NB ZIP -865.1 -864.721-909.922-805.61814.135 0.3162000 142

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Table 4-13. AICs for positively skewed distribution models with a .75 proportion of zeros Model min A IC A IC s A ICsAIC A IC AIC max n Poisson 1951.7 1955.215 1713.523 2183.381 63.85 1.42776 2000 NB Poisson le IP 1750.5 1747.7161630.21001833.61528.76 0.643202000 2000 1828.7 1827.363 1642.875 2002.140 47.74 1.06762 2000 Hurdle 1742.5 1749.705 1632.292 1835.616 28.77 0.64343 2000 NB Hurd 1734.3 1735.402 1617.270 1825.834 28.29 0.63273 2000 Z NB ZIP 1744.3 1743.4421625.2371833.84428.27 0.63215 Table 4-14. Log-likelihood compa p skribu h .9 Model risons for os itively ewed dist t ion wit 0 zerosLL L L mi ma snLLxLLLL LLsn Poisson -546.5 -546.397-653.705-453.19029.33 0.6552000 1 on -511.6 -511.489-594.277-437.84823.20 0.5182000 Hurdle -444.9 -443.180-469.812-408.2419.020 0.2012000 NB Poiss 2 NB Hurdle -443.3 -441.750 -468.410 -410.220 8.848 0.198 1992 Poisson -546.5 -546.397 -653.705 -453.190 29.33 0.655 2000 3 Hurdle -444.9 -443.180 -469.812 -408.241 9.020 0.201 2000 NB Poisson -511.6 -511.489 -594.277 -437.848 23.20 0.518 2000 4 NB Hurd le ZIP -447.9 -446.170-472.716-411.1429.021 0.2012000 -443.3 -441.750 -468.410 -410.220 8.848 0.198 1992 5 NB ZIP -447.3 -445.691-472.314-412.1428.876 0.1982000 143

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Table 4-15. AICs for positively skewed distribution models with a .90 proportion of zeros Model A IC A IC s A ICsAIC A IC AIC min max n Poisson 1099.0 1098.794 912.398 1313.41 58.66007 1.311679 2000 NB Poisson 11 le IP 907.7 904.339834.283957.43218.04285 0.4034502000 399702000 031.2 026.978 879.696 1192.55 46.40218 1.037584 2000 Hurdle 901.8 906.361 836.481 959.623 18.04078 0.403404 2000 NB Hurd 898.7 897.499 834.439 950.819 17.69794 0.396532 1992 Z NB ZIP 908.7 905.381838.283958.62717.75302 0.6 Table 4-16. Log-likelihood com oistritos Model LLparisons f r normal d ibution w h .10 zer L L min maxLL LL LL s LLs n Poisson -1806.7 -1806.38-1834.64-1776.0810.080 0.2252000 1 on -1998.9 -1998.93-2031.97-1961.239.706 0.2172000 -1822.85-1768.77-1822.859.086 0.2032000 NB Poiss Hurdle -1797.1 2 NB Hurdle -1800.2 -1825.6 -1772.63 -1825.60 9.330 0.240 1503 Poisson -1806.7 -1806.38 -1834.64 -1776.08 10.080 0.225 2000 3 Hurdle -1797.1 -1822.85 -1768.77 -1822.85 9.086 0.203 2000 NB Poisson -1998.9 -1998.93 -2031.97 -1961.23 9.706 0.217 2000 4 NB Hurd le ZIP -1800.1 -1799.53-1825.87-1771.789.085 0.2032000 -1800.2 -1799.77 -1825.60 -1772.63 9.330 0.240 1503 5 NB ZIP -1801.1 -1800.63-1826.87-1772.789.042 0.2031982 144

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Table 4-17. AICs for normal distribution models with a .10 proportion of zeros Model A IC A IC s A ICsAIC A ICAIC min max n Poisson 3619.3 3618.760 3558.16 0 3675.27 4 20.17 0.45 2000 NB Poisson 4005.9 4005.863 3930.44071.9 19.410.43 2000 le Hurdle 0.40 1982 5 2 3 4 6 9 Hurd 3606.1 3605.028 3549.5 2 3657.6 5 18.17 0.40 2000 NB 3612.3 3613.546 3559.26 1 3665.19 5 18.66 0.48 1503 ZIP 3612.1 3611.051 3555.56 2 3663.73 3 18.17 0.40 2000 NB ZIP 3616.1 3615.268 3559.56 2 3667.73 9 18.08 TabLooi ormution w zeros el le 4-18. g-likelih od compar sons for n al distrib ith .25 ModLL L L minLL max LLLLs LLs n Poisson -1913.9 -1913.12 -1968.17 -1869.47 13.99 0.312 2000 1 NB Poisson -1931.5 -1930.99 -1979.33 -1889.82 11.77 0.263 2000 Hurdle -1787.6 -1787.24 -1813.49 -1758.97 8.104 0.181 2000 2 NB Hurdle -1790.2 -1789.81 -1816.07 -1764.27 8.152 0.201 1634 Poisson -1913.9 -1913.12 -1968.17 -1869.47 13.999 0.312 2000 3 Hurdle -1787.6 -1787.24 -1813.49 -1758.97 8.104 0.181 2000 NB Poiss on 11.775 -1931.5 -1930.99 -1979.33 -1889.82 0.263 2000 4 NB Hurdle -1790.2-1789.81 -1816.07-1764.278.1520.201 1634 ZIP -1790.6-1790.22 -1816.47-1761.968.1050.181 2000 5 NB ZIP -1791.6-1791.22 -1817.47-1762.978.1050.181 2000 145

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Table 4-19. AICs for normal distribution models with a .25 proportion of zeros Model A IC A ICAICs AIC A IC min max A ICs n Poisson 3833.8 3832.239 3744.945 3942.340 27.99438 0.625973 2000 NB Poisson le IP 3593.1 3592.4363535.9173644.93816.211450.362499 2000 NB ZIP 3597.1 3596.4483539.9443648.93916.211050.362490 2000 3871.0 3869.986 3787.645 3966.654 23.55270 0.526654 2000 Hurdle 3587.2 3586.473 3529.931 3638.976 16.20934 0.362452 2000 NB Hurd 3592.3 3593.629 3542.533 3646.150 16.30441 0.403347 1634 Z 146

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Table 4-20. Log-likelihood comparisons for normal distribution with .50 zeros LLsminLLmaxLLLLs Model LL L L n Poisson -1888.5 -1888.88 -1960.42 -1818.41 20.88 0.466 2000 1 NB Poisson -1723.0 -1723.25 -1769.87 -1674.06 14.60 0.326 2000 Hurdle -1510.1 -1509.76 -1532.97 -1487.29 6.62 0.148 2000 2 NB Hurdle -1511.9 -1509.76 -1532.97 -1487.29 6.62 0.148 1602 Poisson -1888.5 -1888.88 -1960.42 -1818.41 20.88 0.466 2000 3 Hurdle -1510.1 NB -1509.76 -1532.97 -1487.29 6.627 0.148 2000 Poiss on -1723.0 -1723.25-1769.87-1674.0614.60 0.326 2000 4 le NB Hurd -1511.9 -1509.76 -1532.97 -1487.29 6.62 0.148 1602 ZIP -1513.1 -1512.75 -1535.96 -1490.28 6.61 0.147 2000 5 NB 2000 ZIP -1514.1 -1513.75-1536.96-1491.286.61 0.147 Table 4-21. AICs f ode 50 propof ze el or normal distribution m ls with a ortion ros Mod A IC A IC min A IC A maxIC A ICs A IC sn Poisson 3783.0 3783.75 3642.81 3926.846 41.76 0.93387 2000 NB Poisson 3454.0 3454.50 3356.11 3547.741 29.21 0.653052 2000 Hurdle 3023.2 3023.52 2978.57 3069.939 13.25 0.296204 2000 NB Hurdle 3035.8 3039.52 2994.573085.93913.250.296204 1602 ZIP 3038.3 3037.49 2992.553083.92813.230.295936 2000 NB ZIP 3042.3 3041.50 2996.553087.92813.230.295941 2000 147

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Table 4-22. Log-likelihood comparisons for normal distribution with .75 zeros Model LL L Ln x LL miLL maLL s LLs n Poisson -1461.8 -1462.42 -1548.57 -1371.53 27.106 0.606 2000 1 NB Poisson -1275.6 -1276.08 -1332.19 -1214.35 18.507 0.413 2000 Hurdle -970.3 -970.42 -987.29 -953.45 4.779 0.106 2000 2 NB Hurdle -971.2 -971.53 -992.37 -954.37 4.993 0.129 1490 Poisson -1461.8 -1462.42 -1548.57 -1371.53 27.106 0.606 2000 3 Hurdle -970.3 -970.42 -987.29 -953.45 4.779 0.106 2000 NB Poisson -1275.6 -1276.08 -1332.19 -1214.35 18.507 0.413 2000 4 NB Hurdle -971.2 -971.53 -992.37 -954.37 4.993 0.129 1490 ZIP -990.3 -973.30 -956.46 -990.29 4.770 0.106 2000 5 NB ZIP -991.3 -974.30 -957.45 -991.29 4.770 0.106 2000 Table 4 Model -23. As for noral distribution m a .7 proporon of zeros IC m odels with 5 ti n A IC A ICmi n A IC ma x A IC A ICs A ICs Poiss NB on 2929.6 2930.846 2749.052 3103.141 54.21258 1.21223 2000 Poisson 2559.3 2560.165 2436.693 2 le 1 1958.5 1958.852 1924.9221992.589.5575590.21371 2000 NB ZIP 1962.6 1962.866 1928.9181996.5819.5594520.21375 2000 672.373 37.0152 0.82768 2000 Hurdle 1952.6 1952.855 1918.906 1986.58 9.559016 0.21374 2000 NB Hurd ZIP 1954.4 1957.073 1922.741 998.756 9.987633 0.25874 1490 148

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Table 4-24. Log-likelihood comparisons for normal distribution with .90 zeros Model LL L Ln max LLs miLL LL LLs n Poisson -852.6 -855.101 -937.508 -767.641 26.52 0.593 1999 1 NB Poisson -751.1 -753.075 -812.698 -689.054 19.45 0.434 2000 Hurdle -486.4 -486.118 -494.801 -475.228 2.89 0.064 2000 2 NB Hurdle -487.3 -486.659 -498.835 -475.636 2.91 0.074 1537 Poisson -852.6 -855.101 -937.508 -767.641 26.52 0.593 1999 3 Hurdle -486.4 NB -486.118 -494.801 -475.228 2.89 0.064 2000 Poiss on -751.1 -753.075-812.698-689.05419.45 0.4342000 4 le NB Hurd -487.3 -486.659 -498.835 -475.636 2.91 0.074 1537 ZIP -489.4 -489.111 -497.799 -478.242 2.89 0.064 2000 5 -490.4 -490.114-498.799-479.2432.89 0.0642000 NB ZIP Table 4-25. AICs a onith orro for norm l distributi models w a .90 prop tion of ze s Model A IC A IC min A IC max A IC A IC s A IC sn Poisson 1711.1 1716.202 1541.281 1881.015 53.0534 1.18660 1999 NB Poisson 1 NB Hurdle 986.6 987.3187965.27281011.675.82673 0.148621537 ZIP 990.8 990.2214968.48481007.5975.79280 0.129532000 NB ZIP 994.8 994.2271972.48541011.5975.79265 0.129582000 510.2 1514.149 1386.108 1633.397 38.9068 0.86998 2000 Hurdle 984.8 984.2365 962.4563 1001.602 5.79776 0.12964 2000 149

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Table 4-26. Log-likelihood comparisons for ne gatively skewed distribution with .10 zeros Model LL L L miLLn maxLLLLs LLs n Poisson -2019.20 -2019.01 -2035.41 -2001.08 5.26 .12 2000 1 NB Poisson -2277.90 -2277.79 -2296.11 -2258.54 5.97 .13 2000 Hurdle -1928.30 -1928.03 -1948.88 -1907.92 5.82 .13 2000 2 NB Hurdle -1936.10 -1936.16 -1965.98 -1918.98 5.64 .14 1626 Poisson -2019.20 -2019.01-2035.41-2001.085.26 .122000 3 le -1928.30 -1928.03-1948.88-1907.925.82 .132000 on -2277.90 -2277.79-2296.11-2258.545.97 .132000 Hurd NB Poiss 4 le NB Hurd -1936.10 -1936.16 1965.98 -1918.98 5.64 .14 1626 ZIP -1931.30 -1931.03 1951.87 -1910.92 5.82 .13 2000 5 ZIP -1932.30 -1932.041952.87-1911.925.82 NB .132000 Table 4-27ry s el AICs fo negativel kewed models with a .10 proportion of zeros ModAIC A IC min A IC max A ICAICs A ICn s Poisson 4044.50 4044.02 4008.15 4076.81 10.53 .24 2000 NB Poisson 4563.90 4563.58 4525.08 4600.21 11.94 .27 2000 Hurdle 3868.70 3868.06 3827.85 3909.75 11.25 .28 2000 NB urdle 3884.10 3886.31 3851.963945.9711.25.28 1626 ZIP 3874.60 3874.07 3833.843915.7411.65.26 2000 NB ZIP 3878.70 3878.08 3837.853919.7511.65.26 2000 H 150

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Table 4-28. Log-likelihood comparisons for ne gatively skewed distribution with .25 zeros Model LLsminLLmaxLLLLsLL L L n Poisson -2264.3 -2264.24 -2285.03 -2239.14 7.13 0.16 2000 1 NB Poisson -2235.5 -2235.74 -2258.49 -2208.87 7.18 0.16 2000 Hurdle -1897.9 -1897.59 -1917.18 -1880.57 5.24 0.12 2000 2 NB Hurdle -1904.4 -1904.13 -1922.89 -1887.67 5.12 0.13 1517 Poisson -2264.3 -2264.24 -2285.03 -2239.14 7.13 0.16 2000 3 Hurdle -1897.9 -1897.59 -1917.18 -1880.57 5.24 0.12 2000 NB Poiss on -2235.5 -2235.74-2258.49-2208.877.18 0.162000 4 le -1904.4 -1904.13-1922.89-1887.675.12 0.131517 NB Hurd ZIP -1900.9 -1900.58 -1920.18 -1883.57 5.24 0.12 2000 5 -1901.9 -1921.18-1884.57 0.122000 NB ZIP -1901.58 5.24 Table 4-29. AICs for negatively skewed Model mAICodels with a .25 proportion of zeros A IC min A IC max A IC AIC s A ICs n Poisson 4534.7 4534.49 4484.28 4576.05 14.25 0.32 2000 NB Poisson B Hurdle 3820.8 3822.263789.343859.7810.250.26 1517 ZIP 3813.9 3813.163783.143852.3610.480.23 2000 NB ZIP 3817.9 3817.173783.153856.3610.480.23 2000 4479.0 4479.47 4425.75 4524.99 14.37 0.32 2000 Hurdle 3807.9 3807.17 3773.15 3844.37 10.47 0.23 2000 N 151

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Table 4-30. Log-likelihood comparisons for ne gatively skewed distribution with .50 zeros LLs Model LL L Lx minLL maLLLLs n Poisson -2370.5 -2370.81 -2422.16 -2317.09 14.38 0.32 2000 1 NB Poisson -2041.3 -2041.59 -2075.01 -2007.90 9.95 0.22 2000 Hurdle -1583.9 -1583.52 -1597.10 -1570.67 4.44 0.10 2000 2 NB Hurdle -1588.1 -1587.91 -1615.10 -1576.02 4.42 0.11 1672 Poisson -2370.5 -2370.81 -2422.16 -2317.09 14.38 0.32 2000 3 Hurdle -1583.9 -1583.52 -1597.10 -1570.67 4.44 0.10 2000 NB Poisson -2041.3 -2041.59 -2075.01 -2007.90 9.95 0.22 2000 4 NB Hurdle -1588.1 -1587.91 -1615.10 -1576.02 4.42 0.11 1672 ZIP -1586.9 -1586.49 -1600.10 -1573.67 4.43 0.10 2000 5 NB ZIP -1587.9 -1587.49 -1601.10 -1574.67 4.44 0.10 1998 Table 4-31. AICs for negatively skewed models with a .50 proportion of zeros Model A IC A ICmin A ICmax A IC A ICs A ICs n Poisson 4746.9 4747.63 4640.17 4850.32 28.76 0.64 2000 NB Poisson 4090.7 4091.18 4023.80 4158.02 1 ZIP 3185.7 3184.98 3156.343212.208.870.20 2000 NB ZIP 3181.7 3180.98 3155.343208.208.870.20 1998 9.89 0.44 2000 Hurdle 3179.7 3179.05 3153.34 3206.21 8.88 0.20 1672 NB Hurdle 3188.2 3189.83 3166.04 3244.19 8.83 0.22 2000 152

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Table 4-32. Log-likelihood comparisons for ne gatively skewed distribution with .75 zeros Model LL L L minLL maLLLLsx LLs n Poisson -1919.7 -1919.70 -1994.47 -1845.46 21.89 0.49 2000 1 NB Poisson -1567.4 -1567.44 -1614.24 -1519.31 13.88 0.31 2000 Hurdle -1008.1 -1007.74 -1019.17 -998.10 3.14 0.07 2000 2 NB Hurdle -1008.2 -1010.19 -1085.59 -1000.67 4.05 0.10 1562 Poisson -1919.7 -1919.70 -1994.47 -1845.46 21.89 0.49 2000 3 Hurdle -1008.1 -1007.74 -1019.17 -998.10 3.14 0.07 2000 NB Poiss on -1567.4 -1567.44-1614.24-1519.3113.88 0.31 2000 4 le NB Hurd -1008.2 -1010.19 -1085.59 -1000.67 4.05 0.10 1562 ZIP -1011.2 -1010.73 -1022.17 -1001.10 3.13 0.07 2000 5 -1012.2 -1011.73-1023.17-1002.103.13 0.07 1999 NB ZIP Table 4-33. AICs fo of z el r negatively skewed models with a .75 prop rtion o eros Mod A IC A IC min A IC max A IC A IsC A ICs n Poisson 3845.3 3845.40 3696.92 3994.94 43.78 0.98 2000 NB Poisson 3142.8 3142.87 3046.63 3236.48 2 NB Hurdle 2028.3 2034.39 2015.342185.198.110.21 1562 ZIP 2034.3 2033.45 2014.202056.346.260.14 2000 NB ZIP 2038.3 2037.46 2018.202060.346.270.14 1999 7.77 0.62 2000 Hurdle 2028.3 2027.47 2008.20 2050.34 6.27 0.14 2000 153

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Table 4-34. Log-likelihood comparisons for ne gatively skewed distribution with .90 zeros LLsminLLmaxLLLLsModel LL L L n Poisson -1152.1 -1152.54 -1224.16 -1064.51 23.17 0.52 2000 1 NB Poisson -957.3 -957.70 -1005.07 -897.19 15.88 0.36 2000 Hurdle -503.0 -502.73 -510.67 -496.99 2.09 0.05 2000 2 NB Hurdle -503.9 -503.70 -524.86 -498.07 2.31 0.06 1478 Poisson -1152.1 -1152.54 -1224.16 -1064.51 23.17 0.52 2000 3 Hurdle -503.0 -502.73 -510.67 -496.99 2.09 0.05 2000 NB Poisson -957.3 -957.70 -1005.07 -897.19 15.88 0.36 2000 4 NB Hurdle -503.9 -503.70 -524.86 -498.07 2.31 0.06 1478 ZIP -506.0 -505.73 -513.67 -499.99 2.09 0.05 2000 5 NB ZIP -507.0 -506.74 -514.67 -501.00 2.09 0.05 2000 Table 4-35. As for negtively skewed odels with a .90 proportion of zeros IC a m Model AIC A IC min A IC max A IC s AIC A ICsn Poisson 2310.2 2311.08 2135.01 2454.31 46.33 1.04 2000 NB Poiss le on 31.77 le ZIP 1024.1 1023.46 1011.991039.344.180.09 2000 NB ZIP 1028.1 1027.47 1015.991043.344.180.09 2000 1922.5 1923.40 1802.37 2018.13 0.71 2000 Hurd 1018.1 1017.47 1015.99 1033.35 4.18 0.09 2000 NB Hurd 1019.8 1021.39 1010.15 1063.71 4.61 0.12 1478 154

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Table 4-36. Positively skewed distribution: perc entage of simulations favoring complex model. Simple vs. Complex .10 .25 .50 .75 .90 P vs. NBP 0.0 0.0 100.0 100.0 100.0 H vs. NBH 100.0 99.7 98.0 98.1 30.7 P vs. H 100.0 25.2 100.0 100.0 100.0 NBP vs. NBH 100.0 100.0 100.0 100.0 100.0 Z vs. NBZ 0.0 18.6 94.7 65.2 13.3 Table 4-37. AICs: pos itively skewed distri bution (all conditions) Proportion of Zeros Model 0.10 0.25 0.50 0.75 0.90 Poisson 2957.06 2939.65 2672.88 1955.22 1098.79 NB Poisson 3193.30 3020.85 2582.98 1827.36 1026.98 Hurdle 2838.81 2947.90 2601.93 1749.71 906.36 NB Hurdle 2804.88 2921.00 2588.24 1735.40 897.50 ZIP 2967.85 2951.12 2607.84 1747.72 904.34 NB ZIP 2984.79 2956.15 2596.23 1743.44 905.38 Table 4-38. Normal distribu tion: percentage of simulations favoring complex model. Simple vs. Complex .10 .25 .50 .75 .90 P vs. NB P 0.0 0.0 100.0 100.0 100.0 H vs. NBH 0.0 0.0 0.0 0.0 0.0 P vs. H 100.0 100.0 100.0 100.0 100.0 NBP vs. NBH 100.0 100.0 100.0 100.0 100.0 Z vs. NBZ .4 0.0 0.0 0.0 0.0 Table 4-39. AICs: normal distribution (all conditions) Proportion of Zeros Model 0.10 0.25 0.50 0.75 0.90 Poisson 3618.76 3832.24 3783.75 2930.85 1716.20 NB Poisson 4005.86 3869.99 3454.51 2560.17 1514.15 Hurdle 3605.03 3586.47 3023.52 1952.86 984.24 NB Hurdle 3613.55 3593.63 3039.52 1957.07 987.32 ZIP 3611.05 3592.44 3037.49 1958.85 990.22 NB ZIP 3615.27 3596.45 3041.50 1962.87 994.23 155

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Table 4-40. Negatively skewed di stribution: percentage of simu lations favoring complex model. Simple vs. Complex .10 .25 .50 .75 .90 P vs. NB P 0.0 100.0 100.0 100.0 100.0 H vs. NBH 100.0 100.0 99.9 90.7 100.0 P vs. H 100.0 100.0 100.0 100.0 100.0 NBP vs. NBH 100.0 100.0 100.0 100.0 100.0 Z vs. NBZ 0.0 0.0 0.0 0.0 0.0 Table 4-41. AICs: ne gatively skewed distri bution (all conditions) Proportion of Zeros Model 0.10 0.25 0.50 0.75 0.90 Poisson 4044.02 4534.49 4747.63 3845.40 2311.08 NB Poisson 4563.58 4479.47 4091.18 3142.87 1923.40 Hurdle 3868.06 3807.17 3179.05 2027.47 1017.47 NB Hurdle 3886.31 3822.26 3189.83 2034.39 1021.39 ZIP 3874.07 3813.16 3184.98 2033.45 1023.46 NB ZIP 3878.08 3817.17 3180.98 2037.46 1027.47 Table 4-42. Convergence frequencie s: positively skewed distribution Proportion of Zeros Model 0.10 0.25 0.50 0.75 0.90 Poisson 2000 2000 2000 2000 2000 NB Poisson 2000 2000 2000 2000 2000 Hurdle 2000 2000 2000 2000 2000 NB Hurdle 2000 2000 2000 2000 2000 ZIP 1542 1972 2000 2000 2000 NB ZIP 136 1874 2000 2000 1992 Table 4-43. Convergence fre quencies: normal distribution Proportion of Zeros Model 0.10 0.25 0.50 0.75 0.90 Poisson 2000 2000 2000 2000 2000 NB Poisson 2000 2000 2000 2000 2000 Hurdle 1503 2000 2000 2000 2000 NB Hurdle 1982 1634 1602 1490 1537 ZIP 2000 2000 2000 2000 2000 NB ZIP 2000 2000 2000 2000 2000 156

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Table 4-44. Convergence frequencie s: negatively skewed distribution Proportion of Zeros Model 0.10 0.25 0.50 0.75 0.90 Poisson 2000 2000 2000 2000 2000 NB Poisson 2000 2000 2000 2000 2000 Hurdle 2000 2000 2000 2000 2000 NB Hurdle 1627 1517 1672 1562 1478 ZIP 2000 2000 2000 2000 2000 NB ZIP 2000 2000 2000 1999 2000 157

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3250.0000 3000.0000 2750.0000 PosAIC NB ZIP ZIP NB Hurdle Hurdle NB Poisson PoissonModel Figure 4-1. Boxplot of AICs for all models for a .10 proportion of zeros 158

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3100.0000 3000.0000 2900.0000 2800.0000 PosAIC NB ZIP ZIP NB Hurdle Hurdle NB Poisson PoissonModel Figure 4-2. Boxplot of AICs for all models for a .25 proportion of zeros 159

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2800.0000 2600.0000 2400.0000 PosAIC NB ZIP ZIP NB Hurdle Hurdle NB Poisson PoissonModel Figure 4-3. Boxplot of AICs for all models for a .50 proportion of zeros 160

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 2200.0000 2100.0000 2000.0000 1900.0000 1800.0000 1700.0000 1600.0000 PosAIC Figure 4-4. Boxplot of AICs for al l models for a .75 proportion of zeros 161

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model1400.0000 1300.0000 1200.0000 1100.0000 1000.0000 900.0000 800.0000 PosAIC Figure 4-5. Boxplot of AICs for all models for a .90 proportion of zeros 162

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 4100.0000 4000.0000 3900.0000 3800.0000 3700.0000 3600.0000 3500.0000 PosAIC Figure 4-6. Boxplot of AICs for all models for a .10 proportion of zeros 163

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 4000.0000 3800.0000 3600.0000 PosAIC Figure 4-7. Boxplot of AICs for all models for a .25 proportion of zeros 164

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 4000.0000 3800.0000 3600.0000 3400.0000 3200.0000 3000.0000 2800.0000 PosAIC Figure 4-8. Boxplot of AICs for all models for a .50 proportion of zeros 165

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 3000.0000 2700.0000 2400.0000 2100.0000 1800.0000 PosAIC Figure 4-9. Boxplot of AICs for all models for a .75 proportion of zeros 166

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model2000.0000 1800.0000 1600.0000 1400.0000 1200.0000 1000.0000 800.0000 PosAIC Figure 4-10. Boxplot of AICs for al l models for a .90 proportion of zeros 167

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 4600.0000 4400.0000 4200.0000 4000.0000 3800.0000 AIC Figure 4-11. Boxplot of AICs for al l models for a .10 proportion of zeros 168

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 4500.0000 4250.0000 4000.0000 3750.0000 AIC Figure 4-12. Boxplot of AICs for al l models for a .25 proportion of zeros 169

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 5000.0000 4500.0000 4000.0000 3500.0000 3000.0000 AIC Figure 4-13. Boxplot of AICs for al l models for a .50 proportion of zeros 170

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 3900.0000 3600.0000 3300.0000 3000.0000 2700.0000 2400.0000 2100.0000 AIC Figure 4-14. Boxplot of AICs for al l models for a .75 proportion of zeros 171

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NB ZIP ZIP NB Hurdle Hurdle NB Poisson Poisson Model 2400.0000 2200.0000 2000.0000 1800.0000 1600.0000 1400.0000 1200.0000 1000.0000 AIC 54,984 50,641 50,322 48,641 48,322 56,731 58,731 52,731 Figure 4-15. Boxplot of AICs for al l models for a .90 proportion of zeros 172

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0 1 2 3 4 5 6 7 0.10.250.50.750.9Rank Poisson NB Poisson Hurdle NB Hurdle ZIP NB ZIP Figure 4-16. AIC rank order for posi tively skewed distribution models 173

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0 1 2 3 4 5 6 7 0.10.250.50.750.9Rank Poisson NB Pois son Hurdle NB Hurd le ZIP NB ZIP Figure 4-17. AIC rank order fo r normal distribution models 174

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0 1 2 3 4 5 6 7 0.10.250.50.750.9 Proportion of ZerosRan k Poisson NB Pois son Hurdle NB Hurd le ZIP NB ZIP Figure 4-18. AIC rank order for nega tively skewed distribution models 175

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CHAPTER 5 DISCUSSION This research was conducted to determine differences in fit betw een six models under five conditions of zero-infla tion and three conditions of skew for the nonzero distribution.29 A Monte Carlo study was conducted in which samples of size 1,000 were sampled from distributions with prespecified proportions of zeros and prespeci fied proportions for the nonzero count levels. Six models were used to analy ze the samples with each simulated 2,000 times. Each model was analyzed with each combination of the five proportions of zeros combined with the three levels of skew. The research questions to be answered were as follows: Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated log-lik elihood between a) the Negative binomial Poisson model vs. Poiss on model; b) the Hurdle model vs. Poisson model?; c) the Negative binomial Hurdle model vs. negative binomial Poisson model?; d) the Negative binomial Hurdle model vs. Hurdle model; and, e) the Negative binomial ZIP model vs. ZIP model? Given one two-level categorical covariate with known values and one continuous covariate with known values, what is the difference in the estimated AIC between all models? The Impact of the Event Stage Distribution Positively Skewed Event-Stage Distributions The positively skewed event stage distribu tion most resembles that which is normally treated as Poisson or negative binomial Poisson. However, departure from these distributions is possible under different conditions of zero-inflation. Furt her departure may arise depending on the model used to analyze the data. For example, the Hurdle and ZIP formul ations treat the event stage as distributed Poisson while treating th e transition stage as distributed binomial. 29 All conclusions from this research are based on these conditions. Hence, conclusions are provisional as there may be other important determinations (e.g., parameter estimates, standard errors, power) not addressed in this research. 176

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As seen in Figure 5-1, the negative binomial Poisson model displayed better fit than the Poisson model only when the proportion of zeros was .50, .75, or .90. This suggests that overdispersion in a positively skew ed distribution arises in medium to high proportions of zeros. However, an extremely low proportion of zeros offsets this effect leadin g to equidispersion and adequate fit for the Poisson model. In other wo rds, the need for a negative binomial model arises as the proportion of zeros increases hence increasing zero-driven overdispersion. The comparison of the Poisson and Hurdle mo dels presented anomalous findings. At .10 zero-inflation, the Hurdle model displayed s uperior fit supporting its ability to handle zerodeflation. The same results were found at .50, .75, and .90 zero-inflation supporting the Hurdle models ability to accommodate zero-inflation. However, as seen in Figure 5-1, only 25.2% of the simulations favored the Hurdle model over the Poisson model when the proportion of zeros was .25. It is possible that the pr oportions for all counts yields a distribution that is distributed Poisson more than a distribution that tr eats the zeros as distributed binomial. The comparison of the negative binomial Poisson and negative binomial Hurdle yielded similar findings. However, this time it was at th e .50 proportion of zeros that fewer simulations (i.e., 13.3%) favored the more complex model. The reason for this is the addition of the overdispersion parameter for the negative binomial formulations of the model. This parameter accounts for zero-driven overdispersion; however, at .50 zeros, the fit of the overall distribution is reduced by treating the zer os as distributed binomial. The decrease of zero-driven overdispersion at .10 and .25 zeros and the increase of zero-driven overdispersion at .75 and .90 zeros then lend support to the nega tive binomial Hurdle model. This is illustrated in Figure 5-2. The negative binomial Hurdle consistently fit better than the Hurdle model until the proportion of zeros was .90. This can be seen in Figure 5-2; at this proportion, the percent of 177

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simulations favoring the negativ e binomial Hurdle model was 30.7%. For low proportions of zeros, this finding supports Agresti and Mins (2004) assertion that the Hurdle models can accommodate zero-inflation as well as zero-deflation. However, as the proportion of zeros increases, the negative binomial Hurdle would be expected to display better fit due to the increase in zero-driven overdisper sion. One explanation is that th e zero-driven dispersion at high proportions of zeros is counteracted by the reduction in samp le size for the event-stage. For example, if 90 of 100 responses are zero then ther e are certainly far fewe r counts contributing to event-stage model fit. The fit of the ZIP model formulations, which permit zeros in the event stage distribution, were affected by both the proportions of zeros and the positively skew ed distribution. At extremely high or low proportions of zeros (e.g., .10, .90), the majority of the simulations suggest adequate fit for the ZIP model. This is reasonable since, for a .10 proportion of zeros, the probability of a zero in the event stage is low; th e results become similar to that of the Poisson model at a .10 proportion of zeros. As was th e case when comparing the Hurdle and negative binomial Hurdle models, there are not enough zeros in either of the two parts of the model to necessitate a negative binomial formulation to ha ndle zeroand event-driven overdispersion. For a .90 proportion of zeros, a portion of zeros transfe rred to the event-stage l eads to more skew in the event-stage. In between these conditions, over dispersion arises due to the proportion of zeros justifying the negative binomial formulation. In fact, as seen in Figure 5-3, the negative binomial ZIP displayed significantly better fit than the ZIP model for al most 95% of the simulations when the proportion of zeros was .50. An alternative explanation to these findi ngs comparing ZIP and negative binomial ZIP model fit pertains to convergence. As displaye d in Table 4-42, there we re convergence problems 178

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for the ZIP models, especially when the proportion of zeros was 10. Hence, the comparisons are less stable since they are based on a lower number of simulations. This supports Min and Agrestis (2004) findings of supe rior convergence rates for the Hurdle models over the ZIP models. The AICs revealed inconsistent findings compared to those of the deviance tests. This is most likely due to the similarity in AICs offset by their penalty for the number of parameters in the models. For the nested models, the deviance test provides the more valid interpretation of results. The AICs can be useful if a) the res earcher is unsure of the nature of a dual data generating process or b) whether a model refl ecting a dual generating process is necessary. Given this data, the negative binomial Hurdle is superior to the ZIP models. However, the superiority of the other models appears to depend on the proporti on of zeros. At a low proportion of zeros, the Hurdle model was superior to the ZIP models; although, at a high proportion of zeros, the ZIP model was superior to the Hurdle models. There were several situations in which the Po isson models were supe rior to the two-part models suggesting that the two-pa rt models may not be necessary in some conditions. At a very low proportion of zeros (i.e., .10), the Poisson model was superior to both ZIP models. At the .25 proportion of zeros, the Poisson model was superi or to both the ZIP models and the Hurdle model. The negative binomial Pois son model is typically a poor c hoice over the two-part models and the Poisson model with one major exception. At a proportion of zeros of .50, the negative binomial Poisson model can be considered the best fitting model based on the penalty for model complexity provided by the AICs. The results are in line with the findings of Lambert (1992). At a .718 proportion of zeros, her results revealed that the ZIP model was supe rior to the negative binomial Poisson model, 179

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which was superior to the Poisson model. The present study found these results at a .75 proportion of zeros. Overall, the results imply th at zero-inflation and zer o-deflation are best accommodated for by using a negative binomial Hurdle model; however, when half of the data is zeros, the negative binomial Poisson model will suffice. Greene (1994), using a heavily positively sk ewed distribution with a .894 proportion of zeros found similar results for preferring the ne gative binomial ZIP model over the ZIP model. However, for the present study, at a proportion of zeros of .90, this was only true for .133 of the simulations; this may be due to the convergence problems. Mullahy (1986) found adequate f it for the Hurdle model only at a .26 proportion of zeros. At proportions of .41 and .62, the fit was no l onger adequate. This is in agreement with the findings of the present research. At low proportions of zeros with a positively skewed distribution and in terms of AICs, the Hurdle model displayed fit only improved by the negative binomial Hurdle model. However, the fit for the Hurdle model became worse as the proportion of zeros increased. Overall, these results provide suggestions for analyzing data with various proportions of zeros and a positively skewed event-stage distribution. First, if there is no reason to justify a twopart model then it is not necessa rily prudent to assume a nega tive binomial Poisson distribution to accommodate zero-driven overdispersion. The r esults of this study suggest that the negative binomial Poisson model is superior to the Poi sson only when the proportion of zeros is greater than or equal to .50. Second, if justifying separate processes generati ng the zeros and nonzeros then both the Hurdle and nega tive binomial Hurdle should be analyzed and compared at proportions of zeros around .90 since here there appears to be a drop in the need for the negative binomial Hurdle model. Third, it appears that the need for the negative binomial ZIP model 180

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declines as the proportion of zeros increases from .50 to .90. Howe ver, all interpretations should be treated with caution due to the severe convergence pr oblems at the .10 and .25 proportion of zeros conditions. Hence, if the researcher ac knowledges a dual data generating process but is uncertain about the source, the ZIP models are mo re valid since they dont exclude zeros from the event-stage that may be present. However, ther e is a greater probability that the analysis will not converge. This again supports Min and Agrestis (2004) assertion of Hurdle model superiority in the condition of zero-deflation. Finally, under cert ain conditions, two-part models arent necessary. In particular a Poisson model may fit as well as a Hurdle model at .25 zeros, and a negative binomial Poisson model may fit as well as a negative binomial Hurdle model at .50 zeros. Normal Event-Stage Distributions Regardless of the proportion of zeros, the Hu rdle model fits best when the event stage distribution is normal. The negative binomial Hurdle is not necessary. However, the results should be interpreted cautiously approximately since, as displayed in Table 4-43, 25% of simulations failed to converge in the .25 and hi gher proportion of zeros conditions. This is an interesting finding since previous research (Min & Agresti, 2 004) and these findings displayed convergence problems for the posit ively skewed distribution. If opting for a two-part model in which zeros are permitted in the event stage, the superior model is the ZIP model. Hence, the statistical analyses suggest that, regardless of the proportion of zeros, negative binomial formulations are no t necessary for both the ZIP and Hurdle models. If a two-part model is not n ecessary then the model depends on the proportion of zeros. As was the case with a positively skewed distribut ion, negative binomial Poisson models are not necessary when the proportion of zeros is .10 or .25. It is at proportions of .50, .75, and .90 zeros that the negative binomial Poisson model fits the data better than the Poisson model. 181

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It is important to note that model superiority do es not inform researchers as to the extent of that superiority. The descriptive AIC values sugg est better fit for the Hurdle over the other twopart models; however, the AICs for the other tw o-part models do not differ by much. This is most prevalent at the .10 proportion of zeros condition; the relative ad vantage of the Hurdle model over the other two-part m odels declines as the proportion of zeros increases. At the .90 proportion of zeros condition there is barely any distinction between the four two-part models at all. The discrepancy between Poisson model a nd negative binomial Poisson models is consistently large across proporti ons of zeros. The only condition wh ere they approach the AICs of the two-part models is at the .10 proportion of zeros condition. Over all interpretations can then be made. First, the Hurdle model is the be st-fitting model of all two-part models; however, the extent of this fit is great est at low proportions of zeros. Second, negative binomial Poisson models are advantageous at medium to high propo rtions of zeros. Fina lly, when unsure about whether to use a two-part model, a Poisson model will suffice at low proportions of zeros. Slymen, Ayala, Arredondo, and Elder (2006) compared the Po isson model to the negative binomial Poisson model and the ZIP model to th e negative binomial ZI P model. The proportion of zeros was .824, and the event stage distri bution was uniform. The present study, at a .90 proportion of zeros, has an event stage distribution that appears uniform given the reduction in event stage frequencies from a possible 1,000 to 100. The results be tween the studies were similar in that the negative binomial model is su perior to the Poisson model. Also, similar to their findings, there was no drastic distinction between the ZIP and negative binomial ZIP models at this proportion of zeros. 182

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Bhning, Dietz, Schlattmann, Mendona, a nd Kirchner (1999) compared the Poisson and ZIP models under two proportions of zeros both close to .25 and w ith normally distributed event stages. There results are similar to those of the present study. The ZIP model is superior to negative binomial ZIP. However, this isnt surpri sing since the ZIP mode is consistently superior to the negative binomial ZIP model regardless of the proportion of zeros. Only when adjusting for model complexity via AICs do the model become more similar in fit. Table 4-38 summarizes these findings. Negatively Skewed Even t-Stage Distributions No previous research was found in which th e event stage distribution was negatively skewed. It is possible that resear chers begin with the notion of zer o-inflation, which, due to the count nature of the data, lead s to considering the Poisson di stribution. Introductory texts introduce a Poisson distribution with a low me an yielding a positively skewed distribution. Hence, negatively skewed zero-bounded dist ributions receive lit tle consideration. Distributions displaying both zero-inflation and a negatively skewed event stage are certainly feasible. For example, the response vari able may be students number of days absent from an honors course. A proportion of these students should have no absences. Beyond zero, there may be a relationship between enrolling in the honors course and participating in extracurricular activities requiring absences. Thes e students would then have more absences than the others leading to a negatively skewed distribution. The negative binomial Poisson model displaye d significantly better model fit than the Poisson mode for all conditions except when th e proportion of zeros was .10. The Hurdle model fit better than the Poisson model, and the nega tive binomial Hurdle model fit better than the negative binomial Poisson model for all simulations. The negative binomial Hurdle model fit better than the Hurdle model for all simulations except when the proportion of zeros was .75. In 183

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this condition, the negative binomial Hurdle model fit better than the Hurdle model for 90.7% of the simulations. As was found for the normal dist ribution conditions, conclusions regarding the negative binomial Hurdle model should be tr eated cautiously as convergence failures were approximately 25% over all proportion of zeros conditions. Finally, the negative binomial ZIP model never displayed significantly better fit than the ZIP model. These findings are interesting in that, although the proportion of ones is approximately four times the proportion of zer os, the overall distri bution (including the zeros) is negatively skewed. In other words, the serious zero-defla tion may be disguised as part of one negatively skewed distribution. Hence, the implications of these results apply to zero-bounded data that appear to be the inver se of the typical Poisson distribution. The negative binomial Hurdle model could be used to analyze such a distribution pr ovided that the zeros ca n be viewed as arising from a completely different mechanism than the nonzeros. However, the simpler ZIP model could be used if there is reason to consider tw o zero generating processes. Both of these models provide better fit over the Poisson. No previous research was found to address a ne gatively skewed event stage distribution. However, there did appear to be one incongrue nt finding. Several arti cles reported that the Hurdle was advantageous over the ZIP model for zero-deflation since, wh ile both give nearly equivalent results, the ZIP model often has problems with convergence (Min & Agresti, 2004; Min, 2004). For the negatively sk ewed distribution as well as for the normal distribution, the negative binomial ZIP models failed to converge only once out of a total of 20,000 simulations while the Hurdle models failed to converge 3,899 tim es. This is seen in Ta ble 4-44. It is possible that the convergence advantage for the Hurdle m odels only applies to a positively skewed eventstage distribution. However, the present re search only found convergence problems for the 184

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Hurdle model at low proportions of zeros with 112 successful convergences out of 2,000 for the .10 proportion of zeros and 1,850 successful conver gences out of 2,000 for the .25 proportion of zeros. Summary of Findings One of the most interesting findings pertai ned to the Poisson and negative binomial Poisson models. When the event-stage distribution is positively skewed or normal, the negative binomial Poisson model is not necessary until the proportion of zeros is .50. For the negatively skewed distribution, the negative binomial Pois son is not necessary when the proportion of zeros is .10. The results are not surprising or the positively skewed and normal distributions; the proportion of zeros is so low that the event-stage distribut ion not overdispersed. For the negatively distribution, the inte rpretation is suspect since, ig noring the zeros, the event-stage distribution has no resemblance to the Poisson di stribution. However, at least for the positively skewed and normal distributions, these results present some guidelines for selecting either the Poisson or more complicated negative binomial Poisson model. Another interesting finding pertains to the Hu rdle and negative binomial Hurdle models. For the positively skewed distribution, the negative binomial Hu rdle model should be chosen regardless of the proportion of zeros. This was also true for the negatively skewed distribution. However, for the normal distribution, the more complicated negative binomial Hurdle model is not necessary. This provides a guideline for c hoice between the Hurdle and negative binomial Hurdle models for the distributions if willing to trust the statistical conclusion validity of 75% successful convergences for the normal and pos itively skewed distribution comparisons. Finally, this research showed that the nega tive binomial ZIP model is never necessary when the event-stage distribution is normal or negatively skewed. In fact, even for the positively skewed distribution, it may not be necessary except at .10, .25, a nd .90 proportions of zeros. The 185

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negative binomial ZIP model appear s to be superior when the even t-stage distribution is positive and when there is moderate to moderately-hi gh zero-inflation but not extreme zero-inflation. Limitations Discrete Conditions The results generated by this Monte Carlo si mulation derive from prespecified discrete conditions. There were five presp ecified proportions of zeros and three prespecified levels of skew. The former was fixed for each dataset, and the latter was de termined by randomly selecting from prespecified pseudo-population proba bilities for each of the five count levels. The covariates were fixed as one continuous va riable distributed standard normal and one categorical variable with two levels each at n = 500 and alternating cons ecutively in the dataset. The interpretation of results is limited to these discrete conditions. For example, for the negatively skewed event count distribution, the Poisson model fit better than the negative binomial Poisson model at a .10 proportion of ze ros. However, the reverse was true at a .25 proportion of zeros. This is not to say that the Poisson model has better fit until the proportion of zeros is .25. It could be the case that the negative binomial Poisson has better fit at a proportion of .11 zeros. Further, the results are generalizable to the extent that ap plied research maintains the same proportions for event stag e counts (i.e., skew) and covariat e characteristics as used in this study. Convergence and Optimization Optimization is a mathematical process of finding the best solution to a function. Statistically, optimization is achieved through an iterative process until convergence is reached resulting in the optimum values. This topic was previously discu ssed in the context of maximum likelihood estimation. 186

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Different optimization procedures can produce different results and different rates of convergence. The Hurdle model requires complex op timization since there are two models to be optimized. The ZIP model is even more complex since the two models are not independent of one another. The negative binomial ZIP model is most complex since the event stage distribution requires the addition of a dispersion parameter. There is no perfect optimizati on procedure that finds the best solution within the most reasonable amount of time for all sets of data (SAS Institute, 2000). For the SAS programming language, these zero-inflated models would typica lly be analyzed using PROC NLMIXED. The default optimization procedure here is th e Quasi-Newton method which does not require computation of second-order derivatives. Hence, it has the advantage of finding solutions rather quickly for moderately large pr oblems. However, it does not consider the boundary constraints present in the zero-inflated data In fact, the Quasi-Newton met hod is not even included as an option in the R procedures for the Hurdle and ZI P models. Here, the default is the Nelder-Mead Simplex Optimization. This method does not require any derivatives and permits boundary constraints. Its process increases speed of convergence while large sample sizes (e.g., N > 40) increase convergence time and decrease the probab ility of generating a precise result (SAS Institute, 2000). This may then explain some of the discrepancy in fi ndings between this study and that of Min and Agrest i (2004) and Min (2004) who se studies used the NLMIXED procedure with its default Qu asi-Newton optimization method. Underdispersion Strictly speaking, conditions such as a .10 proportion of zeros with a negatively skewed event stage distribution can lead to a variance that is less than its mean. As such, it is expected that the models outperform their negative bi nomial counterparts. However, a dispersion 187

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parameter that accounts for underdispersion may reveal a model that has superior fit to these models. Further, the results are likely to vary be tween the Poisson, Hurdle, and ZIP formulations. Interpreting Model Fit This research used the deviance statistic and Akaikes Information Criterion to examine fit between different models. Th ere are certainly othe r alternatives. Vuong (1989) proposed an inferential likelihood ratio test for comparing nonnested models. Alternatives to the AIC include the Bayesian Information Criterion, which is ty pically used for unbalanced sample sizes and the Schwarz Criterion, which is typica lly used for hierarchical models. Other models ZIP and Hurdle models have become th e most common models for analyzing zeroinflated data. However, as previ ously discussed, there are many othe r models to choose. Some of these include Lamberts (1989) ZIP( ) to accommodate correlation between the zeros and nonzeros, other links functions for the eventstage (e.g., log-gamma, log-log), and Ridout, Demtrio, & Hindes (1998) method for avoidi ng some of the distributional assumptions. Another potential model to analyze zero-inflate d count data is the latent class model. The model is similar to a general linear model in that one may posit group membership. However, in latent class modeling, the groups are unobserved. Hence, it is able to provide estimates for potential classes that may explain heterogeneity in responses (Ding, 2006). It may be very useful for researching zero-inflated count data due to the two potential sources of zeros. In fact, dUva (2006) proposed a Hurdle variation for latent cl ass modeling in health care utilization research. Validity of Model-Fitting and Model-Comparisons The purpose of comparing models is to determ ine which has the best fit. This is done by selecting the model that minimizes th e loss of information. As previ ously mentioned, this is often done by comparing statistics such as the log-li kelihood, deviance, and Akaikes Information 188

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Criterion (AIC). However, any model will fit perfectly or nearly perfectly given enough parameters (Lord, Washington, & Ivan, 2006). Furt her, there is always the possibility that a different set or combination of parameters will yield the same fit statistics. When comparing the Poisson and zero-inflation models, it is important not to lose sight of validity concerns. Ultimately, the best model will not be the one with the best fit. Rather, it will be the one that leads to correct inferences, inte rpretations, and decisions. Although one may not know the mo del specification that will l ead to enhanced statistical conclusion validity, it is possible to maintain several inferential principles as clarified by Burnham and Anderson (2001) and re viewed by Mazerolle (2004). Firs t, the ideal model will be simple and parsimonious in line with Ockhams Razo r, which states that, a ll else being equal, the simplest model is usually the correct model. The use of Akaikes Information Criterion as a method for adjusting for model complexity is one example of striving for parsimony (Lord, Washington, & Ivan, 2006). Another example is favoring a simpler model over a more complex model when the fit is nearly identical: Our ch oice of the ZIP model over the [negative binomial ZIP] model was based on parsimony since they provide similar fit. The ZIP model does not include the random error term that allows the conditional variance of y to exceed the conditional mean. But the interpretations of the model regr ession parameters are the same for both models (Slymen, Ayala, Arredondo, & Elder, 2006, p.8). Second, the researcher should ma intain several hypotheses noting that there is no perfectly true model. Third, since validity is a process of accumulating evidence, conclusions should be substantiated by continuing efforts to confirm the strength of evidence using various techniques such as replication and cross-validation. 189

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Suggestions for Future Research The results of this study provide ample sugge stions for future research. As previously mentioned as limitations, it is suggested that future research consider ot her proportions of zeros and event stage distributions, underdispersion ad justments, different optimization procedures, and other models. This research should also be extended to ordinal data by developing a model that combines the ZIP and Hurdle models w ith a cumulative model (e.g., proportional odds model). The research should also be extended to latent outcomes possibly by specifying a probit distribution. Although recen t research has been extended to incorporate random effects (e.g., Min & Agresti, 2005; Hall & Zhang, 2004; Hall, 2004; Hall, 2002; Olsen, 1999), these findings should be examined in the context of varying pr oportions of zeros and ev ent stage distributions Application in Educational Research Models for zero-inflation have been applied in a wide variety of fi elds from the health sciences to economics to transportation. Howeve r, no research was found to apply models for zero-inflation to the field of education. There ar e plenty of opportunities to analyze data with such methods (e.g., retention, subs tance abuse, disciplinary action). In an era of high-stakes accountability, it is imperative that educational researchers use the most sophisticated methods available. This is especially important si nce the results of these studies lead to decisions direc tly affecting human lives. It wo uld be fallacious to treat zeroinflated count data as purely continuous analy zed via ordinary least squares regression. A higher level of sophistication would l ead to specification of a Poiss on distribution with a log link. However, this would result in responses of zero included with the other responses leading to incorrect inferences due to st ructural and/or ra ndom reasons for the responses of zero. It would be more judicious to first consider the nature of th e zeros and then consider the use of the two-part models. Unfortunately, the application of sophisticated methods tends to 190

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become more pervasive as the methods become si mple to apply. As such, educational research programs should promote the existence of these methods, and software programs should provide prepackaged functions for ease of application. Major Contribution of Findings The most important finding of this dissertation was that the results of the simulation were in agreement with previous fi ndings when the proportion of zeros and event stage distribution were similar. Hence, anomalous findings between previous research ca n be attributed to differences in proportions and differences in the event stage distribution. The intention is that researchers become more aware of how different proporti ons and distributions affect the findings when comparing models such as the Poisson, ZIP, and Hurdle models. For example, as discussed, the negative binomial Poisson model can be superior to the two-part models (e.g., positive event stage distribution with a .50 proporti on of zeros), and the relative fit of the ZIP model and negative binomial ZIP model can depe nd strongly on the proporti on of zeros (e.g., for a negative event stage distribution). Ultimately, the choice should be guided by the data generating process. Data with purely structural zeros should not permit zeros in the even t stage, data with stru ctural and random zeros should not have those zeros completely separate from the counts. The findings of this dissertation help guide researchers when uncertain or when questio ning the need for a particular model 191

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-10 10 30 50 70 90 110 0.100.250.500.750.90 Proportion of Zeros% Simulations Favoring Complex Model P vs NBP P vs H Figure 5-1. Poisson, NB Poisson, and Hurdle over all proportions of zeros 192

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-10 10 30 50 70 90 110 0.100.250.500.750.90 Proportion of Zeros% Simulations Favoring Complex Model H vs NBH NBP vs NBH Figure 5-2. Hurdle, NB Hurdle, and NB Poisson over all proportions of zeros 193

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-10 10 30 50 70 90 110 0.100.250.500.750.90 Proportion of Zeros% Simulations Favoring Complex Model Z vs NBZ H vs NBH Figure 5-3. ZIP, NB ZIP, Hurdle, and NB Hurdle over all proportions of zeros 194

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BIOGRAPHICAL SKETCH Jeffrey Monroe Miller was born in Greensboro, NC and raised in Jacksonville, FL. He earned a Bachelor of Arts degree from the Univ ersity of Wisconsin Eau Claire in 2001 majoring in Psychology with two minors in Music. As an undergraduate, he became fascinated by the methods used to analyze data. In 2002, he en rolled in the Research and Evaluation Methodology program at the University of Florida in the De partment of Educational Psychology (College of Education). In 2004, after author ing and defending a thesis pertai ning to item-order effects in surveys, he received the Master of Arts in Education degree in this program. In 2007, he completed the requirements for the degree of Doctor of Philosophy. 201