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The effect of multidimensionality on unidimensional equating with item response theory

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THE EFFECT OF MULTIDIMENSIONALITY ON
UNIDIMENSIONAL EQUATING WITH
ITEM RESPONSE THEORY












By

PATRICIA DUFFY SPENCE


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


1996




THE EFFECT OF MULTIDIMENSIONALITY ON
UNIDIMENSIONAL EQUATING WITH
ITEM RESPONSE THEORY
By
PATRICIA DUFFY SPENCE
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
1996


This dissertation is dedicated to the memory of my father
James F. Duffy
1929- 1992


ACKNOWLEDGMENTS
An effort of this magnitude always involves many people. The author
wishes to especially thank the chairman of her committee, Dr. M. David Miller,
for his dedication and inspiration. Without his encouragement and good humor,
this dissertation would not have been possible. The author would also like to
thank her committee members for their guidance and patience, particularly Dr.
James Algina and Dr. Linda Crocker. Their suggestions were always correct, if
not always accepted. Also, without the inspiration of Dr. Charles Dziuban of the
University of Central Florida, she would never have pursued studies in this field.
In addition, the author recognizes her colleagues, past and present, at
The Psychological Corporation, Volusia County District Schools, and the Florida
Department of Education for the opportunities to apply her learning in practical
situations. Gratitude is offered to her three parents-Jim, Joan, and Jeanne-
who stressed the importance of learning and doing things well. Thanks also to
special friends: Anne Seraphine for debating the meaning of life and monotonic
curves; Nada Stauffer for quiet friendship; George Suarez for making her laugh;
and Carlos Guffain for demanding her best. But the author is most indebted and
grateful to her husband, Verne, who has supported and encouraged her through
three degrees, and her daughter Cindy who is now left to carry on the Gator
tradition alone.


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES xi
ABSTRACT xii
CHAPTERS
1 INTRODUCTION 1
Purpose 3
Limitations 4
Significance of the Study 4
2 REVIEW OF LITERATURE 6
Test Equating 6
Conditions for Equating 6
Data Collection Designs 7
Single-group designs 7
Equivalent-group designs 9
Anchor-test designs 10
Equating Methods 14
Conventional Methods of Equating 14
Linear equating 16
Equipercentile equating 18
Equating Methods Based on Item Response Theory 21
Item response theory 21
IRT equating 28
IV


Multidimensionality 35
Violation of the Unidimensionality Assumption 35
Multidimensional Models 37
Multidimensionality and Parameter Estimation 45
Multidimensionality and IRT equating 52
3 METHOD 58
Purpose 58
Introduction 58
Research Questions 58
Data Generation 59
Design 59
Model Description 60
Item Parameters 61
Response Data 63
Noncompensatory Data 65
Nonrandom Groups 66
Estimation of Parameters 66
Unidimensional IRT 66
Analytical Estimation 69
Equating 69
Concurrent Calibration 70
Equated bs 70
Characteristic Curve Transformation 71
Evaluation Criteria 73
Comparison Conditions 73
Statistical Criteria 75
Summary 76
4 RESULTS AND DISCUSSION 78
Simulated Data 78
Item Parameters 78
Analytical Estimation 88
Simulated Ability Data 88
Equating Results for Randomly Equivalent Groups 92
Concurrent Calibration 92
Equated bs 99
Characteristic Curve Transformation 103
Equating Results for Nonequivalent Groups 103
Concurrent Calibration 103
Equated bs and Characteristic Curve Transformation 108
v


5 CONCLUSIONS 111
Effects of Multidimensional Model 111
Effects of Equating Method 112
Effects of the Number of Multidimensional Items 112
Effects of Nonequivalent Examinee Groups 115
Implications 116
APPENDIX
ITEM PARAMETER DATA 118
REFERENCES 151
BIOGRAPHICAL SKETCH 158
VI


LIST OF TABLES
Table oaae
1 Summary of Recommendations for a Successful Equating 15
2 Summary of Unidimensional IRT Test Equating Studies 36
3 Summary of Studies of Unidimensional IRT Estimation
with Multidimensional Data 50
4 Summary of Studies of Unidimensional Equating with
Multidimensional Data 57
5 Simulated Compensatory Parameters for MD30, Form A 64
6 Simulated Noncompensatory Parameters for Multidimensional
Items, MD30 Form A 67
7 Summary Statistics for Multidimensional Items in Compensatory
and Noncompensatory Datasets 68
8 Summation of Research Equating Conditions 72
9 Analytical Estimates of the Unidimensional Parameters for
Compensatory MD30, Form A 74
10 Descriptive Statistics for Compensatory Form A Item Parameters ...79
11 Descriptive Statistics for Compensatory Form B Item Parameters 80
12 Descriptive Statistics for Multidimensional Item Parameters in
Noncompensatory Form A 81
13 Descriptive Statistics for Multidimensional Item Parameters in
Noncompensatory Form B 82
14 Descriptive Statistics for Analytical Unidimensional Estimates of
Form A Item Parameters 89
vii


15 Summary Statistics for Analytical Unidimensional Estimates of
Form B Item Parameters 90
16 Descriptive Statistics for Simulated Examinees Taking MD10 91
17 Descriptive Statistics for Simulated Examinees Taking MD20 92
18 Descriptive Statistics for Simulated Examinees Taking MD30 93
19 Descriptive Statistics for Simulated Examinees Taking MD40 94
20 Descriptive Statistics for Simulated Low Ability Examinees 95
21 Summary of Concurrent Calibration Results with Randomly
Equivalent Groups 96
22 Constants for Equated bs Equating of Compensatory Forms with
Randomly Equivalent Groups 100
23 Constants for Equated bs Equating of Noncompensatory Forms
with Randomly Equivalent Groups 101
24 Summary of Equated bs Results with Randomly Equivalent
Groups 102
25 Summary of Characteristic Curve Transformation Results with
Randomly Equivalent Groups 104
26 Summary of Equating Results with Nonequivalent Groups 106
27 Constants for Equated bs Equating of Compensatory Forms with
Nonequivalent Examinee Groups 109
28 Simulated Compensatory Item Parameters for MD10 Form A 119
29 Simulated Compensatory Item Parameters for MD10 Form B 120
30 Simulated Compensatory Item Parameters for MD20 Form A 121
31 Simulated Compensatory Item Parameters for MD20 Form B 122
32 Simulated Compensatory Item Parameters for MD30 Form A 123
33 Simulated Compensatory Item Parameters for MD30 Form B 124
viii


34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
125
126
.127
128
129
130
131
132
133
134
135
136
137
138
139
140
Simulated Compensatory Item Parameters for MD40 Form A...
Simulated Compensatory Item Parameters for MD40 Form B...
Noncompensatory Item Parameters for Multidimensional Items
in MD10 Forms A and B
Noncompensatory Item Parameters for Multidimensional Items
in MD20 Form A
Noncompensatory Item Parameters for Multidimensional Items
in MD20 Form B
Noncompensatory Item Parameters for Multidimensional Items
in MD30 Form A
Noncompensatory Item Parameters for Multidimensional Items
in MD30 Form B
Noncompensatory Item Parameters for Multidimensional Items
in MD40 Form A
Noncompensatory Item Parameters for Multidimensional Items
in MD40 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD10 Form A
Analytical Estimates of Unidimensional Item Parameters for
MD10 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD20 Form A
Analytical Estimates of Unidimensional Item Parameters for
MD20 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD30 Form A
Analytical Estimates of Unidimensional Item Parameters for
MD30 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD40 Form A
IX


50 Analytical Estimates of Unidimensional Item Parameters for
MD40 Form B 141
51 Descriptive Statistics for Compensatory MD10 Linking Items
with Randomly Equivalent Groups 142
52 Descriptive Statistics for Compensatory MD20 Linking Items
with Randomly Equivalent Groups 143
53 Descriptive Statistics for Compensatory MD30 Linking Items
with Randomly Equivalent Groups 144
54 Descriptive Statistics for Compensatory MD40 Linking Items
with Randomly Equivalent Groups 145
55 Descriptive Statistics for Noncompensatory MD10 Linking Items.. 146
56 Descriptive Statistics for Noncompensatory MD20 Linking Items... 147
57 Descriptive Statistics for Noncompensatory MD30 Linking Items... 148
58 Descriptive Statistics for Noncompensatory MD40 Linking Items... 149
59 Descriptive Statistics for Compensatory Linking Items with
Nonequivalent Groups 150
x


LIST OF FIGURES
Figure page
1 An item characteristic curve (ICC) based on the three-
parameter logistic model 23
2 An item response surface (IRS) based on the compensatory
M2PL 40
3 Item response surfaces and contour plots for item 9, MD20,
a = 20 84
4 Item response surfaces and contour plots for item 10, MD20,
a = 30 85
5 Item response surfaces and contour plots for item 11, MD20,
a = 45 86
6 Item response surfaces and contour plots for item 12, MD20,
a = 60 87
xi


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
THE EFFECT OF MULTIDIMENSIONALITY ON
UNIDIMENSIONAL EQUATING WITH
ITEM RESPONSE THEORY
by
Patricia Duffy Spence
May, 1996
Chairman: M. David Miller
Major Department: Foundations of Education
Test publishers apply unidimensional equating techniques to their products
even though tests are expected to be multidimensional to some degree. This
simulation study investigated the effects of ignoring multidimensional data in
applying unidimensional item response theory equating procedures. The
specific effects studied were (a) multidimensional model, (b) type of equating
procedure, (c) number of multidimensional items, and (d) distribution of
examinee ability.
Four test conditions were created by varying the number of multidimensional
items contained in each test. The compensatory multidimensional two-
parameter logistic model was selected for data generation. Four degrees of


multidimensionality were spiraled throughout each test. The data were then
transformed into corresponding noncompensatory items which had the same
probability of success as the compensatory item for a given examinee.
Four tests with 40 items each were simulated with 12 common linking items
and 28 unique items. For each experimental condition and form, responses for
1,000 simulees were generated. To examine the effects of nonrandom groups,
responses for 1,000 less able examinees were also generated.
Three unidimensional IRT equating methods were selected: (a)
concurrent calibration, (b) equated bs, and (c) characteristic curve
transformation. Parameters were calibrated with BILOG386. To evaluate the
results of the research equatings, three comparison conditions were used; (1)
the unidimensional approximations of the multidimensional item parameters
calculated using an analytic procedure; (2) the simulated first ability dimension
only; and (3) the averages of the two simulated abilities. Three statistical
criteria-correlation, standardized differences between means, and standardized
root mean square difference--were applied to the data.
No significant effect on the unidimensional equating results were
attributed to choice of multidimensional model. For randomly equivalent groups,
there was also no effects due to choice of equating procedure. Concurrent
calibration favored low ability examinees when the ability distributions of the two
groups were unequal. When the multidimensional composites described by the
analytical estimation baseline are the data of interest, the number of
xiii


multidimensional items had little effect on the unidimensional equating with
randomly equivalent, normally distributed examinee groups. However, if the
unidimensional factor is the trait of interest, the number of multidimensional
items affected the equating outcomes, with results deteriorating as the number
of multidimensional items increased. When examinee groups were not
equivalent, equating results were affected in all conditions. Caution is advised
in applying unidimensional equating procedures when the examinee groups are
suspected of being from different ability levels.
XIV


CHAPTER 1
INTRODUCTION
In many large testing programs, examinees take one of multiple forms of
the same test. Although the different editions are constructed to be as similar in
content and difficulty as possible, it is inevitable that some differences will exist
among the various forms (Petersen, Cook, & Stocking, 1983). Direct
comparison of scores would, therefore, be unfair to an examinee who
happened to take a more difficult form. Because examinees are often in
competition or are being directly compared, it is important to transform the
scores in some way to make them equivalent.
Equating is the statistical process of establishing equivalent raw or scaled
scores on two or more test forms. Theoretically, the equating process adjusts
for test and item characteristics so the propensity distributions would be the
same regardless of which test form was administered. The application of
equating to real data, however, can be full of problems and complications
(Skaggs & Lissitz, 1986a). In practice, equating requires not only a knowledge
of statistical models, but awareness and consideration of many other issues that
have practical consequences for the use and interpretation of results. Brennan
and Kolen (1987) discussed many of these issues, such as the presence of
equating errors, specification of content, and security breaches.
1


2
Many mathematical procedures have emerged to develop the equating
transformations. Some are based on classical test theory while others arise
from item response theory (IRT). Classical methods, including linear and
equipercentile equating, do not seem robust to departures from optimal
conditions (Cook & Eignor, 1983; Livingston, Dorans, & Wright, 1990; Skaggs &
Lissitz, 1986b). Item response theory procedures, including equated bs,
concurrent calibration, and characteristic curve transformation, present
alternatives. Equating methods based on IRT have been found more accurate
than those based on classical models (Harris & Kolen, 1985; Hills, Subhiyah, &
Hirsch, 1988; Kolen, 1981; Marco, Petersen, & Stewart, 1983; Petersen, Cook,
& Stocking, 1983).
IRT models are grounded on strong assumptions, particularly that the
item responses are unidimensional (Ansley & Forsyth, 1985). The
unidimensionality assumption requires that each of the tests to be equated
measures the same underlying ability. Any other factor that influences an
examinees score-such as guessing, speededness, cheating, item context, or
instructional sensitivity-will violate the unidimensionality assumption. Some of
these violations can be controlled, reduced, or eliminated, but the
unidimensionality assumption will still be violated in many practical testing
situations (Doody-Bogan & Yen, 1983).
Attempts have been made to model multidimensional responses within
the framework of IRT. Although these models describe multidimensional data
more accurately than unidimensional models, estimation of parameters is


3
complex and difficult in practice (Harrison, 1986). Test companies continue to
apply unidimensional equating procedures to their products. The viability of
using unidimensional models with multidimensional data must be explored to
determine the effect on the equating outcomes. An understanding of what effect
multidimensional data have on unidimensional equating results Is of paramount
importance. Empirical studies (Camilli, Wang, & Fesq, 1995; Cook & Eignor,
1988; Dorans & Kingston, 1985; Yen, 1984) indicate that violation of the
unidimensionality assumption, while having some impact on results, may not be
significant. However, each of these studies employed data from a different test
and their content may have influenced findings in an unknown manner. The
number of multidimensional items and the degree of multidimensionality in each
is also unknown. Therefore, the generalization of results are difficult to interpret
across studies (Skaggs & Lissitz, 1986a). It is necessary to design research
studies that permit manipulation of independent variables to understand exactly
how violations of the unidimensionality assumption affect equating. Simulation
studies present a technique to manipulate and control the desired variables.
Purpose
The purpose of the present study was to investigate the effect of
multidimensional data in applying unidimensional IRT equating techniques. The
specific questions to be answered were:
1. Does the number of multidimensional items affect unidimensional
equating results?


4
2. Does the equating procedure affect unidimensional equating
results?
3. Do data simulated by using a compensatory model produce
different unidimensional equating results than data simulated by using a
noncompensatory model?
4. Are unidimensional equating results affected by the ability
distribution of the two examinee groups?
Limitations
Results of this study are applicable only to the research conditions
investigated. Generalizations to other item response theory models or other
equating techniques are not justified.
Significance of the Study
In practice, test publishers today apply unidimensional equating
techniques to their products. Because tests are expected to be
multidimensional to some degree and it is difficult to identify multidimensionality
accurately, it is important to investigate the effect of applying unidimensional
equating techniques to multidimensional data. Previous studies have mainly
explored unidimensional equating with empirical data that was suspected of
being multidimensional. Although the results indicated the impact of violating
the unidimensionality assumption may not be significant, the research designs
did not allow manipulation of independent variables. In addition, the true
multidimensionality of the underlying data was unknown in these empirical
studies.


5
The current simulation study allowed exploration of what effect
multidimensionality had on the results obtained from a variety of unidimensional
equating procedures while providing a means to manipulate variables. The
techniques used to generate the data afforded a mechanism to control the
dimensionality of the items and test forms. The specific questions investigated
were selected as having the most value for current practitioners applying
unidimensional equating procedures.


CHAPTER 2
REVIEW OF LITERATURE
Test Equating
Conditions for Equating
The purpose of equating is to establish a relationship between two test
forms so that it becomes a matter of indifference to the examinee which form is
taken. Petersen, Kolen, and Hoover (1989) stated that equating itself is simply
an empirical procedure which imposes no restrictions on the properties of scores
or on the method used to define the transformation. It is only when the purpose
of equating and the definition of equivalent scores are considered that restrictions
become necessary.
Lord (1980) outlined four conditions that must be met for the successful
equating of two test forms, X and Y. Briefly, the conditions are (a) equity, (b)
population invariance, (c) symmetry, and (d) same ability. To satisfy the equity
condition, it must make no difference to examinees at every ability level, 9, which
form of the test is taken. The conditional frequency distribution fxf) of the score
on form X should be the same as the conditional frequency distribution of the
transformed form Y score, fX(y)i- Lord (1980) added that it is not sufficient for
equity that fx p and fx(y¡ 10 have the same means, but they must also have equal
variances. If the tests are not equally reliable, it is no longer a matter of
6


7
indifference which form is administered. The equity condition requires the
standard error of measurement and the higher moments to be the same after
transformation for examinees of identical ability. To fully satisfy this requirement,
test forms X and Y must be strictly parallel (Kolen, 1981). However, if this
condition is met, equating is no longer necessary.
In practice, it is nearly impossible to construct multiple forms that are
strictly parallel. Therefore, equating is needed. Although the equity condition can
never be met precisely, it serves to keep the purpose of equating in mind and
guide the steps in the process.
The population invariance and symmetry conditions also arise from the
desire to achieve equivalent scores. If the scores from form X and form Y are
equivalent, there is a one-to-one relationship between the two sets of scores.
The transformation must be unique, independent of the groups used to derive the
conversion (Petersen et al., 1989). The purpose of equating also requires that
the equating function be invertible or symmetric. The equating must be the same
regardless of which test is labelled X and which test is labelled Y (Lord, 1980).
The two tests to be equated must also measure the same characteristic,
whether defined as a latent trait, ability, or skill. This condition distinguishes true
equating from scaling. Scores on X and Y can always be placed on the same
scale, but they must measure the same construct to be considered equated
(Dorans, 1990).


8
It is unlikely that all conditions of equating can be met in practice.
However, good approximations to this ideal can be achieved and are usually
fairer to examinees than if no attempt at equating had occurred (Petersen et al.,
1989). Research conducted over the past 20 years serves as a guide in the
application and interpretation of equating transformations.
Data Collection Designs
Every equating consists of two parts--a data collection design and an
analytical method to determine the appropriate transformation. Three basic
sampling designs are most frequently described in the literature (Dorans, 1990;
Dorans & Kingston, 1985; Petersen et al., 1989). The designs are classified as
(a) single-group designs, (b) equivalent-groups designs, and (c) anchor-test
designs.
Single-group designs
In single-group designs, both forms or tests to be equated are given to
the same group of examinees. The difficulty levels of the tests are not
confounded with the differences in the ability levels of the groups taking each
test because the examinees are the same (Hambleton & Swaminathan, 1985).
However, Lord (1980) pointed out that the test administered second is not being
given under typical conditions. Practice effects and fatigue may affect the
equating process. To deal with this threat, the counterbalanced random-groups
design may be employed. The single-group is divided into two random half
groups. Both half-groups then take both tests in counterbalanced order, one


9
group taking the old form first and the other taking the new form first (Petersen
et al., 1989). Scores on both parallel forms are then equally affected by
learning, fatigue, and practice.
Equivalent-groups designs
With single-group designs, it is also important to administer both tests on
the same day so intervening experiences do not affect the results. However, it
is difficult in practice to arrange the required time block. Equivalent-groups
designs are a simple alternative. The two tests to be equated are given to two
different random groups from the same population. However, differences in the
ability distributions of the groups may introduce an unknown degree of bias
(Hambleton & Swaminathan, 1985). Because there are no common data, it is
impossible to adjust for any random differences (Petersen et al., 1989). Several
researchers have studied the effects of these different group ability distributions
on equating results.
Harris and Kolen (1986) investigated the effect of differences in group
ability on the equating of the American College Test (ACT) Math test. Although
their results showed score equivalents somewhat higher for low-ability students
and lower equivalent scores for high-abillty examinees, the differences were not
significant. The authors concluded that the equatings were robust to even large
differences in group ability distributions.
Similar results were found by Angoff and Cowell (1986) when they
studied the population independence of equating transformations using


10
Graduate Record Examination (GRE) data. Some minor discrepancies were
discovered, but the majority were not significant in horizontal equating
situations.
Cook, Eignor, and Taft (1988) hypothesized that differences in ability
were expected when the groups took the two tests to be equated at different
times of the year. Two forms of the Biology achievement test were
administered. One form was given in the fall mainly to high school seniors, and
the other form was administered predominantly to sophomores in the spring.
Two fall administrations were also equated and studied. Because recency of
instruction is important in some parts of this type of achievement test and most
students study Biology in tenth grade, disparate results were attained from the
fall/spring equating. The spring sample, containing mostly students who had
just completed the subject tested, received higher scaled scores than the fall
sample. In this study, the construct measured by the test depended on the
sample of examinees to whom the test was administered. In contrast, the
fall/fall equating was robust to group differences. This study demonstrates the
importance of administering the test forms to be equated at the same time,
especially when the content is instructionally sensitive.
Anchor-test designs
Lord (1980) stated the differences between two samples of examinees
can be measured and controlled by administering to each examinee an anchor
test measuring the same ability as tests X and Y. When an anchor test is used,


11
equating may be carried out even when the two groups are not at the same ability
level. The groups may be random groups from the same population or they may
be nonequivalent or naturally occurring groups. The scores on the anchor test
can be used to estimate the performance of the combined group (Cook &
Petersen, 1987). The anchor test may be an internal part of both tests X and Y,
or it may be an external separate test. If an external anchor test is used, it should
be administered after X or Y to avoid practice effects on the tests to be equated
(Lord, 1980). The anchor-test design, while the most complicated of the data
collection methods, is the most common in real testing situations. Constraints of
time or available samples placed on large testing programs often require its use
(Skaggs & Lissitz, 1986a).
Properties of the anchor test can seriously affect the ensuing equating
results. Klein and Jarjoura (1985) studied the properties and characteristics of
anchor-test items in relation to the total test. A test of 250 items was equated
using three different anchor tests. Although all anchors were similar to the total
test in difficulty, only one of the anchor-tests was representative of the total test
content. The results confirmed the importance of including items on the anchor
test that mirror as nearly as possible the content of the total test.
In addition to content representativeness, the relative position of items in
test books also seems to play an important role in anchor-test design. Kingston
and Dorans (1984) examined relative position effects of items in a version of the
GRE General Test. Although the equatings of the Verbal measure of the test


12
were in close agreement, the Quantitative and Analytical measures showed
sensitivity to relative item position. When possible, it is preferable to include the
anchor items spiralled throughout the test in their operational positions.
The length of the anchor test is another concern and the subject of several
studies. Klein and Kolen (1985) used a certification test to examine the
relationship between anchor test length and accuracy of equating results. The
authors used anchor tests of varying lengths and examinee groups both similar
and dissimilar in ability distribution. They concluded that when groups have
similar ability distributions, the anchor test length has little effect. However, as
group ability distributions become more dissimilar, longer anchor tests work best.
Klein and Kolen also found that anchor tests should correspond closely with the
total test in content representation, difficulty, and discrimination.
The study of Cook et al. (1988) is also pertinent to the question of anchor-
test length. When the groups differ in level of ability, as did the spring and fall
samples, different anchor test lengths yielded disparate results. In contrast, when
the groups have similar ability distributions, like the two fall samples, the
equatings are similar for different anchor test lengths.
When applying item response theory equating methods, anchor items are
usually referred to as linking items. These linking items are used to scale the
item parameter estimates. Equating with IRT requires that the item parameter
estimates for the two test forms be on the same scale before equating. The
quality of the equating depends largely on how well this item scaling is


13
accomplished (Cook & Petersen, 1987). Wingersky and Lord (1984) studied the
problem of the optimal number of linking items in the context of IRT concurrent
calibration. The authors concluded that two linking items with small standard
errors of estimation worked almost as well as a set of 25 linking items with large
standard errors of estimation.
Wingersky, Cook, and Eignor (1986) studied the characteristics of linking
items and their effects on IRT equating. Monte Carlo procedures were used with
parameter values set to imitate those estimated from the Verbal sections of the
College Board Scholastic Aptitude Test (SAT-V). These values were selected to
make the simulation as realistic as possible. Linking test lengths of 10, 20, and
40 items were used as well as variations in the size of the standard errors of
estimation and distributions of examinee ability. Scaling was accomplished by
both concurrent calibration and characteristic curve methods. The results of this
study showed little difference between the two scaling methods, and the accuracy
of the both equating methods improved as the number of linking items increased.
Unlike the findings of Wingersky and Lord (1984), linking items having standard
errors of estimation similar to those found in actual SAT-V items provided slightly
better equating outcomes than those chosen to have small errors of estimation.
The studies reviewed clearly indicate that the properties of an anchor test
are of great concern. Anchor or linking items should remain in the same relative
positions in new and old forms and as many anchor items as possible should be
used (Cook & Eignor, 1988). The question of optimal anchor test length becomes


14
even more important as the ability distribution of the samples used in equating
become more dissimilar. Because anchor test designs are usually used in
situations where ability distributions of the groups may vary to an unknown
degree, the conclusions have important implications. The anchor test must also
closely mirror the total test to be equated in statistical properties and content
representativeness. As the correlation between scores on the anchor test and
the scores on the new and old forms becomes higher, the ensuing equating also
improves (Cook & Petersen, 1987).
Many factors may affect equating results. Because the purpose of
equating is to create a relationship between two tests so it makes no difference to
the examinee which test is administered, each of these factors must be carefully
considered in deciding on the equating design. Some general guidelines to
successful equating are summarized in Table 1. Only after these factors have
been carefully considered and the data have been collected, can a specific
equating method be chosen.
Equating Methods
Conventional Methods of Equating
Once the data have been collected using one of the data collection
designs reviewed, mathematical procedures are applied to the data to develop
the equating transformation. Many such methods exist, some based on classical
test theory and others on item response theory (IRT). The conventional methods,
those arising from classical test theory, may be categorized as linear equating or
equipercentile equating.


15
Table 1
Summary of Recommendations for a Successful Equating
Total Test
Well-defined content specifications
Item selection based on statistical data from field testing
Length of at least 35 items
Examinees
Sample size of at least 500
Better results with groups similar in ability
Administrative
Strictly controlled testing conditions
Security of tests and items is maintained
Scoring is controlled
Anchor Tests
. Representative of the total test in difficulty and discrimination
Similar to the total test in content specifications
Common items are in approximately the same position in the old and
new forms.
Common items are identical in both forms.
About 20% 30% of total test length


16
Linear equating
In horizontal equating, the two tests to be equated are similar in
difficulty. When administered to the same group of examinees, the raw score
distributions are assumed to be different only with respect to the means and
standard deviations (Hambleton & Swaminathan, 1985). Linear equating is
based on this assumption. A transformation is identified such that scores on X
and Y are considered to be equated if they correspond to the same number of
standard deviations above or below the mean in some population. The two
scores are equivalent if
X_^=Y^ (1)
OX
These scores will have the same percentile rank if the distributions are the
same (Crocker & Algina, 1986).
Many variations of linear equating models exist whose details may be
found in the literature (Angoff, 1971; Holland & Rubin, 1982; Marco et al.,
1983). Two of the more commonly used models are the Tucker model and the
Levine equally reliable model. Both of these procedures produce an equating
transformation of the form:
LP(y) Ay B (2)
where Lp (y) is the linear equating function for equating Y to X (Dorans, 1990).
Adaptations of this formula exist for dealing with an anchor test, usually
labelled V, when it is or is not part of the reported score. The difference
between the Tucker model and the Levine equally reliable model lies in their


17
underlying assumptions. Full discussions of these assumptions and
derivations of the appropriate formulas may be found in Dorans (1990).
Many studies have been conducted to assess the accuracy of linear
equating methods. Skaggs and Lissitz (1986b) carried out a simulation study
with an external anchor design. Both difficulty and discrimination values were
manipulated. The authors discovered unacceptable results with linear
equating when the discrimination means were unequal on the two tests.
Marco, Petersen, and Stewart (1983) used 40 different linear equating
models to transform SAT-V data. Both similar and dissimilar samples were
used, as well as variations of anchor test designs and characteristics of the
total tests. Some generalizations reached from the results of this ambitious
study are as follows:
1. When a test is equated to a test or form like itself through a parallel
anchor test and the ability distributions of the samples are identical, a linear
model yields very good results.
2. When a test is equated to a test or form like itself through an easy or
difficult anchor test with random samples, all of the models have a small mean
square error.
3. When samples with dissimilar ability distributions are used, linear
equating does not perform well.
4. When total tests differ in difficulty, linear models yield unsatisfactory
results.


18
Two methods of selecting samples and five methods of equating,
including two linear methods, were combined in a study by Livingston, Dorans,
and Wright (1990). Again, when the samples differed in ability distributions the
linear equatings were inaccurate, showing a large negative bias. Matching the
samples on the basis of the anchor test did little to improve the results. The
authors recommended dealing with ability differences by selecting a
representative sample from each population and choosing an equating method
that does not assume exchangeability for examinees based on their anchor
test scores.
Based on these studies, it can be seen that linear equating methods are
distribution dependent. Although linear equating may perform satisfactorily in
optimal conditions, it is likely to produce bias in real testing situations.
Eauioercentile equating
In equipercentile equating, a transformation is chosen so that raw
scores on two tests are considered to be equated if they have the same
percentile rank (Angoff, 1971). This is based on the definition that score
scales are comparable for two tests if their respective score distributions are
identical in shape for some population (Braun & Holland, 1982). When this is
true, a table of pairs of raw scores can be constructed. Because the pairs of
raw scores are not necessarily numerically equal, it is necessary to transform
one set of scores into the other set or to convert both sets to a new score


19
(Petersen et al., 1989). In mathematical terms, the equipercentile equating
function for equating Y to X on population P is
Epfy) =Fp'(Gr(y)) (3)
where Gp (y) is the cumulative distribution of Y scores and Fp'1 () is the
inverse of the cumulative distribution of X scores, Fp (x). A cumulative
distribution function maps scores onto relative frequencies, while an inverse
cumulative distribution function maps the relative frequencies onto scores
(Dorans, 1990).
As a mathematical model, equipercentile equating makes no
assumptions about the tests to be equated. It simply compresses and
stretches the score units on one test so that its raw score distribution matches
the second test. It is only consideration of the purpose of equating and the
desired condition of population invariance that prevents its application to tests
measuring different constructs (Petersen et al., 1989).
Generally, empirical studies have shown mixed results in assessing the
accuracy of equipercentile equating. Livingston, Dorans, and Wright (1990)
included an equipercentile equating method in their study. A composite of two
equipercentile equatings, the procedure worked well in most situations.
Similarly, the equipercentile equating produced acceptable results in all
combinations of conditions in the Skaggs and Lissitz (1986b) study.
On the other hand, in the investigation conducted by Petersen et al.
(1983) using SAT data, equipercentile equating was studied along with the


20
Tucker Equally Reliable and Levine Unequally Reliable linear models and three
IRT methods. The equipercentile equating produced the worst results of all
the methods investigated. This was especially true for the Verbal Test.
In a 1983 study by Cook and Eignor reported in Skaggs and Lissitz
(1986a), alternate forms of the biology, mathematics, and social studies
achievement tests of the GRE were equated using various procedures. Again,
results varied by test content, but the equipercentile method was inadequate in
all cases. Cook and Eignor felt that equipercentile equating may have suffered
from a lack of data at the extreme scores.
The Cook et al. (1988) equatings with biology achievement test data
also uncovered mixed results. Although the equipercentile equating method
performed adequately with the parallel fall-to-fall samples, it was not sufficiently
robust to the ability differences found in equating the fall and spring samples.
These mixed findings raise some concerns about the application of
equipercentile equating. When raw scores are used, this method does not
meet the conditions for equating. Hambleton and Swaminathan (1985) noted
that a nonlinear transformation is needed to equalize the moments of the two
distributions, resulting in a nonlinear relationship between the raw scores and
the true scores. In turn, this implies that the tests are not equally reliable and it
is no longer a matter of indifference to the examinee which form is taken.
Besides violating the equity condition, the equipercentile equating process is
population dependent.


21
For the past forty years, large scale testing programs publishing multiple
forms of examinations have used an equating process. Until recently, most
have employed one of the conventional linear or equipercentile procedures
described. But recent psychometric developments have presented an
alternative.
Equating Methods Based on Item Response Theory
Item response theory
A brief introduction to item response theory is essential to an
understanding of the following equating procedures. Item response theory
(IRT) is an attempt to model an examinee's performance on a test item as a
function of the characteristics of the item and the examinee's ability on some
unobserved, or latent, trait. The IRT model specifies the relationship between
a latent trait and the observed performance on items designed to measure that
trait.
This relationship can then be depicted graphically by an item
characteristic curve (ICC). The ICC depicts the probability that an examinee at
any given ability level will make a correct response to an item. The graph is
typically an S-shaped curve with ability, symbolized by 0, plotted on the
horizontal axis and the probability of a correct response to item /, P, (9), plotted
on the vertical axis.
Many different mathematical models may be used to depict this
functional relationship. Most common in practice are the logistic class of


22
models due to the ease of estimation. Birnbaum (1968) proposed a two-
parameter logistic model (2PL) of the form
Pl(e) = [1+eD*m>lr1 (4)
where b, is the difficulty value, a, Is the discrimination parameter, and D is a
scaling factor, normally 1.7.
The three-parameter logistic model (3PL) adds a third parameter,
denoted c,, referred to as the lower asymptote. The mathematical form of the
3PL model is written as
Pi(0) = Ci+(1-c)[1+e-*)r1 (5)
with the a,, b,, and D defined as before. The value of c, is typically smaller
than the value that would result if examinees were to make a random response
to the item (Hambleton & Swaminathan, 1985). Figure 1 depicts an ICC based
on the 3PL model.
The one-parameter logistic model, or Rasch model, assumes all items
have equal discrimination and no guessing occurs. This model is written
Pi(9) = [1+eD)r1 (6)
where the parameters are defined as in the previous models.
Cursory examination of the three IRT logistic models may lead to the
conclusion that they form a type of hierarchy from least to most specific.
However, the three models represent very different philosophical perspectives


23
of measurement theory (Skaggs & Lissitz, 1986a). It is these differences that
must be considered when selecting a model for a particular application.
Figure 1. An item characteristic curve (ICC) based on the three-parameter
logistic model
The use of any of the IRT models entails restrictive assumptions about
the item response process. Briefly stated, the major assumptions of IRT are
as follows:
1. The ICC accurately represents the data.
2. The data are unidimensional.
3. Responses are locally independent (Skaggs & Lissitz, 1986a)
An ICC is defined completely when its general form is specified and
when the parameters of a particular item are known (Hambleton &
Swaminathan, 1985). This leads to the basic advantage of IRT models. When
the data fit the model reasonably well, it is possible to demonstrate the


24
invariance of item and ability parameters. When the item parameters are
known, an examinee's ability may be estimated from any subset of the items.
Also, item parameters may be calibrated with any sample drawn from a
sufficiently large population (Skaggs & Lissitz, 1986a). These advantages
cannot be derived from classical test theory and should have tremendous
consequences for equating with item response theory.
All of the practical IRT models are based on the unidimensionality
assumption. This states that the probability of a correct response by
examinees to a set of items can be mathematically modeled by using only one
ability parameter (Kingston & Dorans, 1984). According to Lord (1980), while
ability is probably not normally distributed for most groups of examinees,
unidimensionality is a property of the items and does not cease to exist
because the examinee group is changed in distribution.
Because the items on a test are assumed to measure only one common
trait, for all examinees with the same ability the item responses are
independent of one another. This is the local independence assumption. The
probability of success on any given item depends on the item parameters,
examinee ability, and nothing else In determining the probability of a correct
response to a specific item, success or failure on other items will add no new
information if ability is known (Lord, 1980).
Good estimation of the item and ability parameters is of paramount
importance in describing the data accurately. Many investigators have


25
explored the effect of the number of items and the number of examinees on
parameter estimation for IRT models. The results of these studies varied
according to the estimation procedure used. Available estimation methods
include (a) joint maximum likelihood estimation (JML), (b) conditional maximum
likelihood estimation (CML), (c) marginal maximum likelihood estimation
(MML), and (d) Bayesian estimation (BE). Full explanations of the various
procedures may be found in Hambleton and Swaminathan (1985).
Much of the research on parameter estimation employed the JML
procedure as implemented by the computer program LOGIST (Wood,
Wingersky, & Lord, 1976). These reports will not be reviewed here, but the
interested reader is referred to Harrison (1986), Hulin et al. (1982), Lord
(1968), Ree (1979), Swaminathan and Gifford (1983, 1985), and Wingersky
and Lord (1984). In general, a sample size of at least 1,000 and test length of
50 or more items is required for acceptable estimation with the JML procedure
of LOGIST. One major problem uncovered by these studies is that consistent
estimates of the item parameters cannot be obtained in the presence of
examinee (9) parameters because the latter increase with sample size (Baker,
1990).
This problem can be overcome by using the MML procedure
implemented in the BILOG computer program (Mislevy & Bock, 1987). The
examinee's 9 parameters are removed from item parameter estimation by
integrating them over an assumed unit normal prior distribution. At this point in


26
the procedure, it is not the 6 of each examinee that has been estimated, but
the form of the 0 distribution. The item parameters are first estimated, followed
by the 0 parameters at a later stage (Baker, 1990).
In addition to MML, the BILOG program allows Bayesian maximum a
posteriori estimation (MAP) and Bayesian expected a posteriori estimation
(EAP) of 0 parameters. Mislevy and Stocking (1989) have recommended the
EAP procedure with a unit normal prior for the 0 distribution. Specifying this
prior for abilities limits extreme values of the 6 estimates and the resulting
variances will tend to be smaller than with MML. When the value of the
variance is smaller, the prior distribution becomes more concentrated and pulls
the estimated parameters toward the mean of the distribution.
Yen (1987) compared LOGIST and BILOG for accuracy of item
parameter estimation. Test lengths of 10, 20, and 40 items were simulated
with a sample of 1,000 examinees. The ability distributions examined were
normal, positively skewed, negatively skewed, and symmetric. Item difficulty
was also manipulated. The BILOG estimates were more accurate than those
of LOGIST in almost every situation. The advantage of BILOG was even more
pronounced for the small item set. Although ability distribution had no
substantial effect on the estimation of the ICCs, discrimination and pseudo
chance parameters were somewhat inaccurate with BILOG in the case of the
negatively skewed distribution.


27
In addition to investigating the effect test length had on item and ability
parameter estimates derived from LOGIST and BILOG procedures, Qualls and
Ansley (1985) studied the sample size effect. Sample sizes of 200, 500, and
1,000 examinees with a normal ability distribution were combined with test
lengths of 10, 20, and 30 items. As sample size increased, both procedures
produced estimates more highly correlated with the simulated values. The
BILOG estimates were slightly better in all cases and superior in the
combination of small sample size with 10 items.
Buhr and Algina (1986) used BILOG with four methods of estimation
and sample sizes of 250, 500, 750, and 1,000 to study the similarity of
estimation. The Bayesian procedures were the most robust in dealing with
different ability distributions. Estimation with all procedures improved
substantially as sample size increased to 500, but showed little additional
effect as sample size increased further.
Baker (1990) simulated item response data based on a 45-item test with
500 examinees to study the pattern of estimation results as a function of the
various analysis operations. The data were analyzed under the options
available in BILOG and the obtained parameter estimates were equated back
to the true metric. The equated results were generally very close to the true
parameters. The item parameters were only slightly affected by the
characteristics of various priors. The equated means of the estimated Gs were


28
somewhat higher than the true values, both when priors were and were not
imposed on the item discriminations.
IRT equating
Nothing in IRT contradicts the basic conclusions of classical test theory.
Additional assumptions are made that allow answers not available under
classical test theory (Lord, 1980). The theoretical advantage of IRT models is
that once a set of items have been fitted to an IRT model, it is possible to
estimate the ability of examinees who have taken a different set of items. To
accomplish this, the items must be measuring the same latent trait and must
be on the same scale (Petersen et al., 1989). When this is true and the item
parameters are known, it will make no difference to the examinee what subset
of items is administered. Therefore, in the context of IRT, equating is not
necessary (Hambleton & Swaminathan, 1985).
However, when both item and ability parameters are unknown, it is
necessary to choose an arbitrary metric for either the ability parameter 6 or the
item difficulty b,. Because all the models for P,(9) are functions of the
quantity a* (0 b/), the same constant may be added to every 0 and b, without
changing the item response function P, (0). Additionally, every 0 and b, may
be multiplied by a constant and every a, divided by the same constant without
changing the quantities a, (9 b, ) and P, (0). Therefore, the origin and unit of
measurement of the ability scale are arbitrary and any scale for 0 may be


29
chosen as long as the same scale is chosen for b, (Petersen et al., 1989).
This is referred to as indeterminacy of the parameter scale.
If the parameters of a set of items are estimated separately for two
different groups of examinees, the item parameters may appear to be different
due to the arbitrary fixing of the metric for 9 or b,.. However, the two sets of 9s
and b, s should have a linear relationship to each other (Hambleton &
Swaminathan, 1985). The a, s should be the same except for differences in
unit of measurement and, in the 3PL case, the c, s remain unaffected
(Petersen et al., 1989).
The advantages of IRT equating are most useful in the case where
groups taking the two tests are nonrandom or intact groups (Crocker & Algina,
1986). Consequently, the following discussion will emphasize uses of IRT
equating with an anchor test design. However, item response theory
procedures may also be used with single-group or equivalent groups designs.
An anchor or linking test is one method available to put the parameters
for the two tests on the same scale. Four procedures commonly used with this
method are (a) concurrent calibration, (b) the fixed bs method, (c) the equated
bs method, and (d) the characteristic curve transformation method.
In concurrent calibration, parameters for the two tests are estimated
simultaneously. The linking items, or sometimes common subjects, serve to
unite the two tests and results in item parameter estimates on a common
scale. This allows direct equating of the two tests (Petersen et al., 1989).


30
The parameters of each total test-anchor test combination are
estimated sequentially in the fixed bs method. After the item parameters have
been estimated for one test, the item difficulties of the linking items obtained
from the first calibration are used as input for the estimation of parameters on
the second test. The linking item parameters are not reestimated. The end
result is item parameters for both tests being placed on the same scale
(Petersen, Cook, & Stocking, 1983).
In the equated bs method, the parameters for each test are estimated
separately. Then the means and standard deviations of the difficulties for the
two sets of linking items are set to be equal. Ability estimates could also be
used for this purpose. This linear transformation is then applied to the a,, b,,
and 6 parameters of the second test (Petersen et al., 1989). Several
variations of the transformation, including the mean and sigma method and the
robust mean and sigma method, are described in Hambleton and
Swaminathan (1985). Also, Stocking and Lord (1983) described a modification
which gives lower weights to poorly estimated parameters and outliers.
It is most common in both the fixed bs and equated bs methods to use
only the relationship for item difficulties to obtain the equating function
(Hambleton & Swaminathan, 1985). The characteristic curve method can
prevent the possible loss of information caused by ignoring the discrimination
relationship. For the characteristic curve method, the parameters of each test
are calibrated separately. All parameters are then placed on the same scale


31
by using the two sets of parameter estimates from the common items. A linear
transformation is obtained from minimizing the difference between the true
scores on the linking items. This transformation is then applied to the a,, b,,
and 0 parameters of the second test (Stocking & Lord, 1983). Because it
takes all information into account, this procedure is theoretically an
improvement over the previous methods.
Sometimes the reporting of abilities in terms of 9 is unacceptable. In
these situations, the 9 value from a test may be converted to its corresponding
true score £ through
6-PiO) (7)
¡.1
where n is the number of items on the test. Equating of the true scores on the
two tests is then possible (Hambleton & Swaminathan, 1985). The true score
on one test is said to be equated to the true score on a second test if each
corresponds to the same ability level, or if
5 = Pi(9) t1 = P,(0) (8)
i-1 H
(Skaggs & Lissitz, 1986a). In practice, estimated item parameters are used to
approximate P, (9) and P¡ (9). Paired values of l and q are then computed by
substituting a series of arbitrary values for 9 into Equation 8 and calculating %
and q for each 9. These paired values define £ as a function of q and
constitute an equating of these true scores (Lord, 1980).


32
The relationship between raw scores and true scores on two tests is not
necessarily the same, nor is an equating provided for individuals scoring below
the chance level (Petersen et al., 1989). Observed-score equating provides a
method of predicting the raw-score distribution of a test. This procedure uses
probabilities of correct responses under an IRT model to generate a
hypothetical joint distribution of item responses from all examinees taking both
tests. Conventional equlpercentile equating is then applied to the new
distributions (Skaggs & Lissitz, 1986a). Neither true-score nor observed-score
equating is applied often in practice. Both are complicated to calculate and
expensive to implement.
Many researchers have investigated the accuracy of IRT equating
methods using the various IRT models and procedures. Comparison of IRT
equating with conventional methods is also common. Marco, Petersen, and
Stewart (1983) examined the Rasch and 3PL models along with the 40 linear
and two equipercentile equating methods previously discussed. A variety of
conditions, including random and dissimilar samples, internal and external
anchors, and difficulty levels of the anchor tests were also studied. The two
IRT methods worked well, both with an external anchor test equal in difficulty
to the total test and with an internal anchor. With the external anchor test, the
Rasch results were slightly better than with any of the other equating methods
investigated. Both IRT models were clearly superior to the conventional
equating methods when the samples differed in ability distributions, but neither


33
the Rasch nor the 3PL model showed superiority to the other under the
conditions studied.
Kolen (1981) explored true-score and observed-score equating methods
as well as a linear and an equipercentile equating method. The Rasch, 2PL,
and 3PL models were used for the IRT equatings. The two forms of the Iowa
Test of Educational Development to be equated had no common items. Each
test had been administered to a random sample. The true-score method for
the 3PL model produced the best results. When only quantitative items were
equated, the Rasch true-score combination also worked well.
Kolen and Whitney (1982) used the General Educational Development
Tests (GED) with the Rasch, 2PL,and 3PL IRT models and an equipercentile
equating method. They found with small samples (N < 198) a number of
extreme item parameter estimates were produced by the 3PL model which
seriously affected the equating.
In the Petersen, Cook, and Stocking (1983) study discussed earlier in
the context of conventional equating, a 3PL model was also examined using
concurrent calibration, the fixed bs method, and the characteristic curve
transformation. For the SAT-V, all IRT models and methods outperformed
linear and equipercentile equatings. Both conventional and IRT methods
yielded acceptable results for the mathematics test. Concurrent calibration
with the 3PL model produced the least amount of error.


34
Harris and Kolen (1985) compared conventional equating methods with
IRT 3PL model equating. The sample consisted of high and low ability
examinees. The 3PL model was found to be slightly superior.
The Cook, Eignor, and Taft (1988) study using biology achievement
tests administered at different points in time included a 3PL model with the
characteristic curve transformation in addition to the equipercentile equating
method. The authors concluded that the IRT results, although slightly superior
with the fall-to-spring sample equating, basically paralleled the results obtained
with the conventional method.
A minimum-competency test, Florida's Statewide Student Assessment
Test, Part II (SSAT-II) was equated by Hills, Subhiyah, and Hirsch (1988).
Their purpose was to study the effect of anchor length on equating and
compare different equating methods using a sample with a negatively skewed
distribution. The equating methods investigated were linear, Rasch, and 3PL.
The IRT models were equated with concurrent calibration, fixed bs method,
and equated bs method using robust mean and sigma. The authors concluded
that the 3PL model with concurrent calibration and Rasch models gave similar
good results. Also, when using the 3PL model with concurrent calibration, an
anchor test length of 10 items was found to be sufficient for good equating
outcomes.
Results of these studies indicate that the 3PL model tends to perform
better than conventional and Rasch equating in a variety of situations.


35
Equating with IRT appears to produce better results than conventional
equating methods, especially when the ability distribution of the two groups is
dissimilar. Concurrent calibration and characteristic curve transformation were
the preferred methods of scaling, although fewer linking items are required with
concurrent calibration. Table 2 contains a summary of the equating studies
reviewed here.
Multidimensionalitv
Violation of the Unidimensionalitv Assumption
The mathematical models upon which IRT is based are grounded on
very strong assumptions, particularly that item responses are unidimensional
(Ansley & Forsyth, 1985). The unidimensionality assumption requires that
each of the tests to be equated onto a common scale must measure the same
underlying trait or ability. Any factor that influences an examinee's score, other
than the one assumed latent trait, will violate the unidimensionality assumption.
Although IRT explicitly acknowledges this assumption, other commonly used
procedures that transform scores, such as equipercentile equating, are also
unidimensional even if not stated specifically (Hirsch, 1989). This can be seen
by reviewing the required conditions for equating.
There are many factors that may cause multidimensionality, such as
guessing, speededness, fatigue, cheating, random answering, instructional
sensitivity, or item context and content. Two or more cognitive traits may
influence an examinee's response to an item. For example, reading


Table 2
Summary of Unidimensional IRT Test Equating Studies
Study
Tests
Equating Models
Independent Variables
Cook & Eignor
(1983)
CB-achievement
3PL, equipercentile,
linear
equating models
scaling methods
Cook, Eignor, &
Taft (1988)
Biology achievement
3PL
equipercentile
dissimilar samples
equating models
Harris & Kolen
(1986)
ACT-Math
3PL, equipercentile,
linear
equating models
dissimilar samples
Hills, Subhiyah,
& Hirsch (1988)
SSAT-II
Rasch, 3PL, linear
equating models
negatively skewed distribution
anchor length
scaling models
Kolen (1981)
ITED: Math &
Vocabulary
Rasch, 2PL, 3PL,
equipercentile, linear
equating models
item context
Kolen & Whitney
(1982)
GED
equipercentile
Rasch, 3PL, linear,
equating models
Marco, Petersen,
& Stewart (1983)
SAT-V
Rasch, 3PL, linear,
equipercentile
ability distribution
internal & external anchor
difficulty of anchor
Peterson, Cook, &
Stocking (1983)
SAT-V
SAT-Q
3PL, linear,
equipercentile
equating models
scaling models (3PL)


37
skill may be required to correctly answer a mathematical item. Some of these
violations can be controlled, reduced, or eliminated, but the unidimensionality
assumption will still be violated in many practical situations (Doody-Bogan &
Yen, 1983). Achievement tests are not constructed using methods that yield
factor pure instruments. Instead, a table of specifications is customarily
developed and items are written to match the specifications. These items
rarely measure a single trait (Reckase, 1979). Due to the many possible
causes leading to violation of the unidimensionality assumption, it can be
concluded that dimensionality is a joint property of both the item set and the
particular sample of examinees (Hattie, 1985).
Multidimensional Models
Recently, attempts have been made to model multidimensional
responses within the framework of IRT. Several multidimensional item
response theory (MIRT) models have been proposed. Although
multidimensional versions of all three logistic parameter IRT models have been
derived, only the multidimensional two-parameter logistic (M2PL) model will be
discussed.
Doody-Bogan and Yen (1983) described a multidimensional model of
the form
Pii(Qh) =
1 + exp[-D Xa(e,h bjh)]
h=1
(9)


38
where On, is the ability parameter for person i for dimension h; a,h is the
discrimination parameter for item j for dimension h; by/, is the difficulty
parameter for item j for dimension h; and D is the scaling constant, 1.7.
Another model discussed by Sympson (1978) is defined
PtfW-- (10)
ri(1+exp[-D ajh[6ih bjJ])
h-1
where all parameters are defined as above.
These two models can be distinguished by comparing their
denominators. The Doody-Bogan and Yen model contains no product of
probabilities in the denominator as does the Sympson model. Equation 9 can
be classified as a compensatory model that permits high ability on one
dimension to compensate for low ability on another dimension in terms of the
probability of a correct response. If dimensionality is considered in the context
of factor analysis, a two-dimensional test has a group of items measuring each
dimension. A compensatory model seems reasonable because the test is
being considered as a whole (Ansley and Forsyth, 1985).
The second model, defined by Equation 10, is called a
noncompensatory model where high abilities on one factor are not allowed to
supplement low abilities on the second factor. When a two-dimensional test is
considered as one that requires simultaneous application of the two abilities to
answer each item correctly, the noncompensatory model seems more
appropriate (Ansley and Forsyth, 1985).


39
Reckase (1985) has alternately defined the compensatory M2PL to
provide a simple framework for specifying and generating multidimensional
item response data. This model defines the probability of a correct response
as
P(Xj = l|a,d¡,ej) =
EXP^, § + dj)
1 + EXP(a¡', ft + dj)
(11)
where §j is a vector of discrimination parameters; dj is related to item difficulty;
andjj is a vector of ability parameters. The exponent can also be written as
3jh(fth bjh) (12)
h=l
where m is the number of dimensions; ajln is an element of a¡; 0¡h is an element
of ftj; and d¡= -lajhbjh- When this form is used, the relationship to the more
familiar expression in Equation 9 can be seen.
The data described by a multidimensional IRT model can be depicted
graphically by an item response surface (IRS). Figure 2 presents an IRS for
an M2PL item. The IRS increases monotonically as the elements of 0;
increase (Reckase, 1985).
To identify the multidimensional item difficulty (MID) for an item, the
point in the IRS where the Item is most discriminating must be found. This
point, which provides the maximum information about an examinee, will have
the greatest slope. Because the slope along the IRS can differ according to


40
the direction taken, Reckase (1985) determined the slope using the direction
from the origin of the 0 space to the point of highest discrimination.
Figure 2. An item response surface (IRS) based on the compensatory M2PL.
To accomplish this analysis, the model given in equation 11 is
translated to polar coordinates, replacing each eih by 0¡ cos aih, where 0¡ is the
distance from the origin to 0¡ and aih is the angle from the h,h axis to the
maximum information point (Reckase, 1985). In a two-dimensional item, the
value of ocjh can range between 0 and 90 depending on the degree to which
the item measures the two traits. If the item only measures the first trait, cm
equals 0, while an = 90 would depict an item measuring only the second trait.
The relationship between aih and discrimination element aih can then be stated
as


41
COS oiih
The MID parameters can now be expressed as
(13)
MID, = ~d (14)
JZ(ah)2
V h=1
Finally, an item that requires two abilities for a correct response can be
represented as a vector in the two-dimensional latent ability space. The length
of the vector for an item is equal to the degree of multidimensional
discrimination (MDISC) (Ackerman, 1991). Reckase (1985) expressed MDISC
as
MDISC, = 05)
These equations provide an excellent framework for manipulating conditions
during generation of multidimensional data.
Many indices have been developed to assess the dimensionality of a
test and test items Hattie (1985) examined over 30 of these indices which
were grouped into methods based on (a) answer patterns, (b) reliability, (c)
principal components, (d) factor analysis, and (e) latent traits. Hattie
concluded that none of the indices were satisfactory and only four could even


42
distinguish unidimensional from multidimensional data sets. A major problem
encountered by Hattie in assessing the indices was that unidimensionality was
often confused with reliability, internal consistency, and homogeneity.
More recently, other procedures have been developed to assess the
dimensionality of latent traits. Roznowski, Tucker, and Humphreys (1991)
explored several of these indices. Procedures based on the shape of the
curve of successive eigenvalues were found to be unsatisfactory under most
conditions. A pattern index of second factor loadings was accurate except with
high obliqueness. The most accurate index in this study was based on local
independence. The use of this index is particularly recommended with large
samples and many items.
Linear factor analysis has been widely used to assess dimensionality of
dichotomous items. However, use of phi correlations often leads to
overestimation of the number of factors underlying the responses by
confounding factor coefficients with item difficulties (Bock, Gibbons, & Muraki,
1988; Hambleton & Swaminathan, 1985). Tetrachoric correlations may be
substituted, but may still be confounded with item difficulty or guessing in real
data (Camilli, 1992). Bock, Gibbons, and Muraki (1988) have developed a
maximum likelihood full information factor analysis procedure as an attempt to
deal with these problems.
Another approach to dimensionality taken by Stout (1990) replaced the
strong assumptions of unidimensionality and local independence with less


43
restrictive assumptions of essential unidimensionality and essential
independence Stout contended that a dominant dimension results when an
attribute overlaps many items and other dimensions common to only a few
items are unavoidable in reality, but are also not significant. These minor
dimensions are rarely discussed in IRT literature, but are a frequent theme in
classical factor analysis. While the IRT definition of dimensionality would take
all factors, major and minor, into account, essential dimensionality is a
mathematical conceptualization of the number of dominant dimensions with
minor dimensions ignored. An essentially unidimensional test is therefore any
set of items selected from an infinite item pool that measures exactly one
major dimension. When essential unidimensionality is assumed, latent ability
is unique in an ordinal scaling sense and this unique latent ability is estimated
consistently. Stout presented theorems and proofs to show that dimensions
distributed nondensely over items or dimensions that have a minor influence
on possibly many items do not necessarily negate essential unidimensionality.
He continued to present guidelines for development of essentially
unidimensional tests. Among the recommendations are limiting the number of
abilities per item; keeping the number of items dependent on the same ability,
other than the intended-to-be-measured 9, small; and controlling the number of
item pairs assigned to the same ability other than 9. These conditions are
usually met with the carefully designed tests usually found in practice.


44
Nandakumar (1991) used simulations to investigate Stout's statistical
test of essential unidimensionality. When one dominant trait and one or more
minor dimensions having little influence on item scores were present, Stout's
test performed well in indicating essential unidimensionality. The test is more
likely to reject the hypothesis of essential unidimensionality as the effect of the
minor dimensions increases.
To facilitate application of the test of essential unidimensionality, Stout
developed the computer program DIMTEST. An investigation of the program
revealed problems when a test consisted of difficult, highly discriminating items
where guessing was also present (Nandakumar & Stout, 1993). Refinements
were subsequently made to the program to make it more robust and beneficial
to the measurement practitioner.
Nandakumar (1994) studied three commonly used methodologies for
assessing dimensionality in a set of item responses. The three procedures-
DIMTEST, Holland and Rosenbaums approach, and nonlinear factor anaiysis-
-were unreliable in detecting lack of unidimensionality in real data sets.
Although the more recent procedures based on local independence, full
information factor analysis, and essential unidimensionality offer promise for
assessing the dimensionality of dichotomous data, especially with large
datasets, a satisfactory method has not yet been agreed upon by
measurement researchers. Because of the current lack of an acceptable index
to detect multidimensionality, it becomes even more urgent to understand


45
exactly what effect violation of the unidimensionality assumption may have on
IRT applications. When a test measures several dimensions, examinees'
scores will be influenced by all of these factors. As a result, systematic and
unsystematic errors of equating might be expected from scaling and equating
procedures that are applied to multidimensional tests (Yen, 1984). The
estimation of ability and item parameters is likely to be affected also.
Multidimensionalitv and Parameter Estimation
Violation of the unidimensionality assumption has been suggested as a
problem in the estimation of item and ability parameters, the first step in IRT
equating procedures. Thus, it is important to determine how robust estimation
procedures are to this violation.
Ansley and Forsyth (1985) used a noncompensatory M3PL model to
simulate a two-dimensional dataset. The two discrimination parameters were
set to have respective means of 1.23 and .49 and respective standard
deviations of .34 and .11. The b values were scaled to reflect fairly easy items
(Pbi = 33, obi = .82, nb2 = -1.03, Ob2 = -82). The c parameter was set to .2. A
bivariate normal distribution was selected to generate the 0 vectors with both
dimensions scaled to have mean 0 and standard deviation 1.0. The
correlation p(6i, 62) was varied with values of 0.0, .3, .6, .9, and .95 simulated.
Four combinations of sample size (1,000 and 2,000) and test length (30, 60)
were examined. Corresponding unidimensional datasets were also simulated.
Correlations of the estimated and simulated parameters showed the a,


46
estimates appeared to be averages of the true ai and a2 values. The b,
estimates overestimated the true bi values. The 6 estimates were highly
related to the averages of the true 0 values. The authors concluded that item
parameter estimation was affected by violation of the unidimensionality
assumption, but as the 0 vectors became more highly correlated, the
estimations derived from the two-dimensional dataset approached results
obtained from the unidimensional data. Sample size and test length had little
effect on any of the relationships.
Reckase (1979) studied five forms of the Missouri State Testing
Program and five datasets simulated to match various factor structures to
determine what characteristics are estimated by the unidimensional Rasch and
3PL models when the data are multidimensional. Reckase concluded that for
tests with several equally strong dimensions, the Rasch estimates should be
considered as a sum or average of the abilities required for each dimension.
For data with a dominant first factor, the Rasch and 3PL difficulty estimates
were highly correlated with the scores for that factor. With the 3PL model and
more than two potent factors, the b, estimates correlated with just one of the
common factors. The author concluded good ability estimates can be obtained
from unidimensional estimation procedures when the first factor accounts for at
least 20 percent of the test variance, as is likely in practice.
Yen (1984) used data simulated with a compensatory M3PL model and
data from the Comprehensive Test of Basic Skills, Form U (CTBS/U) to study


47
unidimensional parameter estimation of multidimensional data. A variety of a,
parameters were configured and p(9i, 82) was set at .5 or .6. When
multidimensionality was present, the a, and b, parameter estimates were
larger than those of unidimensional sets of items. The unidimensional
estimates of both a, and 9 parameters appeared to be a combination of the
respective two-dimensional parameters.
Data simulated from a hierarchical factor model was used in a study by
Drasgow and Parsons (1983). Item responses were generated from five
oblique common factors. Loadings were varied producing diversity in
correlations between the common factors. Each simulated dataset consisted
of 50-item tests and 1,000 simulees. The general latent trait was recovered
well when the correlations between the common factors were .46 or higher.
Harrison (1986) also used a hierarchical factor model to simulate data.
The strength of the second-order general factor, the number of first-order
common factors, the distribution of items loading on the common factors, and
the number of test items were manipulated. The effect of test length was
significant. As the number of items increased, the general trait was recovered
more effectively regardless of the latent structure, distribution of items across
common factors, or the number of common factors. Estimation of the b,
parameters was found to be robust to violations of unidimensionality. The
estimation of both the a, and b/ parameters improved as test length and
strength of the general factor increased. In general, Harrison found


48
unidimensional parameter estimation procedures to be robust in the presence
of multidimensional data.
The studies reviewed indicate that IRT parameters implied by the
general factor are recovered well when the common factors have sufficiently
high correlations. Reckase, Ackerman, and Carlson (1988) used both
simulated and empirical data to demonstrate that Items can be selected to
construct a test that meets the unidimensionality assumption even though
more than one ability is required for a correct response. The authors showed
that the unidimensionality assumption only requires the items in a test to
measure the same composite of abilities. This seems to have been met in the
previous investigations. Based on this study, it appears as if the
unidimensionality assumption is not as restrictive as formerly thought.
Although these studies explored the effect of multidimensionality on
unidimensional parameter estimation, it is also important to understand what
effect the choice between compensatory and noncompensatory
multidimensional models may have on estimation. Ackerman (1989) simulated
two-dimensional data using both compensatory and noncompensatory M2PL
models. Forty two-dimensional items were generated using the compensatory
model. Difficulty was confounded with dimensionality and p(6i, 02) was
selected at 0.0, .3, .6, and .9. For each compensatory item, a corresponding
noncompensatory item was created using a least-squares approach to
minimize the quantity


49
1000
ZlfPcl e¡, a, B-PncI e> a, b)f (16)
j=1
where Pc is a given compensatory item's probability of a correct response and
Pnc is the noncompensatory item's probability of a correct response which
varies as a function of a and b given 0. The unidimensional 2PL model was
used to estimate parameters using both BILOG and LOGIST. The authors
discovered minimal differences in the IRS for each model when the parameters
are matched. The confounding of difficulty with dimensionality was only
detected by BILOG. For both models, as p(0i, 02) increased, the response
data became more unidimensional and estimation of all parameters improved.
Way, Ansley, and Forsyth (1988) also compared compensatory and
noncompensatory models with simulated data. The values assigned p(0i, 02)
ranged from 0.0 to .95. Results showed the number-right distributions for the
two models were comparable. In the noncompensatory model, the
unidimensional a, estimates appeared to be averages of the a¡ and a2 values,
while the compensatory model provided a, estimates best considered as sums
of ai and a2. The b, estimates for the noncompensatory data were greater
than b, values, while the compensatory model seemed to average the bi and
b2 values. For both models, the 0 estimates were related to the average of the
two 0 parameters.
A summary of the studies investigating the effect of multidimensional
data on unidimensional IRT parameter estimation is presented in Table 3.
Generally, parameters appear to be recovered adequately with data fit


Table 3
Summary of Studies of Unidmensional IRT Estimation with Multidimensional Data
Study
Tests
Model for
Simulating
Estimation
Model
Number of
Dimensions
Independent Variables
Ackerman (1989)
Simulation
Least-squares
conversion
M2PL, Comp.
2PL
2
p(e,, e2)
Difficulty confounded with dimensionality
Comp, vs noncomp, models
BILOG vs LOGIST
Ansley & Forsyth
(1985)
Simulation
M3PL, Noncomp.
3PL
2
p(e,. e2)
Sample size
Test length
Drasgow & Parsons Simulation
(1983)
Hierarchical
factor model
2PL
1 5 p(8i, e2...en)
General factor strength
Harrison (1986)
Simulation
Hierarchical
factor model
2PL
1-8
General factor strength
# of common factors
Test length
Reckase (1979)
Simulation,
Missouri
Linear factor
analysis
Rasch
3PL
2-9
# of dimensions
Estimation methods
Reckase, Ackerman,Simulation,
& Carlson (1988) ACT
M2PL, Comp.
2PL
2
Violation of unidimensionality
Yen (1984)
Simulation,
M3PL, Comp.
3PL
2
p(0i. 62)
a parameters
Note. Comp. = Compensatory model, Noncomp. = Noncompensatory model.


conditions usually found in practice. Both compensatory and
noncompensatory models are apparently viable as MIRT models. Determining
the adequacy of unidimensional parameter estimation of multidimensional data
has important consequences for equating multidimensional tests.
In addition to the estimation procedures discussed, the relationship
between multidimensional and unidimensional IRT models can also be
approached from an analytical framework. Wang (1986), as reported in
Ackerman (1988) and Oshima and Miller (1990), determined explicit algebraic
relationships between unidimensional estimates and the true multidimensional
parameters for the case in which the underlying response process is modeled
by the compensatory M2PL model and the unidimensional 2PL model. Using
the results for unidimensional estimation of a multidimensional data matrix,
Wang concluded that the unidimensional item parameter estimates are
obtained as a weighted composite of the underlying traits. The weights are a
function of the discrimination vectors for the items, the correlations among the
latent traits, and the difficulty parameters of the items. For group g who can be
described as having a diagonal variance-covariance structure fig and a mean
ability vector y, the 2PL item parameters for two-dimensional item j can be
approximated by


52
V
di-% tj
a)Qi
(18)
where a¡ is the discrimination vector for the M2PL model; d\¡ is the difficulty
parameter for the M2PL model; and Q2 are the first and second
standardized eigenvectors of the matrix I'A'AX where A is the matrix of
discrimination parameters for all items in the test and VX = Q Therefore,
when the means, standard deviations, and item parameters of a two-
dimensional distribution are known, the corresponding 2PL unidimensional
item parameters can be approximated.
Multidimensionalitv and IRT Equating
In practice, test equating almost exclusively assumes unidimensionality.
A single score from one test is transformed to a single score from another test.
An understanding of what effect the presence of multidimensional data has on
these unidimensional equating results is of paramount importance.
Dorans and Kingston (1985) equated four forms of the Verbal GRE
Aptitude Test using the 3PL model and an equated bs procedure. Two data
collection designs, equivalent groups and anchor-test, were investigated as
well as several variations in calibration procedures. Dimensionality was
assessed through factor analyses conducted at the item level on interitem
tetrachoric correlations. Two highly related verbal dimensions were identified.


53
To examine their results, the researchers first calibrated the whole test,
then divided the test items into two homogeneous subgroups. The subgroups
were recalibrated separately and placed on the same scale as the original test.
They were then recombined back into an entire test and their corresponding
ICCs were compared. The authors discovered that differences in magnitude of
discrimination parameter estimates had an impact on IRT equating results,
affecting the symmetry of the equating. However, the different research
combinations yielded very similar equatings, leading the authors to conclude
that IRT equating may be sufficiently robust to the dimensionality displayed in
their data.
Cook and Eignor (1988) used SAT data that was suspected to be
multidimensional to examine the robustness of 3PL model concurrent
calibration and the characteristic curve transformation procedures. Scale drift
was used as the criterion for evaluating equating results. Cook and Eignor
concluded that both IRT equating methods produced acceptable results
despite the multidimensionality present in the tests being studied.
In addition to studying parameter estimation, Yen (1984) equated the
LOGIST trait estimates for both real (CTBS/U) and simulated data. Several
statistics were used to evaluate the results: (1) the correlation r; (2)
standardized difference between means (SDM); (3) ratio of standard
deviations; and (4) standardized root mean squared difference (SRMSD).
Trait estimates based on items that measured different dimensions had lower


54
correlations and higher SDMs and SRMSDs. That is, when tests measuring
different dimensions were equated, large unsystematic errors occurred.
Systematic errors were found only when the tests measured several
dimensions that differed in difficulty and were likely to be taught sequentially,
as in a vertical equating situation.
Camilli, Wang, and Fesq (1995) adapted the methodology of Dorans
and Kingston (1985) to examine how multidimensionality may affect the
equating of the Law School Admission Test (LSAT). Two dimensions of the
LSAT were identified using primary and secondary factor analyses, and the
stability of the dimensions was established over six administrations. The test
was divided into two homogeneous subtests to study the effect of
multidimensionality on IRT true-score test equating. Item calibration was done
with BILOG. The authors found very small differences in the equatings except
at the ends of the raw score distribution. They concluded that, for the LSAT,
IRT true-score equating was robust to the presence of multidimensionality.
These empirical studies indicate that violations of the unidimensionality
assumption, while having some impact on results, may not be significant.
However, different tests were used in this research and their content may have
affected findings in an unknown manner. Therefore, the generalization of
results are difficult to interpret across studies (Skaggs & Lissitz, 1986a). Also,
because indices designed to detect multidimensionality are generally
unsatisfactory, it is necessary to design research studies that permit


55
manipulation of independent variables to understand exactly how violations of
the unidimensionality assumption affect equating. Simulation studies present a
technique to manipulate and control the desired variables.
There has been little simulation research on the effects of
multidimensionality on unidimensional IRT equating. One notable exception is
a study by Doody-Bogan and Yen (1983). The main purpose of this paper was
to examine the stability of several chi-square statistics for their ability to detect
multidimensionality in vertical equating, but the findings are significant in the
context of unidimensional equating with multidimensional data. Four
multidimensional data configurations were simulated with the compensatory
M3PL model described in Equation 9. One unidimensional 3PL dataset was
also generated. Three differences in mean ability between the two tests to be
equated were simulated with parameter estimates for all data modelled after
the CTBS for realism. Correlations, standardized difference between means
(SDM), and standardized root mean square differences (SRMSD) were used to
evaluate results The findings of this study were mixed. When the correlations
were examined, the results of the equatings, both horizontal and vertical, were
as good for the tests with multidimensional configurations as for the
unidimensional tests. On the other hand, when the means were used as the
criterion for comparison, the multidimensional tests provided worse equatings
than the unidimensional data, especially when the tests differed in difficulty.


56
Another concern raised was that the equatings might deteriorate if the factors
loaded differently on the two tests.
More recently, attempts have been made to develop a multidimensional
equating procedure. Hirsch (1989) conducted a study in which real and
simulated data were equated with a multidimensional method. The procedure
involves (a) estimating item parameters and abilities on both dimensions for
both tests, (b) identifying common basis vectors, (c) aligning basis vectors
through Procustes rotation, and (d) equating means and standard deviations of
the ability estimates for each dimension of the two tests. Results of this
preliminary research indicated that effective equating was possible with these
techniques, but the instability of the ability estimates make it impractical at this
time. While work on development of MIRT equating is continuing (Hirsch &
Miller, 1991), the procedure has little current value for the equating needs of
testing companies. The results of the studies of unidimensional equating with
multidimensional data are summarized in Table 4.
The emphasis of the present study was to examine the effect of
multidimensional data on unidimensional IRT equating through the use of a
simulation study. The research questions chosen were those considered to be
of most value to the practitioner.


Table 4
Summary of Studies of Unidimensional Equating with Multidimensional Data
Study
Tests
Model
Equating
Method
Number of
Dimensions
Independent Variables
Evaluation
Criterion
Camilli, Wang
& Fesq (1995)
LSAT
3PL
true-score
2
test dimensionality
equating method
split test
Cook & Eignor
(1988)
SAT
3PL
concurrent
calibration
characteristic
curve trans.
unknown
equating methods
scale drift
Doody-Bogan
& Yen (1983)
Simulation
M3PL
3PL
equated bs
2
criterion measures
P correlation
SDM
SRMSD
Dorans &
Kingston
(1985)
GRE-V
3PL
equated bs
2
calibration procedures
data collection design
split test
Yen (1984)
CTBS/U
3PL
Simulation
equated bs CTBS-unknown
Sim.- 2
a & b parameters
p(e,,02)
correlation
SDM
SRMSD
ratio of a


CHAPTER 3
METHOD
Purpose
Introduction
The purpose of this study was to examine the effects of
multidimensional data on unidimensional equating procedures. The effects of
the number of multidimensional items, type of multidimensional model, and
choice of equating procedure were investigated. Most investigations were
conducted with randomly equivalent, normally distributed examinee groups
having mean 0 and standard deviation 1. In addition, data from examinee
groups of lower ability ( X1 = -0.8, SD, = 0.6) were equated to results obtained
from the randomly equivalent groups.
The methods applied to investigate these effects are described in this
chapter. The methodology is discussed in the following sections: (a) data
generation, (b) estimation of parameters, (c) equating, and (d) criteria for
evaluation.
Research Questions
The specific questions to be answered in the present study were:
1. Does the number of multidimensional items in a test affect
unidimensional equating results?
58


59
2. Does the equating procedure affect unidimensional equating results?
3. Do data simulated by using a compensatory multidimensional model
produce different unidimensional equating results than data simulated using a
noncompensatory model?
4. Are unidimensional equating results affected by differing ability
distributions of the two examinee groups?
Data Generation
Design
Data for two parallel forms, A and B, of each test condition were
simulated. Four test conditions were created by varying the number of
multidimensional items contained in each test. These conditions were created
to mirror what might be found in published tests. For example, in a test of
mathematics problem solving, all items might be multidimensional to some
degree if reading skill were also required. However, relatively few
multidimensional items might be found in a reading comprehension test
containing only one graph-reading passage that also needed a math skill for
completion. In the present study, 10, 20, 30, and 40 items of an 40 item test
were two-dimensional. These conditions are referred to as MD10, MD20,
MD30, and MD40 respectively.
In addition to modifying the number of multidimensional items, the
strength of each multidimensional item's first factor was manipulated. This
was done within each test condition because it is unreasonable to expect a
published test to contain multidimensional items which all have an identical


60
factor structure. The angle of item direction was varied to 20, 30, 45, and
60 to reflect items that predominantly measure the first trait (20 and 30),
both traits equally (45), and the second trait (60).
Finally, data were originally generated using a compensatory
multidimensional model. To investigate any variations due to the difference in
modeling, each compensatory dataset was transformed into its corresponding
noncompensatory parameters through application of the least-squares
approach used by Ackerman (1989) and described in Chapter 2.
Noncompensatory parameters were considered corresponding if the probability
of a correct response was the same as for the compensatory parameters. This
was accomplished through the NLIN procedure in the Statistical Analysis
System (SAS.1989). Specific methodology is discussed later in this chapter.
Model Description
To avoid problems associated with estimating the lower asymptote, the
compensatory multidimensional two-parameter logistic (M2PL) model
(Reckase, 1985) was selected for data generation. Because this is a
compensatory model, high abilities on one ability trait are allowed to
compensate for lower abilities on the second ability trait.
The multidimensional item difficulty (MID,) parameter was defined by
Reckase as in equation 14 where a* is the kth element of a and m is the
number of dimensions. The data of interest in this study were considered to
be two-dimensional, so m equaled 2. Multidimensional item difficulty is the


61
distance from the origin of the multidimensional ability space to the point where
the item provides maximum examinee information, or where the IRS has the
steepest slope. A line joins these points at angle a*. In a two-dimensional
item, the value of a* can range between 0 and 90 depending on the degree
to which the item measures the two traits. If the item only measures the first
trait, a,) equals 0, while an = 90 would depict an item measuring only the
second trait. For this study, an was set to either 0, 20, 30, 45, or 60.
Item Parameters
Four tests with 40 items each were simulated using the compensatory
M2PL model described above. Forty items were selected as sufficient to
provide good equating results. An anchor test design was chosen for data
collection as it is widely used by practitioners.(Skaggs & Lissitz, 1886a). Each
test consisted of two forms with 12 common linking items and 28 unique items.
The difficulty values were selected to be reasonable for published tests. Lord
(1968) found difficulties ranging from -1.5 to 2.5 ( X=0.58, SD=0.87) on SAT
Verbal data. Doody-Bogan and Yen (1983) employed a range of b¡ of -2.0 to
1.52 ( X =-0.028, SD=0.818) in a simulation designed to imitate CTBS-U data.
In a study using multidimensional data, Ackerman (1988) reported MID values
ranging form -0.73 through 1.87 on an ACT Mathematics test. Oshima and
Miller (1990) used MID values in the interval -2.0 to 2.0. For the purpose of
this investigation, multidimensional item difficulty parameters (MID) were
generated using the RANNOR function of SAS. Values were chosen randomly


62
from a normal distribution within the range of-2.0 through 2.0 and to have
mean 0 and standard deviation 1.0.
The multidimensional discrimination parameters (MDISC) defined by
equation 15 were randomly selected from a lognormal distribution. A majority
of MDISC values lay between .5 and 2.5 with mean 1.15 and standard
deviation .60. These values correspond to those reported by Doody-Bogan
and Yen (1983) of .5 to 2.00 with mean 1.03 and standard deviation .3387.
Ackerman (1988) found an MDISC range of .58 through 2.39.
To create two 40 item test forms, 68 items were generated for each test
condition. The first 12 items in each set were identified as the linking items
and were common to both forms. Items 13 through 40 were unique items for
Form A and items 41 through 68 were unique to Form B. In order to simulate
two-dimensional items, the values of an as expressed in Equation 13 varied.
In the case of unidimensional items, an was set to 0. For two-dimensional
items, an was either 20, 30, 45, or 60. Those items with an = 20 or 30,
primarily measured the first trait. Items having an = 45 measured both traits
equally, and those with an = 60 discriminated on the second factor more
heavily. More multidimensional items in this study predominantly measured
the first factor because it is reasonable to anticipate this to occur in a well-
designed commercial test. These four an values were spiraled throughout the
items in each dataset. To illustrate, in MD40 an was 20 for item 1, 30 for


63
item 2, 45 for item 3, and 60 for item 4. This pattern then repeated for the 64
remaining items.
For datasets containing both unidimensional and two-dimensional items,
the last 3, 6, and 9 linking items were multidimensional for MD10, MD20, and
MD30 respectively. Thus the linking test had the same proportion of
unidimensional items as did the coresponding unique items in each condition..
The last 7, 14, and 21 unique items for each of Forms A and B were also
multidimensional. Table 5 presents the item parameters for Form A of MD30
with 75% of the items in each form being two-dimensional.
Response Data
For each experimental condition and form, response vectors for 1,000
simulees were generated. This sample size was selected as being adequate
to provide stable parameter estimates. The ability values were randomly
generated through the normal distribution RANNOR function of SAS to range
from approximately -3.00 to 3.00. The theta values were assumed to be
uncorrelated. Probabilities of correctly answering an item were then calculated
for each simulee through application of Equation 11. Finally, the SAS function
RANUNI was used to produce a random number from the uniform distribution
between 0 and 1. If this number was less than or equal to P(X¡j = 1 |a¡,d¡, 0j),
the simulee passed the item. If the random number was greater, the simulee
failed. To increase confidence in results, twenty sets of response data were
generated for each condition and form.


64
Table 5
Simulated Compensatory Parameters for MD30. Form A
Item
Form
Oil
a,
a2
d,
MDISC
MID
1
A,B
0
0.475
0.000
-0.584
0.475
1.231
2
A,B
0
0.563
0.000
-0.173
0.563
0.308
3
A,B
0
0.515
0.000
0.652
0.515
-1.266
4
A,B
60
0.736
1.275
1.199
1.472
-0.814
5
A,B
20
1.159
0.422
0.681
1.234
-0.552
6
A,B
30
0.706
0.407
-0.054
0.815
0.066
7
A.B
45
0.936
0.936
-0.939
1.323
0.709
8
A,B
60
0.291
0.504
-0.618
0.582
1.062
9
A,B
20
0.684
0.249
-0.599
0.728
0.822
10
A,B
30
0.882
0.510
1.652
1.019
-1.621
11
A,B
45
1.129
1.129
2.676
1.597
-1.675
12
A.B
60
0.881
1.526
-1.018
1.763
0.578
13
A
0
0.973
0.000
0.549
0.973
-0.565
14
A
0
1.358
0.000
-0.324
1.358
0.239
15
A
0
1.857
0.000
1.417
1.857
-0.763
16
A
0
0.860
0.000
-0.524
0.860
0.609
17
A
0
1.448
0.000
1.538
1.448
-1.062
18
A
0
1.517
0.000
-0.448
1.517
0.295
19
A
0
0.663
0.000
-0.142
0.663
0.214
20
A
60
0.480
0.832
0.723
0.961
-0.753
21
A
20
0.648
0.236
-0.550
0.689
0.798
22
A
30
1.944
1.122
0.992
2.244
-0.442
23
A
45
1.120
1.120
0.654
1.584
-0.413
24
A
60
0.268
0.464
-0.122
0.535
0.228
25
A
20
0.790
0.288
0.295
0.841
-0.351
26
A
30
0.442
0.255
0.159
0.510
-0.313
27
A
45
1.452
1.452
0.019
2.053
-0.009
28
A
60
0.328
0.568
-0.243
0.656
0.370
29
A
20
0.744
0.271
0.055
0.792
-0.070
30
A
30
0.398
0.230
0.315
0.460
-0.686
31
A
45
0.355
0.355
0.924
0.502
-1.840
32
A
60
0.465
0.806
-1.060
0.930
1.140
33
A
20
1 442
0.525
-1.014
1.535
0.661
34
A
30
1.031
0.595
-0.284
1.191
0.238
35
A
45
0.879
0.879
1.320
1.244
-1.061
36
A
60
0.431
0.747
-0.965
0.862
1.119
37
A
20
0.589
0.214
0.533
0.627
-0.850
38
A
30
1.144
0.661
2.296
1.321
-1.738
39
A
45
0.810
0.810
1.050
1.145
-0.917
40
A
60
0.147
0.254
-0.135
0.293
0.461


65
Noncompensatory Data
For each compensatory item generated, a corresponding
noncompensatory item was created. A noncompensatory item was considered
corresponding if it had the same probability of success as the compensatory
item (Ackerman, 1989). To accomplish this, the NLIN procedure of SAS was
applied to Equation 16. Specifically, the compensatory probability was
calculated for each case and became the dependent variable. The
independent variable in the NLIN model statement was the noncompensatory
probability function. Only multidimensional items were transformed as the
compensatory/noncompensatory question was not applicable to
unidimensional items. Starting values for noncompensatory parameter
estimation were set to equal the compensatory parameters. The 1,000 theta
vectors generated for the first of each compensatory response set were
treated as known values. To ensure the program was producing unique local
minima, starting values were changed for several items in each set and
reestimated. Any differences which appeared in the parameter estimates were
contained in the fourth or fifth decimal place. For approximately 10% of the
items in each dataset, the convergence criterion was not met within 40
iterations. In these cases, the final parameter estimates were substituted for
the starting values and the program rerun. In all such cases, convergence was
achieved with the second attempt.


66
Response vectors were generated by applying Equation 10 and using
the same (01,02) combinations utilized to produce the corresponding
compensatory responses. Twenty response sets were simulated for each
noncompensatory dataset. The item parameters for the multidimensional Form
A items of noncompensatory MD30 are shown in Table 6. Summary statistics
for datasets of both models are displayed in Table 7.
Noneauivalent Groups
One of the strongest theoretical advantages of IRT is its usefulness with
groups of subjects who differ in abilities. One case where this may occur is
when a second form of a test, such as a high school proficiency test, is
administered only to examinees who failed to pass the first attempt. To
examine the effect of data from a lower ability group being equated to data
gathered from a normally distributed group, sets of 1,000 less able simulees
were generated. Scores on 0i for the lower group ranged between -3.00 and
0.00 with mean -0.80 and standard deviation 0.6. Abilities on the second
dimension were normally distributed with mean 0 and standard deviation 1.
Five replications of scores were generated for all four compensatory test
conditions.
Estimation of Parameters
Unidimensional IRT
The responses of the 1,000 simulated examinees in each response set
were analyzed by the computer program BILOG (Mislevy & Bock, 1990) to
estimate the unidimensional item discrimination and difficulty parameters.


67
Table 6
Simulated Noncompensatory Parameters for Multidimensional Items. MD30 Form A
Item
Form
Oil
a,
a2
bi
b2
4
A,B
60
0.664
0.888
-0.945
0.309
5
A,B
20
0.778
0.528
0.236
-2.081
6
A,B
30
0.528
0.447
-0.713
-2.092
7
A,B
45
0.705
0.698
-1.534
-1.596
8
A,B
60
0.352
0.395
-3.164
-1.776
9
A,B
20
0.478
0.390
-1.175
-3.624
10
A,B
30
0.638
0.494
0.834
-0.661
11
A,B
45
0.849
0.844
0.555
0.565
12
A,B
60
0.728
0.942
-1.999
-0.964
20
A
60
0.496
0.606
-1.268
0.047
21
A
20
0.184
0.149
-1.188
-4.872
22
A
30
2.256
0.235
-0.426
0.705
23
A
45
0.830
0.792
-0.481
-0.491
24
A
60
0.692
0.248
-4.282
0.835
25
A
20
0.545
0.413
-0.089
-2.602
26
A
30
0.344
0.306
-0.745
-2.472
27
A
45
0.957
0.910
-0.750
-0.786
28
A
60
0.381
0.436
-2.495
-1.136
29
A
20
0.516
0.400
-0.361
-2.876
30
A
30
0.310
0.276
-0.561
-2.430
31
A
45
0.312
0.315
-0.297
-0.388
32
A
60
0.499
0.585
-2.774
-1.633
33
A
20
0.918
0.610
-0.856
-2.870
34
A
30
0.725
0.578
-0.701
-1.944
35
A
45
0.698
0.677
-0.060
-0.058
36
A
60
0.474
0.551
-2.809
-1.638
37
A
20
0.412
0.322
0.187
-2.772
38
A
30
0.814
0.584
1.073
-0.399
39
A
45
0.653
0.636
-0.219
-0.231
40
A
60
0.096
0.207
-4.957
-3.588


68
Table 7
Summary Statistics for Multidimensional Items in Compensatory and
Noncompensatory Datasets
MD10
MD20
MD30
MD40
Parameter
c
NC
C
NC
C
NC
C
NC
Mean
84
.67
.37
.60
86
.66
.91
.69
at
SD
.55
.33
.80
.28
.50
.35
.50
.37
Mean
.71
.61
.75
.56
.68
.55
.70
.62
a2
SD
.36
.23
.47
.30
.39
.23
.38
.27
Mean
.11
.00
.25
.14
d|
SD
.97
1.12
1.11
1.07
b,
Mean
-1.09
-1.08
-1.02
-.88
SD
1.03
1.09
1.28
1.16
Mean
-1.37
-1.54
-1.35
-1.23
b2
SD
94
1.80
1.37
1.18
Note. C = Compensatory item parameters; NC = Noncompensatory item parameters


69
Program default values were used in the calibration of the two-parameter
logistic model item parameters. Specifically, this involved marginal maximum
likelihood estimation procedures, no priors specified for difficulties, and
lognormal priors for discrimination parameters. For the randomly equivalent
groups, each of the 160 response sets-20 replications each for four
compensatory and four noncompensatory multidimensional conditions-was
analyzed twice. The procedure was repeated for the nonequivalent groups.
First the responses for combined Forms A and B for each dataset were
analyzed simultaneously. Then each form was analyzed separately. This
resulted in a total of 520 BILOG runs.
Analytical Estimation
Unidimensional estimation of the multidimensional item parameters for
the eight datasets was performed analytically using Wangs (1986) procedure.
The SAS IML procedure was employed to determine the unidimensional
estimates of the two-dimensional item parameters for each of the eight
conditions.
Equating
In IRT, because the ICCs are population independent, item parameter
estimates from two BILOG runs should theoretically be identical. However,
P,{6) in the 2PL model is a function of the quantity a, (6 b,). As such, the
origin and the unit of 6 and b, measurement are arbitrary or indeterminant.
Any scale may be selected for 6 as long as the same scale is chosen for b,.
Estimated abilities and item difficulties from two calibration runs should have a


linear relationship to each other (Petersen et al., 1989). Equating is a
procedure used to place the item parameters from two tests on the same
scale.
Three unidimensional IRT equating methods were selected for this
study: (a) concurrent calibration, (b) equated bs, and (c) characteristic curve
transformation.
Concurrent Calibration
Concurrent calibration is the simplest of the IRT methods of equating to
implement. A common group of examinees or items is required to tie the
information from the two tests together. For this study, the parameters of both
forms were estimated simultaneously by BILOG. Twelve common items in
each dataset served to link the forms and the resulting item parameter
estimates were therefore on the same scale. This process was repeated for
each of the response sets in each condition.
Equated bs
The equated bs method is based on determining the linear relationship that
exists between item difficulties estimated in two separate BILOG calibration
runs, one for each form. The means and standard deviations of the b,s for
each set of linking items from Form A and B were calculated. The linear
transformation was determined by


71
Once the slope (A) and intercept (B) of the linear transformation were found,
they were applied to all ability and item estimates for Form B, yielding
/t>; = Ab, + B (20)
9*=A6a + B (22)
All parameters were now transformed to the same scale. Although item
discrimination or ability estimates could have been used to determine the linear
transformation, item difficulty estimates are usually used in practice because
they yield the most stable parameter estimates (Cook & Eignor, 1991).
Characteristic Curve Transformation
The parameter estimates computed separately for Form A and Form B
were also used in the characteristic curve transformation. This equating
method used both a¡ and b¡ estimates from the linking items to derive a linear
transformation through an iterative process that minimized the difference
between the item parameter estimates of the linking items. The process is
based on the assumption that if the estimates were free of error, choosing the
proper linear transformation would cause the true-score estimates of the
linking items to correspond (Petersen et al., 1989; Stocking & Lord, 1983).
The resulting transformation was then applied to all Form B parameters to
create estimates on the same scale. The EQUATE (Baker, Al-Karni, & Al-
Dosary, 1991) computer program was used to accomplish this. Data were


72
examined at 80 points along the ICC and the transformation was generally
identified after approximately 8-10 iterations.
All three equating procedures described were applied to each of the
replications for each of the twelve data conditions. This resulted in 660
equatings for this study. A summation of the research equating conditions is
presented in Table 8.
Table 8
Summation of Research Equating Conditions
Equating Method
Dataset
Concurrent
Calibration
Equated
bs
Characteristic
Curve
Compensatory,
Randomly Equivalent Groups
MD10
V
V
V
MD20
<
V
V
MD30
V
V
V
MD40
V
V
V
Noncompensatory, Randomly Equivalent
Groups
MD10
V
V
V
MD20
V
V
V
MD30
V
V
V
MD40
V
V
V
Compensatory,
Nonequivalent Groups
MD10
V
V
V
MD20
V
V
V
MD30
V
V
V
MD40
V
V
V


73
Evaluation Criteria
To establish a foundation for evaluating the results of the research
equatings, the three comparison conditions described below were used. In
addition, three statistical criteria-correlation, standardized mean difference,
and standardized root mean square difference--were applied to the data.
Comparison Conditions
For the first comparison condition the unidimensional approximations
of the multidimensional item parameters were calculated using the analytic
procedure described by equations 17 and 18 (Wang, 1986). To compute
these approximations for the eight research conditions, the SAS IML procedure
was applied to each of the simulated parameter sets. The means and
standard deviations of the responses for each condition were determined for
inclusion in the formula. The resulting sets of unidimensional comparison item
parameters were weighted composites of the item parameters for the two traits
(Ackerman, 1988). Table 9 presents the analytical unidimensional item
parameter approximations for compensatory MD30, Form A. The resulting
analytical item parameter estimates were then fixed in BILOG 386 and all
compensatory and noncompensatory response sets were analyzed to establish
the comparison ability estimates.
For the next comparison condition, the second dimension of each
multidimensional item was ignored. This would be reasonable if arguing that
most published tests were designed to measure only the first factor. For


74
Table 9
Analytical Estimates of the Unidimensional Parameters for Compensatory MD30,
Form A
Item
Discrimination
Difficulty
1
0.242
1.408
2
0.286
0.352
3
0.262
-1.449
4
0.679
-0.949
5
0.712
-0.559
6
0.479
0.066
7
0.733
0.737
8
0.289
1.237
9
0.422
0.833
10
0.599
-1.622
11
0.876
-1.742
12
0.786
0.673
13
0.482
-0.646
14
0.650
0.273
15
0.842
-0.874
16
0.429
0.698
17
0.687
-1.216
18
0.715
0.338
19
0.335
0.245
20
0.466
-0.877
21
0.400
0.808
22
1.320
-0.442
23
0.868
-0.429
24
0.267
0.265
25
0.487
-0.355
26
0.300
-0.312
27
1.103
-0.010
28
0.325
0.432
29
0.459
-0.070
30
0.270
-0.685
31
0.283
-1.913
32
0.452
1.327
33
0.882
0.670
34
0.700
0.239
35
0.690
-1.104
36
0.421
1.304
37
0.363
-0.862
38
0.777
-1.734
39
0.637
-0.953
40
0.148
0.536


75
example, although mathematics problem solving requires reading skills to
understand the prompts, the reading level Is usually well below the grade level
being tested. In this study, the simulated ability parameters of the first
dimension only from each compensatory and noncompensatory dataset were
utilized. This comparison criterion would enable evaluation of how well the
dominant first factor was recovered in the equatings.
A third comparison condition was created which employed the
averages of the two true 6 values. This condition was based on the parameter
estimation studies of Yen (1984) and Ansley and Forsyth (1985) in which the
unidimensional estimates of the 9 parameters appeared to be combinations of
the true multidimensional abilities.
Statistical Criteria
Correlation coefficients between the simulated 6 and the equated 6
estimates were computed to establish the relationship between the comparison
criterion and the research equatings for each condition. For concurrent
calibration, the appropriate simulated 0 parameters were correlated to the
corresponding estimated ability parameters for both Form A and Form B. Only
the equated form, Form B, was compared to the comparison conditions for all
other equating procedures.
The standardized difference between means (SDM) is the difference in
mean scores for the two sets of ability traits divided by a pooled estimate of the
standard deviation


76
s.^L3) where Sf and are the variances of the two sets of abilities (Yen, 1984).
The means of the estimated ability parameters were subtracted from the
means of each comparison condition to calculate this statistic.
The standardized root mean square difference is the square root of the
mean squared difference between examinees trait estimates, divided by S.
Again, the estimated 0 parameter values were subtracted from the appropriate
comparison values to derive the criterion value.
Summary
Four test conditions with differing numbers of multidimensional items
were simulated using the compensatory M2PL item response theory model.
The item direction for multidimensional items was varied within each test.
Comparable noncompensatory datasets were then created for each condition.
Two 40 item forms were constructed for each situation consisting of 12 linking
and 28 unique items. Responses for 1,000 normally distributed simulated
examinees were generated through application of the appropriate probability
equation and replicated 20 times. The same (81,62) combinations were used to
generate corresponding compensatory and noncompensatory response sets.
In addition, responses for 1,000 low ability examinees were generated with 5
replications for each compensatory test condition.


77
Parameter estimation was executed on all conditions using both
unidimensional IRT procedures and analytical estimation. For the IRT
parameter estimates, equating was performed through through techniques: (a)
concurrent calibration, (b) equated bs, and (c) characteristic curve
transformation.
Three comparison conditions--the first simulated theta, the average of
theta 1 and theta 2, and the analytical estimations of the unidimensional
parameters-were selected for comparison with equated abillity estimates.
Finally, the three statistical procedures of correlation, standardized mean
difference, and standardized root mean square difference were applied to
examine the comparisons.


CHAPTER 4
RESULTS AND DISCUSSION
Simulated Data
Item Parameters
Item parameters for two 40 item forms of a test were generated with a
compensatory multidimensional 2PL model. Four conditions were created with
either 10, 20, 30, or 40 multidimensional items in each form. Four degrees of
dimensionality were spiraled throughout each test and form. Each form
contained twelve linking items that mirrored the total test in psychometric
properties. Additionally, Forms A and B were designed to be randomly parallel.
Examination of the simulated compensatory item parameters confirms
this was accomplished. Descriptive statistics for the four compensatory Form A
conditions are presented in Table 10 and Form B data are shown in Table 11.
All generated values are within the limits found in published tests and described
in previous empirical studies (Doody-Bogan & Yen, 1883; Ackerman, 1988).
For both forms and across all conditions, the means of the d¡ parameters
approach 0.0 with standard deviations of approximately 1.0. The means and
standard deviations of all item parameters for both forms are similar.
The multidimensional compensatory item parameters were then
transformed into their noncompensatory correlates. Descriptive statistics for
78


79
Table 10
Descriptive Statistics for Compensatory Form A Item Parameters
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.29
3.49
1.15
0.8
20
0.30
2.41
0.89
04
30
0.15
1.94
0.84
0.4
40
0.28
2.45
0.98
0.6
a2
10
0.00
1.22
0.17
0.3
20
0.00
1.87
0.42
0.6
30
0.00
1.53
0.49
0.4
40
0.21
1.63
0.71
0.3
d
10
-2.27
2.18
0.08
1.1
20
-2.44
2.76
0.20
1.1
30
-1.06
2.68
0.25
0.9
40
-2.90
2.78
0.17
1.2
MDISC
10
0.41
3.49
1.23
0.8
20
0.30
2.41
1.08
0.5
30
0.29
2.24
1.04
0.5
40
0.57
2.61
1.25
0.6
MID
10
-1.94
1.62
-0.11
0.9
20
-1.86
1.83
-0.09
0.9
30
-1.84
1.23
-0.17
0.9
40
-1.43
1.73
-0.10
0.8
Note. N = 40 items in each condition.
Form A conditions are presented in Table 12 and Form B information is given in
Table 13. The item parameter values calculated from the noncompensatory
transformations are within the ranges given by Ackerman (1989). For all


80
Table 11
Descriptive Statistics for Compensatory Form B Item Parameters
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.37
3.65
1.14
0.8
20
0.27
2.41
0.96
0.5
30
0.15
2.11
0.94
0.5
40
0.27
2.45
0.88
0.5
a2
10
0.00
1.37
0.20
0.4
20
0.00
1.58
0.36
0.5
30
0.00
2.11
0.55
0.5
40
0.18
2.27
0.71
0.4
d
10
-2.55
6.23
0.30
1.5
20
-1.87
2.76
0.00
1.1
30
-3.30
4.65
0.10
1.3
40
-2.90
2.78
0.20
1.2
MDISC
10
0.39
3.65
1.23
0.8
20
0.32
2.41
1.12
0.5
30
0.30
2.98
1.16
0.6
40
0.42
2.62
1.18
0.5
MID
10
-1.71
1.96
-0.13
0.9
20
-1.88
1.58
0.08
0.9
30
-1.68
1.77
-0.02
0.9
40
-1.79
1.94
-0.09
0.8
Note. N = 40 items in each condition.
conditions and in both forms, b2 is slightly less difficult than bi, and a2 is less
discriminating than a*.
In all cases, the noncompensatory b¡ parameters are lower than the MID¡
for the corresponding item. This may be explained by considering the method


81
Table 12
Form A
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.27
0.99
0.63
0.3
20
0.10
1.10
0.60
0.3
30
0.10
2.26
0.63
0.4
40
0.33
2.65
0.76
0.4
a2
10
0.00
1.22
0.17
0.3
20
0.15
0.94
0.52
0.2
30
0.00
1.53
0.49
0.4
40
0.32
1.14
0.62
0.2
b1
10
-3.44
0.86
-1.20
1.1
20
-2.94
1.68
-0.79
1.2
30
-4.96
1.07
-1.07
1.4
40
-3.55
1.93
-0.92
1.2
b2
10
-3.01
-0.54
-1.43
0.8
20
-5.75
2.34
-1.29
2.0
30
-4.87
0.84
-1.45
1.4
40
-3.10
3.98
-1.12
1.2
Note. The number of multidimensional items is the same as the condition number.
used to calculate the transformations. A compensatory and a noncompensatory
item were considered corresponding if, for each 9-1,02 combination, the
probability of a correct response was the same on both items. Because the
noncompensatory model does not allow a high ability on one trait to compensate
for a low ability on the other dimension, the b¡ parameters on a


82
Table 13
Form B
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.33
1.54
0.69
0.4
20
0.13
1.10
0.59
0.3
30
0.33
1.33
0.69
0.3
40
0.10
1.58
0.64
0.3
a2
10
0.29
0.97
0.64
0.3
20
0.10
1.12
0.57
0.3
30
0.16
0.83
0.63
0.4
40
0.25
1.82
0.64
0.3
b1
10
-2.32
0.57
-0.96
0.8
20
-3.22
0.27
-1.21
0.9
30
-3.30
1.65
0.10
1.3
40
-4.06
1.86
-0.79
1.1
b2
10
-3.04
0.33
-1.25
1.0
20
-4.90
0.69
-1.53
1.5
30
-3.62
1.03
-1.24
1.3
40
-3.51
0.82
-1.32
1.1
Note. The number of multidimensional items is the same as the condition number.
noncompensatory item must be smaller than the MID] parameterof the
compensatory item if the condition for items to be corresponding is to be met.
The differences between the compensatory and noncompensatory M2PL
models can also be shown graphically. Because the probability of a correct
response varies as a function of the 0 in each model, the item response
surfaces (IRS) and contour plots of matched items should differ. The


83
compensatory and corresponding noncompensatory model IRS and contour plot
for an item of each degree of dimensionality are shown in Figures 4 through 7.
In Figure 4, a matched item that discriminates predominantly on 0i (a = 20)
is pictured. The differences between the two IRSs are minor. A similarity also
exists in the two conditions where the degree of dimensionality is 15 from
equally discriminating. Figure 5 shows the IRS for a = 30, which discriminates
slightly more on 01 than on 02. Conversely, Figure 7 presents the graphs for a =
60, which discriminates slightly more on 02 than on 0i. Although differences
exist in the baselines, the curves of the IRSs remain similar. This is true both
within each of the two matched sets and between the items with a = 30 and
a=60. In Figure 6, where a = 45, the corresponding compensatory and
noncompensatory items discriminate equally along 01 and 02, and there is a
sharp contrast between corresponding curves.
Similar conclusions can be drawn from examination of the equiprobability
lines of the contour plots. For the compensatory model, parallel lines join the
01,02 combinations that have an equal probability of a correct response. The
incline of these lines is a function of the discrimination parameters. However,
because the noncompensatory model does not allow a high ability on one
dimension to compensate for a low ability on another dimension, the lines
connecting the 0i,02 combinations are curvilinear. The direction of these lines
in the noncompensatory model is a function of the items difficulty parameters


84
(a) Compensatory IRS
(a,=.732, a2=.266, d=-.104)
(c) Compensatory Contour Plot
(b) Noncompensatory IRS
(a,=.526, a2=.378, b,=-.595, b2=-2.961)
(d) Noncompensatory Contour Plot
Figure 3. Item response surfaces and contour plots for item 9, MD20,

85
(a) Compensatory IRS
(a,=.934, a?= 539, d=.650)
(c) Compensatory Contour Plot
(b) Noncompensatory IRS
(a,=.709, a2=.526, b,=-.092, b2=-1.177)
(d) Noncompensatory Contour Plot
Figure 4. Item response surfaces and contour plots for item 10, MD20, a=30'


Full Text
127
Table 36
Noncompensatory Item Parameters for Multidimensional Items in MD10 Forms
A and B
Item
Form
OCi1
ai
a2
bi
b2
10
A,B
30
0.352
0.303
-0.247
-1.859
11
A,B
45
0.923
0.916
-1.199
-1.192
12
A,B
60
0.605
0.774
-1.609
-0.538
34
A
30
0.625
0.499
-1.826
-3.009
35
A
45
0.741
0.736
-0.738
-0.731
36
A
60
0.359
0.402
-3.443
-2.080
37
A
20
0.993
0.628
0.859
-1.192
38
A
30
0.926
0.671
-0.741
-1.749
39
A
45
0.268
0.272
-1.294
-1.329
40
A
60
0.552
0.694
-1.727
-0.620
62
B
30
0.327
0.288
-1.429
-3.039
63
B
45
1.006
0.971
-0.808
-0.801
64
B
60
0.430
0.512
-2.320
-1.079
65
B
20
1.535
0.804
0.568
-1.301
66
B
30
0.591
0.477
-1.525
-2.707
67
B
45
0.465
0.476
-0.280
-0.328
68
B
60
0.669
0.900
-0.762
0.327


90
Table 15
Summary Statistics for Analytical Unidimensional Estimates of Form B Item
Parameters
Parameter
Condition
Minimum
Maximum
Mean
SD
Compensatory Parameters
a
10
0.23
2.10
0.66
0.4
20
0.17
1.13
0.57
0.2
30
0.15
1.52
0.62
0.3
40
0.24
1.35
0.64
0.3
b
10
-1.71
1.96
-0.14
1.0
20
-2.30
1.93
0.07
1.0
30
-1.74
2.03
-0.01
1.0
40
-1.86
2.12
-0.10
0.9
Noncompensatory Parameters
a
10
0.23
2.10
0.68
0.4
20
0.10
2.10
0.56
0.3
30
0.19
1.10
0.52
0.2
40
0.14
0.96
0.52
0.2
b
10
-3.08
1.96
-0.47
1.1
20
-3.97
1.80
-1.07
1.4
30
-3.49
1.45
-1.15
1.3
40
-4.53
1.18
-1.45
1.2
Note. N = 40 in all conditions.
Additional simulee sets were generated to represent low ability
examinees for all conditions. Five replications of each response set were
generated. The summary statistics for these data are contained in Table 20.
For 01, the values ranged from approximately -3.5 to 0.0 with all means around


CHAPTER 1
INTRODUCTION
In many large testing programs, examinees take one of multiple forms of
the same test. Although the different editions are constructed to be as similar in
content and difficulty as possible, it is inevitable that some differences will exist
among the various forms (Petersen, Cook, & Stocking, 1983). Direct
comparison of scores would, therefore, be unfair to an examinee who
happened to take a more difficult form. Because examinees are often in
competition or are being directly compared, it is important to transform the
scores in some way to make them equivalent.
Equating is the statistical process of establishing equivalent raw or scaled
scores on two or more test forms. Theoretically, the equating process adjusts
for test and item characteristics so the propensity distributions would be the
same regardless of which test form was administered. The application of
equating to real data, however, can be full of problems and complications
(Skaggs & Lissitz, 1986a). In practice, equating requires not only a knowledge
of statistical models, but awareness and consideration of many other issues that
have practical consequences for the use and interpretation of results. Brennan
and Kolen (1987) discussed many of these issues, such as the presence of
equating errors, specification of content, and security breaches.
1


35
Equating with IRT appears to produce better results than conventional
equating methods, especially when the ability distribution of the two groups is
dissimilar. Concurrent calibration and characteristic curve transformation were
the preferred methods of scaling, although fewer linking items are required with
concurrent calibration. Table 2 contains a summary of the equating studies
reviewed here.
Multidimensionalitv
Violation of the Unidimensionalitv Assumption
The mathematical models upon which IRT is based are grounded on
very strong assumptions, particularly that item responses are unidimensional
(Ansley & Forsyth, 1985). The unidimensionality assumption requires that
each of the tests to be equated onto a common scale must measure the same
underlying trait or ability. Any factor that influences an examinee's score, other
than the one assumed latent trait, will violate the unidimensionality assumption.
Although IRT explicitly acknowledges this assumption, other commonly used
procedures that transform scores, such as equipercentile equating, are also
unidimensional even if not stated specifically (Hirsch, 1989). This can be seen
by reviewing the required conditions for equating.
There are many factors that may cause multidimensionality, such as
guessing, speededness, fatigue, cheating, random answering, instructional
sensitivity, or item context and content. Two or more cognitive traits may
influence an examinee's response to an item. For example, reading


98
agreement of the means of the two forms. Negative numbers indicate the
estimated 6s were slightly higher than those in the baseline conditions. No
discernible pattern emerges from examination of this data.
Results calculated from the noncompensatory conditions display similar
trends. Strengths of correlations decrease and SRMSDs increase as the
number of multidimensional items increase. The analytical estimation results
are approximately equal and high for all conditions. Like the compensatory
outcomes, the SDM statistic is around 0.0 for all conditions. No differences are
seen between forms.
Comparisons between the compensatory and noncompensatory models
reveal some variations. For MD10, the correlations with 0i are equal for both
models. For MD20 and MD30, the correlation is slightly higher for the
noncompensatory model than for the compensatory model, but is slightly lower
for MD40. However, for all tests, the correlations with (0i + 02)/2 are slightly
lower with the noncompensatory model than with the compensatory model. The
greatest difference is found in MD10 with .88 for the compensatory model and
.80 for the noncompensatory model. The correlations with analytical estimations
are equal for comparable conditions in both models.
The SRMSD results show similar variations but in opposite directions.
The SDM is approximately equal for corresponding conditions in both the
compensatory and noncompensatory conditions.


4
2. Does the equating procedure affect unidimensional equating
results?
3. Do data simulated by using a compensatory model produce
different unidimensional equating results than data simulated by using a
noncompensatory model?
4. Are unidimensional equating results affected by the ability
distribution of the two examinee groups?
Limitations
Results of this study are applicable only to the research conditions
investigated. Generalizations to other item response theory models or other
equating techniques are not justified.
Significance of the Study
In practice, test publishers today apply unidimensional equating
techniques to their products. Because tests are expected to be
multidimensional to some degree and it is difficult to identify multidimensionality
accurately, it is important to investigate the effect of applying unidimensional
equating techniques to multidimensional data. Previous studies have mainly
explored unidimensional equating with empirical data that was suspected of
being multidimensional. Although the results indicated the impact of violating
the unidimensionality assumption may not be significant, the research designs
did not allow manipulation of independent variables. In addition, the true
multidimensionality of the underlying data was unknown in these empirical
studies.


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and Is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
M. David Miller, Chair
Associate Professor of
Foundations of Education
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
James J. Algina
Projtessor of Foun
Education
itions of
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
Linda M. Crocker
Professor of Foundations of
Education
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is ftjlly adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
n
John Newell
Professor of Foundations of
Education


149
Table 58
Descriptive Statistics for Noncompensatory MD40 Linking Items
Replication
Form A
Form B
Mean
SD
Mean
SD
1
0.336
1.115
0.321
1.246
2
0.341
1.203
0.369
1.074
3
0.384
1.090
0.286
1.007
4
0.330
1.060
0.315
1.092
5
0.324
1.097
0.298
1.207
6
0.303
1.212
0.325
1.051
7
0.335
1.070
0.416
1.119
8
0.431
1.128
0.279
1.061
9
0.291
1.083
0.397
1.095
10
0.403
1.080
0.270
1.041
11
0.323
1.084
0.448
1.024
12
0.468
1.089
0.295
1.000
13
0.316
1.010
0.362
1.076
14
0.398
1.111
0.265
1.043
15
0.282
1.048
0.336
1.091
16
0.370
1.118
0.303
1.106
17
0.315
1.151
0.306
1.076
18
0.324
1.127
0.277
1.151
19
0.303
1.163
0.301
1.027
20
0.297
1.039
0.311
1.039


146
Table 55
Descriptive Statistics for Noncompensatory MD10 Linking Items
Form A Form B
Replication
Mean
SD
Mean
SD
1
0.080
0.785
0.051
0.756
2
0.102
0.777
0.124
0.744
3
0.136
0.768
0.192
0.817
4
0.206
0.804
0.125
0.765
5
0.069
0.777
0.041
0.779
6
0.059
0.778
0.130
0.796
7
0.130
0.812
0.083
0.745
8
0.117
0.779
0.172
0.792
9
0.156
0.800
0.172
0.834
10
0.163
0.860
0.158
0.780
11
0.151
0.776
0.178
0.764
12
0.214
0.775
0.135
0.827
13
0.104
0.830
0.110
0.785
14
0.100
0.816
0.165
0.833
15
0.187
0.846
0.111
0.785
16
0.147
0.781
0.099
0.764
17
0.067
0.763
0.125
0.858
18
0.134
0.888
0.160
0.825
19
0.197
0.840
0.093
0.732
20
0.102
0.739
0.066
0.765


15 Summary Statistics for Analytical Unidimensional Estimates of
Form B Item Parameters 90
16 Descriptive Statistics for Simulated Examinees Taking MD10 91
17 Descriptive Statistics for Simulated Examinees Taking MD20 92
18 Descriptive Statistics for Simulated Examinees Taking MD30 93
19 Descriptive Statistics for Simulated Examinees Taking MD40 94
20 Descriptive Statistics for Simulated Low Ability Examinees 95
21 Summary of Concurrent Calibration Results with Randomly
Equivalent Groups 96
22 Constants for Equated bs Equating of Compensatory Forms with
Randomly Equivalent Groups 100
23 Constants for Equated bs Equating of Noncompensatory Forms
with Randomly Equivalent Groups 101
24 Summary of Equated bs Results with Randomly Equivalent
Groups 102
25 Summary of Characteristic Curve Transformation Results with
Randomly Equivalent Groups 104
26 Summary of Equating Results with Nonequivalent Groups 106
27 Constants for Equated bs Equating of Compensatory Forms with
Nonequivalent Examinee Groups 109
28 Simulated Compensatory Item Parameters for MD10 Form A 119
29 Simulated Compensatory Item Parameters for MD10 Form B 120
30 Simulated Compensatory Item Parameters for MD20 Form A 121
31 Simulated Compensatory Item Parameters for MD20 Form B 122
32 Simulated Compensatory Item Parameters for MD30 Form A 123
33 Simulated Compensatory Item Parameters for MD30 Form B 124
viii


25
explored the effect of the number of items and the number of examinees on
parameter estimation for IRT models. The results of these studies varied
according to the estimation procedure used. Available estimation methods
include (a) joint maximum likelihood estimation (JML), (b) conditional maximum
likelihood estimation (CML), (c) marginal maximum likelihood estimation
(MML), and (d) Bayesian estimation (BE). Full explanations of the various
procedures may be found in Hambleton and Swaminathan (1985).
Much of the research on parameter estimation employed the JML
procedure as implemented by the computer program LOGIST (Wood,
Wingersky, & Lord, 1976). These reports will not be reviewed here, but the
interested reader is referred to Harrison (1986), Hulin et al. (1982), Lord
(1968), Ree (1979), Swaminathan and Gifford (1983, 1985), and Wingersky
and Lord (1984). In general, a sample size of at least 1,000 and test length of
50 or more items is required for acceptable estimation with the JML procedure
of LOGIST. One major problem uncovered by these studies is that consistent
estimates of the item parameters cannot be obtained in the presence of
examinee (9) parameters because the latter increase with sample size (Baker,
1990).
This problem can be overcome by using the MML procedure
implemented in the BILOG computer program (Mislevy & Bock, 1987). The
examinee's 9 parameters are removed from item parameter estimation by
integrating them over an assumed unit normal prior distribution. At this point in


62
from a normal distribution within the range of-2.0 through 2.0 and to have
mean 0 and standard deviation 1.0.
The multidimensional discrimination parameters (MDISC) defined by
equation 15 were randomly selected from a lognormal distribution. A majority
of MDISC values lay between .5 and 2.5 with mean 1.15 and standard
deviation .60. These values correspond to those reported by Doody-Bogan
and Yen (1983) of .5 to 2.00 with mean 1.03 and standard deviation .3387.
Ackerman (1988) found an MDISC range of .58 through 2.39.
To create two 40 item test forms, 68 items were generated for each test
condition. The first 12 items in each set were identified as the linking items
and were common to both forms. Items 13 through 40 were unique items for
Form A and items 41 through 68 were unique to Form B. In order to simulate
two-dimensional items, the values of an as expressed in Equation 13 varied.
In the case of unidimensional items, an was set to 0. For two-dimensional
items, an was either 20, 30, 45, or 60. Those items with an = 20 or 30,
primarily measured the first trait. Items having an = 45 measured both traits
equally, and those with an = 60 discriminated on the second factor more
heavily. More multidimensional items in this study predominantly measured
the first factor because it is reasonable to anticipate this to occur in a well-
designed commercial test. These four an values were spiraled throughout the
items in each dataset. To illustrate, in MD40 an was 20 for item 1, 30 for


46
estimates appeared to be averages of the true ai and a2 values. The b,
estimates overestimated the true bi values. The 6 estimates were highly
related to the averages of the true 0 values. The authors concluded that item
parameter estimation was affected by violation of the unidimensionality
assumption, but as the 0 vectors became more highly correlated, the
estimations derived from the two-dimensional dataset approached results
obtained from the unidimensional data. Sample size and test length had little
effect on any of the relationships.
Reckase (1979) studied five forms of the Missouri State Testing
Program and five datasets simulated to match various factor structures to
determine what characteristics are estimated by the unidimensional Rasch and
3PL models when the data are multidimensional. Reckase concluded that for
tests with several equally strong dimensions, the Rasch estimates should be
considered as a sum or average of the abilities required for each dimension.
For data with a dominant first factor, the Rasch and 3PL difficulty estimates
were highly correlated with the scores for that factor. With the 3PL model and
more than two potent factors, the b, estimates correlated with just one of the
common factors. The author concluded good ability estimates can be obtained
from unidimensional estimation procedures when the first factor accounts for at
least 20 percent of the test variance, as is likely in practice.
Yen (1984) used data simulated with a compensatory M3PL model and
data from the Comprehensive Test of Basic Skills, Form U (CTBS/U) to study


55
manipulation of independent variables to understand exactly how violations of
the unidimensionality assumption affect equating. Simulation studies present a
technique to manipulate and control the desired variables.
There has been little simulation research on the effects of
multidimensionality on unidimensional IRT equating. One notable exception is
a study by Doody-Bogan and Yen (1983). The main purpose of this paper was
to examine the stability of several chi-square statistics for their ability to detect
multidimensionality in vertical equating, but the findings are significant in the
context of unidimensional equating with multidimensional data. Four
multidimensional data configurations were simulated with the compensatory
M3PL model described in Equation 9. One unidimensional 3PL dataset was
also generated. Three differences in mean ability between the two tests to be
equated were simulated with parameter estimates for all data modelled after
the CTBS for realism. Correlations, standardized difference between means
(SDM), and standardized root mean square differences (SRMSD) were used to
evaluate results The findings of this study were mixed. When the correlations
were examined, the results of the equatings, both horizontal and vertical, were
as good for the tests with multidimensional configurations as for the
unidimensional tests. On the other hand, when the means were used as the
criterion for comparison, the multidimensional tests provided worse equatings
than the unidimensional data, especially when the tests differed in difficulty.


67
Table 6
Simulated Noncompensatory Parameters for Multidimensional Items. MD30 Form A
Item
Form
Oil
a,
a2
bi
b2
4
A,B
60
0.664
0.888
-0.945
0.309
5
A,B
20
0.778
0.528
0.236
-2.081
6
A,B
30
0.528
0.447
-0.713
-2.092
7
A,B
45
0.705
0.698
-1.534
-1.596
8
A,B
60
0.352
0.395
-3.164
-1.776
9
A,B
20
0.478
0.390
-1.175
-3.624
10
A,B
30
0.638
0.494
0.834
-0.661
11
A,B
45
0.849
0.844
0.555
0.565
12
A,B
60
0.728
0.942
-1.999
-0.964
20
A
60
0.496
0.606
-1.268
0.047
21
A
20
0.184
0.149
-1.188
-4.872
22
A
30
2.256
0.235
-0.426
0.705
23
A
45
0.830
0.792
-0.481
-0.491
24
A
60
0.692
0.248
-4.282
0.835
25
A
20
0.545
0.413
-0.089
-2.602
26
A
30
0.344
0.306
-0.745
-2.472
27
A
45
0.957
0.910
-0.750
-0.786
28
A
60
0.381
0.436
-2.495
-1.136
29
A
20
0.516
0.400
-0.361
-2.876
30
A
30
0.310
0.276
-0.561
-2.430
31
A
45
0.312
0.315
-0.297
-0.388
32
A
60
0.499
0.585
-2.774
-1.633
33
A
20
0.918
0.610
-0.856
-2.870
34
A
30
0.725
0.578
-0.701
-1.944
35
A
45
0.698
0.677
-0.060
-0.058
36
A
60
0.474
0.551
-2.809
-1.638
37
A
20
0.412
0.322
0.187
-2.772
38
A
30
0.814
0.584
1.073
-0.399
39
A
45
0.653
0.636
-0.219
-0.231
40
A
60
0.096
0.207
-4.957
-3.588


41
COS oiih
The MID parameters can now be expressed as
(13)
MID, = ~d (14)
JZ(ah)2
V h=1
Finally, an item that requires two abilities for a correct response can be
represented as a vector in the two-dimensional latent ability space. The length
of the vector for an item is equal to the degree of multidimensional
discrimination (MDISC) (Ackerman, 1991). Reckase (1985) expressed MDISC
as
MDISC, = 05)
These equations provide an excellent framework for manipulating conditions
during generation of multidimensional data.
Many indices have been developed to assess the dimensionality of a
test and test items Hattie (1985) examined over 30 of these indices which
were grouped into methods based on (a) answer patterns, (b) reliability, (c)
principal components, (d) factor analysis, and (e) latent traits. Hattie
concluded that none of the indices were satisfactory and only four could even


93
Table 18
Descriptive Statistics for Simulated Examinees Taking MD30
Rep
THETA 1
THETA 2
Low
High
Mean
SD
Low
High
Mean
SD
1
-3.79
3.40
-0.06
1.03
-3.23
2.81
-0.03
0.99
2
-3.40
3.00
0.00
1.02
-3.34
3.88
-0.06
0.99
3
-3.28
3.15
-0.02
1.04
-3.07
2.93
-0.02
0.99
4
-3.07
3.09
0.02
1.01
-3.14
3.48
0.04
1.00
5
-3.76
3.32
-0.04
1.02
-3.08
3.29
0.02
1.01
6
-2.82
4.18
-0.01
1.01
-2.85
3.03
0.08
0.98
7
-3.35
3.13
0.03
0.97
-3.55
3.12
0.02
0.99
8
-2.51
3.54
-0.02
1.02
-3.11
3.24
0.03
0.98
9
-3.60
4.23
-0.04
1.03
-3.06
3.38
0.03
1.05
10
-3.08
2.84
0.01
0.96
-2.96
3.03
-0.02
1.03
11
-2.70
3.21
0.04
1.00
-3.04
3.28
-0.01
0.99
12
-3.04
3.25
0.01
0.99
-3.05
3.19
0.07
0.99
13
-3.26
3.91
-0.02
1.02
-4.00
3.22
0.04
0.98
14
-3.09
4.19
0.03
1.01
-3.04
3.15
0.06
0.98
15
-3.22
2.46
0.02
0.98
-3.54
3.84
-0.01
1.01
16
-3.84
3.41
-0.02
1.00
-3.16
2.97
-0.01
0.97
17
-3.26
3.08
0.03
0.98
-3.14
2.83
0.01
0.98
18
-2.67
3.26
-0.01
1.01
-3.47
3.06
0.02
1.02
19
-4.00
3.15
0.01
1.07
-3.73
2.92
-0.05
1.05
20
-3.23
3.37
0.01
1.04
-2.81
2.91
-0.05
0.98
Note. N = 1000 for each replication
Twelve common items linked each form resulting in parameter estimates on the
same scale. The equated ability estimates were then compared with the three
comparison conditionsthe simulated 01t the average of the simulated 9i and 02,
and the analytical unidimensional estimates. Correlations (p), standardized
differences between means (SDM), and standardized root mean squared


72
examined at 80 points along the ICC and the transformation was generally
identified after approximately 8-10 iterations.
All three equating procedures described were applied to each of the
replications for each of the twelve data conditions. This resulted in 660
equatings for this study. A summation of the research equating conditions is
presented in Table 8.
Table 8
Summation of Research Equating Conditions
Equating Method
Dataset
Concurrent
Calibration
Equated
bs
Characteristic
Curve
Compensatory,
Randomly Equivalent Groups
MD10
V
V
V
MD20
<
V
V
MD30
V
V
V
MD40
V
V
V
Noncompensatory, Randomly Equivalent
Groups
MD10
V
V
V
MD20
V
V
V
MD30
V
V
V
MD40
V
V
V
Compensatory,
Nonequivalent Groups
MD10
V
V
V
MD20
V
V
V
MD30
V
V
V
MD40
V
V
V


109
Table 27
Constants for Equated bs Equating of Compensatory Forms with Noneauivalent
Examinee Gtouds
MD10
MD20
MD30
MD40
Rep
Slope
Intercept
Slope
Intercept
Slope
Intercept
Slope
Intercept
1
0.68
-0.81
0.65
-0.71
0.71
-0.63
0.70
-0.71
2
0.64
-0.70
0.62
-0.74
0.67
-0.57
0.70
-0.62
3
0.58
-0.71
0.65
-0.68
0.76
-0.65
0.69
-0.68
4
0.59
-0.76
0.59
-0.68
0.73
-0.62
0.62
-0.66
5
0.69
-0.89
0.45
-0.80
0.79
-0.62
0.63
-0.59
comparison conditions. The results are substantial and positive values,
indicating the estimated abilities were lower than the average of the two
simulated traits. This is opposite to the results for the concurrent calibration.
The SRMSDs are also similar for equated bs and characteristic curve
transformation procedures. When correlated using 81, the SRMSD generally
increases with an increase in the number of multidimensional items, but
decreases when calculated by using the average of 6, and 02. When calculated
by using the analytical estimates, the SRMSDs are fairly consistent across
conditions for equated bs, although more variability is noted with the
characteristic curve transformation data.
In general, when performed on data from nonequivalent examinee
groups, the equating procedures studied produced less than optimal equating


LIST OF TABLES
Table oaae
1 Summary of Recommendations for a Successful Equating 15
2 Summary of Unidimensional IRT Test Equating Studies 36
3 Summary of Studies of Unidimensional IRT Estimation
with Multidimensional Data 50
4 Summary of Studies of Unidimensional Equating with
Multidimensional Data 57
5 Simulated Compensatory Parameters for MD30, Form A 64
6 Simulated Noncompensatory Parameters for Multidimensional
Items, MD30 Form A 67
7 Summary Statistics for Multidimensional Items in Compensatory
and Noncompensatory Datasets 68
8 Summation of Research Equating Conditions 72
9 Analytical Estimates of the Unidimensional Parameters for
Compensatory MD30, Form A 74
10 Descriptive Statistics for Compensatory Form A Item Parameters ...79
11 Descriptive Statistics for Compensatory Form B Item Parameters 80
12 Descriptive Statistics for Multidimensional Item Parameters in
Noncompensatory Form A 81
13 Descriptive Statistics for Multidimensional Item Parameters in
Noncompensatory Form B 82
14 Descriptive Statistics for Analytical Unidimensional Estimates of
Form A Item Parameters 89
vii


108
an ordinal sense, the scaling of the two sets of data is different. It appears that
the concurrent calibration, because it included the 1,000 examinees of normally
distributed abilities, raised the ability estimates of the combined groups. This
would be an advantage to an examinee of low ability.
The one exception to this pattern of results is found in the MD40
condition. For the test containing all multidimensional items, the SRMSD is
smaller than in the other three conditions. This is also true with the ability
average comparison condition. Furthermore, the SDMs are closer to zero for
MD40. This may imply that when the multidimensionality becomes more
pervasive, the concurrent calibration of abilities was slightly more accurate.
Equated bs and Characteristic Curve Transformation
As in the case with equivalent group equating, the results for the equated bs
and characteristic curve transformation procedures are nearly identical so
they will be discussed together. Constants for the equated bs linking items are
shown in Table 27.
The correlation coefficients of equated abilities with 9i decrease and
those with the average of 0i and 02 increase as the amount of
multidimensionality in a test increases. The correlations with the analytical
estimates are again high and consistent. For the nonequivalent examinee
groups, the SDMs for the 0i and analytical estimation comparison are closer to
zero than they were for concurrent calibration. An unusual occurrence is seen
with the standardized difference between means (SDM) for the theta average


60
factor structure. The angle of item direction was varied to 20, 30, 45, and
60 to reflect items that predominantly measure the first trait (20 and 30),
both traits equally (45), and the second trait (60).
Finally, data were originally generated using a compensatory
multidimensional model. To investigate any variations due to the difference in
modeling, each compensatory dataset was transformed into its corresponding
noncompensatory parameters through application of the least-squares
approach used by Ackerman (1989) and described in Chapter 2.
Noncompensatory parameters were considered corresponding if the probability
of a correct response was the same as for the compensatory parameters. This
was accomplished through the NLIN procedure in the Statistical Analysis
System (SAS.1989). Specific methodology is discussed later in this chapter.
Model Description
To avoid problems associated with estimating the lower asymptote, the
compensatory multidimensional two-parameter logistic (M2PL) model
(Reckase, 1985) was selected for data generation. Because this is a
compensatory model, high abilities on one ability trait are allowed to
compensate for lower abilities on the second ability trait.
The multidimensional item difficulty (MID,) parameter was defined by
Reckase as in equation 14 where a* is the kth element of a and m is the
number of dimensions. The data of interest in this study were considered to
be two-dimensional, so m equaled 2. Multidimensional item difficulty is the


Table 28
Simulated Compensatory Item Parameters for MD10 Form A
Item
Form
CXii
ai
a2
d,
MDISC
MID
1
A,B
0
1.307
0.000
0.438
1.307
-0.335
2
A,B
0
1.328
0.000
-0.630
1.328
0.475
3
A,B
0
0.929
0.000
0.476
0.929
-0.512
4
A,B
0
0.631
0.000
-0.076
0.631
0.120
5
A,B
0
0.660
0.000
0.627
0.660
-0.950
6
A.B
0
1.928
0.000
0.031
1.928
-0.016
7
A,B
0
0.456
0.000
0.076
0.456
-0.166
8
A,B
0
1.642
0.000
-2.100
1.642
1.279
9
A,B
0
1.865
0.000
2.179
1.865
-1.168
10
A,B
30
0.448
0.259
0.514
0.518
-0.992
11
A,B
45
1.223
1.223
-0.873
1.730
0.504
12
A.B
60
0.598
1.036
-0.060
1.197
0.050
13
A
0
1.450
0.000
1.746
1.450
-1.205
14
A
0
0.603
0.000
0.020
0.603
-0.034
15
A
0
0.685
0.000
1.331
0.685
-1.943
16
A
0
0.589
0.000
0.596
0.589
-1.012
17
A
0
1.232
0.000
0.552
1.232
-0.448
18
A
0
3.491
0.000
1.678
3.491
-0.481
19
A
0
0.511
0.000
0.053
0.511
-0.105
20
A
0
1.556
0.000
-1.732
1.556
1.114
21
A
0
0.441
0.000
-0.243
0.441
0.550
22
A
0
0.618
0.000
-0.138
0.618
0.223
23
A
0
3.254
0.000
-2.267
3.254
0.697
24
A
0
1.569
0.000
-2.037
1.569
1.298
25
A
0
0.962
0.000
0.742
0.962
-0.771
26
A
0
0.707
0.000
-0.450
0.707
0.637
27
A
0
0.490
0.000
0.276
0.490
-0.563
28
A
0
0.414
0.000
0.387
0.414
-0.935
29
A
0
0.817
0.000
-1.327
0.817
1.624
30
A
0
2.963
0.000
1.193
2.963
-0.403
31
A
0
1.110
0.000
1.827
1.110
-1.646
32
A
0
2.238
0.000
-0.674
2.238
0.301
33
A
0
1.903
0.000
1.475
1.903
-0.775
34
A
30
0.806
0.465
-1.366
0.930
1.468
35
A
45
0.897
0.897
0.227
1.269
-0.179
36
A
60
0.285
0.494
-0.865
0.570
1.517
37
A
20
1.462
0.532
1.910
1.556
-1.227
38
A
30
1.268
0.732
-0.499
1.464
0.341
39
A
45
0.286
0.286
0.378
0.405
-0.933
40
A
60
0.525
0.910
-0.092
1.050
0.088
119


123
Table 32
Simulated Compensatory Item Parameters for MD30 Form A
Item
Form
Oh
3i
a2
d,
MDISC
MID
1
A,B
0
0.475
0.000
-0.584
0.475
1.231
2
A,B
0
0.563
0.000
-0.173
0.563
0.308
3
A,B
0
0.515
0.000
0.652
0.515
-1.266
4
A,B
60
0.736
1.275
1.199
1.472
-0.814
5
A,B
20
1.159
0.422
0.681
1.234
-0.552
6
A,B
30
0.706
0.407
-0.054
0.815
0.066
7
A,B
45
0.936
0.936
-0.939
1.323
0.709
8
A.B
60
0.291
0.504
-0.618
0.582
1.062
9
A,B
20
0.684
0.249
-0.599
0.728
0.822
10
A,B
30
0.882
0.510
1.652
1.019
-1.621
11
A,B
45
1.129
1.129
2.676
1.597
-1.675
12
A,B
60
0.881
1.526
-1.018
1.763
0.578
13
A
0
0.973
0.000
0.549
0.973
-0.565
14
A
0
1.358
0.000
-0.324
1.358
0.239
15
A
0
1.857
0.000
1.417
1.857
-0.763
16
A
0
0.860
0.000
-0.524
0.860
0.609
17
A
0
1.448
0.000
1.538
1.448
-1.062
18
A
0
1.517
0.000
-0.448
1.517
0.295
19
A
0
0.663
0.000
-0.142
0.663
0.214
20
A
60
0.480
0.832
0.723
0.961
-0.753
21
A
20
0.648
0.236
-0.550
0.689
0.798
22
A
30
1.944
1.122
0.992
2.244
-0.442
23
A
45
1.120
1.120
0.654
1.584
-0.413
24
A
60
0.268
0.464
-0.122
0.535
0.228
25
A
20
0.790
0.288
0.295
0.841
-0.351
26
A
30
0.442
0.255
0.159
0.510
-0.313
27
A
45
1.452
1.452
0.019
2.053
-0.009
28
A
60
0.328
0.568
-0.243
0.656
0.370
29
A
20
0.744
0.271
0.055
0.792
-0.070
30
A
30
0.398
0.230
0.315
0.460
-0.686
31
A
45
0.355
0.355
0.924
0.502
-1.840
32
A
60
0.465
0.806
-1.060
0.930
1.140
33
A
20
1.442
0.525
-1.014
1.535
0.661
34
A
30
1.031
0.595
-0.284
1.191
0.238
35
A
45
0.879
0.879
1.320
1.244
-1.061
36
A
60
0.431
0.747
-0.965
0.862
1.119
37
A
20
0.589
0.214
0.533
0.627
-0.850
38
A
30
1.144
0.661
2.296
1.321
-1.738
39
A
45
0.810
0.810
1.050
1.145
-0.917
40
A
60
0.147
0.254
-0.135
0.293
0.461


113
results varied according to the comparison condition used. A review of these
comparison conditions is required before proceeding.
Two baseline conditions employed the simulated ability parameters. The
first, 0i, was used because most tests are designed to measure only one
dominant factor. In this case, the first ability would be the trait of interest in the
test. The second comparison condition was the average of the simulated 0i and
02- This condition was based on the work of Yen (1984) and Ansley and Forsyth
(1985) in which the unidimensional estimates of the ability parameters appeared
to be combinations of the simulated multidimensional traits. The third comparison
condition was calculated using the unidimensional approximation of the
multidimensional item parameters described by Wang (1986). The resulting
analytical estimations used all available item information to define the item
parameters. This procedure was used because it provided an alternative method
to define the composite of multidimensional traits.
This discussion about the effects due to the number of multidimensional
items applies only to the equatings performed on data from examinee groups with
equivalent ability levels. The effects for nonequivalent examinee groups will be
presented in the next section. A change in the number of multidimensional items
yield results that vary greatly, but consistently, depending on the comparison
condition viewed. Comparisons with Gi for all equating methods produce
correlation coefficients that decrease in strength as the number of
multidimensional items in a test increase. The SRMSD increases, revealing


14
even more important as the ability distribution of the samples used in equating
become more dissimilar. Because anchor test designs are usually used in
situations where ability distributions of the groups may vary to an unknown
degree, the conclusions have important implications. The anchor test must also
closely mirror the total test to be equated in statistical properties and content
representativeness. As the correlation between scores on the anchor test and
the scores on the new and old forms becomes higher, the ensuing equating also
improves (Cook & Petersen, 1987).
Many factors may affect equating results. Because the purpose of
equating is to create a relationship between two tests so it makes no difference to
the examinee which test is administered, each of these factors must be carefully
considered in deciding on the equating design. Some general guidelines to
successful equating are summarized in Table 1. Only after these factors have
been carefully considered and the data have been collected, can a specific
equating method be chosen.
Equating Methods
Conventional Methods of Equating
Once the data have been collected using one of the data collection
designs reviewed, mathematical procedures are applied to the data to develop
the equating transformation. Many such methods exist, some based on classical
test theory and others on item response theory (IRT). The conventional methods,
those arising from classical test theory, may be categorized as linear equating or
equipercentile equating.


28
somewhat higher than the true values, both when priors were and were not
imposed on the item discriminations.
IRT equating
Nothing in IRT contradicts the basic conclusions of classical test theory.
Additional assumptions are made that allow answers not available under
classical test theory (Lord, 1980). The theoretical advantage of IRT models is
that once a set of items have been fitted to an IRT model, it is possible to
estimate the ability of examinees who have taken a different set of items. To
accomplish this, the items must be measuring the same latent trait and must
be on the same scale (Petersen et al., 1989). When this is true and the item
parameters are known, it will make no difference to the examinee what subset
of items is administered. Therefore, in the context of IRT, equating is not
necessary (Hambleton & Swaminathan, 1985).
However, when both item and ability parameters are unknown, it is
necessary to choose an arbitrary metric for either the ability parameter 6 or the
item difficulty b,. Because all the models for P,(9) are functions of the
quantity a* (0 b/), the same constant may be added to every 0 and b, without
changing the item response function P, (0). Additionally, every 0 and b, may
be multiplied by a constant and every a, divided by the same constant without
changing the quantities a, (9 b, ) and P, (0). Therefore, the origin and unit of
measurement of the ability scale are arbitrary and any scale for 0 may be


112
Effects of Equating Method
Concurrent calibration, equated bs, and the characteristic curve
transformation were Investigated. Discussion of results for randomly equivalent
examinee groups will be presented here. Results for groups differing in mean
ability will be presented in another section.
Few differences are seen in the results obtained for each of the three
methods. The standardized difference between means (SDM) statistics are
almost identical for corresponding comparison conditions at each level of
multidimensionality for all equating methods. The same similarities are also found
with the standardized root mean squared difference (SRMSD) statistics.
All results for equated bs and characteristic curve transformation equatings
are nearly identical. These results are somewhat unexpected. Because the
characteristic curve transformation procedure, unlike the equated bs, includes
more information by using the item discriminations, it seemed reasonable to
believe it would be more sensitive to the presence of multidimensional data. It is
probable that the unidimensional parameter estimation process with BILOG386,
the first step in both equating methodologies, concealed the multidimensionality
in the items. Each procedure was then using the same basic data for the
equating process.
Effects of the Number of Multidimensional Items
Forty-item tests were generated with 10, 20, 30 or 40 multidimensional
items. The effects of the number of multidimensional items on the equating


54
correlations and higher SDMs and SRMSDs. That is, when tests measuring
different dimensions were equated, large unsystematic errors occurred.
Systematic errors were found only when the tests measured several
dimensions that differed in difficulty and were likely to be taught sequentially,
as in a vertical equating situation.
Camilli, Wang, and Fesq (1995) adapted the methodology of Dorans
and Kingston (1985) to examine how multidimensionality may affect the
equating of the Law School Admission Test (LSAT). Two dimensions of the
LSAT were identified using primary and secondary factor analyses, and the
stability of the dimensions was established over six administrations. The test
was divided into two homogeneous subtests to study the effect of
multidimensionality on IRT true-score test equating. Item calibration was done
with BILOG. The authors found very small differences in the equatings except
at the ends of the raw score distribution. They concluded that, for the LSAT,
IRT true-score equating was robust to the presence of multidimensionality.
These empirical studies indicate that violations of the unidimensionality
assumption, while having some impact on results, may not be significant.
However, different tests were used in this research and their content may have
affected findings in an unknown manner. Therefore, the generalization of
results are difficult to interpret across studies (Skaggs & Lissitz, 1986a). Also,
because indices designed to detect multidimensionality are generally
unsatisfactory, it is necessary to design research studies that permit


135
Table 44
Analytical Estimates of Unidimensional Item Parameters for MD10 Form B
Item
Form
CXi1
Compensatory
Noncompensatory
a
b
a
b
1
A.B
0
0.763
-0.336
0.763
-0.336
2
A.B
0
0.775
0.476
0.775
0.476
3
A,B
0
0.543
-0.514
0.543
-0.514
4
A,B
0
0.369
0.120
0.369
0.120
5
A,B
0
0.386
-0.954
0.386
-0.954
6
A,B
0
1.123
-0.016
1.123
-0.016
7
A,B
0
0.267
-0.167
0.267
-0.167
8
A,B
0
0.957
1.284
0.957
1.284
9
A,B
0
1.086
-1.173
1.086
-1.173
10
A,B
30
0.274
-1.093
0.273
-1.399
11
A,B
45
0.656
0.655
0.762
-1.694
12
A,B
60
0.352
0.086
0.563
-1.441
41
B
0
0.830
1.800
0.830
1.800
42
B
0
0.576
-0.127
0.576
-0.127
43
B
0
0.701
-1.235
0.701
-1.235
44
B
0
0.334
-0.117
0.334
-0.117
45
B
0
0.495
0.296
0.495
0.296
46
B
0
0.485
0.388
0.485
0.388
47
B
0
0.226
0.450
0.226
0.450
48
B
0
0.393
0.992
0.393
0.992
49
B
0
2.096
-1.714
2.096
-1.714
50
B
0
0.602
-0.222
0.602
-0.222
51
B
0
0.427
-0.560
0.427
-0.560
52
B
0
0.344
0.179
0.344
0.179
53
B
0
1.003
-0.364
1.003
-0.364
54
B
0
0.535
-1.041
0.535
-1.041
55
B
0
0.282
-1.403
0.282
-1.403
56
B
0
0.512
1.499
0.512
1.499
57
B
0
1.216
-0.734
1.216
-0.734
58
B
0
0.741
-0.884
0.741
-0.884
59
B
0
0.759
1.964
0.759
1.964
60
B
0
1.995
-0.544
1.995
-0.544
61
B
0
0.700
-1.341
0.700
-1.341
62
B
30
0.245
0.468
0.256
-3.081
63
B
45
0.708
0.102
0.820
-1.139
64
B
60
0.238
0.776
0.387
-2.347
65
B
20
1.361
-0.814
0.965
-0.103
66
B
30
0.454
1.188
0.446
-2.888
67
B
45
0.327
-1.564
0.389
-0.431
68
B
60
0.399
-1.559
0.637
-0.196


157
Wingersky, M. S., Cook, L. L, & Eignor, D. R. (1986, April). Specifying
the characteristics of linking items used for item response theory item
calibration. Paper presented at the annual meeting of the American Educational
Research Association, San Francisco.
Wingersky, M. S., & Lord, F. M. (1984). An investigation of methods for
reducing sampling error in certain IRT procedures. Applied Psychological
Measurement. 8, 347-364.
Wood, R. L, Wingersky, M. S & Lord, F. M. (1976). LOGIST A
computer program for estimating examinee ability and item characteristic curve
parameters. (Research Memorandum 76-6). Princeton, NJ: Educational
Testing Service.
Yen, W. (1984). Effects of local item dependence on the fit and
equating performance of the three-parameter logistic model. Applied
Psychological Measurement. 8, 125-145.
Yen, W. (1987). A comparison of the efficiency and accuracy of BILOG
and LOGIST. Psvchometrika. 52, 275-291.


144
Table 53
DescrlDtive Statistics for Compensatory MD30 Linkina Items with Randomly
Eauivalent Gtouds
Replication
Form A
Form B
Mean
SD
Mean
SD
1
-0.012
1.096
-0.040
1.181
2
-0.052
1.130
-0.070
1.159
3
-0.111
1.175
-0.167
1.141
4
-0.159
1.105
-0.013
1.064
5
-0.045
1.083
-0.180
1.125
6
-0.182
1.102
-0.143
1.256
7
-0.147
1.207
-0.079
1.141
8
-0.068
1.111
-0.063
1.042
9
-0.074
1.037
-0.117
1.084
10
-0.107
1.077
-0.156
1.090
11
-0.203
1.100
-0.109
1.212
12
-0.073
1.157
-0.096
1.125
13
-0.119
1.120
-0.218
1.180
14
-0.232
1.206
-0.114
1.208
15
-0.113
1.191
-0.091
1.149
16
-0.075
1.126
-0.100
1.163
17
-0.104
1.167
-0.121
1.134
18
-0.107
1.126
-0.080
1.152
19
-0.102
1.143
-0.097
1.112
20
-0.107
1.122
0.029
1.099


APPENDIX
ITEM PARAMETER DATA


128
Table 37
Noncompensatory Item Parameters for Multidimensional Items in MD20 Form A
Item
Form
Otil
ai
a2
bi
b2
7
A,B
45
1.100
1.116
0.127
0.282
8
A,B
60
0.644
0.834
-1.234
-0.030
9
A,B
20
0.526
0.378
-0.595
-2.961
10
A,B
30
0.709
0.526
-0.092
-1.177
11
A,B
45
0.132
0.100
0.271
0.408
12
A,B
60
0.542
0.664
-1.764
-0.538
27
A
45
0.585
0.602
-2.523
-2.482
28
A
60
0.373
0.430
-2.941
-1.515
29
A
20
0.123
0.125
-1.570
-5.268
30
A
30
0.537
0.421
-0.273
-1.475
31
A
45
0.970
0.933
-0.951
-0.913
32
A
60
1.012
0.249
-1.899
2.344
33
A
20
0.401
0.306
-0.806
-3.516
34
A
30
0.476
0.382
-0.424
-1.697
35
A
45
0.100
0.107
0.546
0.583
36
A
60
0.784
1.098
-2.475
-1.459
37
A
20
0.390
0.257
1.677
-5.746
38
A
30
0.966
0.667
0.146
-0.840
39
A
45
0.799
0.785
0.065
0.222
40
A
60
0.846
1.247
-1.062
0.072


81
Table 12
Form A
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.27
0.99
0.63
0.3
20
0.10
1.10
0.60
0.3
30
0.10
2.26
0.63
0.4
40
0.33
2.65
0.76
0.4
a2
10
0.00
1.22
0.17
0.3
20
0.15
0.94
0.52
0.2
30
0.00
1.53
0.49
0.4
40
0.32
1.14
0.62
0.2
b1
10
-3.44
0.86
-1.20
1.1
20
-2.94
1.68
-0.79
1.2
30
-4.96
1.07
-1.07
1.4
40
-3.55
1.93
-0.92
1.2
b2
10
-3.01
-0.54
-1.43
0.8
20
-5.75
2.34
-1.29
2.0
30
-4.87
0.84
-1.45
1.4
40
-3.10
3.98
-1.12
1.2
Note. The number of multidimensional items is the same as the condition number.
used to calculate the transformations. A compensatory and a noncompensatory
item were considered corresponding if, for each 9-1,02 combination, the
probability of a correct response was the same on both items. Because the
noncompensatory model does not allow a high ability on one trait to compensate
for a low ability on the other dimension, the b¡ parameters on a


26
the procedure, it is not the 6 of each examinee that has been estimated, but
the form of the 0 distribution. The item parameters are first estimated, followed
by the 0 parameters at a later stage (Baker, 1990).
In addition to MML, the BILOG program allows Bayesian maximum a
posteriori estimation (MAP) and Bayesian expected a posteriori estimation
(EAP) of 0 parameters. Mislevy and Stocking (1989) have recommended the
EAP procedure with a unit normal prior for the 0 distribution. Specifying this
prior for abilities limits extreme values of the 6 estimates and the resulting
variances will tend to be smaller than with MML. When the value of the
variance is smaller, the prior distribution becomes more concentrated and pulls
the estimated parameters toward the mean of the distribution.
Yen (1987) compared LOGIST and BILOG for accuracy of item
parameter estimation. Test lengths of 10, 20, and 40 items were simulated
with a sample of 1,000 examinees. The ability distributions examined were
normal, positively skewed, negatively skewed, and symmetric. Item difficulty
was also manipulated. The BILOG estimates were more accurate than those
of LOGIST in almost every situation. The advantage of BILOG was even more
pronounced for the small item set. Although ability distribution had no
substantial effect on the estimation of the ICCs, discrimination and pseudo
chance parameters were somewhat inaccurate with BILOG in the case of the
negatively skewed distribution.


120
Table 29
Simulated Compensatory Item Parameters for MD10 Form B
Item
Form
an
a,
a2
d,
MDISC
MID
1
A.B
0
1.307
0.000
0.438
1.307
-0.335
2
A,B
0
1.328
0.000
-0.630
1.328
0.475
3
A.B
0
0.929
0.000
0.476
0.929
-0.512
4
A.B
0
0.631
0.000
-0.076
0.631
0.120
5
A.B
0
0.660
0.000
0.627
0.660
-0.950
6
A.B
0
1.928
0.000
0.031
1.928
-0.016
7
A.B
0
0.456
0.000
0.076
0.456
-0.166
8
A.B
0
1.642
0.000
-2.100
1.642
1.279
9
A.B
0
1.865
0.000
2.179
1.865
-1.168
10
A,B
30
0.448
0.259
0.514
0.518
-0.992
11
A.B
45
1.223
1.223
-0.873
1.730
0.504
12
A.B
60
0.598
1.036
-0.060
1.197
0.050
41
B
0
1.422
0.000
-2.549
1.422
1.793
42
B
0
0.985
0.000
0.125
0.985
-0.127
43
B
0
1.201
0.000
1.477
1.201
-1.230
44
B
0
0.571
0.000
0.067
0.571
-0.117
45
B
0
0.847
0.000
-0.250
0.847
0.296
46
B
0
0.830
0.000
-0.321
0.830
0.387
47
B
0
0.386
0.000
-0.173
0.386
0.448
48
B
0
0.673
0.000
-0.665
0.673
0.989
49
B
0
3.650
0.000
6.230
3.650
-1.707
50
B
0
1.030
0.000
0.228
1.030
-0.222
51
B
0
0.731
0.000
0.408
0.731
-0.558
52
B
0
0.589
0.000
-0.105
0.589
0.178
53
B
0
1.720
0.000
0.625
1.720
-0.364
54
B
0
0.916
0.000
0.950
0.916
-1.037
55
B
0
0.483
0.000
0.675
0.483
-1.396
56
B
0
0.876
0.000
-1.308
0.876
1.494
57
B
0
2.090
0.000
1.529
2.090
-0.732
58
B
0
1.269
0.000
1.118
1.269
-0.881
59
B
0
1.300
0.000
-2.543
1.300
1.956
60
B
0
3.467
0.000
1.880
3.467
-0.542
61
B
0
1.199
0.000
1.602
1.199
-1.336
62
B
30
0.401
0.232
-0.197
0.463
0.426
63
B
45
1.368
1.368
-0.152
1.935
0.078
64
B
60
0.372
0.645
-0.334
0.744
0.448
65
B
20
2.407
0.876
2.017
2.562
-0.787
66
B
30
0.753
0.434
-0.939
0.869
1.080
67
B
45
0.532
0.532
0.906
0.753
-1.203
68
B
60
0.711
1.231
1.282
1.422
-0.902


multidimensional items had little effect on the unidimensional equating with
randomly equivalent, normally distributed examinee groups. However, if the
unidimensional factor is the trait of interest, the number of multidimensional
items affected the equating outcomes, with results deteriorating as the number
of multidimensional items increased. When examinee groups were not
equivalent, equating results were affected in all conditions. Caution is advised
in applying unidimensional equating procedures when the examinee groups are
suspected of being from different ability levels.
XIV


139
Table 48
Analytical Estimates of Unidimensional Item Parameters for MD30 Form B
Item
Form
(Xi1
Compensatory
Noncompensatory
a
b
a
b
1
A.B
0
0.242
1.408
0.242
-1.408
2
A,B
0
0.286
0.352
0.286
-0.352
3
A.B
0
0.262
-1.449
0.262
1.449
4
A,B
60
0.679
-0.949
0.619
-0.330
5
A.B
20
0.712
-0.559
0.551
-0.975
6
A,B
30
0.479
0.066
0.406
-1.897
7
A,B
45
0.733
0.737
0.577
-2.229
8
A,B
60
0.289
1.237
0.305
-3.490
9
A,B
20
0.422
0.833
0.362
-3.202
10
A.B
30
0.599
-1.627
0.474
0.254
11
A.B
45
0.875
-1.742
0.696
0.798
12
A,B
60
0.790
0.673
0.667
-2.051
41
B
0
0.751
0.665
0.751
-0.665
42
B
0
0.387
-0.579
0.387
0.579
43
B
0
0.436
-0.475
0.436
0.475
44
B
0
0.328
-1.221
0.328
1.221
45
B
0
0.614
2.027
0.614
-2.027
46
B
0
0.882
1.914
0.882
-1.914
47
B
0
0.554
-1.395
0.554
1.395
48
B
60
0.458
-0.624
0.440
-1.061
49
B
20
1.084
-0.798
0.770
-0.373
50
B
30
0.547
-0.681
0.448
-0.898
51
B
45
1.519
-1.625
1.102
0.953
52
B
60
0.549
-1.131
0.515
-0.346
53
B
20
1.267
-0.188
0.847
-0.978
54
B
30
0.566
-1.068
0.457
-0.424
55
B
45
0.318
-0.070
0.301
-2.118
56
B
60
0.386
-0.028
0.383
-1.861
57
B
20
1.014
0.418
0.716
-1.890
58
B
30
0.588
0.573
0.478
-2.303
59
B
45
0.816
1.084
0.617
-2.642
60
B
60
0.346
0.707
0.352
-2.730
61
B
20
0.740
-0.191
0.569
-1.370
62
B
30
0.883
-0.922
0.651
-0.194
63
B
45
0.998
0.821
0.699
-2.234
64
B
60
0.151
1.680
0.190
-1.793
65
B
20
0.847
0.498
0.640
-2.021
66
B
30
0.571
-0.112
0.466
-1.507
67
B
45
0.372
0.853
0.349
-2.868
68
B
60
0.310
0.604
0.321
-2.778


110
results. The differences between the equated ability estimates and the
simulated abilities in the comparison conditions were larger for nonequivalent
examinee groups than for randomly equivalent groups. The concurrent
calibration procedure, due to the presence of the normally distributed group, led
to departures from the SDM and SRMSD comparison conditions, although the
correlations indicated the ranking of examinees was still fairly similar.


CHAPTER 2
REVIEW OF LITERATURE
Test Equating
Conditions for Equating
The purpose of equating is to establish a relationship between two test
forms so that it becomes a matter of indifference to the examinee which form is
taken. Petersen, Kolen, and Hoover (1989) stated that equating itself is simply
an empirical procedure which imposes no restrictions on the properties of scores
or on the method used to define the transformation. It is only when the purpose
of equating and the definition of equivalent scores are considered that restrictions
become necessary.
Lord (1980) outlined four conditions that must be met for the successful
equating of two test forms, X and Y. Briefly, the conditions are (a) equity, (b)
population invariance, (c) symmetry, and (d) same ability. To satisfy the equity
condition, it must make no difference to examinees at every ability level, 9, which
form of the test is taken. The conditional frequency distribution fxf) of the score
on form X should be the same as the conditional frequency distribution of the
transformed form Y score, fX(y)i- Lord (1980) added that it is not sufficient for
equity that fx p and fx(y¡ 10 have the same means, but they must also have equal
variances. If the tests are not equally reliable, it is no longer a matter of
6


CHAPTER 4
RESULTS AND DISCUSSION
Simulated Data
Item Parameters
Item parameters for two 40 item forms of a test were generated with a
compensatory multidimensional 2PL model. Four conditions were created with
either 10, 20, 30, or 40 multidimensional items in each form. Four degrees of
dimensionality were spiraled throughout each test and form. Each form
contained twelve linking items that mirrored the total test in psychometric
properties. Additionally, Forms A and B were designed to be randomly parallel.
Examination of the simulated compensatory item parameters confirms
this was accomplished. Descriptive statistics for the four compensatory Form A
conditions are presented in Table 10 and Form B data are shown in Table 11.
All generated values are within the limits found in published tests and described
in previous empirical studies (Doody-Bogan & Yen, 1883; Ackerman, 1988).
For both forms and across all conditions, the means of the d¡ parameters
approach 0.0 with standard deviations of approximately 1.0. The means and
standard deviations of all item parameters for both forms are similar.
The multidimensional compensatory item parameters were then
transformed into their noncompensatory correlates. Descriptive statistics for
78


3
complex and difficult in practice (Harrison, 1986). Test companies continue to
apply unidimensional equating procedures to their products. The viability of
using unidimensional models with multidimensional data must be explored to
determine the effect on the equating outcomes. An understanding of what effect
multidimensional data have on unidimensional equating results Is of paramount
importance. Empirical studies (Camilli, Wang, & Fesq, 1995; Cook & Eignor,
1988; Dorans & Kingston, 1985; Yen, 1984) indicate that violation of the
unidimensionality assumption, while having some impact on results, may not be
significant. However, each of these studies employed data from a different test
and their content may have influenced findings in an unknown manner. The
number of multidimensional items and the degree of multidimensionality in each
is also unknown. Therefore, the generalization of results are difficult to interpret
across studies (Skaggs & Lissitz, 1986a). It is necessary to design research
studies that permit manipulation of independent variables to understand exactly
how violations of the unidimensionality assumption affect equating. Simulation
studies present a technique to manipulate and control the desired variables.
Purpose
The purpose of the present study was to investigate the effect of
multidimensional data in applying unidimensional IRT equating techniques. The
specific questions to be answered were:
1. Does the number of multidimensional items affect unidimensional
equating results?


33
the Rasch nor the 3PL model showed superiority to the other under the
conditions studied.
Kolen (1981) explored true-score and observed-score equating methods
as well as a linear and an equipercentile equating method. The Rasch, 2PL,
and 3PL models were used for the IRT equatings. The two forms of the Iowa
Test of Educational Development to be equated had no common items. Each
test had been administered to a random sample. The true-score method for
the 3PL model produced the best results. When only quantitative items were
equated, the Rasch true-score combination also worked well.
Kolen and Whitney (1982) used the General Educational Development
Tests (GED) with the Rasch, 2PL,and 3PL IRT models and an equipercentile
equating method. They found with small samples (N < 198) a number of
extreme item parameter estimates were produced by the 3PL model which
seriously affected the equating.
In the Petersen, Cook, and Stocking (1983) study discussed earlier in
the context of conventional equating, a 3PL model was also examined using
concurrent calibration, the fixed bs method, and the characteristic curve
transformation. For the SAT-V, all IRT models and methods outperformed
linear and equipercentile equatings. Both conventional and IRT methods
yielded acceptable results for the mathematics test. Concurrent calibration
with the 3PL model produced the least amount of error.


103
Characteristic Curve Transformation
Item parameters for Form A and Form B which had been calibrated
separately by BILOG386 were also analyzed by an iterative process that
minimized the differences in the item parameters of the linking items. The
resulting transformation was applied to the Form B ability estimates, placing
them on the same scale as the Form A abilities. Because this process includes
information from the discrimination parameters as well as the difficulties, it was
anticipated that the transformation would be more affected by the presence of
multidimensionality than either of the two preceding equating procedures.
Table 25 presents the results for the characteristic curve transformation
equatings. Contrary to expectations, the results for this method are almost
identical to those for the equated bs. The patterns and comparisons found for
the equated bs method also occur for the characteristic curve transformation.
Equating Results for Noneauivalent Groups
The ability level of an examinee group taking a second form of a test may
differ significantly from that of the original testing group. This may occur, for
example, in s state testing program where only those examinees failing their first
attempt at a proficiency test will take the second form. The distribution of
abilities is thus lower for this second group. One of the theoretical advantages
of IRT is its application with nonequivalent groups.
For all four multidimensional conditions, response data for the
compensatory model were generated to simulate two examinee groups that


45
exactly what effect violation of the unidimensionality assumption may have on
IRT applications. When a test measures several dimensions, examinees'
scores will be influenced by all of these factors. As a result, systematic and
unsystematic errors of equating might be expected from scaling and equating
procedures that are applied to multidimensional tests (Yen, 1984). The
estimation of ability and item parameters is likely to be affected also.
Multidimensionalitv and Parameter Estimation
Violation of the unidimensionality assumption has been suggested as a
problem in the estimation of item and ability parameters, the first step in IRT
equating procedures. Thus, it is important to determine how robust estimation
procedures are to this violation.
Ansley and Forsyth (1985) used a noncompensatory M3PL model to
simulate a two-dimensional dataset. The two discrimination parameters were
set to have respective means of 1.23 and .49 and respective standard
deviations of .34 and .11. The b values were scaled to reflect fairly easy items
(Pbi = 33, obi = .82, nb2 = -1.03, Ob2 = -82). The c parameter was set to .2. A
bivariate normal distribution was selected to generate the 0 vectors with both
dimensions scaled to have mean 0 and standard deviation 1.0. The
correlation p(6i, 62) was varied with values of 0.0, .3, .6, .9, and .95 simulated.
Four combinations of sample size (1,000 and 2,000) and test length (30, 60)
were examined. Corresponding unidimensional datasets were also simulated.
Correlations of the estimated and simulated parameters showed the a,


76
s.^L3) where Sf and are the variances of the two sets of abilities (Yen, 1984).
The means of the estimated ability parameters were subtracted from the
means of each comparison condition to calculate this statistic.
The standardized root mean square difference is the square root of the
mean squared difference between examinees trait estimates, divided by S.
Again, the estimated 0 parameter values were subtracted from the appropriate
comparison values to derive the criterion value.
Summary
Four test conditions with differing numbers of multidimensional items
were simulated using the compensatory M2PL item response theory model.
The item direction for multidimensional items was varied within each test.
Comparable noncompensatory datasets were then created for each condition.
Two 40 item forms were constructed for each situation consisting of 12 linking
and 28 unique items. Responses for 1,000 normally distributed simulated
examinees were generated through application of the appropriate probability
equation and replicated 20 times. The same (81,62) combinations were used to
generate corresponding compensatory and noncompensatory response sets.
In addition, responses for 1,000 low ability examinees were generated with 5
replications for each compensatory test condition.


9
group taking the old form first and the other taking the new form first (Petersen
et al., 1989). Scores on both parallel forms are then equally affected by
learning, fatigue, and practice.
Equivalent-groups designs
With single-group designs, it is also important to administer both tests on
the same day so intervening experiences do not affect the results. However, it
is difficult in practice to arrange the required time block. Equivalent-groups
designs are a simple alternative. The two tests to be equated are given to two
different random groups from the same population. However, differences in the
ability distributions of the groups may introduce an unknown degree of bias
(Hambleton & Swaminathan, 1985). Because there are no common data, it is
impossible to adjust for any random differences (Petersen et al., 1989). Several
researchers have studied the effects of these different group ability distributions
on equating results.
Harris and Kolen (1986) investigated the effect of differences in group
ability on the equating of the American College Test (ACT) Math test. Although
their results showed score equivalents somewhat higher for low-ability students
and lower equivalent scores for high-abillty examinees, the differences were not
significant. The authors concluded that the equatings were robust to even large
differences in group ability distributions.
Similar results were found by Angoff and Cowell (1986) when they
studied the population independence of equating transformations using


130
Table 39
Noncompensatory Item Parameters for Multidimensional Items in MD30 Form A
Item
Form
Oil
ai
32
bi
th
4
A,B
60
0.664
0.888
-0.945
0.309
5
A,B
20
0.778
0.528
0.236
-2.081
6
A,B
30
0.528
0.447
-0.713
-2.092
7
A,B
45
0.705
0.698
-1.534
-1.596
8
A,B
60
0.352
0.395
-3.164
-1.776
9
A,B
20
0.478
0.390
-1.175
-3.624
10
A,B
30
0.638
0.494
0.834
-0.661
11
A,B
45
0.849
0.844
0.555
0.565
12
A,B
60
0.728
0.942
-1.999
-0.964
20
A
60
0.496
0.606
-1.268
0.047
21
A
20
0.184
0.149
-1.188
-4.872
22
A
30
2.256
0.235
-0.426
0.705
23
A
45
0.830
0.792
-0.481
-0.491
24
A
60
0.692
0.248
-4.282
0.835
25
A
20
0.545
0.413
-0.089
-2.602
26
A
30
0.344
0.306
-0.745
-2.472
27
A
45
0.957
0.910
-0.750
-0.786
28
A
60
0.381
0.436
-2.495
-1.136
29
A
20
0.516
0.400
-0.361
-2.876
30
A
30
0.310
0.276
-0.561
-2.430
31
A
45
0.312
0.315
-0.297
-0.388
32
A
60
0.499
0.585
-2.774
-1.633
33
A
20
0.918
0.610
-0.856
-2.870
34
A
30
0.725
0.578
-0.701
-1.944
35
A
45
0.698
0.677
-0.060
-0.058
36
A
60
0.474
0.551
-2.809
-1.638
37
A
20
0.412
0.322
0.187
-2.772
38
A
30
0.814
0.584
1.073
-0.399
39
A
45
0.653
0.636
-0.219
-0.231
40
A
60
0.096
0.207
-4.957
-3.588


129
Table 38
Noncompensatory Item Parameters for Multidimensional Items In MD20 Form B
Item
Form
0)1
at
a2
bi
b2
7
A,B
45
1.100
1.116
0.127
0.282
8
A,B
60
0.644
0.834
-1.234
-0.030
9
A,B
20
0.526
0.378
-0.595
-2.961
10
A,B
30
0.709
0.526
-0.092
-1.177
11
A,B
45
0.132
0.100
0.271
0.408
12
A,B
60
0.542
0.664
-1.764
-0.538
55
B
45
0.263
0.260
-2.007
-1.924
56
B
60
0.672
0.865
-1.578
-0.434
57
B
20
1.044
0.622
-0.830
-2.740
58
B
30
0.728
0.571
-1.263
-2.366
59
B
45
0.803
0.828
-0.097
-0.090
60
B
60
0.722
0.972
-1.220
-0.057
61
B
20
0.767
0.548
-1.711
-3.651
62
B
30
0.478
0.400
-1.288
-2.559
63
B
45
0.475
0.478
-2.396
-2.358
64
B
60
0.354
0.430
-0.922
0.689
65
B
20
0.250
0.202
-1.119
-4.901
66
B
30
0.884
0.644
-0.697
-1.755
67
B
45
0.326
0.326
-2.632
-2.577
68
B
60
0.463
0.564
-3.223
-1.960


CHAPTER 5
CONCLUSIONS
The purpose of this study was to investigate the effects of ignoring the
presence of multidimensional items when applying unidimensional IRT equating
procedures. The specific effects of interest were (a) data generated with the
compensatory and noncompensatory models, (b) the IRT equating method
chosen, (c) the number of multidimensional items present in the test, and (d) the
equivalence of the ability levels of examinee groups. Data were simulated and
equated and the results were compared against three comparison criteria using
three statistical indicators.
Effects of Multidimensional Model
Data for the four multidimensional conditions were simulated using a
compensatory M2PL model (Reckase, 1985). Each compensatory item set was
transformed to corresponding noncompensatory item parameters so the
differences in the probability of a correct response for the two models by a given
examinee were minimized (Ackerman, 1989). Very similar results were achieved
for both models with all three IRT equating procedures using data from randomly
equivalent examinee groups. When the multidimensional data are matched so
the probability of a correct response is equal, there is little effect of model on the
ensuing equatings, even though the IRSs may differ for individual items.
111


69
Program default values were used in the calibration of the two-parameter
logistic model item parameters. Specifically, this involved marginal maximum
likelihood estimation procedures, no priors specified for difficulties, and
lognormal priors for discrimination parameters. For the randomly equivalent
groups, each of the 160 response sets-20 replications each for four
compensatory and four noncompensatory multidimensional conditions-was
analyzed twice. The procedure was repeated for the nonequivalent groups.
First the responses for combined Forms A and B for each dataset were
analyzed simultaneously. Then each form was analyzed separately. This
resulted in a total of 520 BILOG runs.
Analytical Estimation
Unidimensional estimation of the multidimensional item parameters for
the eight datasets was performed analytically using Wangs (1986) procedure.
The SAS IML procedure was employed to determine the unidimensional
estimates of the two-dimensional item parameters for each of the eight
conditions.
Equating
In IRT, because the ICCs are population independent, item parameter
estimates from two BILOG runs should theoretically be identical. However,
P,{6) in the 2PL model is a function of the quantity a, (6 b,). As such, the
origin and the unit of 6 and b, measurement are arbitrary or indeterminant.
Any scale may be selected for 6 as long as the same scale is chosen for b,.
Estimated abilities and item difficulties from two calibration runs should have a


131
Table 40
Noncompensatory Item Parameters for Multidimensional Items in MD30 Form B
Item
Form
orn
a1
a2
b,
b2
4
A.B
60
0.664
0.888
-0.945
0.309
5
A,B
20
0.778
0.528
0.236
-2.081
6
A,B
30
0.528
0.447
-0.713
-2.092
7
A,B
45
0.705
0.698
-1.534
-1.596
8
A,B
60
0.352
0.395
-3.164
-1.776
9
A,B
20
0.478
0.390
-1.175
-3.624
10
A,B
30
0.638
0.494
0.834
-0.661
11
A,B
45
0.849
0.844
0.555
0.565
12
A,B
60
0.728
0.942
-1.999
-0.964
48
B
60
0.492
0.595
-1.443
-0.150
49
B
20
1.143
0.674
0.583
-1.718
50
B
30
0.592
0.481
-0.009
-1.413
51
B
45
1.329
1.388
0.690
0.645
52
B
60
0.562
0.717
-0.967
0.331
53
B
20
1.294
0.705
0.019
-2.054
54
B
30
0.609
0.485
0.338
-1.106
55
B
45
0.365
0.366
-1.455
-1.516
56
B
60
0.435
0.508
-1.994
-0.692
57
B
20
1.043
0.647
-0.594
-2.610
58
B
30
0.625
0.523
-1.062
-2.318
59
B
45
0.751
0.750
-1.823
-1.884
60
B
60
0.403
0.462
-2.594
-1.291
61
B
20
0.803
0.545
-0.087
-2.307
62
B
30
0.899
0.647
0.429
-0.929
63
B
45
0.856
0.844
-1.536
-1.601
64
B
60
0.333
0.100
-2.658
3.029
65
B
20
0.934
0.577
-0.713
-2.671
66
B
30
0.614
0.503
-0.471
-1.803
67
B
45
0.423
0.425
-1.983
-2.039
68
B
60
0.368
0.418
-2.663
-1.290


137
Table 46
Analytical Estimates of Unidimensional Item Parameters for MD20 Form B
Item
Form
(Xii
Compensatory
Noncompensatory
a
b
a
b
1
A,8
0
0.320
1.178
0.320
1.178
2
A,B
0
0.771
-0.937
0.771
-0.937
3
A.B
0
1.104
-1.262
1.104
-1.262
4
A,B
0
0.703
0.092
0.703
0.092
5
A,B
0
0.414
-1.817
0.414
-1.817
6
A,B
0
0.359
-2.053
0.359
-2.053
7
A,B
45
1.126
-1.180
0.921
0.290
8
A,B
60
0.584
-0.589
0.610
-0.786
9
A,B
20
0.456
0.134
0.376
-2.233
10
A,B
30
0.630
-0.605
0.513
-0.781
11
A,B
45
0.901
-0.735
0.096
0.465
12
A,B
60
0.470
-0.013
0.499
-1.543
41
B
0
0.512
1.219
0.830
1.800
42
B
0
0.578
0.685
0.576
-0.127
43
B
0
0.629
0.047
0.701
-1.235
44
B
0
0.632
0.815
0.334
-0.117
45
B
0
0.691
0.776
0.495
0.296
46
B
0
0.620
-1.351
0.485
0.388
47
B
0
0.592
1.617
0.226
0.450
48
B
0
0.458
0.897
0.393
0.992
49
B
0
0.186
-0.305
2.096
-1.714
50
B
0
0.548
-0.056
0.602
-0.222
51
B
0
0.660
1.244
0.427
-0.560
52
B
0
0.782
0.278
0.344
0.179
53
B
0
0.576
-0.136
1.003
-0.364
54
B
0
0.168
0.084
0.535
-1.041
55
B
45
0.213
-0.154
0.217
-2.780
56
B
60
0.613
0.002
0.634
-1.324
57
B
20
0.986
0.645
0.687
-2.172
58
B
30
0.664
0.847
0.540
-2.466
59
B
45
0.786
-0.985
0.677
-0.132
60
B
60
0.676
-0.468
0.697
-0.783
61
B
20
0.701
1.564
0.546
-3.552
62
B
30
0.408
0.614
0.365
-2.636
63
B
45
0.402
1.527
0.396
-3.362
64
B
60
0.310
-2.295
0.325
-0.054
65
B
20
0.213
0.149
0.188
-3.965
66
B
30
0.831
0.259
0.634
-1.611
67
B
45
0.261
1.281
0.271
-3.683
68
B
60
0.390
1.928
0.425
-3.584


This dissertation is dedicated to the memory of my father
James F. Duffy
1929- 1992


116
With the nonequivalent groups, multidimensionality becomes more of a
consideration. The dimensionality of a given test is not merely a function of the
item parameters, but an interaction between the items and the examinees (Stout,
1990). This appears to be occurring in the data investigated in this study. Even if
the test itself is considered essentially unidimensional, the introduction of the
nonequivalent groups seems to have negated any tendency for the equatings to
perform well. Concurrent calibration seems most affected by the interaction of
the multidimensional items and dissimilar examinee groups.
Implications
The results of this study failed to reveal any effect on unidimensional
equating results associated with the choice of multidimensional model. With
randomly equivalent groups, there was little difference attributable to
unidimensional equating procedure. The number of multidimensional items of
had an effect if either the dominant first factor or the composite of traits defined
by the average is of interest. If the composite defined by the analytical
approximations is of interest, the number of multidimensional items had no effect
on equating results. With two groups who differ in ability levels, there was an
effect of both equating procedure and number of multidimensional items. Further
investigation is needed with other combinations of ability levels to determine what
differences can be tolerated. In this study, the effect of model was not
investigated. Although no differences were found due to model with randomly
equivalent groups, further research is needed to determine the effect of model on
nonequivalent groups.


132
Table 41
Noncompensatory Item Parameters for Multidimensional Items in MD40 Form A
Item
Form
Oil
ai
a2
b,
b2
1
A,B
20
1.023
0.595
1.034
-1.132
2
A,B
30
0.932
0.687
-0.250
-1.413
3
A.B
45
0.796
0.844
-1.730
-1.621
4
A,B
60
0.743
1.141
-0.440
0.815
5
A.B
20
0.403
0.324
-0.410
-3.057
6
A,B
30
0.712
0.583
-0.621
-1.668
7
A,B
45
0.582
0.598
-0.650
-0.591
8
A,B
60
0.467
0.583
-1.664
-0.358
9
A,B
20
1.576
0.752
0.841
-1.235
10
A,B
30
0.935
0.825
-2.037
-2.758
11
A,B
45
0.468
0.482
-1.432
-1.358
12
A,B
60
0.519
0.668
-1.123
0.164
13
A
20
0.726
0.499
0.142
-1.977
14
A
30
0.605
0.491
0.490
-0.776
15
A
45
0.613
0.633
-0.543
-0.488
16
A
60
0.458
0.569
-2.129
-0.821
17
A
20
1.205
0.662
0.363
-1.623
18
A
30
0.994
0.754
-0.601
-1.553
19
A
45
1.030
1.095
-0.145
-0.112
20
A
60
0.336
0.403
-2.146
-0.601
21
A
20
0.602
0.420
0.645
-1.720
22
A
30
0.845
0.708
-1.442
-2.314
23
A
45
0.530
0.544
-1.227
-1.153
24
A
60
0.533
0.750
-3.361
-1.948
25
A
20
1.185
0.721
-0.475
-2.198
26
A
30
0.939
0.728
-0.655
-1.610
27
A
45
0.630
0.644
-1.116
-1.037
28
A
60
0.341
0.409
-2.834
-1.313
29
A
20
0.496
0.398
-0.870
-3.100
30
A
30
0.688
0.559
-0.087
-1.219
31
A
45
0.547
0.565
-1.453
-1.372
32
A
60
0.334
0.400
-2.929
-1.391
33
A
20
1.374
0.867
-1.095
-2.547
34
A
30
0.477
0.406
-0.071
-1.394
35
A
45
0.495
0.508
-0.461
-0.417
36
A
60
0.339
0.410
-3.552
-1.999
37
A
20
2.651
0.528
-2.881
3.975
38
A
30
1.001
0.317
1.926
0.677
39
A
45
0.644
0.669
-0.344
-0.300
40
A
60
0.683
0.932
-1.394
-0.232


THE EFFECT OF MULTIDIMENSIONALITY ON
UNIDIMENSIONAL EQUATING WITH
ITEM RESPONSE THEORY
By
PATRICIA DUFFY SPENCE
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
1996


30
The parameters of each total test-anchor test combination are
estimated sequentially in the fixed bs method. After the item parameters have
been estimated for one test, the item difficulties of the linking items obtained
from the first calibration are used as input for the estimation of parameters on
the second test. The linking item parameters are not reestimated. The end
result is item parameters for both tests being placed on the same scale
(Petersen, Cook, & Stocking, 1983).
In the equated bs method, the parameters for each test are estimated
separately. Then the means and standard deviations of the difficulties for the
two sets of linking items are set to be equal. Ability estimates could also be
used for this purpose. This linear transformation is then applied to the a,, b,,
and 6 parameters of the second test (Petersen et al., 1989). Several
variations of the transformation, including the mean and sigma method and the
robust mean and sigma method, are described in Hambleton and
Swaminathan (1985). Also, Stocking and Lord (1983) described a modification
which gives lower weights to poorly estimated parameters and outliers.
It is most common in both the fixed bs and equated bs methods to use
only the relationship for item difficulties to obtain the equating function
(Hambleton & Swaminathan, 1985). The characteristic curve method can
prevent the possible loss of information caused by ignoring the discrimination
relationship. For the characteristic curve method, the parameters of each test
are calibrated separately. All parameters are then placed on the same scale


Table 3
Summary of Studies of Unidmensional IRT Estimation with Multidimensional Data
Study
Tests
Model for
Simulating
Estimation
Model
Number of
Dimensions
Independent Variables
Ackerman (1989)
Simulation
Least-squares
conversion
M2PL, Comp.
2PL
2
p(e,, e2)
Difficulty confounded with dimensionality
Comp, vs noncomp, models
BILOG vs LOGIST
Ansley & Forsyth
(1985)
Simulation
M3PL, Noncomp.
3PL
2
p(e,. e2)
Sample size
Test length
Drasgow & Parsons Simulation
(1983)
Hierarchical
factor model
2PL
1 5 p(8i, e2...en)
General factor strength
Harrison (1986)
Simulation
Hierarchical
factor model
2PL
1-8
General factor strength
# of common factors
Test length
Reckase (1979)
Simulation,
Missouri
Linear factor
analysis
Rasch
3PL
2-9
# of dimensions
Estimation methods
Reckase, Ackerman,Simulation,
& Carlson (1988) ACT
M2PL, Comp.
2PL
2
Violation of unidimensionality
Yen (1984)
Simulation,
M3PL, Comp.
3PL
2
p(0i. 62)
a parameters
Note. Comp. = Compensatory model, Noncomp. = Noncompensatory model.


143
Table 52
DescriDtive Statistics for ComDensatorv MD20 Linkina Items with Randomly
Eauivalent Gtouds
Replication
Form A
Form B
Mean
SD
Mean
SD
1
-0.646
0.885
-0.661
0.824
2
-0.666
0.892
-0.652
0.931
3
-0.653
0.949
-0.631
0.836
4
-0.601
0.862
-0.643
0.807
5
-0.623
0.823
-0.570
0 843
6
-0.600
0.882
-0.651
0.886
7
-0.619
0.918
-0.686
0.889
8
-0.674
0.934
-0.660
0.877
9
-0.664
0.907
-0.656
0.858
10
-0.631
0.877
-0.651
0.962
11
-0.652
0.972
-0.687
0.870
12
-0.751
0.909
-0.653
0.837
13
-0.613
0.831
-0.626
0.895
14
-0.630
0.943
-0.665
0.875
15
-0.608
0.866
-0.718
0.924
16
-0.698
0.950
-0.647
0.907
17
-0.639
0.904
-0.645
0.905
18
-0.671
0.958
-0.239
1.019
19
-0.575
0.869
-0.713
1.045
20
-0.682
1.059
-0.651
0.891


97
increases. Conversely, the strength of the correlation between the estimated 9s
and the average of the simulated 0iS and 82s increases as the number of
multidimensional items increases. The same pattern of decreasing desirability is
repeated in the SRMSD values. This agrees with earlier findings that as
multidimensionality in a test becomes more predominant, the abilities should be
viewed as a composite of the underlying traits (Ackerman, 1989; Way, et al.,
1988; Yen, 1984). The average of the abilities seems to recover this composite
more accurately than simply using the dominant first dimension to define the
multidimensional data. There is virtually no difference in the results for Forms A
and B
When examining the results of comparisons with the analytical
estimations, outcomes are quite different. For all conditions, correlations are
similar and very high at .97 or .98. Similarly, the SRMSD values are
approximately equal across conditions and are relatively low. Although the data
were simulated in a manner that conforms to the theory behind the analytical
approximation procedure, differences in the amount of multidimensionality may
have had an effect on analytical estimates. For concurrent calibration, the
analytical estimation procedure appears to describe the unidimensional equating
with multidimensional data well, and it is not affected by the amount of
dimensionality in the test. Results are almost identical for both forms.
The application of the SDM statistic provided comparable results across
all conditions and baselines. All values are close to 0.0 denoting close


150
Table 59
Descriptive Statistics for Compensatory Linking Items with Noneauivalent
Examinee Groups
Form A
(Average Ability)
Form B
(Low Ability)
Replication
Mean
S.D.
Mean
S.D.
MD10
1
-0.20
0.79
0.90
1.15
2
-0.10
0.79
0.93
1.25
3
-0.12
0.80
1.03
1.39
4
-0.17
0.81
1.01
1.37
5
-0.17
0.89
1.05
1.28
MD20
1
-0.65
0.85
0.10
1.31
2
-0.67
0.85
0.11
1.39
3
-0.65
0.85
0.04
1.32
4
-0.60
0.82
0.13
1.39
5
-0.62
0.82
0.39
1.83
MD30
1
-0.01
1.08
0.88
1.53
2
-0.05
1.09
0.77
1.62
3
-0.11
1.09
0.71
1.44
4
-0.16
1.07
0.63
1.47
5
-0.04
1.08
0.73
1.37
MD40
1
-0.31
0.97
0.57
1.38
2
-0.32
0.96
0.43
1.37
3
-0.31
0.95
0.54
1.37
4
-0.33
0.94
0.53
1.51
5
-0.25
0.95
0.54
1.51


39
Reckase (1985) has alternately defined the compensatory M2PL to
provide a simple framework for specifying and generating multidimensional
item response data. This model defines the probability of a correct response
as
P(Xj = l|a,d¡,ej) =
EXP^, § + dj)
1 + EXP(a¡', ft + dj)
(11)
where §j is a vector of discrimination parameters; dj is related to item difficulty;
andjj is a vector of ability parameters. The exponent can also be written as
3jh(fth bjh) (12)
h=l
where m is the number of dimensions; ajln is an element of a¡; 0¡h is an element
of ftj; and d¡= -lajhbjh- When this form is used, the relationship to the more
familiar expression in Equation 9 can be seen.
The data described by a multidimensional IRT model can be depicted
graphically by an item response surface (IRS). Figure 2 presents an IRS for
an M2PL item. The IRS increases monotonically as the elements of 0;
increase (Reckase, 1985).
To identify the multidimensional item difficulty (MID) for an item, the
point in the IRS where the Item is most discriminating must be found. This
point, which provides the maximum information about an examinee, will have
the greatest slope. Because the slope along the IRS can differ according to


134
Table 43
Analytical Estimates of Unidimensional Item Parameters for MD10 Form A
Item
Form
Oil
Compensatory
Noncompensatory
a
b
a
b
1
A.B
0
0.763
-0.336
0.763
-0.336
2
A,B
0
0.775
0.476
0.775
0.476
3
A,B
0
0.543
-0.514
0.543
-0.514
4
A,B
0
0.369
0.120
0.369
0.120
5
A,B
0
0.386
-0.954
0.386
-0.954
6
A,B
0
1.123
-0.016
1.123
-0.016
7
A,B
0
0.267
-0.167
0.267
-0.167
8
A,B
0
0.957
1.284
0.957
1.284
9
A,B
0
1.086
-1.173
1.086
-1.173
10
A,B
30
0.274
-1.093
0.273
-1.399
11
A.B
45
0.656
0.655
0.762
-1.694
12
A.B
60
0.352
0.086
0.563
-1.441
13
A
0
0.846
-1.209
0.846
-1.209
14
A
0
0.352
-0.033
0.352
-0.033
15
A
0
0.400
-1.951
0.400
-1.951
16
A
0
0.344
-1.016
0.344
-1.016
17
A
0
0.719
-0.449
0.719
-0.449
18
A
0
2.008
-0.482
2.008
-0.482
19
A
0
0.299
-0.104
0.299
-0.104
20
A
0
0.908
0.117
0.908
0.117
21
A
0
0.258
0.553
0.258
0.553
22
A
0
0.361
0.224
0.361
0.224
23
A
0
1.876
0.699
1.876
0.699
24
A
0
0.915
1.303
0.915
1.303
25
A
0
0.562
-0.774
0.562
-0.774
26
A
0
0.413
0.639
0.413
0.639
27
A
0
0.286
-0.565
0.286
-0.565
28
A
0
0.242
-0.938
0.242
-0.938
29
A
0
0.478
1.631
0.478
1.631
30
A
0
1.713
-0.404
1.713
-0.404
31
A
0
0.648
-1.653
0.648
-1.653
32
A
0
1.301
0.302
1.301
0.302
33
A
0
1.108
-0.778
1.108
-0.778
34
A
30
0.484
1.615
0.469
-3.307
35
A
45
0.518
-0.232
0.612
-1.041
36
A
60
0.186
2.625
0.314
-3.876
37
A
20
0.862
-1.269
0.679
0.089
38
A
30
0.736
0.375
0.668
-1.632
39
A
45
0.180
-1.214
0.223
-1.860
40
A
60
0.318
0.151
0.510
-1.587


42
distinguish unidimensional from multidimensional data sets. A major problem
encountered by Hattie in assessing the indices was that unidimensionality was
often confused with reliability, internal consistency, and homogeneity.
More recently, other procedures have been developed to assess the
dimensionality of latent traits. Roznowski, Tucker, and Humphreys (1991)
explored several of these indices. Procedures based on the shape of the
curve of successive eigenvalues were found to be unsatisfactory under most
conditions. A pattern index of second factor loadings was accurate except with
high obliqueness. The most accurate index in this study was based on local
independence. The use of this index is particularly recommended with large
samples and many items.
Linear factor analysis has been widely used to assess dimensionality of
dichotomous items. However, use of phi correlations often leads to
overestimation of the number of factors underlying the responses by
confounding factor coefficients with item difficulties (Bock, Gibbons, & Muraki,
1988; Hambleton & Swaminathan, 1985). Tetrachoric correlations may be
substituted, but may still be confounded with item difficulty or guessing in real
data (Camilli, 1992). Bock, Gibbons, and Muraki (1988) have developed a
maximum likelihood full information factor analysis procedure as an attempt to
deal with these problems.
Another approach to dimensionality taken by Stout (1990) replaced the
strong assumptions of unidimensionality and local independence with less


75
example, although mathematics problem solving requires reading skills to
understand the prompts, the reading level Is usually well below the grade level
being tested. In this study, the simulated ability parameters of the first
dimension only from each compensatory and noncompensatory dataset were
utilized. This comparison criterion would enable evaluation of how well the
dominant first factor was recovered in the equatings.
A third comparison condition was created which employed the
averages of the two true 6 values. This condition was based on the parameter
estimation studies of Yen (1984) and Ansley and Forsyth (1985) in which the
unidimensional estimates of the 9 parameters appeared to be combinations of
the true multidimensional abilities.
Statistical Criteria
Correlation coefficients between the simulated 6 and the equated 6
estimates were computed to establish the relationship between the comparison
criterion and the research equatings for each condition. For concurrent
calibration, the appropriate simulated 0 parameters were correlated to the
corresponding estimated ability parameters for both Form A and Form B. Only
the equated form, Form B, was compared to the comparison conditions for all
other equating procedures.
The standardized difference between means (SDM) is the difference in
mean scores for the two sets of ability traits divided by a pooled estimate of the
standard deviation


147
Table 56
Descriptive Statistics for Noncompensatory MD20 Linking Items
Replica ticn
Form A
Form B
Mean
SD
Mean
SD
1
0.961
1.386
0.988
1.374
2
0.969
1.393
0.924
1.405
3
0.893
1.356
0.903
1.211
4
0.905
1.217
1.047
1.386
5
1.013
1.327
1.043
1.336
6
0.995
1.244
0.974
1.229
7
1.002
1.355
0.970
1.396
8
0.889
1.248
0.966
1.290
9
0.946
1.308
0.883
1.090
10
0.894
1.080
1.055
1.512
11
0.953
1.362
1.072
1.770
12
1.042
1.717
1.001
1.397
13
0.970
1.411
1.262
1.784
14
1.184
1.663
0.748
1.134
15
0.759
1.145
0.951
1.323
16
0.936
1.278
1.016
1.359
17
1.033
1.314
1.030
1.532
18
0.987
1.455
1.078
1.434
19
1.042
1.445
0.930
1.174
20
0.926
1.190
1.008
1.523


10
Graduate Record Examination (GRE) data. Some minor discrepancies were
discovered, but the majority were not significant in horizontal equating
situations.
Cook, Eignor, and Taft (1988) hypothesized that differences in ability
were expected when the groups took the two tests to be equated at different
times of the year. Two forms of the Biology achievement test were
administered. One form was given in the fall mainly to high school seniors, and
the other form was administered predominantly to sophomores in the spring.
Two fall administrations were also equated and studied. Because recency of
instruction is important in some parts of this type of achievement test and most
students study Biology in tenth grade, disparate results were attained from the
fall/spring equating. The spring sample, containing mostly students who had
just completed the subject tested, received higher scaled scores than the fall
sample. In this study, the construct measured by the test depended on the
sample of examinees to whom the test was administered. In contrast, the
fall/fall equating was robust to group differences. This study demonstrates the
importance of administering the test forms to be equated at the same time,
especially when the content is instructionally sensitive.
Anchor-test designs
Lord (1980) stated the differences between two samples of examinees
can be measured and controlled by administering to each examinee an anchor
test measuring the same ability as tests X and Y. When an anchor test is used,


34
Harris and Kolen (1985) compared conventional equating methods with
IRT 3PL model equating. The sample consisted of high and low ability
examinees. The 3PL model was found to be slightly superior.
The Cook, Eignor, and Taft (1988) study using biology achievement
tests administered at different points in time included a 3PL model with the
characteristic curve transformation in addition to the equipercentile equating
method. The authors concluded that the IRT results, although slightly superior
with the fall-to-spring sample equating, basically paralleled the results obtained
with the conventional method.
A minimum-competency test, Florida's Statewide Student Assessment
Test, Part II (SSAT-II) was equated by Hills, Subhiyah, and Hirsch (1988).
Their purpose was to study the effect of anchor length on equating and
compare different equating methods using a sample with a negatively skewed
distribution. The equating methods investigated were linear, Rasch, and 3PL.
The IRT models were equated with concurrent calibration, fixed bs method,
and equated bs method using robust mean and sigma. The authors concluded
that the 3PL model with concurrent calibration and Rasch models gave similar
good results. Also, when using the 3PL model with concurrent calibration, an
anchor test length of 10 items was found to be sufficient for good equating
outcomes.
Results of these studies indicate that the 3PL model tends to perform
better than conventional and Rasch equating in a variety of situations.


Table 4
Summary of Studies of Unidimensional Equating with Multidimensional Data
Study
Tests
Model
Equating
Method
Number of
Dimensions
Independent Variables
Evaluation
Criterion
Camilli, Wang
& Fesq (1995)
LSAT
3PL
true-score
2
test dimensionality
equating method
split test
Cook & Eignor
(1988)
SAT
3PL
concurrent
calibration
characteristic
curve trans.
unknown
equating methods
scale drift
Doody-Bogan
& Yen (1983)
Simulation
M3PL
3PL
equated bs
2
criterion measures
P correlation
SDM
SRMSD
Dorans &
Kingston
(1985)
GRE-V
3PL
equated bs
2
calibration procedures
data collection design
split test
Yen (1984)
CTBS/U
3PL
Simulation
equated bs CTBS-unknown
Sim.- 2
a & b parameters
p(e,,02)
correlation
SDM
SRMSD
ratio of a


8
It is unlikely that all conditions of equating can be met in practice.
However, good approximations to this ideal can be achieved and are usually
fairer to examinees than if no attempt at equating had occurred (Petersen et al.,
1989). Research conducted over the past 20 years serves as a guide in the
application and interpretation of equating transformations.
Data Collection Designs
Every equating consists of two parts--a data collection design and an
analytical method to determine the appropriate transformation. Three basic
sampling designs are most frequently described in the literature (Dorans, 1990;
Dorans & Kingston, 1985; Petersen et al., 1989). The designs are classified as
(a) single-group designs, (b) equivalent-groups designs, and (c) anchor-test
designs.
Single-group designs
In single-group designs, both forms or tests to be equated are given to
the same group of examinees. The difficulty levels of the tests are not
confounded with the differences in the ability levels of the groups taking each
test because the examinees are the same (Hambleton & Swaminathan, 1985).
However, Lord (1980) pointed out that the test administered second is not being
given under typical conditions. Practice effects and fatigue may affect the
equating process. To deal with this threat, the counterbalanced random-groups
design may be employed. The single-group is divided into two random half
groups. Both half-groups then take both tests in counterbalanced order, one


122
Table 31
Simulated Compensatory Item Parameters for MD20 Form B
Item
Form
Oil
ai
a2
di
MDISC
MID
1
A,B
0
0.607
0.000
-0.649
0.607
1.069
2
A,B
0
1.547
0.000
1.315
1.547
-0.850
3
A,B
0
2.411
0.000
2.761
2.411
-1.145
4
A,B
0
1.394
0.000
-0.117
1.394
0.084
5
A,B
0
0.791
0.000
1.304
0.791
-1.649
6
A,B
0
0.682
0.000
1.270
0.682
-1.862
7
A,B
45
1.581
1.581
2.479
2.236
-1.108
8
A,B
60
0.666
1.153
0.642
1.331
-0.482
9
A,B
20
0.732
0.266
-0.104
0.779
0.134
10
A,B
30
0.934
0.539
0.650
1.079
-0.603
11
A,B
45
1.223
1.223
1.194
1.730
-0.690
12
A,B
60
0.516
0.894
0.011
1.032
-0.010
41
B
0
0.990
0.000
-1.095
0.990
1.106
42
B
0
1.125
0.000
-0.699
1.125
0.621
43
B
0
1.232
0.000
-0.053
1.232
0.043
44
B
0
1.240
0.000
-0.917
1.240
0.740
45
B
0
1.368
0.000
-0.964
1.368
0.705
46
B
0
1.214
0.000
1.488
1.214
-1.226
47
B
0
1.154
0.000
-1.693
1.154
1.467
48
B
0
0.878
0.000
-0.715
0.878
0.814
49
B
0
0.350
0.000
0.097
0.350
-0.278
50
B
0
1.063
0.000
0.054
1.063
-0.051
51
B
0
1.299
0.000
-1.466
1.299
1.128
52
B
0
1.574
0.000
-0.398
1.574
0.253
53
B
0
1.121
0.000
0.139
1.121
-0.124
54
B
0
0.315
0.000
-0.024
0.315
0.076
55
B
45
0.273
0.273
0.056
0.386
-0.145
56
B
60
0.706
1.223
-0.003
1.413
0.002
57
B
20
1.587
0.578
-1.087
1.689
0.644
58
B
30
0.983
0.567
-0.958
1.135
0.844
59
B
45
1.050
1.050
1.374
1.485
-0.925
60
B
60
0.798
1.382
0.612
1.596
-0.383
61
B
20
1.126
0.410
-1.868
1.198
1.560
62
B
30
0.604
0.349
-0.427
0.697
0.612
63
B
45
0.520
0.520
-1.055
0.735
1.435
64
B
60
0.330
0.571
1.239
0.659
-1.881
65
B
20
0.341
0.124
-0.054
0.363
0.148
66
B
30
1.231
0.711
-0.368
1.422
0.259
67
B
45
0.335
0.335
-0.570
0.474
1.202
68
B
60
0.421
0.730
-1.329
0.842
1.578


13
accomplished (Cook & Petersen, 1987). Wingersky and Lord (1984) studied the
problem of the optimal number of linking items in the context of IRT concurrent
calibration. The authors concluded that two linking items with small standard
errors of estimation worked almost as well as a set of 25 linking items with large
standard errors of estimation.
Wingersky, Cook, and Eignor (1986) studied the characteristics of linking
items and their effects on IRT equating. Monte Carlo procedures were used with
parameter values set to imitate those estimated from the Verbal sections of the
College Board Scholastic Aptitude Test (SAT-V). These values were selected to
make the simulation as realistic as possible. Linking test lengths of 10, 20, and
40 items were used as well as variations in the size of the standard errors of
estimation and distributions of examinee ability. Scaling was accomplished by
both concurrent calibration and characteristic curve methods. The results of this
study showed little difference between the two scaling methods, and the accuracy
of the both equating methods improved as the number of linking items increased.
Unlike the findings of Wingersky and Lord (1984), linking items having standard
errors of estimation similar to those found in actual SAT-V items provided slightly
better equating outcomes than those chosen to have small errors of estimation.
The studies reviewed clearly indicate that the properties of an anchor test
are of great concern. Anchor or linking items should remain in the same relative
positions in new and old forms and as many anchor items as possible should be
used (Cook & Eignor, 1988). The question of optimal anchor test length becomes


94
Table 19
Descriptive Statistics for Simulated Examinees Taking MD40
THETA 1 THETA 2
Rep
Low
High
Mean
SD
Low
High
Mean
SD
1
-2.85
3.79
0.02
0.96
-3.26
2.73
-0.04
0.98
2
-3.21
2.99
0.00
1.02
-3.57
3.18
0.00
1.00
3
-3.40
2.87
-0.03
1.03
-3.73
3.47
0.03
0.97
4
-3.30
3.17
0.01
1.03
-2.89
3.18
0.03
1.02
5
-3.12
3.59
-0.01
1.03
-2.78
2.81
-0.01
0.97
6
-3.58
3.54
-0.02
1.00
-3.27
3.14
0.05
1.01
7
-3.09
3.26
-0.04
1.00
-3.09
3.13
0.01
1.05
8
-3.40
2.75
-0.02
1.01
-2.81
3.37
-0.04
1.01
9
-3.60
4.23
-0.04
1.03
-3.06
3.38
0.03
1.05
10
-3.51
3.69
-0.05
1.02
-2.77
2.87
-0.01
0.97
11
-3.24
2.83
0.03
1.05
-3.45
2.74
-0.01
1.00
12
-3.81
2.66
-0.04
1.02
-2.99
3.28
0.00
0.98
13
-3.10
2.77
-0.02
1.02
-2.91
2.56
0.00
1.00
14
-3.05
3.44
-0.03
1.00
-3.27
3.29
0.00
1.02
15
-3.44
3.13
0.04
1.02
-3.82
3.43
0.00
1.01
16
-3.03
3.37
-0.02
0.98
-3.59
2.88
-0.03
0.99
17
-3.68
3.63
0.02
1.03
-3.51
3.96
0.02
0.97
18
-2.73
3.17
0.01
0.98
-2.83
3.27
0.01
1.01
19
-3.59
3.37
0.02
1.03
-2.99
3.35
0.01
0.96
20
-2.69
2.59
-0.02
0.95
-3.29
3.51
-0.01
1.04
Note. N = 1000 for each replication
difference (SRMSD) statistics were calculated for each form. Table 21 presents
results summarized across the 20 replications for compensatory data. For both
Forms A and B, the strength of the correlation between the estimated Os and the
simulated Ots decreases as the number of multidimensional items in the test


47
unidimensional parameter estimation of multidimensional data. A variety of a,
parameters were configured and p(9i, 82) was set at .5 or .6. When
multidimensionality was present, the a, and b, parameter estimates were
larger than those of unidimensional sets of items. The unidimensional
estimates of both a, and 9 parameters appeared to be a combination of the
respective two-dimensional parameters.
Data simulated from a hierarchical factor model was used in a study by
Drasgow and Parsons (1983). Item responses were generated from five
oblique common factors. Loadings were varied producing diversity in
correlations between the common factors. Each simulated dataset consisted
of 50-item tests and 1,000 simulees. The general latent trait was recovered
well when the correlations between the common factors were .46 or higher.
Harrison (1986) also used a hierarchical factor model to simulate data.
The strength of the second-order general factor, the number of first-order
common factors, the distribution of items loading on the common factors, and
the number of test items were manipulated. The effect of test length was
significant. As the number of items increased, the general trait was recovered
more effectively regardless of the latent structure, distribution of items across
common factors, or the number of common factors. Estimation of the b,
parameters was found to be robust to violations of unidimensionality. The
estimation of both the a, and b/ parameters improved as test length and
strength of the general factor increased. In general, Harrison found


100
Table 22
Constants for Equated bs Equating of Compensatory Forms with Randomly Equivalent
Groups
MD10 MD20 MD30 MD40
Rep
Slope
Intercept
Slope
Intercept
Slope
Intercept
Slope Intercept
1
1.04
-0.08
1.07
0.06
0.93
0.03
1.08
0.07
2
0.94
-0.04
0.94
-0.04
0.94
-0.04
0.94
-0.04
3
1.07
0.09
1.14
0.06
1.03
0.06
1.06
0.04
4
0.90
-0.01
1.07
0.09
1.04
-0.15
1.02
-0.06
5
1.13
-0.01
0.98
-0.07
0.96
0.13
0.96
0.08
6
0.94
0.04
1.00
0.05
0.88
-0.06
1.02
-0.14
7
1.09
0.00
1.03
0.09
1.06
-0.06
1.02
0.04
8
0.90
-0.06
1.07
0.03
1.07
0.00
0.94
0.04
9
1.09
-0.02
1.06
0.03
0.96
0.04
0.99
-0.13
10
1.02
0.02
0.91
-0.04
0.99
0.05
1.01
0.08
11
1.04
-0.01
1.12
0.12
0.91
-0.10
1.04
-0.03
12
0.98
0.06
1.09
-0.04
1.03
0.03
1.02
0.01
13
1.01
0.00
0.93
-0.03
0.95
0.09
1.02
0.00
14
1.01
-0.04
1.08
0.09
1.00
-0.12
1.10
0.10
15
1.04
0.02
0.94
0.06
1.04
-0.02
0.91
-0.09
16
1.01
0.09
1.05
-0.02
0.97
0.02
1.06
0.04
17
1.05
-0.04
1.00
0.00
1.03
0.02
1.04
0.01
18
0.90
-0.01
0.94
-0.45
0.98
-0.03
0.98
0.04
19
1.17
0.06
0.83
0.02
1.03
0.00
1.08
-0.07
20
1.00
0.05
1.19
0.09
1.02
-0.14
0.93
0.05
In comparing the results of the equated bs procedure with those of
concurrent calibration, some small differences can be noted but they do not form
a consistent pattern. In general, the results are similar to those found for
concurrent calibration. The results are also similar for compensatory and
noncompensatory data.


22
models due to the ease of estimation. Birnbaum (1968) proposed a two-
parameter logistic model (2PL) of the form
Pl(e) = [1+eD*m>lr1 (4)
where b, is the difficulty value, a, Is the discrimination parameter, and D is a
scaling factor, normally 1.7.
The three-parameter logistic model (3PL) adds a third parameter,
denoted c,, referred to as the lower asymptote. The mathematical form of the
3PL model is written as
Pi(0) = Ci+(1-c)[1+e-*)r1 (5)
with the a,, b,, and D defined as before. The value of c, is typically smaller
than the value that would result if examinees were to make a random response
to the item (Hambleton & Swaminathan, 1985). Figure 1 depicts an ICC based
on the 3PL model.
The one-parameter logistic model, or Rasch model, assumes all items
have equal discrimination and no guessing occurs. This model is written
Pi(9) = [1+eD)r1 (6)
where the parameters are defined as in the previous models.
Cursory examination of the three IRT logistic models may lead to the
conclusion that they form a type of hierarchy from least to most specific.
However, the three models represent very different philosophical perspectives



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PAGE 175

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24
invariance of item and ability parameters. When the item parameters are
known, an examinee's ability may be estimated from any subset of the items.
Also, item parameters may be calibrated with any sample drawn from a
sufficiently large population (Skaggs & Lissitz, 1986a). These advantages
cannot be derived from classical test theory and should have tremendous
consequences for equating with item response theory.
All of the practical IRT models are based on the unidimensionality
assumption. This states that the probability of a correct response by
examinees to a set of items can be mathematically modeled by using only one
ability parameter (Kingston & Dorans, 1984). According to Lord (1980), while
ability is probably not normally distributed for most groups of examinees,
unidimensionality is a property of the items and does not cease to exist
because the examinee group is changed in distribution.
Because the items on a test are assumed to measure only one common
trait, for all examinees with the same ability the item responses are
independent of one another. This is the local independence assumption. The
probability of success on any given item depends on the item parameters,
examinee ability, and nothing else In determining the probability of a correct
response to a specific item, success or failure on other items will add no new
information if ability is known (Lord, 1980).
Good estimation of the item and ability parameters is of paramount
importance in describing the data accurately. Many investigators have


44
Nandakumar (1991) used simulations to investigate Stout's statistical
test of essential unidimensionality. When one dominant trait and one or more
minor dimensions having little influence on item scores were present, Stout's
test performed well in indicating essential unidimensionality. The test is more
likely to reject the hypothesis of essential unidimensionality as the effect of the
minor dimensions increases.
To facilitate application of the test of essential unidimensionality, Stout
developed the computer program DIMTEST. An investigation of the program
revealed problems when a test consisted of difficult, highly discriminating items
where guessing was also present (Nandakumar & Stout, 1993). Refinements
were subsequently made to the program to make it more robust and beneficial
to the measurement practitioner.
Nandakumar (1994) studied three commonly used methodologies for
assessing dimensionality in a set of item responses. The three procedures-
DIMTEST, Holland and Rosenbaums approach, and nonlinear factor anaiysis-
-were unreliable in detecting lack of unidimensionality in real data sets.
Although the more recent procedures based on local independence, full
information factor analysis, and essential unidimensionality offer promise for
assessing the dimensionality of dichotomous data, especially with large
datasets, a satisfactory method has not yet been agreed upon by
measurement researchers. Because of the current lack of an acceptable index
to detect multidimensionality, it becomes even more urgent to understand


86
(a) Compensatory IRS
(a,=1.223, a2= 1.223, d=1.994)
(b) Noncompensatory IRS
(a,=.970, a2=. 933, b,=-.951, b2=-,913)
(c) Compensatory Contour Plot (d) Noncompensatory Contour Plot
Figure 5. Item response surfaces and contour plots for item 11, MD20, a=45'


REFERENCES
Ackerman, T. A. (1988, April). An explanation of differential item
functioning from a multidimensional perspective. A paper presented at the
annual meeting of the American Educational Research Association, New
Orleans.
Ackerman, T. A. (1989). Unidimensional IRT calibration of compensatory
and noncompensatory multidimensional items. Applied Psychological
Measurement. 13, 113-127.
Ackerman, T. A. (1992). A didactic explanation of item bias, item impact,
and item validity from a multidimensional perspective. Journal of Educational
Measurement. 29, 67-91.
Angoff, W. H. (1971). Norms, scales, and equivalent scores. In R. L.
Thorndike (Ed.), Educational measurement (2nd ed.). Washington, D C.:
American Council on Education.
Angoff, W. H. (1988). Proposals for theoretical and applied development
in measurement. Applied Measurement in Education. 1, 215-222.
Angoff, W. H., & Cowell, W. R. (1986). A examination of the assumption
that the equating of parallel forms of a test is population-independent. Journal
of Educational Measurement. 23, 327-345.
Ansley, T. N., & Forsyth, R. A. (1985). An examination of the
characteristics of unidimensional IRT parameter estimates derived from two-
dimensional data. Applied Psychological Measurement. 9, 37-48.
Baker, F. B. (1990). Some observations on the metric of PC-BILOG
results. Applied Psychological Measurement. 14. 139-150.
Baker, F. B., Al-Kami, A., & Al-Dosary, I. M. (1991). EQUATE: A
computer program for the test characteristic curve method of IRT eouatino.
Madison, Wl: University of Wisconsin.
Birnbaum, A. (1968). Some latent trait models and their use in inferring
an examinee's ability. In F. M. Lord and M. R. Novick, Statistical theories of
mental test scores (pp. 397-479). Reading, MA: Addison-Wesley.
151


21
For the past forty years, large scale testing programs publishing multiple
forms of examinations have used an equating process. Until recently, most
have employed one of the conventional linear or equipercentile procedures
described. But recent psychometric developments have presented an
alternative.
Equating Methods Based on Item Response Theory
Item response theory
A brief introduction to item response theory is essential to an
understanding of the following equating procedures. Item response theory
(IRT) is an attempt to model an examinee's performance on a test item as a
function of the characteristics of the item and the examinee's ability on some
unobserved, or latent, trait. The IRT model specifies the relationship between
a latent trait and the observed performance on items designed to measure that
trait.
This relationship can then be depicted graphically by an item
characteristic curve (ICC). The ICC depicts the probability that an examinee at
any given ability level will make a correct response to an item. The graph is
typically an S-shaped curve with ability, symbolized by 0, plotted on the
horizontal axis and the probability of a correct response to item /, P, (9), plotted
on the vertical axis.
Many different mathematical models may be used to depict this
functional relationship. Most common in practice are the logistic class of


43
restrictive assumptions of essential unidimensionality and essential
independence Stout contended that a dominant dimension results when an
attribute overlaps many items and other dimensions common to only a few
items are unavoidable in reality, but are also not significant. These minor
dimensions are rarely discussed in IRT literature, but are a frequent theme in
classical factor analysis. While the IRT definition of dimensionality would take
all factors, major and minor, into account, essential dimensionality is a
mathematical conceptualization of the number of dominant dimensions with
minor dimensions ignored. An essentially unidimensional test is therefore any
set of items selected from an infinite item pool that measures exactly one
major dimension. When essential unidimensionality is assumed, latent ability
is unique in an ordinal scaling sense and this unique latent ability is estimated
consistently. Stout presented theorems and proofs to show that dimensions
distributed nondensely over items or dimensions that have a minor influence
on possibly many items do not necessarily negate essential unidimensionality.
He continued to present guidelines for development of essentially
unidimensional tests. Among the recommendations are limiting the number of
abilities per item; keeping the number of items dependent on the same ability,
other than the intended-to-be-measured 9, small; and controlling the number of
item pairs assigned to the same ability other than 9. These conditions are
usually met with the carefully designed tests usually found in practice.


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
THE EFFECT OF MULTIDIMENSIONALITY ON
UNIDIMENSIONAL EQUATING WITH
ITEM RESPONSE THEORY
by
Patricia Duffy Spence
May, 1996
Chairman: M. David Miller
Major Department: Foundations of Education
Test publishers apply unidimensional equating techniques to their products
even though tests are expected to be multidimensional to some degree. This
simulation study investigated the effects of ignoring multidimensional data in
applying unidimensional item response theory equating procedures. The
specific effects studied were (a) multidimensional model, (b) type of equating
procedure, (c) number of multidimensional items, and (d) distribution of
examinee ability.
Four test conditions were created by varying the number of multidimensional
items contained in each test. The compensatory multidimensional two-
parameter logistic model was selected for data generation. Four degrees of


7
indifference which form is administered. The equity condition requires the
standard error of measurement and the higher moments to be the same after
transformation for examinees of identical ability. To fully satisfy this requirement,
test forms X and Y must be strictly parallel (Kolen, 1981). However, if this
condition is met, equating is no longer necessary.
In practice, it is nearly impossible to construct multiple forms that are
strictly parallel. Therefore, equating is needed. Although the equity condition can
never be met precisely, it serves to keep the purpose of equating in mind and
guide the steps in the process.
The population invariance and symmetry conditions also arise from the
desire to achieve equivalent scores. If the scores from form X and form Y are
equivalent, there is a one-to-one relationship between the two sets of scores.
The transformation must be unique, independent of the groups used to derive the
conversion (Petersen et al., 1989). The purpose of equating also requires that
the equating function be invertible or symmetric. The equating must be the same
regardless of which test is labelled X and which test is labelled Y (Lord, 1980).
The two tests to be equated must also measure the same characteristic,
whether defined as a latent trait, ability, or skill. This condition distinguishes true
equating from scaling. Scores on X and Y can always be placed on the same
scale, but they must measure the same construct to be considered equated
(Dorans, 1990).


138
Table 47
Analytical Estimates of Unidimensional Item Parameters for MD30 Form A
Item
Form
Oil
Compensatory
Noncompensatory
a
b
a
b
1
A,B
0
0.242
1.408
0.242
-1.408
2
A,B
0
0.286
0.352
0.286
-0.352
3
A.B
0
0.262
-1.449
0.262
1.449
4
A,B
60
0.679
-0.949
0.619
-0.330
5
A,B
20
0.712
-0.559
0.551
-0.975
6
A.B
30
0.479
0.066
0.406
-1.897
7
A.B
45
0.733
0.737
0.577
-2.229
8
A,B
60
0.289
1.237
0.305
-3.490
9
A,B
20
0.422
0.833
0.362
-3.202
10
A,B
30
0.599
-1.627
0.474
0.254
11
A.B
45
0.875
-1.742
0.696
0.798
12
A,B
60
0.790
0.673
0.667
-2.051
13
A
0
0.482
-0.646
0.482
0.646
14
A
0
0.650
0.273
0.650
-0.273
15
A
0
0.842
-0.874
0.842
0.874
16
A
0
0.429
0.698
0.429
-0.698
17
A
0
0.687
-1.216
0.687
1.216
18
A
0
0.715
0.338
0.715
-0.338
19
A
0
0.335
0.245
0.335
-0.245
20
A
60
0.466
-0.877
0.446
-0.786
21
A
20
0.400
0.808
0.139
-3.990
22
A
30
1.320
-0.442
0.928
-0.412
23
A
45
0.868
-0.429
0.669
-0.690
24
A
60
0.267
0.265
0.407
-3.941
25
A
20
0.487
-0.355
0.402
-1.642
26
A
30
0.300
-0.312
0.270
-2.204
27
A
45
1.103
-0.010
0.770
-1.090
28
A
60
0.325
0.432
0.333
-2.545
29
A
20
0.459
-0.070
0.384
-2.047
30
A
30
0.270
-0.685
0.243
-2.039
31
A
45
0.283
-1.913
0.258
-0.488
32
A
60
0.452
1.327
0.440
-3.108
33
A
20
0.882
0.669
0.646
-2.307
34
A
30
0.700
0.239
0.545
-1.760
35
A
45
0.690
-1.104
0.567
-0.083
36
A
60
0.421
1.304
0.417
-3.137
37
A
20
0.363
-0.862
0.307
-1.559
38
A
30
0.777
-1.738
0.588
0.639
39
A
45
0.637
-0.953
0.531
-0.320
40
A
60
0.148
0.536
0.118
-6.017


107
lower, and the SRMSDs much higher than the corresponding results for
concurrent calibration equating with the randomly equivalent examinee groups.
Correlations of equated abilities with the average of simulated 01 and 02
increase with the number of multidimensional items. The SDM values do not
show as serious a departure from zero with the average of simulated @i and 02
as is found with the 0i comparison. Furthermore, the increase in SRMSDs is
not as marked with the ability average comparison condition. Examination of
the data for nonequivalent examinee groups reveals lower correlations, lower
SDMs, and higher SRMSDs across all conditions than those found for
equivalent groups. This seems to suggest that of the two conditions comparing
the equated ability estimates with simulated thetas, the average of the two
abilities is a better descriptor of the underlying multidimensional relationship
than is 01.
Inspection of the concurrent calibration outcomes in comparison with the
analytic estimates produces some striking results. As was found with the
equivalent groups, the correlation coefficients are very high and consistent
across all multidimensionality conditions. Unlike the equivalent group results,
the SDMs reveal large differences between the means of the equated ability
sets and means of sets of analytical estimates. In all cases, the equated ability
estimates are higher. Similarly, the SRMSDs are higher for the nonequivalent
examinee groups. This may indicate that although the relationship between the
analytical estimates and the concurrent calibration estimates is almost perfect in


114
greater differences between the comparison condition and equated abilities for
individual examinees as the number of items increases. The SDM values remain
approximately 0.0 for all conditions and comparison conditions showing equal
mean abilities for the two sets of data.
When the average of 01 and 02 is used as the condition for comparison,
the differences between multidimensional conditions are in the opposite direction.
Correlations increase and the SRMSDs decrease as the amount of
multidimensionality increases. These findings are consistent across all equating
methods and for both multidimensional IRT models.
The results seen for these two comparison conditions seem reasonable.
As the multidimensionality becomes more pervasive, the second ability trait
becomes more important. The average of the two abilities then describes the
relationship more accurately than the single dominant trait. However, even
though the correlation with 01 decreases in the presence of more
multidimensional items, it remains moderately high (p = .78 or .74) in even the
most extreme condition, MD40. This implies the first factor retains a dominant
position in the multidimensional relationship.
Comparisons against the analytical estimations are perhaps even more
revealing. The correlations yield coefficients of .97 or .98 for all multidimensional
conditions and equating methods. The SRMSD statistics are consistent and
comparatively low across all levels of multidimensionality.
If the dominant first factor is considered of main interest, the number of
multidimensional items in a test does have an effect. As the number of


115
multidimensional items increases, the equated traits appear discriminate lesson
the factor of interest. However, if a composite of the traits measured by a test are
of primary interest, the number of multidimensional items may not be as critical. If
the composite is defined as the average of the 9s, more multidimensional items in
a test lead to better equated traits. There does not appear to be any effect of
multidimensional items on unidimensional equating when the criterion is the
analytical approximations. For data generated through application of a
compensatory multidimensional IRT model, the analytical approximations define
the 9 composite equally well in all research conditions.
Effects of Noneauivalent Examinee Groups
No significant effects of equating method or number of multidimensional
items were found with the data from randomly equivalent groups of normally
distributed examinees. Results from the equatings of data from nonequivalent
examinee groups raise some questions. Comparisons of the correlations with
the 9i baseline for differing ability groups reveal the same direction of change as
was found for the equivalent groups, but the strength of these relationships is
noticeably lower. However, the correlations between the equated abilities and the
analytical estimations remain strong and consistent across all multidimensionality
conditions. The SDM statistics are of interest for concurrent calibration because
they are so much lower than was previously seen for all equivalent group
conditions, and they are also lower than the corresponding results for the other
two equating methods.


145
Table 54
DescriDtive Statistics for ComDensatorv MD40 Llnkina Items with Randomly
Eauivalent GrouDS
Replication
Form A
Form B
Mean
SD
Mean
SD
1
-0.314
1.071
-0.358
0.993
2
-0.323
1.036
-0.320
0.997
3
-0.310
1.008
-0.332
0.954
4
-0.329
0.967
-0.258
0.944
5
-0.255
0.954
-0.343
0.989
6
-0.360
0.981
-0.221
0.965
7
-0.251
0.993
-0.285
0.973
8
-0.288
0.988
-0.348
1.046
9
-0.360
1.035
-0.236
1.049
10
-0.242
1.038
-0.320
1.025
11
-0.318
1.048
-0.277
1.008
12
-0.272
1.030
-0.275
1.010
13
-0.257
1.030
-0.252
1.015
14
-0.274
1.049
-0.337
0.957
15
-0.343
0.956
-0.282
1.048
16
-0.264
1.088
-0.291
1.024
17
-0.299
1.044
-0.300
1.001
18
-0.284
1.020
-0.331
1.036
19
-0.348
1.072
-0.261
0.990
20
-0.245
1.003
-0.317
1.074


59
2. Does the equating procedure affect unidimensional equating results?
3. Do data simulated by using a compensatory multidimensional model
produce different unidimensional equating results than data simulated using a
noncompensatory model?
4. Are unidimensional equating results affected by differing ability
distributions of the two examinee groups?
Data Generation
Design
Data for two parallel forms, A and B, of each test condition were
simulated. Four test conditions were created by varying the number of
multidimensional items contained in each test. These conditions were created
to mirror what might be found in published tests. For example, in a test of
mathematics problem solving, all items might be multidimensional to some
degree if reading skill were also required. However, relatively few
multidimensional items might be found in a reading comprehension test
containing only one graph-reading passage that also needed a math skill for
completion. In the present study, 10, 20, 30, and 40 items of an 40 item test
were two-dimensional. These conditions are referred to as MD10, MD20,
MD30, and MD40 respectively.
In addition to modifying the number of multidimensional items, the
strength of each multidimensional item's first factor was manipulated. This
was done within each test condition because it is unreasonable to expect a
published test to contain multidimensional items which all have an identical


5
The current simulation study allowed exploration of what effect
multidimensionality had on the results obtained from a variety of unidimensional
equating procedures while providing a means to manipulate variables. The
techniques used to generate the data afforded a mechanism to control the
dimensionality of the items and test forms. The specific questions investigated
were selected as having the most value for current practitioners applying
unidimensional equating procedures.


56
Another concern raised was that the equatings might deteriorate if the factors
loaded differently on the two tests.
More recently, attempts have been made to develop a multidimensional
equating procedure. Hirsch (1989) conducted a study in which real and
simulated data were equated with a multidimensional method. The procedure
involves (a) estimating item parameters and abilities on both dimensions for
both tests, (b) identifying common basis vectors, (c) aligning basis vectors
through Procustes rotation, and (d) equating means and standard deviations of
the ability estimates for each dimension of the two tests. Results of this
preliminary research indicated that effective equating was possible with these
techniques, but the instability of the ability estimates make it impractical at this
time. While work on development of MIRT equating is continuing (Hirsch &
Miller, 1991), the procedure has little current value for the equating needs of
testing companies. The results of the studies of unidimensional equating with
multidimensional data are summarized in Table 4.
The emphasis of the present study was to examine the effect of
multidimensional data on unidimensional IRT equating through the use of a
simulation study. The research questions chosen were those considered to be
of most value to the practitioner.


148
Table 57
Descriptive Statistics for Noncompensatory MD30 Linking Items
Replication
Form A
Form B
Mean
SD
Mean
SD
1
0.147
1.150
0.170
1.234
2
0.178
1.236
-0.019
1.155
3
0.004
1.136
0.046
1.226
4
0.041
1.172
0.099
1.280
5
0.068
1.237
0.001
1.179
6
-0.002
1.151
0.139
1.280
7
0.160
1.268
0.050
1.252
8
0.039
1.238
0.041
1.222
9
0.043
1.200
0.204
1.298
10
0.205
1.241
0.077
1.144
11
0.062
1.109
0.097
1.237
12
0.131
1.200
0.082
1.228
13
0.112
1.242
0.082
1.228
14
0.052
1.171
0.169
1.320
15
0.146
1.244
0.106
1.255
16
0.101
1.230
0.054
1.186
17
0.032
1.126
0.065
1.355
18
0.073
1.284
0.071
1.125
19
0.065
1.091
0.125
1.226
20
0.139
1.227
0.147
1.183


74
Table 9
Analytical Estimates of the Unidimensional Parameters for Compensatory MD30,
Form A
Item
Discrimination
Difficulty
1
0.242
1.408
2
0.286
0.352
3
0.262
-1.449
4
0.679
-0.949
5
0.712
-0.559
6
0.479
0.066
7
0.733
0.737
8
0.289
1.237
9
0.422
0.833
10
0.599
-1.622
11
0.876
-1.742
12
0.786
0.673
13
0.482
-0.646
14
0.650
0.273
15
0.842
-0.874
16
0.429
0.698
17
0.687
-1.216
18
0.715
0.338
19
0.335
0.245
20
0.466
-0.877
21
0.400
0.808
22
1.320
-0.442
23
0.868
-0.429
24
0.267
0.265
25
0.487
-0.355
26
0.300
-0.312
27
1.103
-0.010
28
0.325
0.432
29
0.459
-0.070
30
0.270
-0.685
31
0.283
-1.913
32
0.452
1.327
33
0.882
0.670
34
0.700
0.239
35
0.690
-1.104
36
0.421
1.304
37
0.363
-0.862
38
0.777
-1.734
39
0.637
-0.953
40
0.148
0.536


17
underlying assumptions. Full discussions of these assumptions and
derivations of the appropriate formulas may be found in Dorans (1990).
Many studies have been conducted to assess the accuracy of linear
equating methods. Skaggs and Lissitz (1986b) carried out a simulation study
with an external anchor design. Both difficulty and discrimination values were
manipulated. The authors discovered unacceptable results with linear
equating when the discrimination means were unequal on the two tests.
Marco, Petersen, and Stewart (1983) used 40 different linear equating
models to transform SAT-V data. Both similar and dissimilar samples were
used, as well as variations of anchor test designs and characteristics of the
total tests. Some generalizations reached from the results of this ambitious
study are as follows:
1. When a test is equated to a test or form like itself through a parallel
anchor test and the ability distributions of the samples are identical, a linear
model yields very good results.
2. When a test is equated to a test or form like itself through an easy or
difficult anchor test with random samples, all of the models have a small mean
square error.
3. When samples with dissimilar ability distributions are used, linear
equating does not perform well.
4. When total tests differ in difficulty, linear models yield unsatisfactory
results.


105
were unequal In ability. Equating was performed with all three procedures. The
normally distributed examinee group was assigned to Form A and the low ability
group was matched with Form B. Comparison conditions and evaluation criteria
remained the same. To establish the analytical ability estimates for comparison,
the item parameters previously calculated for the four compensatory conditions
were fixed in BILOG386 and used to calibrate the ability parameters for the low
group. Because the standard deviations across the twenty replications for the
randomly equivalent groups were so small, only five repetitions were conducted
for each experimental condition with nonequivalent groups. Table 26 presents a
summary of the results of the three equatings with data from nonequivalent
examinee groups.
Concurrent Calibration
The correlations of the equated ability estimates with the simulated 0iS
decrease as the number of multidimensional items increase. The difference is
especially noticeable between MD10 and MD20. The SDM is negative and
substantially different from zero for all conditions. Because this statistics
subtracts the equated ability estimate from the simulated ability, a negative
value indicates the mean of the equated ability estimates is much higher than
the comparison condition. A high positive SRMSD value is found in all
conditions, another indication of the large difference in the comparison
conditions and equated ability estimates associated with each individual
examinee. In all cases, the correlation coefficients were lower, the SDMs much


Table 24
Summary of Equated bs Results with Randomly Equivalent Groups
Correlation
SDM
SRMSD
Condition
01
01+02
2
AEa
01 + 02
01 2
AEa
01
01+02
2
AEa
Compensatory
MD 10
0.93(.01)
0.75(.02)
0.97(00)
-0.02( 05) -0.02( 05)
0.03(04)
0.37(.02)
0.76( 03)
0.29(02)
MD 20
0.88(.01)
0.84(01)
0.97(.01)
-0.03( 06) -0.03( 07)
0.04(04)
0.50( 02)
0.63(.04)
0.30(.02)
MD 30
0.82(01)
0.91(.01)
0.97(.00)
0.01(.07) 0.01( 08)
0.05(05)
0.61(02)
0.51(03)
0.28(04)
MD 40
0.74(01)
0.94(00)
0.98(01)
-0.01( 05) -0.01 (.05)
0.03(.03)
0.73(.02
0.47(.03)
0.25(02)
Noncompensatory
MD 10
0.93(.01)
0.71(.01)
0.96(.01)
0.01(.05) 0.01(06)
0.03(04)
0.38(02)
0.80(.02)
0.30(02)
MD 20
0 90(01)
0.78(01)
0.96(.01)
0.01(,10) 0.01(09)
0.03(.07)
0.50(.04)
0.71(.04)
0.29(03)
MD 30
0.84(01)
0.84(.01)
0.97(.01)
0.02(.09) 0.02(.05)
0.05(04)
0.56(02)
0.62(03)
0.29( 04)
MD 40
0.70(02)
0.90(01)
0.98(01)
-0.01( 05) -0.01(.07)
0.04(05)
0.78(.03)
0.53(04)
0.26(.03)
Note. Means and (standard deviations) of 20 replications for each condition,
a = Analytical Estimation
102


conditions usually found in practice. Both compensatory and
noncompensatory models are apparently viable as MIRT models. Determining
the adequacy of unidimensional parameter estimation of multidimensional data
has important consequences for equating multidimensional tests.
In addition to the estimation procedures discussed, the relationship
between multidimensional and unidimensional IRT models can also be
approached from an analytical framework. Wang (1986), as reported in
Ackerman (1988) and Oshima and Miller (1990), determined explicit algebraic
relationships between unidimensional estimates and the true multidimensional
parameters for the case in which the underlying response process is modeled
by the compensatory M2PL model and the unidimensional 2PL model. Using
the results for unidimensional estimation of a multidimensional data matrix,
Wang concluded that the unidimensional item parameter estimates are
obtained as a weighted composite of the underlying traits. The weights are a
function of the discrimination vectors for the items, the correlations among the
latent traits, and the difficulty parameters of the items. For group g who can be
described as having a diagonal variance-covariance structure fig and a mean
ability vector y, the 2PL item parameters for two-dimensional item j can be
approximated by


99
Equated bs
The means and standard deviations of the bs for the twelve linking items
on each form were calculated. These were used in equations 20 and 21 in
Chapter 3 to derive the slope and intercept constants presented in Tables 22
and 23. Their values were similar with slopes not deviating far from 1.00 and
intercepts close to 0.0. The linear transformations were applied to the Form B
estimated abilities and comparisons to the Form A baselines were made.
Because a new potential source of error was being introduced with the equating
constants, results for this procedure were expected to be less precise than
those produced by concurrent calibration.
Results for the equated bs equatings for equivalent groups are shown in
Table 24. The outcomes of this procedure generated criteria patterns similar to
those found with concurrent calibration. As the number of multidimensional
items in a test form increases, the strength of the correlation of equated ability
estimates with 9i decreases and their relationship with the average abilities
increases. The correlations of the analytical estimations with the equated bs
data are similar for all conditions and are very high (pAE, 0 = 97 or .98). The
SRMSD statistic also produces patterns like those found for concurrent
calibration, increasing in comparisons with 01 as multidimensionality increases,
and decreasing in comparison with the 9avg comparison condition. Correlations
are approximately equal across conditions when the analytical estimations are
used as the comparison.


155
Mislevy, R. J., & Stocking, M. L. (1989). A consumer's guide to LOGIST
and BILOG. Applied Psychological Measurement. 13, 57-75.
Nandakumar, R. (1991). Traditional dimensionality versus essential
dimensionality. Journal of Educational Measurement. 28. 99-117.
Nandakumar, R. (1994). Assessing dimensionality of a set of item
responses-comparison of different approaches. Journal of Educational
Measurement. 31. 17-35.
Nandakumar, R. & Stout, W. (1993). Refinements of Stouts procedure
for assessing latent trait unidimensionality. Journal of Educational Statistics. 18,
41-68.
Oshima, T. C., & Miller, M. D. (1990). Multidimensionality and IRT-
based item invariance indices: The effect of between group variation in trait
correlation. Journal of Educational Measurement. 27, 273-283.
Petersen, N. S., Cook, L. L, & Stocking, M. L. (1983). IRT versus
conventional equating methods: A comparative study of scale stability. Journal
of Educational Statistics. 8, 137-156.
Petersen, N. S., Kolen, M., & Hoover, H. D. (1989). Scaling, norming,
and equating. In R. L. Linn (Ed.), Educational measurement (3rd ed.).
Washington, D C.: American Council on Education.
Qualls, A. L, & Ansley, T. N. (1985, April). A comparison of item and
ability parameter estimates derived from LOGIST and BILOG. Paper presented
at the annual meeting of the National Council on Measurement in Education,
Chicago.
Reckase, M. D. (1979). Unifactor latent trait models applied to
multifactor tests: Results and implications. Journal of Educational Statistics. 4,
207-230.
Reckase, M. D. (1985). The difficulty of test items that measure more
than one ability. Applied Psychological Measurement. 9. 401-412.
Reckase, M. D., Ackerman, T. A., & Carlson, J. E. (1988). Building a
unidimensional test using multidimensional items. Journal of Educational
Measurement. 25, 193-203.
Ree, M. J. (1979). Estimating item characteristic curves. Applied
Psychological Measurement. 3, 371-385.


40
the direction taken, Reckase (1985) determined the slope using the direction
from the origin of the 0 space to the point of highest discrimination.
Figure 2. An item response surface (IRS) based on the compensatory M2PL.
To accomplish this analysis, the model given in equation 11 is
translated to polar coordinates, replacing each eih by 0¡ cos aih, where 0¡ is the
distance from the origin to 0¡ and aih is the angle from the h,h axis to the
maximum information point (Reckase, 1985). In a two-dimensional item, the
value of ocjh can range between 0 and 90 depending on the degree to which
the item measures the two traits. If the item only measures the first trait, cm
equals 0, while an = 90 would depict an item measuring only the second trait.
The relationship between aih and discrimination element aih can then be stated
as


12
were in close agreement, the Quantitative and Analytical measures showed
sensitivity to relative item position. When possible, it is preferable to include the
anchor items spiralled throughout the test in their operational positions.
The length of the anchor test is another concern and the subject of several
studies. Klein and Kolen (1985) used a certification test to examine the
relationship between anchor test length and accuracy of equating results. The
authors used anchor tests of varying lengths and examinee groups both similar
and dissimilar in ability distribution. They concluded that when groups have
similar ability distributions, the anchor test length has little effect. However, as
group ability distributions become more dissimilar, longer anchor tests work best.
Klein and Kolen also found that anchor tests should correspond closely with the
total test in content representation, difficulty, and discrimination.
The study of Cook et al. (1988) is also pertinent to the question of anchor-
test length. When the groups differ in level of ability, as did the spring and fall
samples, different anchor test lengths yielded disparate results. In contrast, when
the groups have similar ability distributions, like the two fall samples, the
equatings are similar for different anchor test lengths.
When applying item response theory equating methods, anchor items are
usually referred to as linking items. These linking items are used to scale the
item parameter estimates. Equating with IRT requires that the item parameter
estimates for the two test forms be on the same scale before equating. The
quality of the equating depends largely on how well this item scaling is


CHAPTER 3
METHOD
Purpose
Introduction
The purpose of this study was to examine the effects of
multidimensional data on unidimensional equating procedures. The effects of
the number of multidimensional items, type of multidimensional model, and
choice of equating procedure were investigated. Most investigations were
conducted with randomly equivalent, normally distributed examinee groups
having mean 0 and standard deviation 1. In addition, data from examinee
groups of lower ability ( X1 = -0.8, SD, = 0.6) were equated to results obtained
from the randomly equivalent groups.
The methods applied to investigate these effects are described in this
chapter. The methodology is discussed in the following sections: (a) data
generation, (b) estimation of parameters, (c) equating, and (d) criteria for
evaluation.
Research Questions
The specific questions to be answered in the present study were:
1. Does the number of multidimensional items in a test affect
unidimensional equating results?
58


80
Table 11
Descriptive Statistics for Compensatory Form B Item Parameters
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.37
3.65
1.14
0.8
20
0.27
2.41
0.96
0.5
30
0.15
2.11
0.94
0.5
40
0.27
2.45
0.88
0.5
a2
10
0.00
1.37
0.20
0.4
20
0.00
1.58
0.36
0.5
30
0.00
2.11
0.55
0.5
40
0.18
2.27
0.71
0.4
d
10
-2.55
6.23
0.30
1.5
20
-1.87
2.76
0.00
1.1
30
-3.30
4.65
0.10
1.3
40
-2.90
2.78
0.20
1.2
MDISC
10
0.39
3.65
1.23
0.8
20
0.32
2.41
1.12
0.5
30
0.30
2.98
1.16
0.6
40
0.42
2.62
1.18
0.5
MID
10
-1.71
1.96
-0.13
0.9
20
-1.88
1.58
0.08
0.9
30
-1.68
1.77
-0.02
0.9
40
-1.79
1.94
-0.09
0.8
Note. N = 40 items in each condition.
conditions and in both forms, b2 is slightly less difficult than bi, and a2 is less
discriminating than a*.
In all cases, the noncompensatory b¡ parameters are lower than the MID¡
for the corresponding item. This may be explained by considering the method


Multidimensionality 35
Violation of the Unidimensionality Assumption 35
Multidimensional Models 37
Multidimensionality and Parameter Estimation 45
Multidimensionality and IRT equating 52
3 METHOD 58
Purpose 58
Introduction 58
Research Questions 58
Data Generation 59
Design 59
Model Description 60
Item Parameters 61
Response Data 63
Noncompensatory Data 65
Nonrandom Groups 66
Estimation of Parameters 66
Unidimensional IRT 66
Analytical Estimation 69
Equating 69
Concurrent Calibration 70
Equated bs 70
Characteristic Curve Transformation 71
Evaluation Criteria 73
Comparison Conditions 73
Statistical Criteria 75
Summary 76
4 RESULTS AND DISCUSSION 78
Simulated Data 78
Item Parameters 78
Analytical Estimation 88
Simulated Ability Data 88
Equating Results for Randomly Equivalent Groups 92
Concurrent Calibration 92
Equated bs 99
Characteristic Curve Transformation 103
Equating Results for Nonequivalent Groups 103
Concurrent Calibration 103
Equated bs and Characteristic Curve Transformation 108
v


77
Parameter estimation was executed on all conditions using both
unidimensional IRT procedures and analytical estimation. For the IRT
parameter estimates, equating was performed through through techniques: (a)
concurrent calibration, (b) equated bs, and (c) characteristic curve
transformation.
Three comparison conditions--the first simulated theta, the average of
theta 1 and theta 2, and the analytical estimations of the unidimensional
parameters-were selected for comparison with equated abillity estimates.
Finally, the three statistical procedures of correlation, standardized mean
difference, and standardized root mean square difference were applied to
examine the comparisons.


29
chosen as long as the same scale is chosen for b, (Petersen et al., 1989).
This is referred to as indeterminacy of the parameter scale.
If the parameters of a set of items are estimated separately for two
different groups of examinees, the item parameters may appear to be different
due to the arbitrary fixing of the metric for 9 or b,.. However, the two sets of 9s
and b, s should have a linear relationship to each other (Hambleton &
Swaminathan, 1985). The a, s should be the same except for differences in
unit of measurement and, in the 3PL case, the c, s remain unaffected
(Petersen et al., 1989).
The advantages of IRT equating are most useful in the case where
groups taking the two tests are nonrandom or intact groups (Crocker & Algina,
1986). Consequently, the following discussion will emphasize uses of IRT
equating with an anchor test design. However, item response theory
procedures may also be used with single-group or equivalent groups designs.
An anchor or linking test is one method available to put the parameters
for the two tests on the same scale. Four procedures commonly used with this
method are (a) concurrent calibration, (b) the fixed bs method, (c) the equated
bs method, and (d) the characteristic curve transformation method.
In concurrent calibration, parameters for the two tests are estimated
simultaneously. The linking items, or sometimes common subjects, serve to
unite the two tests and results in item parameter estimates on a common
scale. This allows direct equating of the two tests (Petersen et al., 1989).


37
skill may be required to correctly answer a mathematical item. Some of these
violations can be controlled, reduced, or eliminated, but the unidimensionality
assumption will still be violated in many practical situations (Doody-Bogan &
Yen, 1983). Achievement tests are not constructed using methods that yield
factor pure instruments. Instead, a table of specifications is customarily
developed and items are written to match the specifications. These items
rarely measure a single trait (Reckase, 1979). Due to the many possible
causes leading to violation of the unidimensionality assumption, it can be
concluded that dimensionality is a joint property of both the item set and the
particular sample of examinees (Hattie, 1985).
Multidimensional Models
Recently, attempts have been made to model multidimensional
responses within the framework of IRT. Several multidimensional item
response theory (MIRT) models have been proposed. Although
multidimensional versions of all three logistic parameter IRT models have been
derived, only the multidimensional two-parameter logistic (M2PL) model will be
discussed.
Doody-Bogan and Yen (1983) described a multidimensional model of
the form
Pii(Qh) =
1 + exp[-D Xa(e,h bjh)]
h=1
(9)


27
In addition to investigating the effect test length had on item and ability
parameter estimates derived from LOGIST and BILOG procedures, Qualls and
Ansley (1985) studied the sample size effect. Sample sizes of 200, 500, and
1,000 examinees with a normal ability distribution were combined with test
lengths of 10, 20, and 30 items. As sample size increased, both procedures
produced estimates more highly correlated with the simulated values. The
BILOG estimates were slightly better in all cases and superior in the
combination of small sample size with 10 items.
Buhr and Algina (1986) used BILOG with four methods of estimation
and sample sizes of 250, 500, 750, and 1,000 to study the similarity of
estimation. The Bayesian procedures were the most robust in dealing with
different ability distributions. Estimation with all procedures improved
substantially as sample size increased to 500, but showed little additional
effect as sample size increased further.
Baker (1990) simulated item response data based on a 45-item test with
500 examinees to study the pattern of estimation results as a function of the
various analysis operations. The data were analyzed under the options
available in BILOG and the obtained parameter estimates were equated back
to the true metric. The equated results were generally very close to the true
parameters. The item parameters were only slightly affected by the
characteristics of various priors. The equated means of the estimated Gs were


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES xi
ABSTRACT xii
CHAPTERS
1 INTRODUCTION 1
Purpose 3
Limitations 4
Significance of the Study 4
2 REVIEW OF LITERATURE 6
Test Equating 6
Conditions for Equating 6
Data Collection Designs 7
Single-group designs 7
Equivalent-group designs 9
Anchor-test designs 10
Equating Methods 14
Conventional Methods of Equating 14
Linear equating 16
Equipercentile equating 18
Equating Methods Based on Item Response Theory 21
Item response theory 21
IRT equating 28
IV


106
Table 26
Summary of Equating Results with Noneauivalent Groups
Correlation
SDM
SRMSD
Condition 6,
01+02
2
AEa
01
01 + 02 AE"
e,
01 + 02
2
AE
Concurrent Calibration
10 0.84
0.58
0.94
-0.79
-0.20 -0.75
0.85
0.84
0.66
20 0.67
0.85
0.95
-0.84
-0.28 -0.72
0.89
0.64
0.64
30 0.60
0.88
0.95
-0.86
-0.29 -0.62
0.91
0.63
0.60
40 0.57
0.91
0.97
-0.73
-0.19 -0.46
0.96
0.57
0.53
Equated bs
10 0.85
0.63
0.95
-0.03
0.47 -0.03
0.57
0.89
0.43
20 0.65
0.85
0.95
-0.10
0.47 -0.01
0.65
0.80
0.45
30 0.58
0.90
0.95
-0.13
0.34 -0.07
0.77
0.57
0.43
40 0.51
0.92
0.96
-0.20
0.44 0.01
0.89
0.59
0.42
Characteristic Curve Transformation
10 0.85
0.63
0.95
-0.03
0.47 -0.03
0.57
0.90
041
20 0.65
0.85
0.94
-0.11
0.45 -0.01
0.74
0.78
0.39
30 0.58
0.90
0.95
-0.12
0.33 -0.10
0.84
0.55
0.46
40 0.51
0.92
0.96
-0.18
0.45 0.02
0.97
0.61
0.37
Note. Means of 5 replications for each condition,
a = Analytical Estimation


34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
125
126
.127
128
129
130
131
132
133
134
135
136
137
138
139
140
Simulated Compensatory Item Parameters for MD40 Form A...
Simulated Compensatory Item Parameters for MD40 Form B...
Noncompensatory Item Parameters for Multidimensional Items
in MD10 Forms A and B
Noncompensatory Item Parameters for Multidimensional Items
in MD20 Form A
Noncompensatory Item Parameters for Multidimensional Items
in MD20 Form B
Noncompensatory Item Parameters for Multidimensional Items
in MD30 Form A
Noncompensatory Item Parameters for Multidimensional Items
in MD30 Form B
Noncompensatory Item Parameters for Multidimensional Items
in MD40 Form A
Noncompensatory Item Parameters for Multidimensional Items
in MD40 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD10 Form A
Analytical Estimates of Unidimensional Item Parameters for
MD10 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD20 Form A
Analytical Estimates of Unidimensional Item Parameters for
MD20 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD30 Form A
Analytical Estimates of Unidimensional Item Parameters for
MD30 Form B
Analytical Estimates of Unidimensional Item Parameters for
MD40 Form A
IX


50 Analytical Estimates of Unidimensional Item Parameters for
MD40 Form B 141
51 Descriptive Statistics for Compensatory MD10 Linking Items
with Randomly Equivalent Groups 142
52 Descriptive Statistics for Compensatory MD20 Linking Items
with Randomly Equivalent Groups 143
53 Descriptive Statistics for Compensatory MD30 Linking Items
with Randomly Equivalent Groups 144
54 Descriptive Statistics for Compensatory MD40 Linking Items
with Randomly Equivalent Groups 145
55 Descriptive Statistics for Noncompensatory MD10 Linking Items.. 146
56 Descriptive Statistics for Noncompensatory MD20 Linking Items... 147
57 Descriptive Statistics for Noncompensatory MD30 Linking Items... 148
58 Descriptive Statistics for Noncompensatory MD40 Linking Items... 149
59 Descriptive Statistics for Compensatory Linking Items with
Nonequivalent Groups 150
x


38
where On, is the ability parameter for person i for dimension h; a,h is the
discrimination parameter for item j for dimension h; by/, is the difficulty
parameter for item j for dimension h; and D is the scaling constant, 1.7.
Another model discussed by Sympson (1978) is defined
PtfW-- (10)
ri(1+exp[-D ajh[6ih bjJ])
h-1
where all parameters are defined as above.
These two models can be distinguished by comparing their
denominators. The Doody-Bogan and Yen model contains no product of
probabilities in the denominator as does the Sympson model. Equation 9 can
be classified as a compensatory model that permits high ability on one
dimension to compensate for low ability on another dimension in terms of the
probability of a correct response. If dimensionality is considered in the context
of factor analysis, a two-dimensional test has a group of items measuring each
dimension. A compensatory model seems reasonable because the test is
being considered as a whole (Ansley and Forsyth, 1985).
The second model, defined by Equation 10, is called a
noncompensatory model where high abilities on one factor are not allowed to
supplement low abilities on the second factor. When a two-dimensional test is
considered as one that requires simultaneous application of the two abilities to
answer each item correctly, the noncompensatory model seems more
appropriate (Ansley and Forsyth, 1985).


152
Bock, R. D., Gibbons, R., & Muraki, E. (1988). Full-information item
factor analysis. Applied Psychological Measurement. 12, 261-280.
Braun, H. I., & Holland, P. W. (1982). Observed-score test equating: A
mathematical analysis of some ETS equating procedures. In P. W. Holland &
D. B. Rubin (Eds.), Test equating (pp. 9-49). New York: Academic Press.
Brennan, R. L, & Kolen, M. J. (1987). Some practical issues in
equating. Applied Psychological Measurement. 11, 279-290.
Buhr, D. C., & Algina, J. (1986, April). A comparison of item parameter
estimates and ability parameter estimates obtained bv different methods
implemented bv BILOG. Paper presented at the annual meeting of the
American Educational Research Association, San Francisco.
Camilli, G. (1992). A conceptual analysis of differential item functioning
in terms of a multidimensional item response model. Applied Psychological
Measurement. 16, 129-147.
Camilli, G., Wang, M., & Fesq, J. (1995). The effects of dimensionality
on equating the Law School Admission Test. Journal of Educational
Measurement. 32, 79-96.
Cook, L. L., & Eignor, D. R. (1983, April). An investigation of the
feasibility of applying item response theory to equate achievement tests. Paper
presented at the annual meeting of the American Educational Research
Association, Montreal.
Cook, L. L., & Eignor, D. R. (1988). Using item response theory in test
score equating. International Journal of Educational Research. 23. 161-173.
Cook, L. L, Eignor, D. R., & Taft, H. L. (1988). A comparative study of
the effects of recency of instruction on the stability of IRT and conventional item
parameter estimates. Journal of Educational Measurement. 25. 31-45.
Cook, L. L., & Petersen, N. S. (1987). Problems related to the use of
conventional and item response theory equating methods in less than optimal
circumstances. Applied Psychological Measurement. H, 225-244.
Crocker, L., & Algina, J. (1986). Introduction to classical and modern
test theory. New York: Holt, Rinehart, and Winston.


linear relationship to each other (Petersen et al., 1989). Equating is a
procedure used to place the item parameters from two tests on the same
scale.
Three unidimensional IRT equating methods were selected for this
study: (a) concurrent calibration, (b) equated bs, and (c) characteristic curve
transformation.
Concurrent Calibration
Concurrent calibration is the simplest of the IRT methods of equating to
implement. A common group of examinees or items is required to tie the
information from the two tests together. For this study, the parameters of both
forms were estimated simultaneously by BILOG. Twelve common items in
each dataset served to link the forms and the resulting item parameter
estimates were therefore on the same scale. This process was repeated for
each of the response sets in each condition.
Equated bs
The equated bs method is based on determining the linear relationship that
exists between item difficulties estimated in two separate BILOG calibration
runs, one for each form. The means and standard deviations of the b,s for
each set of linking items from Form A and B were calculated. The linear
transformation was determined by


63
item 2, 45 for item 3, and 60 for item 4. This pattern then repeated for the 64
remaining items.
For datasets containing both unidimensional and two-dimensional items,
the last 3, 6, and 9 linking items were multidimensional for MD10, MD20, and
MD30 respectively. Thus the linking test had the same proportion of
unidimensional items as did the coresponding unique items in each condition..
The last 7, 14, and 21 unique items for each of Forms A and B were also
multidimensional. Table 5 presents the item parameters for Form A of MD30
with 75% of the items in each form being two-dimensional.
Response Data
For each experimental condition and form, response vectors for 1,000
simulees were generated. This sample size was selected as being adequate
to provide stable parameter estimates. The ability values were randomly
generated through the normal distribution RANNOR function of SAS to range
from approximately -3.00 to 3.00. The theta values were assumed to be
uncorrelated. Probabilities of correctly answering an item were then calculated
for each simulee through application of Equation 11. Finally, the SAS function
RANUNI was used to produce a random number from the uniform distribution
between 0 and 1. If this number was less than or equal to P(X¡j = 1 |a¡,d¡, 0j),
the simulee passed the item. If the random number was greater, the simulee
failed. To increase confidence in results, twenty sets of response data were
generated for each condition and form.


136
Table 45
Analytical Estimates of Unidimensional Item Parameters for MD2Q Form A
Item
Form
Ctii
Compensatory
Noncompensatory
a
b
a
b
1
A,B
0
0.320
1.178
0.320
1.178
2
A,B
0
0.771
-0.937
0.771
-0.937
3
A.B
0
1.104
-1.262
1.104
-1.262
4
A,B
0
0.703
0.092
0.703
0.092
5
A.B
0
0.414
-1.817
0.414
-1.817
6
A.B
0
0.359
-2.053
0.359
-2.053
7
A.B
45
1.126
-1.180
0.921
0.290
8
A,B
60
0.584
-0.589
0.610
-0.786
9
A,B
20
0.456
0.134
0.376
-2.233
10
A.B
30
0.630
-0.605
0.513
-0.781
11
A.B
45
0.901
-0.735
0.096
0.465
12
A,B
60
0.470
-0.013
0.499
-1.543
13
A
0
0.527
0.594
0.527
0.594
14
A
0
0.874
-0.039
0.874
-0.039
15
A
0
0.495
-1.130
0.495
-1.130
16
A
0
0.417
0.294
0.417
0.294
17
A
0
0.286
0.772
0.286
0.772
18
A
0
0.272
-0.364
0.272
-0.364
19
A
0
0.413
0.588
0.413
0.588
20
A
0
0.493
0.416
0.493
0.416
21
A
0
0.451
2.016
0.451
2.016
22
A
0
0.240
0.601
0.240
0.601
23
A
0
0.389
-0.020
0.389
-0.020
24
A
0
0.400
-0.971
0.400
-0.971
25
A
0
0.396
1.314
0.396
1.314
26
A
0
0.160
-0.665
0.160
-0.665
27
A
45
0.529
1.856
0.493
-3.540
28
A
60
0.295
1.048
0.333
-3.084
29
A
20
0.337
1.138
0.103
-4.857
30
A
30
0.466
-0.590
0.398
-1.130
31
A
45
1.002
0.190
0.791
-1.318
32
A
60
0.386
-1.710
0.507
-1.483
33
A
20
0.344
0.215
0.294
-2.791
34
A
30
0.409
-0.508
0.357
-1.398
35
A
45
0.619
1.308
0.086
0.799
36
A
60
0.828
1.410
0.772
-2.670
37
A
20
0.558
-1.141
0.269
-1.791
38
A
30
0.906
-0.722
0.677
-0.361
39
A
45
0.750
-1.321
0.658
0.201
40
A
60
0.839
-0.546
0.852
-0.548


BIOGRAPHICAL SKETCH
Patricia Duffy Spence was born in Plainfield, New Jersey, where she
continued to live until graduating from high school. She moved with her family
to Daytona Beach, Florida, and received her Bachelor of Arts degree in
elementary education from the University of Central Florida, Orlando. While
teaching elementary and middle school children in Volusia County, Patricia
completed requirements for the Master of Education degree in educational
leadership, also at the University of Central Florida.
After completing her doctoral classes at the University of Florida in
Gainesville, Patricia moved to San Antonio to work as Project Director for
Customized State Testing Programs at The Psychological Corporation. She
returned to Florida where she is currently employed as the Research,
Evaluation, and School Improvement Specialist for the Florida Region III Title I
Technical Assistance Center in Orlando. She also teaches graduate research
and measurement courses as an adjunct at the University of Central Florida.
Patricia and her husband, Verne Spence, have a daughter Cindy,
currently an undergraduate majoring in art, also at the University of Florida.
158


53
To examine their results, the researchers first calibrated the whole test,
then divided the test items into two homogeneous subgroups. The subgroups
were recalibrated separately and placed on the same scale as the original test.
They were then recombined back into an entire test and their corresponding
ICCs were compared. The authors discovered that differences in magnitude of
discrimination parameter estimates had an impact on IRT equating results,
affecting the symmetry of the equating. However, the different research
combinations yielded very similar equatings, leading the authors to conclude
that IRT equating may be sufficiently robust to the dimensionality displayed in
their data.
Cook and Eignor (1988) used SAT data that was suspected to be
multidimensional to examine the robustness of 3PL model concurrent
calibration and the characteristic curve transformation procedures. Scale drift
was used as the criterion for evaluating equating results. Cook and Eignor
concluded that both IRT equating methods produced acceptable results
despite the multidimensionality present in the tests being studied.
In addition to studying parameter estimation, Yen (1984) equated the
LOGIST trait estimates for both real (CTBS/U) and simulated data. Several
statistics were used to evaluate the results: (1) the correlation r; (2)
standardized difference between means (SDM); (3) ratio of standard
deviations; and (4) standardized root mean squared difference (SRMSD).
Trait estimates based on items that measured different dimensions had lower


2
Many mathematical procedures have emerged to develop the equating
transformations. Some are based on classical test theory while others arise
from item response theory (IRT). Classical methods, including linear and
equipercentile equating, do not seem robust to departures from optimal
conditions (Cook & Eignor, 1983; Livingston, Dorans, & Wright, 1990; Skaggs &
Lissitz, 1986b). Item response theory procedures, including equated bs,
concurrent calibration, and characteristic curve transformation, present
alternatives. Equating methods based on IRT have been found more accurate
than those based on classical models (Harris & Kolen, 1985; Hills, Subhiyah, &
Hirsch, 1988; Kolen, 1981; Marco, Petersen, & Stewart, 1983; Petersen, Cook,
& Stocking, 1983).
IRT models are grounded on strong assumptions, particularly that the
item responses are unidimensional (Ansley & Forsyth, 1985). The
unidimensionality assumption requires that each of the tests to be equated
measures the same underlying ability. Any other factor that influences an
examinees score-such as guessing, speededness, cheating, item context, or
instructional sensitivity-will violate the unidimensionality assumption. Some of
these violations can be controlled, reduced, or eliminated, but the
unidimensionality assumption will still be violated in many practical testing
situations (Doody-Bogan & Yen, 1983).
Attempts have been made to model multidimensional responses within
the framework of IRT. Although these models describe multidimensional data
more accurately than unidimensional models, estimation of parameters is


88
(Ackerman, 1989). Examination of Figures 4 through 7 indicates that, like the
IRS, as a multidimensional noncompensatory item approaches equal
discrimination on the two abilities, the curves of the equiprobability lines
increase greatly.
Analytical Estimation
Unidimensional estimates of the multidimensional item parameters for
both compensatory and noncompensatory conditions were calculated using the
procedure described by Wang (1986). Although this procedure was developed
for the compensatory model, the purpose of this research merited its use with
noncompensatory data also. Descriptive statistics are presented in Table 14 for
Form A and Table 15 for Form B. For the compensatory conditions, the
analytical difficulty estimates approximate the simulated MID values. However,
variation is found in the discrimination parameters. This occurs because the
analytical solution projects the discriminations onto a reference composite
vector. This same pattern is repeated for the noncompensatory conditions.
Simulated Ability Data
For each condition, 1,000 examinees were simulated from a normal
distribution with mean 0.0 and standard deviation 1.0 on both 0i and 02. The
two thetas were assumed to be uncorrelated. The thetas were then applied in
the appropriate compensatory or noncompensatory probability equation and
responses were generated. Twenty response sets were generated for each
condition. Inspection of the descriptive statistics contained in Tables 16


52
V
di-% tj
a)Qi
(18)
where a¡ is the discrimination vector for the M2PL model; d\¡ is the difficulty
parameter for the M2PL model; and Q2 are the first and second
standardized eigenvectors of the matrix I'A'AX where A is the matrix of
discrimination parameters for all items in the test and VX = Q Therefore,
when the means, standard deviations, and item parameters of a two-
dimensional distribution are known, the corresponding 2PL unidimensional
item parameters can be approximated.
Multidimensionalitv and IRT Equating
In practice, test equating almost exclusively assumes unidimensionality.
A single score from one test is transformed to a single score from another test.
An understanding of what effect the presence of multidimensional data has on
these unidimensional equating results is of paramount importance.
Dorans and Kingston (1985) equated four forms of the Verbal GRE
Aptitude Test using the 3PL model and an equated bs procedure. Two data
collection designs, equivalent groups and anchor-test, were investigated as
well as several variations in calibration procedures. Dimensionality was
assessed through factor analyses conducted at the item level on interitem
tetrachoric correlations. Two highly related verbal dimensions were identified.


140
Table 49
Analytical Estimates of Unidimensional Item Parameters for MD40 Form A
Item
Form
On
Compensatory
Noncompensatory
a
b
a
b
1
A,B
20
0.863
-1.485
0.680
-0.329
2
A,B
30
0.842
-0.194
0.679
-1.041
3
A,B
45
0.881
0.971
0.675
-2.382
4
A,B
60
0.925
-1.475
0.746
0.462
5
A,B
20
0.336
-0.243
0.304
-2.235
6
A,B
30
0.651
0.075
0.541
-1.537
7
A,B
45
0.566
-0.549
0.487
-0.881
8
A,B
60
0.461
-0.366
0.429
-1.345
9
A,B
20
1.352
-1.110
0.962
0.234
10
A,B
30
0.973
1.743
0.733
-3.353
11
A.B
45
0.428
0.201
0.392
-1.981
12
A,B
60
0.537
-0.952
0.483
-0.572
13
A
20
0.619
-0.567
0.515
-1.007
14
A
30
0.547
-1.363
0.458
-0.108
15
A
45
0.606
-0.628
0.514
-0.732
16
A
60
0.445
0.232
0.420
-2.012
17
A
20
1.050
-0.641
0.783
-0.472
18
A
30
0.994
0.191
0.732
-1.419
19
A
45
1.301
-0.721
0.873
-0.182
20
A
60
0.311
-0.414
0.303
-1.864
21
A
20
0.512
-1.278
0.429
-0.457
22
A
30
0.815
1.088
0.648
-2.591
23
A
45
0.497
0.077
0.444
-1.690
24
A
60
0.596
1.848
0.519
-3.651
25
A
20
1.063
0.291
0.800
-1.566
26
A
30
0.924
0.233
0.698
-1.505
27
A
45
0.618
0.096
0.526
-1.528
28
A
60
0.309
0.594
0.307
-2.867
29
A
20
0.413
0.462
0.373
-2.619
30
A
30
0.628
-0.571
0.521
-0.836
31
A
45
0.516
0.396
0.459
-2.006
32
A
60
0.302
0.677
0.301
-2.991
33
A
20
1.284
0.990
0.938
-2.306
34
A
30
0.420
-0.887
0.368
-0.957
35
A
45
0.470
-0.971
0.414
-0.623
36
A
60
0.304
1.562
0.307
-3.866
37
A
20
0.725
0.057
1.106
-2.338
38
A
30
1.280
-0.519
0.555
2.208
39
A
45
0.649
-0.824
0.542
-0.457
40
A
60
0.773
-0.234
0.651
-1.040


19
(Petersen et al., 1989). In mathematical terms, the equipercentile equating
function for equating Y to X on population P is
Epfy) =Fp'(Gr(y)) (3)
where Gp (y) is the cumulative distribution of Y scores and Fp'1 () is the
inverse of the cumulative distribution of X scores, Fp (x). A cumulative
distribution function maps scores onto relative frequencies, while an inverse
cumulative distribution function maps the relative frequencies onto scores
(Dorans, 1990).
As a mathematical model, equipercentile equating makes no
assumptions about the tests to be equated. It simply compresses and
stretches the score units on one test so that its raw score distribution matches
the second test. It is only consideration of the purpose of equating and the
desired condition of population invariance that prevents its application to tests
measuring different constructs (Petersen et al., 1989).
Generally, empirical studies have shown mixed results in assessing the
accuracy of equipercentile equating. Livingston, Dorans, and Wright (1990)
included an equipercentile equating method in their study. A composite of two
equipercentile equatings, the procedure worked well in most situations.
Similarly, the equipercentile equating produced acceptable results in all
combinations of conditions in the Skaggs and Lissitz (1986b) study.
On the other hand, in the investigation conducted by Petersen et al.
(1983) using SAT data, equipercentile equating was studied along with the


133
Table 42
Noncompensatory Item Parameters for Multidimensional Items in MD40 Form B
Item
Form
Cti1
a1
a2
b,
b2
1
A,B
20
1.023
0.595
1.034
-1.132
2
A,B
30
0.932
0.687
-0.250
-1.413
3
A,B
45
0.796
0.844
-1.730
-1.621
4
A,B
60
0.743
1.141
-0.440
0.815
5
A,B
20
0.403
0.324
-0.410
-3.057
6
A,B
30
0.712
0.583
-0.621
-1.668
7
A,B
45
0.582
0.598
-0.650
-0.591
8
A.B
60
0.467
0.583
-1.664
-0.358
9
A,B
20
1.576
0.752
0.841
-1.235
10
A,B
30
0.935
0.825
-2.037
-2.758
11
A.B
45
0.468
0.482
-1.432
-1.358
12
A,B
60
0.519
0.668
-1.123
0.164
41
B
20
1.089
0.255
0.000
0.000
42
B
30
0.100
0.254
1.863
0.380
43
B
45
0.904
0.959
-0.548
-0.481
44
B
60
0.585
0.785
-0.830
0.437
45
B
20
0.361
0.299
-0.731
-3.511
46
B
30
0.304
0.278
-1.119
-2.776
47
B
45
0.626
0.644
-1.185
-1.106
48
B
60
0.834
1.439
-0.883
0.736
49
B
20
0.836
0.618
-1.162
-2.849
50
B
30
0.525
0.444
-0.240
-1.485
51
B
45
0.619
0.523
-0.763
-1.850
52
B
60
0.606
0.821
-2.144
-0.966
53
B
20
0.894
0.595
-0.326
-2.212
54
B
30
0.247
1.823
0.689
-2.552
55
B
45
0.369
0.378
-1.217
-1.163
56
B
60
0.435
0.537
-2.208
-0.872
57
B
20
0.559
0.439
-0.762
-2.881
58
B
30
0.289
0.266
-1.169
-2.893
59
B
45
0.678
0.700
-0.771
-0.700
60
B
60
0.769
1.141
-1.824
-0.723
61
B
20
0.614
0.409
1.123
-1.304
62
B
30
0.570
0.476
-0.086
-1.306
63
B
45
0.565
0.580
-0.673
-0.615
64
B
60
0.479
0.602
-2.139
-0.862
65
B
20
1.031
0.641
-0.202
-2.065
66
B
30
0.807
0.642
-0.494
-1.513
67
B
45
0.392
0.402
-1.061
-1.009
68
B
60
0.321
0.388
-4.059
-2.426


32
The relationship between raw scores and true scores on two tests is not
necessarily the same, nor is an equating provided for individuals scoring below
the chance level (Petersen et al., 1989). Observed-score equating provides a
method of predicting the raw-score distribution of a test. This procedure uses
probabilities of correct responses under an IRT model to generate a
hypothetical joint distribution of item responses from all examinees taking both
tests. Conventional equlpercentile equating is then applied to the new
distributions (Skaggs & Lissitz, 1986a). Neither true-score nor observed-score
equating is applied often in practice. Both are complicated to calculate and
expensive to implement.
Many researchers have investigated the accuracy of IRT equating
methods using the various IRT models and procedures. Comparison of IRT
equating with conventional methods is also common. Marco, Petersen, and
Stewart (1983) examined the Rasch and 3PL models along with the 40 linear
and two equipercentile equating methods previously discussed. A variety of
conditions, including random and dissimilar samples, internal and external
anchors, and difficulty levels of the anchor tests were also studied. The two
IRT methods worked well, both with an external anchor test equal in difficulty
to the total test and with an internal anchor. With the external anchor test, the
Rasch results were slightly better than with any of the other equating methods
investigated. Both IRT models were clearly superior to the conventional
equating methods when the samples differed in ability distributions, but neither



PAGE 1

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PAGE 175

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82
Table 13
Form B
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.33
1.54
0.69
0.4
20
0.13
1.10
0.59
0.3
30
0.33
1.33
0.69
0.3
40
0.10
1.58
0.64
0.3
a2
10
0.29
0.97
0.64
0.3
20
0.10
1.12
0.57
0.3
30
0.16
0.83
0.63
0.4
40
0.25
1.82
0.64
0.3
b1
10
-2.32
0.57
-0.96
0.8
20
-3.22
0.27
-1.21
0.9
30
-3.30
1.65
0.10
1.3
40
-4.06
1.86
-0.79
1.1
b2
10
-3.04
0.33
-1.25
1.0
20
-4.90
0.69
-1.53
1.5
30
-3.62
1.03
-1.24
1.3
40
-3.51
0.82
-1.32
1.1
Note. The number of multidimensional items is the same as the condition number.
noncompensatory item must be smaller than the MID] parameterof the
compensatory item if the condition for items to be corresponding is to be met.
The differences between the compensatory and noncompensatory M2PL
models can also be shown graphically. Because the probability of a correct
response varies as a function of the 0 in each model, the item response
surfaces (IRS) and contour plots of matched items should differ. The


18
Two methods of selecting samples and five methods of equating,
including two linear methods, were combined in a study by Livingston, Dorans,
and Wright (1990). Again, when the samples differed in ability distributions the
linear equatings were inaccurate, showing a large negative bias. Matching the
samples on the basis of the anchor test did little to improve the results. The
authors recommended dealing with ability differences by selecting a
representative sample from each population and choosing an equating method
that does not assume exchangeability for examinees based on their anchor
test scores.
Based on these studies, it can be seen that linear equating methods are
distribution dependent. Although linear equating may perform satisfactorily in
optimal conditions, it is likely to produce bias in real testing situations.
Eauioercentile equating
In equipercentile equating, a transformation is chosen so that raw
scores on two tests are considered to be equated if they have the same
percentile rank (Angoff, 1971). This is based on the definition that score
scales are comparable for two tests if their respective score distributions are
identical in shape for some population (Braun & Holland, 1982). When this is
true, a table of pairs of raw scores can be constructed. Because the pairs of
raw scores are not necessarily numerically equal, it is necessary to transform
one set of scores into the other set or to convert both sets to a new score


79
Table 10
Descriptive Statistics for Compensatory Form A Item Parameters
Parameter
Condition
Minimum
Maximum
Mean
SD
a1
10
0.29
3.49
1.15
0.8
20
0.30
2.41
0.89
04
30
0.15
1.94
0.84
0.4
40
0.28
2.45
0.98
0.6
a2
10
0.00
1.22
0.17
0.3
20
0.00
1.87
0.42
0.6
30
0.00
1.53
0.49
0.4
40
0.21
1.63
0.71
0.3
d
10
-2.27
2.18
0.08
1.1
20
-2.44
2.76
0.20
1.1
30
-1.06
2.68
0.25
0.9
40
-2.90
2.78
0.17
1.2
MDISC
10
0.41
3.49
1.23
0.8
20
0.30
2.41
1.08
0.5
30
0.29
2.24
1.04
0.5
40
0.57
2.61
1.25
0.6
MID
10
-1.94
1.62
-0.11
0.9
20
-1.86
1.83
-0.09
0.9
30
-1.84
1.23
-0.17
0.9
40
-1.43
1.73
-0.10
0.8
Note. N = 40 items in each condition.
Form A conditions are presented in Table 12 and Form B information is given in
Table 13. The item parameter values calculated from the noncompensatory
transformations are within the ranges given by Ackerman (1989). For all


23
of measurement theory (Skaggs & Lissitz, 1986a). It is these differences that
must be considered when selecting a model for a particular application.
Figure 1. An item characteristic curve (ICC) based on the three-parameter
logistic model
The use of any of the IRT models entails restrictive assumptions about
the item response process. Briefly stated, the major assumptions of IRT are
as follows:
1. The ICC accurately represents the data.
2. The data are unidimensional.
3. Responses are locally independent (Skaggs & Lissitz, 1986a)
An ICC is defined completely when its general form is specified and
when the parameters of a particular item are known (Hambleton &
Swaminathan, 1985). This leads to the basic advantage of IRT models. When
the data fit the model reasonably well, it is possible to demonstrate the


5 CONCLUSIONS 111
Effects of Multidimensional Model 111
Effects of Equating Method 112
Effects of the Number of Multidimensional Items 112
Effects of Nonequivalent Examinee Groups 115
Implications 116
APPENDIX
ITEM PARAMETER DATA 118
REFERENCES 151
BIOGRAPHICAL SKETCH 158
VI


ACKNOWLEDGMENTS
An effort of this magnitude always involves many people. The author
wishes to especially thank the chairman of her committee, Dr. M. David Miller,
for his dedication and inspiration. Without his encouragement and good humor,
this dissertation would not have been possible. The author would also like to
thank her committee members for their guidance and patience, particularly Dr.
James Algina and Dr. Linda Crocker. Their suggestions were always correct, if
not always accepted. Also, without the inspiration of Dr. Charles Dziuban of the
University of Central Florida, she would never have pursued studies in this field.
In addition, the author recognizes her colleagues, past and present, at
The Psychological Corporation, Volusia County District Schools, and the Florida
Department of Education for the opportunities to apply her learning in practical
situations. Gratitude is offered to her three parents-Jim, Joan, and Jeanne-
who stressed the importance of learning and doing things well. Thanks also to
special friends: Anne Seraphine for debating the meaning of life and monotonic
curves; Nada Stauffer for quiet friendship; George Suarez for making her laugh;
and Carlos Guffain for demanding her best. But the author is most indebted and
grateful to her husband, Verne, who has supported and encouraged her through
three degrees, and her daughter Cindy who is now left to carry on the Gator
tradition alone.


95
Table 20
Descriptive Statistics for Simulated Low Ability Examinees
THETA 1 THETA 2
Rep
Low
High
Mean
SD
Low
High
Mean
SD
MD10
1
-3.19
0.09
-0.73
0.60
-3.80
3.13
-0.05
1.02
2
-3.62
-0.03
-0.82
0.58
-3.19
2.98
0.01
1.04
3
-3.13
-0.02
-0.79
0.58
-2.91
3.16
0.02
0.98
4
-2.98
0.04
-0.80
0.59
-3.46
4.03
0.02
1.00
5
-4.02
-0.02
-0.81
0.61
-2.96
3.15
-0.00
1.02
MD20
1
-3.13
0.00
-0.78
0.61
-3.30
3.00
-0.06
0.96
2
-3.30
0.02
-0.80
0.62
-2.92
3.42
-0.02
1.01
3
-3.18
0.00
-0.77
0.59
-3.15
3.73
0.04
1.02
4
-4.34
-0.02
-0.82
0.60
-4.05
2.87
-0.04
1.00
5
-3.35
-0.04
-0.86
0.60
-2.83
2.98
0.02
1.00
MD30
1
-3.79
0.01
-0.84
0.64
-3.24
3.88
-0.05
1.02
2
-3.28
-0.01
-0.82
0.62
-3.14
3.48
0.01
1.01
3
-3.76
-0.03
-0.82
0.64
-3.08
3.29
0.06
0.98
4
-3.35
0.01
-0.79
0.58
-3.55
3.24
0.05
1.00
5
-3.60
0.02
-0.80
0.60
-3.01
3.03
-0.02
1.02
MD40
1
-3.10
-0.02
-0.80
0.59
-2.85
2.13
-0.01
1.02
2
-3.44
0.02
-0.78
0.60
-2.97
2.75
0.01
1.03
3
-3.51
0.01
-0.81
0.62
-3.82
2.27
-0.02
0.98
4
-3.24
0.01
-0.80
0.61
-3.47
2.38
-0.04
1.00
5
-3.81
0.02
-0.83
0.62
-2.99
2.91
0.04
0.96
Note. N = 1000 for each replication


multidimensionality were spiraled throughout each test. The data were then
transformed into corresponding noncompensatory items which had the same
probability of success as the compensatory item for a given examinee.
Four tests with 40 items each were simulated with 12 common linking items
and 28 unique items. For each experimental condition and form, responses for
1,000 simulees were generated. To examine the effects of nonrandom groups,
responses for 1,000 less able examinees were also generated.
Three unidimensional IRT equating methods were selected: (a)
concurrent calibration, (b) equated bs, and (c) characteristic curve
transformation. Parameters were calibrated with BILOG386. To evaluate the
results of the research equatings, three comparison conditions were used; (1)
the unidimensional approximations of the multidimensional item parameters
calculated using an analytic procedure; (2) the simulated first ability dimension
only; and (3) the averages of the two simulated abilities. Three statistical
criteria-correlation, standardized differences between means, and standardized
root mean square difference--were applied to the data.
No significant effect on the unidimensional equating results were
attributed to choice of multidimensional model. For randomly equivalent groups,
there was also no effects due to choice of equating procedure. Concurrent
calibration favored low ability examinees when the ability distributions of the two
groups were unequal. When the multidimensional composites described by the
analytical estimation baseline are the data of interest, the number of
xiii


92
Table 17
Descriptive Statistics for Simulated Examinees Taking MD20
Rep
THETA 1
THETA 2
Low
High
Mean
SD
Low
High
Mean
SD
1
-2.96
3.50
0.00
0.98
-3.30
3.00
-0.05
0.99
2
-3.30
3.58
0.02
1.02
-2.97
3.06
0.02
1.01
3
-3.18
3.91
0.02
0.98
-2.98
3.73
0.02
1.03
4
-3.38
3.24
-0.02
1.00
-3.05
3.64
-0.02
0.99
5
-3.35
3.42
-0.05
1.01
-3.27
2.98
0.01
1.01
6
-3.07
3.22
-0.04
1.01
-2.77
3.89
0.01
0.96
7
-4.34
3.34
-0.02
1.00
-4.05
3.17
0.01
1.03
8
-3.14
3.28
0.01
0.99
-3.15
3.16
0.05
1.01
9
-3.09
3.30
-0.02
0.98
-2.50
3.72
-0.03
1.00
10
-3.13
3.50
0.00
0.98
-4.55
2.89
-0.02
1.00
11
-3.12
2.80
0.01
0.97
-3.32
3.52
-0.05
0.98
12
-2.88
3.48
0.04
1.01
-2.95
2.98
0.00
1.00
13
-3.33
2.97
-0.01
1.01
-3.47
3.86
0.01
1.05
14
-3.20
3.75
-0.05
0.98
-3.08
2.73
-0.01
0.98
15
-3.96
2.93
0.04
0.99
-3.49
3.31
-0.01
1.02
16
-2.84
3.71
0.00
1.00
-2.97
3.27
0.00
0.97
17
-3.17
3.54
-0.01
0.98
-3.71
3.92
-0.04
1.04
18
-2.98
2.94
0.01
0.97
-3.54
3.37
-0.04
0.97
19
-3.62
3.11
-0.04
0.98
-3.17
3.22
-0.02
1.03
20
-2.84
3.44
0.00
0.97
-3.11
3.47
-0.04
0.97
Note. N = 1000 for each replication
Equating Results for Randomly Equivalent Groups
Concurrent Calibration
Response data for all equivalent groups were used by BILOG386 to calibrate
the item and ability parameters for both Forms A and B simultaneously.


11
equating may be carried out even when the two groups are not at the same ability
level. The groups may be random groups from the same population or they may
be nonequivalent or naturally occurring groups. The scores on the anchor test
can be used to estimate the performance of the combined group (Cook &
Petersen, 1987). The anchor test may be an internal part of both tests X and Y,
or it may be an external separate test. If an external anchor test is used, it should
be administered after X or Y to avoid practice effects on the tests to be equated
(Lord, 1980). The anchor-test design, while the most complicated of the data
collection methods, is the most common in real testing situations. Constraints of
time or available samples placed on large testing programs often require its use
(Skaggs & Lissitz, 1986a).
Properties of the anchor test can seriously affect the ensuing equating
results. Klein and Jarjoura (1985) studied the properties and characteristics of
anchor-test items in relation to the total test. A test of 250 items was equated
using three different anchor tests. Although all anchors were similar to the total
test in difficulty, only one of the anchor-tests was representative of the total test
content. The results confirmed the importance of including items on the anchor
test that mirror as nearly as possible the content of the total test.
In addition to content representativeness, the relative position of items in
test books also seems to play an important role in anchor-test design. Kingston
and Dorans (1984) examined relative position effects of items in a version of the
GRE General Test. Although the equatings of the Verbal measure of the test


126
Table 35
Simulated Compensatory Item Parameters for MD40 Form B
Item
Form
Cti1
a.
a2
d.
MDISC
MID
1
A,B
20
1.486
0.541
2.255
1.581
-1.426
2
A,B
30
1.254
0.724
0.280
1.449
-0.193
3
A,B
45
1.082
1.082
-1.469
1.530
0.960
4
A,B
60
0.940
1.629
2.542
1.881
-1.352
5
A,B
20
0.562
0.205
0.140
0.598
-0.234
6
A,B
30
0.968
0.559
-0.084
1.118
0.075
7
A.B
45
0.691
0.691
0.531
0.977
-0.543
8
A,B
60
0.438
0.758
0.294
0.876
-0.336
9
A,B
20
2.453
0.893
2.783
2.611
-1.066
10
A,B
30
1.451
0.838
-2.904
1.676
1.733
11
A,B
45
0.522
0.522
-0.147
0.739
0.199
12
A,B
60
0.513
0.889
0.895
1.027
-0.872
41
B
20
1.013
0.369
-1.083
1.078
1.004
42
B
30
1.696
0.979
1.507
1.959
-0.769
43
B
45
1.305
1.305
0.598
1.845
-0.324
44
B
60
0.617
1.069
1.348
1.235
-1.092
45
B
20
0.500
0.182
-0.039
0.532
0.073
46
B
30
0.381
0.220
0.033
0.440
-0.075
47
B
45
0.750
0.750
-0.189
1.060
0.178
48
B
60
1.312
2.272
2.665
2.623
-1.016
49
B
20
1.243
0.452
-1.267
1.322
0.958
50
B
30
0.688
0.397
0.466
0.794
-0.587
51
B
45
0.642
0.642
-1.220
0.908
1.344
52
B
60
0.647
1.120
-0.741
1.293
0.573
53
B
20
1.328
0.483
-0.078
1.414
0.055
54
B
30
0.822
0.475
-0.157
0.949
0.165
55
B
45
0.402
0.402
0.242
0.568
-0.427
56
B
60
0.395
0.684
-0.183
0.789
0.232
57
B
20
0.791
0.288
-0.320
0.841
0.380
58
B
30
0.361
0.209
0.034
0.417
-0.082
59
B
45
0.841
0.841
0.307
1.189
-0.258
60
B
60
0.993
1.721
-0.789
1.987
0.397
61
B
20
0.882
0.321
1.679
0.939
-1.788
62
B
30
0.756
0.436
0.614
0.873
-0.703
63
B
45
0.665
0.665
0.514
0.940
-0.546
64
B
60
0.449
0.777
-0.267
0.897
0.297
65
B
20
1.562
0.568
0.071
1.662
-0.042
66
B
30
1.129
0.652
0.014
1.303
-0.011
67
B
45
0.432
0.432
0.322
0.611
-0.527
68
B
60
0.266
0.461
-1.032
0.532
1.940


125
Table 34
Simulated Compensatory Item Parameters for MD40 Form A
Item
Form
Oil
a,
a2
d,
MDISC
MID
1
A,B
20
1.486
0.541
2.255
1.581
-1.426
2
A,B
30
1.254
0.724
0.280
1.449
-0.193
3
A,B
45
1.082
1.082
-1.469
1.530
0.960
4
A,B
60
0.940
1.629
2.542
1.881
-1.352
5
A,B
20
0.562
0.205
0.140
0.598
-0.234
6
A,B
30
0.968
0.559
-0.084
1.118
0.075
7
A,B
45
0.691
0.691
0.531
0.977
-0.543
8
A,B
60
0.438
0.758
0.294
0.876
-0.336
9
A,B
20
2.453
0.893
2.783
2.611
-1.066
10
A,B
30
1.451
0.838
-2.904
1.676
1.733
11
A,B
45
0.522
0.522
-0.147
0.739
0.199
12
A.B
60
0.513
0.889
0.895
1.027
-0.872
13
A
20
1.049
0.382
0.608
1.116
-0.544
14
A
30
0.813
0.469
1.272
0.939
-1.355
15
A
45
0.741
0.741
0.651
1.048
-0.621
16
A
60
0.422
0.731
-0.180
0.845
0.214
17
A
20
1.838
0.669
1.205
1.956
-0.616
18
A
30
1.482
0.856
-0.326
1.712
0.190
19
A
45
1.615
1.615
1.629
2.284
-0.713
20
A
60
0.292
0.506
0.222
0.584
-0.381
21
A
20
0.862
0.314
1.126
0.917
-1.228
22
A
30
1.213
0.700
-1.515
1.401
1.082
23
A
45
0.607
0.607
-0.066
0.858
0.077
24
A
60
0.574
0.994
-1.943
1.148
1.692
25
A
20
1.864
0.678
-0.554
1.983
0.279
26
A
30
1.376
0.795
-0.369
1.589
0.232
27
A
45
0.755
0.755
-0.102
1.068
0.095
28
A
60
0.290
0.503
-0.316
0.581
0.544
29
A
20
0.694
0.253
-0.328
0.739
0.444
30
A
30
0.934
0.539
0.613
1.078
-0.569
31
A
45
0.630
0.630
-0.349
0.891
0.392
32
A
60
0.283
0.490
-0.351
0.566
0.619
33
A
20
2.308
0.840
-2.334
2.456
0.951
34
A
30
0.623
0.360
0.635
0.719
-0.882
35
A
45
0.574
0.574
0.780
0.812
-0.960
36
A
60
0.285
0.494
-0.816
0.570
1.432
37
A
20
1.236
0.450
-0.073
1.315
0.055
38
A
30
1.916
1.106
1.143
2.212
-0.517
39
A
45
0.794
0.794
0.915
1.122
-0.816
40
A
60
0.763
1.321
0.328
1.525
-0.215


31
by using the two sets of parameter estimates from the common items. A linear
transformation is obtained from minimizing the difference between the true
scores on the linking items. This transformation is then applied to the a,, b,,
and 0 parameters of the second test (Stocking & Lord, 1983). Because it
takes all information into account, this procedure is theoretically an
improvement over the previous methods.
Sometimes the reporting of abilities in terms of 9 is unacceptable. In
these situations, the 9 value from a test may be converted to its corresponding
true score £ through
6-PiO) (7)
¡.1
where n is the number of items on the test. Equating of the true scores on the
two tests is then possible (Hambleton & Swaminathan, 1985). The true score
on one test is said to be equated to the true score on a second test if each
corresponds to the same ability level, or if
5 = Pi(9) t1 = P,(0) (8)
i-1 H
(Skaggs & Lissitz, 1986a). In practice, estimated item parameters are used to
approximate P, (9) and P¡ (9). Paired values of l and q are then computed by
substituting a series of arbitrary values for 9 into Equation 8 and calculating %
and q for each 9. These paired values define £ as a function of q and
constitute an equating of these true scores (Lord, 1980).


85
(a) Compensatory IRS
(a,=.934, a?= 539, d=.650)
(c) Compensatory Contour Plot
(b) Noncompensatory IRS
(a,=.709, a2=.526, b,=-.092, b2=-1.177)
(d) Noncompensatory Contour Plot
Figure 4. Item response surfaces and contour plots for item 10, MD20, a=30'


142
Table 51
Descriptive Statistics for Compensatory MD10 Linking Items with Randomly
Equivalent Groups
Replication
Form A
Form B
Mean
SD
Mean
SD
1
-0.196
0.801
-0.107
0.771
2
-0.105
0.788
-0.070
0.835
3
-0.117
0.827
-0.197
0.774
4
-0.169
0.763
-0.170
0.846
5
-0.166
0.889
-0.141
0.784
6
-0.104
0.761
-0.154
0.806
7
-0.164
0.823
-0.146
0.752
8
-0.170
0.754
-0.123
0.833
9
-0.125
0.828
-0.097
0.762
10
-0.084
0.753
-0.102
0.740
11
-0.071
0.779
-0.057
0.748
12
-0.059
0.750
-0.119
0.765
13
-0.128
0.780
-0.127
0.769
14
-0.135
0.779
-0.096
0.774
15
-0.083
0.793
-0.102
0.766
16
-0.109
0.778
-0.200
0.772
17
-0.167
0.777
-0.121
0.742
18
-0.115
0.753
-0.122
0.834
19
-0.132
0.851
-0.167
0.729
20
-0.178
0.751
-0.232
0.740


73
Evaluation Criteria
To establish a foundation for evaluating the results of the research
equatings, the three comparison conditions described below were used. In
addition, three statistical criteria-correlation, standardized mean difference,
and standardized root mean square difference--were applied to the data.
Comparison Conditions
For the first comparison condition the unidimensional approximations
of the multidimensional item parameters were calculated using the analytic
procedure described by equations 17 and 18 (Wang, 1986). To compute
these approximations for the eight research conditions, the SAS IML procedure
was applied to each of the simulated parameter sets. The means and
standard deviations of the responses for each condition were determined for
inclusion in the formula. The resulting sets of unidimensional comparison item
parameters were weighted composites of the item parameters for the two traits
(Ackerman, 1988). Table 9 presents the analytical unidimensional item
parameter approximations for compensatory MD30, Form A. The resulting
analytical item parameter estimates were then fixed in BILOG 386 and all
compensatory and noncompensatory response sets were analyzed to establish
the comparison ability estimates.
For the next comparison condition, the second dimension of each
multidimensional item was ignored. This would be reasonable if arguing that
most published tests were designed to measure only the first factor. For


87
(a) Compensatory IRS (b) Noncompensatory IRS
(a,=.516, a2=.894, d=.011) (a,=.542, a2=.664, b,=-1.764, b2=-.538)
(c) Compensatory Contour Plot (d) Noncompensatory Contour Plot
Figure 6. Item response surfaces and contour plots for item 12, MD20, a=60'


61
distance from the origin of the multidimensional ability space to the point where
the item provides maximum examinee information, or where the IRS has the
steepest slope. A line joins these points at angle a*. In a two-dimensional
item, the value of a* can range between 0 and 90 depending on the degree
to which the item measures the two traits. If the item only measures the first
trait, a,) equals 0, while an = 90 would depict an item measuring only the
second trait. For this study, an was set to either 0, 20, 30, 45, or 60.
Item Parameters
Four tests with 40 items each were simulated using the compensatory
M2PL model described above. Forty items were selected as sufficient to
provide good equating results. An anchor test design was chosen for data
collection as it is widely used by practitioners.(Skaggs & Lissitz, 1886a). Each
test consisted of two forms with 12 common linking items and 28 unique items.
The difficulty values were selected to be reasonable for published tests. Lord
(1968) found difficulties ranging from -1.5 to 2.5 ( X=0.58, SD=0.87) on SAT
Verbal data. Doody-Bogan and Yen (1983) employed a range of b¡ of -2.0 to
1.52 ( X =-0.028, SD=0.818) in a simulation designed to imitate CTBS-U data.
In a study using multidimensional data, Ackerman (1988) reported MID values
ranging form -0.73 through 1.87 on an ACT Mathematics test. Oshima and
Miller (1990) used MID values in the interval -2.0 to 2.0. For the purpose of
this investigation, multidimensional item difficulty parameters (MID) were
generated using the RANNOR function of SAS. Values were chosen randomly


Table 21
Summary of Concurrent Calibration Results with Randomly Equivalent Groups
Correlation
SDM
SRMSD
Condition
e,
01+02
AE8
01
01 + 02
AEa
01
01 + 02
AE8
2
2
2
Compensatory, Form A
MD10
0.94(.00)
0.73(01)
0.98(00)
0.00(.04)
-0.01(.03)
0.01 (.02)
0.34(01)
0.77(.02)
0.27(.01)
MD20
0.83(02)
0.88(01)
0.98(.00)
-0.01 (.05)
-0.01 (.03)
0.01(03)
0.58(03)
0.54(.03)
0.28(01)
MD30
0.81 (.02)
0.90(01)
0.97(.00)
-0.01(05)
0.00(.03)
0.04(.03)
0.63(03)
0.50(.03)
0.29(.01)
MD40
0.78(02)
0.93(01)
0.98(01)
-0.01(02)
-0.01(02)
0.03(03)
0.66(.03)
0.47(02)
0.27(02)
Compensatory, Form B
MD10
0.94(01)
0.73(.02)
0.98(.01)
0.01 (.04)
0.00(03)
0.02(02)
0.34(.01)
0.77(.03)
0.27(01)
MD20
0.83(01)
0.88(.01)
0.98(01)
-0.01(03)
-0.01 (.03)
0.01(03)
0.59(.02)
0.55(.03)
0.28(,01)
MD30
0.81 (.02)
0.90(.01)
0.97(.00)
0.00(.03)
0.01 (.04)
0.05(04)
0.63(02)
0.50(.02)
0,29(.01)
MD40
0.78(02)
0.93(.01)
0.98(00)
-0.01 (.04)
-0.01(03)
0.03(.03)
0.67(.02)
0.46(02)
0.27(.01)
Noncompensatory, Form A
MD10
0.94(.00)
0.71 (.02)
0.98(01)
0.00(.04)
0.00(04)
0.01(03)
0.34(01)
0.80(02)
0.30(02)
MD20
0.86(.01)
0.80( 01)
0.97(01)
-0.01 (.04)
-0.01(04)
0.02(02)
0.54(02)
0.66(.02)
0.32(01)
MD30
0.82(02)
0.85(01)
0.97(.00)
0.00(05)
0.01 (.03)
0.02(02)
0.61(02)
0.59(.03)
0.31(01)
MD40
0.75(03)
0.89(01)
0.97(.00)
-0.01 (.03)
0.00(02)
0.04(03)
0.72(03)
0.53(03)
0.28(.02)
Noncompensatory, Form B
MD10
0.94(01)
0.71(03)
0.98(01)
0.00(04)
0.00(03)
0.01(02)
0.35(02)
0.79(.03)
0.31 (.02)
MD20
0.86(01)
0.80(02)
0.97(.01)
-0.01(03)
-0.01(03)
0.01(02)
0.54(02)
0.66(03)
0.32(.02)
MD30
0.83(01)
0.85(.01)
0.97(.00)
0.00(03)
0.00(04)
0.03(03)
0.60(02)
0.59(02)
0.31 (.01)
MD40
0.75(.02)
0.89(01)
0.97(00)
0.00(.04)
0.00(.03)
0.04(02)
0.71(03)
0.53(.02)
0,29(01)
Note. Means and (standard deviations) of 20 replications for each condition,
a = Analytical Estimation


65
Noncompensatory Data
For each compensatory item generated, a corresponding
noncompensatory item was created. A noncompensatory item was considered
corresponding if it had the same probability of success as the compensatory
item (Ackerman, 1989). To accomplish this, the NLIN procedure of SAS was
applied to Equation 16. Specifically, the compensatory probability was
calculated for each case and became the dependent variable. The
independent variable in the NLIN model statement was the noncompensatory
probability function. Only multidimensional items were transformed as the
compensatory/noncompensatory question was not applicable to
unidimensional items. Starting values for noncompensatory parameter
estimation were set to equal the compensatory parameters. The 1,000 theta
vectors generated for the first of each compensatory response set were
treated as known values. To ensure the program was producing unique local
minima, starting values were changed for several items in each set and
reestimated. Any differences which appeared in the parameter estimates were
contained in the fourth or fifth decimal place. For approximately 10% of the
items in each dataset, the convergence criterion was not met within 40
iterations. In these cases, the final parameter estimates were substituted for
the starting values and the program rerun. In all such cases, convergence was
achieved with the second attempt.


20
Tucker Equally Reliable and Levine Unequally Reliable linear models and three
IRT methods. The equipercentile equating produced the worst results of all
the methods investigated. This was especially true for the Verbal Test.
In a 1983 study by Cook and Eignor reported in Skaggs and Lissitz
(1986a), alternate forms of the biology, mathematics, and social studies
achievement tests of the GRE were equated using various procedures. Again,
results varied by test content, but the equipercentile method was inadequate in
all cases. Cook and Eignor felt that equipercentile equating may have suffered
from a lack of data at the extreme scores.
The Cook et al. (1988) equatings with biology achievement test data
also uncovered mixed results. Although the equipercentile equating method
performed adequately with the parallel fall-to-fall samples, it was not sufficiently
robust to the ability differences found in equating the fall and spring samples.
These mixed findings raise some concerns about the application of
equipercentile equating. When raw scores are used, this method does not
meet the conditions for equating. Hambleton and Swaminathan (1985) noted
that a nonlinear transformation is needed to equalize the moments of the two
distributions, resulting in a nonlinear relationship between the raw scores and
the true scores. In turn, this implies that the tests are not equally reliable and it
is no longer a matter of indifference to the examinee which form is taken.
Besides violating the equity condition, the equipercentile equating process is
population dependent.


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
Paul George
Professor of Edqi
Leadership
This dissertation was submitted to the Graduate Faculty of the College of
Education and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
May, 1996
Chairman, Foundations of
Education
D
i, College of Education1
Dean, Graduate School


154
Hulin, C. L, Lissak, R. J., & Drasgow, F. (1982). Recovery of two- and
three-parameter item characteristic curves: A monte carlo study. Applied
Psychological Measurement. 6, 249-260.
Kingston, N. M., & Dorans, N. J. (1984). Item location effects and their
implications for IRT equating and adaptive testing. Applied Psychological
Measurement. 8, 147-154.
Klein, L. W., & Jarjoura, D. (1985). The importance of content
representation for common-item equating with nonrandom groups. Journal of
Educational Measurement. 22, 197-206.
Klein, L. W., & Kolen, M. J. (1985, April). Effects of number of common
items in common-item eouatino with nonrandom groups. Paper presented at the
annual meeting of the American Educational Research Association, Chicago.
Kolen, M. J. (1981). Comparison of traditional and item response theory
methods for equating tests. Journal of Educational Measurement. 18, 1-11.
Kolen, M. J., & Whitney, D. R. (1982). Comparison of four procedures
for equating the tests of General Educational Development. Journal of
Educational Measurement. 19, 279-293.
Livingston, S. A., Dorans, N. J & Wright, N. K. (1990). What
combination of sampling and equating methods works best? Applied
Measurement in Education. 3, 73-95.
Lord, F. M. (1968). An analysis of the Verbal Scholastic Aptitude Test
using Birnbaum's three-parameter logistic model. Educational and
Psychological Measurement. 28, 989-1020.
Lord, F. M. (1980). Applications of item response theory to practical
testing problems. Hillsdale, NJ: Lawrence Erlbaum.
Marco, G. L., Petersen, N. S & Stewart, E. E. (1983). A test of the
adequacy of curvilinear score equating models. In D. L. Weiss (Ed ), New
horizons in testing: Latent trait theory and computerized adaptive testing (pp.
147-177). New York: Academic.
Mislevy, R. J., & Bock, R. D. (1987). PC-BILOG maximum likelihood
item analysis and test scoring: Logistic model. Mooresville, IN: Scientific
Software.
Mislevy, R. J., & Bock, R. D. (1990). BILOG 3: Item analysis and test
scoring with binary logistic models. Mooresville, IN: Scientific Software.


117
The analytical estimation comparison condition produced excellent
consistency and performance throughout the variety of conditions. The
unidimensional approximations of the multidimensional item parameters seems to
provide an excellent description of the underlying multidimensional relationship
between the discriminations, difficulties, and latent traits. This procedure needs
to be studied further to determine if it truly recovers the nature of the
multidimensional data, or it is simply mimicking the BILOG estimations.
The large mean differences displayed in the concurrent calibration of
nonequivalent groups suggests another area for further study. Because the
analytical estimates correlated highly with the equated abilities, even though the
SDMs were low, there is reason to believe the ranking of the examinees
remained relatively similar. Perhaps the calibration procedure was affected by
the vast differences in traits, resulting in ability estimates on a different scale but
in the same rank order.
Results of this study are not meant to generalize to other IRT models or
equating methods. The degree and number of multidimensional Items were
selected to be reasonable for practical testing situations, but they do not cover all
possible contingencies. Although there were no effects on unidimensional
equating found due to procedure or amount of multidimensionality present in the
data, caution in applying these unidimensional IRT equating methods on data
gathered from examinee groups unequal in ability is advised.


16
Linear equating
In horizontal equating, the two tests to be equated are similar in
difficulty. When administered to the same group of examinees, the raw score
distributions are assumed to be different only with respect to the means and
standard deviations (Hambleton & Swaminathan, 1985). Linear equating is
based on this assumption. A transformation is identified such that scores on X
and Y are considered to be equated if they correspond to the same number of
standard deviations above or below the mean in some population. The two
scores are equivalent if
X_^=Y^ (1)
OX
These scores will have the same percentile rank if the distributions are the
same (Crocker & Algina, 1986).
Many variations of linear equating models exist whose details may be
found in the literature (Angoff, 1971; Holland & Rubin, 1982; Marco et al.,
1983). Two of the more commonly used models are the Tucker model and the
Levine equally reliable model. Both of these procedures produce an equating
transformation of the form:
LP(y) Ay B (2)
where Lp (y) is the linear equating function for equating Y to X (Dorans, 1990).
Adaptations of this formula exist for dealing with an anchor test, usually
labelled V, when it is or is not part of the reported score. The difference
between the Tucker model and the Levine equally reliable model lies in their


89
Table 14
Descriptive Statistics for Analytical Unidimensional Estimates of Form A Item
Parameters
Parameter
Condition
Minimum
Maximum
Mean
SD
Compensatory Model
a
10
0.18
2.01
0.67
0.5
20
0.16
1.13
0.55
0.2
30
0.15
1.32
0.55
0.3
40
0.30
1.35
0.69
0.3
b
10
-1.95
2.63
-0.11
1.0
20
-2.05
2.02
-0.09
1.0
30
-1.91
1.41
-0.16
0.9
40
-1.49
1.85
-0.11
0.9
Noncompensatory Model
a
10
0.22
2.01
0.68
0.4
20
0.09
1.10
0.48
0.2
30
0.12
0.93
0.47
0.2
40
0.30
1.11
0.57
0.2
b
10
-4.04
1.63
-0.69
1.3
20
-4.86
2.02
-0.75
1.4
30
-6.02
1.45
-1.29
1.6
40
-3.87
2.21
-1.41
1.2
Note. N = 40 in all conditions.
through 19 verify the success of the data generation. Most ability values are
between -3.00 and +3.00 with the mean and standard deviation specified. The
correlation between 0, and 02 is approximately zero in all conditions.


66
Response vectors were generated by applying Equation 10 and using
the same (01,02) combinations utilized to produce the corresponding
compensatory responses. Twenty response sets were simulated for each
noncompensatory dataset. The item parameters for the multidimensional Form
A items of noncompensatory MD30 are shown in Table 6. Summary statistics
for datasets of both models are displayed in Table 7.
Noneauivalent Groups
One of the strongest theoretical advantages of IRT is its usefulness with
groups of subjects who differ in abilities. One case where this may occur is
when a second form of a test, such as a high school proficiency test, is
administered only to examinees who failed to pass the first attempt. To
examine the effect of data from a lower ability group being equated to data
gathered from a normally distributed group, sets of 1,000 less able simulees
were generated. Scores on 0i for the lower group ranged between -3.00 and
0.00 with mean -0.80 and standard deviation 0.6. Abilities on the second
dimension were normally distributed with mean 0 and standard deviation 1.
Five replications of scores were generated for all four compensatory test
conditions.
Estimation of Parameters
Unidimensional IRT
The responses of the 1,000 simulated examinees in each response set
were analyzed by the computer program BILOG (Mislevy & Bock, 1990) to
estimate the unidimensional item discrimination and difficulty parameters.


156
Roznowski, M., Tucker, L., & Humphreys, L. (1991). Three approaches
to determining the dimensionality of binary items. Applied Psychological
Measurement. 15, 109-127.
SAS Institute Inc. (1990). Statistical analysis system (6.071. Cary, NC:
SAS Institute.
Skaggs, G., & Lissitz, R. W. (1986a). IRT test equating: Relevant issues
and a review of recent research. Review of Educational Research. 56, 495-529.
Skaggs, G., & Lissitz, R. W. (1986b). An exploration of the robustness
of four test equating models. Applied Psychological Measurement. 10, 303-
317.
Stocking, M. L., & Lord, F. M. (1983). Developing a common metric in
item response theory. Applied Psychological Measurement. 7. 201-210.
Stout, W. F. (1990). A new item response theory modeling approach
with applications to unidimensionality assessment and ability estimation.
Psvchometrika. 55, 293-325.
Swaminathan, H., & Gifford, J. A. (1983). Estimation of parameters in
the three-parameter latent trait model. In D. Weiss (Ed.), New horizons in
testing, (pp. 13-30). New York: Academic Press.
Swaminathan, H., & Gifford, J. A. (1985). Bayesian estimation in the
two-parameter model. Psvchometrika. 50, 349-364.
Sympson, J. B. (1978). A model fortesting with multidimensional items.
In D. J. Weiss (Ed.), Proceedings of the 1982 item response theory/
computerized adaptive testing conference (pp. 151-177). Minneapolis, MN:
University of Minnesota, Department of Psychology.
Wang, M. (1986, April). Fitting a unidimensional model to
multidimensional item response data. Paper presented at the ONR contractors
conference, Gatlinburg, TN.
Way, W. D., Ansley, T. N., & Forsyth, R. A. (1988). The comparative
effects of compensatory and noncompensatory two-dimensional data on
unidimensional IRT estimates. Applied Psychological Measurement. 12, 239-
252.


84
(a) Compensatory IRS
(a,=.732, a2=.266, d=-.104)
(c) Compensatory Contour Plot
(b) Noncompensatory IRS
(a,=.526, a2=.378, b,=-.595, b2=-2.961)
(d) Noncompensatory Contour Plot
Figure 3. Item response surfaces and contour plots for item 9, MD20,

LD
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141
Table 50
Analytical Estimates of Unidimensional Item Parameters for MD40 Form B
Compensatory Noncompensatory
Item
Form
Oil
a
b
a
b
1
A,B
20
0.863
-1.485
0.680
-0.329
2
A,B
30
0.842
-0.194
0.679
-1.041
3
A,B
45
0.881
0.971
0.675
-2.382
4
A,B
60
0.925
-1.475
0.746
0.462
5
A,B
20
0.336
-0.243
0.304
-2.235
6
A,B
30
0.651
0.075
0.541
-1.537
7
A,B
45
0.566
-0.549
0.487
-0.881
8
A,B
60
0.461
-0.366
0.429
-1.345
9
A,B
20
1.352
-1.110
0.962
0.234
10
A.B
30
0.973
1.743
0.733
-3.353
11
A,B
45
0.428
0.201
0.392
-1.981
12
A,B
60
0.537
-0.952
0.483
-0.572
41
B
20
0.599
1.046
0.562
0.000
42
B
30
1.135
-0.774
0.140
1.178
43
B
45
1.059
-0.327
0.767
-0.730
44
B
60
0.638
-1.192
0.555
-0.149
45
B
20
0.299
0.076
0.275
-2.802
46
B
30
0.257
-0.075
0.242
-2.701
47
B
45
0.614
0.180
0.524
-1.627
48
B
60
1.201
-1.109
0.873
0.206
49
B
20
0.729
0.998
0.609
-2.633
50
B
30
0.463
-0.590
0.404
-1.142
51
B
45
0.526
1.359
0.476
-1.776
52
B
60
0.666
0.625
0.578
-2.108
53
B
20
0.776
0.057
0.626
-1.506
54
B
30
0.554
0.166
0.648
-3.289
55
B
45
0.330
-0.430
0.309
-1.690
56
B
60
0.418
0.252
0.397
-2.105
57
B
20
0.470
0.396
0.417
-2.380
58
B
30
0.243
-0.081
0.230
-2.822
59
B
45
0.687
-0.261
0.569
-1.044
60
B
60
0.968
0.433
0.759
-1.683
61
B
20
0.523
-1.864
0.430
0.213
62
B
30
0.509
-0.707
0.436
-0.903
63
B
45
0.544
-0.552
0.473
-0.914
64
B
60
0.473
0.324
0.441
-2.046
65
B
20
0.904
-0.044
0.703
-1.274
66
B
30
0.759
-0.010
0.606
-1.329
67
B
45
0.354
-0.533
0.328
-1.470
68
B
60
0.284
2.117
0.291
-4.530


91
Table 16
Descriptive Statistics for Simulated Examinees Taking MD10
THETA 1 THETA 2
Rep
Low
High
Mean
SD
Low
High
Mean
SD
1
-2.89
3.02
0.07
0.96
-3.80
3.13
-0.01
1.01
2
-3.19
3.15
0.04
1.00
-2.94
3.03
-0.03
1.02
3
-3.62
2.79
-0.06
1.00
-3.02
2.62
0.02
1.00
4
-2.87
3.33
-0.01
0.99
-3.26
2.98
0.05
1.03
5
-2.98
3.65
0.03
0.99
-3.05
3.16
-0.01
0.99
6
-3.13
2.75
-0.02
1.00
-3.05
3.56
0.01
0.99
7
-2.98
3.63
0.01
0.98
-3.46
2.77
-0.01
0.97
8
-2.97
3.19
0.01
1.03
-3.42
4.03
0.03
0.98
9
-3.76
3.24
-0.02
0.97
-2.96
3.15
0.00
0.99
10
-4.02
2.96
-0.01
1.02
-2.68
3.33
0.00
1.03
11
-3.97
3.28
-0.02
1.03
-3.11
3.58
-0.03
1.03
12
-3.29
3.05
-0.06
1.02
-3.28
3.23
0.01
1.01
13
-3.11
3.57
-0.01
0.96
-3.38
4.00
-0.01
1.06
14
-2.86
3.07
0.03
0.97
-3.20
3.15
-0.02
1.03
15
-3.91
3.39
-0.01
0.99
-3.43
3.22
-0.01
1.03
16
-3.53
2.94
0.01
0.99
-4.28
3.08
-0.03
1.01
17
-3.18
2.76
0.01
0.97
-3.32
3.01
0.03
1.00
18
-4.30
3.71
-0.01
1.03
-3.62
3.26
-0.02
1.02
19
-3.61
2.80
-0.02
0.99
-3.27
3.35
0.00
1.00
20
-2.84
2.97
0.03
1.02
-2.97
3.18
0.00
1.00
Note. N = 1000 for each replication
-0.8 and standard deviations of 0.6. The 02 values were normally distributed
and ranged from -3.00 to +3.00 with mean 0.0 and standard deviation 1. This
would be expected with uncorrelated theta abilities.


Table 25
Summary of Characteristic Curve Transformation Results with Randomly Equivalent Groups
Correlation
SDM
SRMSD
Condition
e,
01+02
2
AEa
e,
01+02
2
AEa
01
01+02
2
AEa
Compensatory
MD 10
0.94(01)
0.75(01)
0.97(,01)
-0.01(05)
-0.01(04)
0.01 (.03)
0.37(.02)
0.76(.02)
0.30( 02)
MD 20
0.88(01)
0.84(.01)
0.97(01)
-0.01(08)
-0.01(.09)
-0.02(.05)
0.49(02)
0.62(02)
0.30( 02)
MD 30
0.82(.02)
0.91(01)
0.97(.01)
0.00(07)
0.01(07)
0.04(05)
0.61(.02)
0.51(.03)
0.27(.03)
MD 40
0.74(.02)
0.94(01)
0.98(01)
-0.01(05)
-0.01(.05)
0.03(.03)
0.73(.03)
0.47(03)
0.26(02)
Noncompensatory
MD 10
0.93(.01)
0.71(.01)
0.97(.01)
0.00(02)
0.01(04)
0.00(.02)
0.38(.02)
0.81(.03)
0.31(.02)
MD 20
0.90(01)
0.78(01)
0.97(.01)
0.00(07)
-0.01(09)
0.00(06)
0.47(01)
0.70(02)
0.30(.02)
MD 30
0.84(01)
0.84(.01)
0.97(.01)
0.00(07)
0.00(.08)
0.02(05)
0.58(02)
0.61(.02)
0.28(02)
MD 40
0.70(02)
0.90(01)
0.98(.00)
0.00(04)
0.00(06)
0.02(04)
0.78(03)
0.53(.03)
0.26(02)
Note. Means and (standard deviations) of 20 replications for each condition,
a = Analytical Estimation
104


83
compensatory and corresponding noncompensatory model IRS and contour plot
for an item of each degree of dimensionality are shown in Figures 4 through 7.
In Figure 4, a matched item that discriminates predominantly on 0i (a = 20)
is pictured. The differences between the two IRSs are minor. A similarity also
exists in the two conditions where the degree of dimensionality is 15 from
equally discriminating. Figure 5 shows the IRS for a = 30, which discriminates
slightly more on 01 than on 02. Conversely, Figure 7 presents the graphs for a =
60, which discriminates slightly more on 02 than on 0i. Although differences
exist in the baselines, the curves of the IRSs remain similar. This is true both
within each of the two matched sets and between the items with a = 30 and
a=60. In Figure 6, where a = 45, the corresponding compensatory and
noncompensatory items discriminate equally along 01 and 02, and there is a
sharp contrast between corresponding curves.
Similar conclusions can be drawn from examination of the equiprobability
lines of the contour plots. For the compensatory model, parallel lines join the
01,02 combinations that have an equal probability of a correct response. The
incline of these lines is a function of the discrimination parameters. However,
because the noncompensatory model does not allow a high ability on one
dimension to compensate for a low ability on another dimension, the lines
connecting the 0i,02 combinations are curvilinear. The direction of these lines
in the noncompensatory model is a function of the items difficulty parameters


48
unidimensional parameter estimation procedures to be robust in the presence
of multidimensional data.
The studies reviewed indicate that IRT parameters implied by the
general factor are recovered well when the common factors have sufficiently
high correlations. Reckase, Ackerman, and Carlson (1988) used both
simulated and empirical data to demonstrate that Items can be selected to
construct a test that meets the unidimensionality assumption even though
more than one ability is required for a correct response. The authors showed
that the unidimensionality assumption only requires the items in a test to
measure the same composite of abilities. This seems to have been met in the
previous investigations. Based on this study, it appears as if the
unidimensionality assumption is not as restrictive as formerly thought.
Although these studies explored the effect of multidimensionality on
unidimensional parameter estimation, it is also important to understand what
effect the choice between compensatory and noncompensatory
multidimensional models may have on estimation. Ackerman (1989) simulated
two-dimensional data using both compensatory and noncompensatory M2PL
models. Forty two-dimensional items were generated using the compensatory
model. Difficulty was confounded with dimensionality and p(6i, 02) was
selected at 0.0, .3, .6, and .9. For each compensatory item, a corresponding
noncompensatory item was created using a least-squares approach to
minimize the quantity


153
Doody-Bogan, E., & Yen, W. M. (1983, April). Detecting
multidimensionalitv and examining its effect on vertical equating with the three-
parameter logistic model. A paper presented at the annual meeting of the
American Educational Research Association, Montreal.
Dorans, N. J. (1990). Equating methods and sampling designs. Applied
Measurement in Education. 3, 3-17.
Dorans, N. J., & Kingston, N. M. (1985). The effects of violations of
unidimensionality on the estimation of item and ability parameters and on the
item response theory equating of the GRE verbal scale. Journal of Educational
Measurement. 22, 249-262.
Drasgow, F., & Parsons, C. K. (1983). Application of unidimensional item
response theory models to multidimensional data. Applied Psychological
Measurement. 7, 189-199.
Hambleton, R. K., & Swaminathan, H. (1985). Item response theory.
Boston: Kluwer-Nijhoff.
Harris, D. J., & Kolen, M. J. (1986). Effect of examinee group on
equating relationships. Applied Psychological Measurement. 10, 35-43.
Harrison, D. A. (1986). Robustness of IRT parameter estimation to
violation of the unidimensionality assumption. Journal of Educational Statistics.
11, 91-115.
Hattie, J. (1985). Methodology review: Assessing unidimensionality of
tests and items. Applied Psychological Measurement. 9. 139-164.
Hills, J. R., Subhlyah, R. G., & Hirsch, T. M. (1988). Equating minimum
competency tests: Comparison of methods. Journal of Educational
Measurement. 25, 221-231.
Hirsch, T. M. (1989). Multidimensional equating. Journal of Educational
Measurement. 26, 337-349.
Hirsch, T. M., & Miller, T. R. (1991). Comparison of rotational methods
applied to multidimensional item response theory item-parameter estimates.
Paper presented at the annual meeting of the American Educational Research
Association, Chicago.
Holland, P. W., & Rubin, D. B. (Eds ). (1982). Test equating. New
York: Academic Press.


15
Table 1
Summary of Recommendations for a Successful Equating
Total Test
Well-defined content specifications
Item selection based on statistical data from field testing
Length of at least 35 items
Examinees
Sample size of at least 500
Better results with groups similar in ability
Administrative
Strictly controlled testing conditions
Security of tests and items is maintained
Scoring is controlled
Anchor Tests
. Representative of the total test in difficulty and discrimination
Similar to the total test in content specifications
Common items are in approximately the same position in the old and
new forms.
Common items are identical in both forms.
About 20% 30% of total test length


LD
1780
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SITfJf


Table 2
Summary of Unidimensional IRT Test Equating Studies
Study
Tests
Equating Models
Independent Variables
Cook & Eignor
(1983)
CB-achievement
3PL, equipercentile,
linear
equating models
scaling methods
Cook, Eignor, &
Taft (1988)
Biology achievement
3PL
equipercentile
dissimilar samples
equating models
Harris & Kolen
(1986)
ACT-Math
3PL, equipercentile,
linear
equating models
dissimilar samples
Hills, Subhiyah,
& Hirsch (1988)
SSAT-II
Rasch, 3PL, linear
equating models
negatively skewed distribution
anchor length
scaling models
Kolen (1981)
ITED: Math &
Vocabulary
Rasch, 2PL, 3PL,
equipercentile, linear
equating models
item context
Kolen & Whitney
(1982)
GED
equipercentile
Rasch, 3PL, linear,
equating models
Marco, Petersen,
& Stewart (1983)
SAT-V
Rasch, 3PL, linear,
equipercentile
ability distribution
internal & external anchor
difficulty of anchor
Peterson, Cook, &
Stocking (1983)
SAT-V
SAT-Q
3PL, linear,
equipercentile
equating models
scaling models (3PL)


68
Table 7
Summary Statistics for Multidimensional Items in Compensatory and
Noncompensatory Datasets
MD10
MD20
MD30
MD40
Parameter
c
NC
C
NC
C
NC
C
NC
Mean
84
.67
.37
.60
86
.66
.91
.69
at
SD
.55
.33
.80
.28
.50
.35
.50
.37
Mean
.71
.61
.75
.56
.68
.55
.70
.62
a2
SD
.36
.23
.47
.30
.39
.23
.38
.27
Mean
.11
.00
.25
.14
d|
SD
.97
1.12
1.11
1.07
b,
Mean
-1.09
-1.08
-1.02
-.88
SD
1.03
1.09
1.28
1.16
Mean
-1.37
-1.54
-1.35
-1.23
b2
SD
94
1.80
1.37
1.18
Note. C = Compensatory item parameters; NC = Noncompensatory item parameters


71
Once the slope (A) and intercept (B) of the linear transformation were found,
they were applied to all ability and item estimates for Form B, yielding
/t>; = Ab, + B (20)
9*=A6a + B (22)
All parameters were now transformed to the same scale. Although item
discrimination or ability estimates could have been used to determine the linear
transformation, item difficulty estimates are usually used in practice because
they yield the most stable parameter estimates (Cook & Eignor, 1991).
Characteristic Curve Transformation
The parameter estimates computed separately for Form A and Form B
were also used in the characteristic curve transformation. This equating
method used both a¡ and b¡ estimates from the linking items to derive a linear
transformation through an iterative process that minimized the difference
between the item parameter estimates of the linking items. The process is
based on the assumption that if the estimates were free of error, choosing the
proper linear transformation would cause the true-score estimates of the
linking items to correspond (Petersen et al., 1989; Stocking & Lord, 1983).
The resulting transformation was then applied to all Form B parameters to
create estimates on the same scale. The EQUATE (Baker, Al-Karni, & Al-
Dosary, 1991) computer program was used to accomplish this. Data were


121
Table 30
Simulated Compensatory Item Parameters for MD20 Form A
Item
Form
a¡i
ai
a2
d¡
MDISC
MID
1
A.B
0
0,607
0.000
-0.649
0.607
1.069
2
A,B
0
1.547
0.000
1.315
1.547
-0.850
3
A,B
0
2.411
0.000
2.761
2.411
-1.145
4
A,B
0
1.394
0.000
-0.117
1.394
0.084
5
A,B
0
0.791
0.000
1.304
0.791
-1.649
6
A.B
0
0.682
0.000
1.270
0.682
-1.862
7
A,B
45
1.581
1.581
2.479
2.236
-1.108
8
A,B
60
0.666
1.153
0.642
1.331
-0.482
9
A,B
20
0.732
0.266
-0.104
0.779
0.134
10
A,B
30
0.934
0.539
0.650
1.079
-0.603
11
A,B
45
1.223
1.223
1.194
1.730
-0.690
12
A,B
60
0.516
0.894
0.011
1.032
-0.010
13
A
0
1.019
0.000
-0.549
1.019
0.539
14
A
0
1.792
0.000
0.064
1.792
-0.036
15
A
0
0.953
0.000
0.977
0.953
-1.025
16
A
0
0.797
0.000
-0.213
0.797
0.267
17
A
0
0.541
0.000
-0.379
0.541
0.701
18
A
0
0.514
0.000
0.170
0.514
-0.331
19
A
0
0.789
0.000
-0.421
0.789
0.534
20
A
0
0.950
0.000
-0.359
0.950
0.378
21
A
0
0.865
0.000
-1.582
0.865
1.829
22
A
0
0.453
0.000
-0.247
0.453
0.545
23
A
0
0.741
0.000
0.014
0.741
-0.019
24
A
0
0.764
0.000
0.673
0.764
-0.881
25
A
0
0.756
0.000
-0.901
0.756
1.191
26
A
0
0.300
0.000
0.181
0.300
-0.603
27
A
45
0.690
0.690
-1.701
0.976
1.742
28
A
60
0.313
0.542
-0.537
0.626
0.858
29
A
20
0.541
0.197
-0.653
0.576
1.135
30
A
30
0.689
0.398
0.468
0.796
-0.588
31
A
45
1.378
1.378
-0.348
1.949
0.179
32
A
60
0.417
0.721
1.166
0.833
-1.400
33
A
20
0.552
0.201
-0.126
0.587
0.214
34
A
30
0.605
0.350
0.354
0.699
-0.506
35
A
45
0.813
0.813
-1.413
1.149
1.229
36
A
60
1.057
1.830
-2.439
2.113
1.154
37
A
20
0.896
0.326
1.084
0.953
-1.137
38
A
30
1.344
0.776
1.117
1.551
-0.720
39
A
45
0.998
0.998
1.752
1.411
-1.241
40
A
60
1.079
1.869
0.965
2.158
-0.447


49
1000
ZlfPcl e¡, a, B-PncI e> a, b)f (16)
j=1
where Pc is a given compensatory item's probability of a correct response and
Pnc is the noncompensatory item's probability of a correct response which
varies as a function of a and b given 0. The unidimensional 2PL model was
used to estimate parameters using both BILOG and LOGIST. The authors
discovered minimal differences in the IRS for each model when the parameters
are matched. The confounding of difficulty with dimensionality was only
detected by BILOG. For both models, as p(0i, 02) increased, the response
data became more unidimensional and estimation of all parameters improved.
Way, Ansley, and Forsyth (1988) also compared compensatory and
noncompensatory models with simulated data. The values assigned p(0i, 02)
ranged from 0.0 to .95. Results showed the number-right distributions for the
two models were comparable. In the noncompensatory model, the
unidimensional a, estimates appeared to be averages of the a¡ and a2 values,
while the compensatory model provided a, estimates best considered as sums
of ai and a2. The b, estimates for the noncompensatory data were greater
than b, values, while the compensatory model seemed to average the bi and
b2 values. For both models, the 0 estimates were related to the average of the
two 0 parameters.
A summary of the studies investigating the effect of multidimensional
data on unidimensional IRT parameter estimation is presented in Table 3.
Generally, parameters appear to be recovered adequately with data fit


124
Table 33
Simulated Compensatory Item Parameters for MD30 Form B
Item
Form
Cth
a,
a2
di
MDISC
MID
1
A,B
0
0.475
0.000
-0.584
0.475
1.231
2
A,B
0
0.563
0.000
-0.173
0.563
0.308
3
A,B
0
0.515
0.000
0.652
0.515
-1.266
4
A.B
60
0.736
1.275
1.199
1.472
-0.814
5
A.B
20
1.159
0.422
0.681
1.234
-0.552
6
A,B
30
0.706
0.407
-0.054
0.815
0.066
7
A.B
45
0.936
0.936
-0.939
1.323
0.709
8
A,B
60
0.291
0.504
-0.618
0.582
1.062
9
A.B
20
0.684
0.249
-0.599
0.728
0.822
10
A,B
30
0.882
0.510
1.652
1.019
-1.621
11
A,B
45
1.129
1.129
2.676
1.597
-1.675
12
A,B
60
0.881
1.526
-1.018
1.763
0.578
41
B
0
1.609
0.000
-0.935
1.609
0.581
42
B
0
0.771
0.000
0.390
0.771
-0.506
43
B
0
0.875
0.000
0.363
0.875
-0.415
44
B
0
0.650
0.000
0.693
0.650
-1.065
45
B
0
1.272
0.000
-2.252
1.272
1.771
46
B
0
1.971
0.000
-3.295
1.971
1.672
47
B
0
1.133
0.000
1.381
1.133
-1.219
48
B
60
0.471
0.816
0.505
0.943
-0.536
49
B
20
1.781
0.648
1.493
1.896
-0.788
50
B
30
0.805
0.465
0.633
0.929
-0.681
51
B
45
2.105
2.105
4.653
2.977
-1.563
52
B
60
0.576
0.998
1.119
1.152
-0.971
53
B
20
2.093
0.762
0.413
2.227
-0.185
54
B
30
0.834
0.481
1.028
0.963
-1.068
55
B
45
0.399
0.399
0.038
0.564
-0.067
56
B
60
0.393
0.681
0.019
0.787
-0.024
57
B
20
1.663
0.605
-0.731
1.770
0.413
58
B
30
0.866
0.500
-0.573
1.000
0.573
59
B
45
1.048
1.048
-1.546
1.483
1.042
60
B
60
0.350
0.607
-0.425
0.700
0.606
61
B
20
1.206
0.439
0.242
1.283
-0.189
62
B
30
1.300
0.751
1.384
1.501
-0.922
63
B
45
1.301
1.301
-1.453
1.839
0.790
64
B
60
0.150
0.259
-0.432
0.299
1.445
65
B
20
1.383
0.503
-0.724
1.471
0.492
66
B
30
0.841
0.485
0.109
0.971
-0.113
67
B
45
0.467
0.467
-0.542
0.660
0.822
68
B
60
0.313
0.541
-0.324
0.625
0.518


101
Table 23
Constants for Equated bs Equaling of Noncompensatory Forms with Randomly
Equivalent Groups
MD10 MD20 MD30 MD40
Rep
Slope
Intercept
Slope
Intercept
Slope
Intercept
Slope Intercept
1
1.04
0.03
1.01
-0.04
0.93
-0.01
0.89
0.05
2
0.94
-0.04
0.94
-0.04
0.94
-0.04
0.94
-0.04
3
0.94
-0.04
1.12
-0.12
0.93
-0.04
1.08
0.07
4
1.05
0.07
0.88
-0.01
0.92
-0.05
0.97
0.02
5
1.00
0.03
0.99
-0.02
1.05
0.07
0.91
0.05
6
0.98
-0.07
1.01
0.01
0.90
-0.13
1.15
-0.07
7
1.09
0.04
0.97
0.06
1.01
0.11
0.96
-0.06
8
0.98
-0.05
0.97
-0.05
1.01
0.00
1.06
0.13
9
0.96
-0.01
1.20
-0.11
0.92
-0.15
0.99
-0.10
10
1.10
-0.01
0.71
0.14
1.09
0.12
1.04
0.12
11
1.02
-0.03
0.77
0.13
0.90
-0.02
1.06
-0.15
12
0.94
0.09
1.23
-0.19
0.98
0.05
1.09
0.15
13
1.06
-0.01
0.79
-0.03
1.01
0.03
0.94
-0.02
14
0.98
-0.06
1.47
0.09
0.89
-0.10
1.07
0.12
15
1.08
0.07
0.87
-0.06
0.99
0.04
0.96
-0.04
16
1.02
0.05
0.94
-0.02
1.04
0.04
1.01
0.06
17
0.89
-0.04
0.86
0.15
0.83
-0.02
1.07
-0.01
18
1.08
-0.04
1.01
-0.11
1.14
-0.01
0.98
0.05
19
1.15
0.09
1.23
-0.10
0.89
-0.05
1.13
-0.04
20
0.97
0.04
0.78
0.14
1.04
-0.01
1.00
-0.01


LIST OF FIGURES
Figure page
1 An item characteristic curve (ICC) based on the three-
parameter logistic model 23
2 An item response surface (IRS) based on the compensatory
M2PL 40
3 Item response surfaces and contour plots for item 9, MD20,
a = 20 84
4 Item response surfaces and contour plots for item 10, MD20,
a = 30 85
5 Item response surfaces and contour plots for item 11, MD20,
a = 45 86
6 Item response surfaces and contour plots for item 12, MD20,
a = 60 87
xi


64
Table 5
Simulated Compensatory Parameters for MD30. Form A
Item
Form
Oil
a,
a2
d,
MDISC
MID
1
A,B
0
0.475
0.000
-0.584
0.475
1.231
2
A,B
0
0.563
0.000
-0.173
0.563
0.308
3
A,B
0
0.515
0.000
0.652
0.515
-1.266
4
A,B
60
0.736
1.275
1.199
1.472
-0.814
5
A,B
20
1.159
0.422
0.681
1.234
-0.552
6
A,B
30
0.706
0.407
-0.054
0.815
0.066
7
A.B
45
0.936
0.936
-0.939
1.323
0.709
8
A,B
60
0.291
0.504
-0.618
0.582
1.062
9
A,B
20
0.684
0.249
-0.599
0.728
0.822
10
A,B
30
0.882
0.510
1.652
1.019
-1.621
11
A,B
45
1.129
1.129
2.676
1.597
-1.675
12
A.B
60
0.881
1.526
-1.018
1.763
0.578
13
A
0
0.973
0.000
0.549
0.973
-0.565
14
A
0
1.358
0.000
-0.324
1.358
0.239
15
A
0
1.857
0.000
1.417
1.857
-0.763
16
A
0
0.860
0.000
-0.524
0.860
0.609
17
A
0
1.448
0.000
1.538
1.448
-1.062
18
A
0
1.517
0.000
-0.448
1.517
0.295
19
A
0
0.663
0.000
-0.142
0.663
0.214
20
A
60
0.480
0.832
0.723
0.961
-0.753
21
A
20
0.648
0.236
-0.550
0.689
0.798
22
A
30
1.944
1.122
0.992
2.244
-0.442
23
A
45
1.120
1.120
0.654
1.584
-0.413
24
A
60
0.268
0.464
-0.122
0.535
0.228
25
A
20
0.790
0.288
0.295
0.841
-0.351
26
A
30
0.442
0.255
0.159
0.510
-0.313
27
A
45
1.452
1.452
0.019
2.053
-0.009
28
A
60
0.328
0.568
-0.243
0.656
0.370
29
A
20
0.744
0.271
0.055
0.792
-0.070
30
A
30
0.398
0.230
0.315
0.460
-0.686
31
A
45
0.355
0.355
0.924
0.502
-1.840
32
A
60
0.465
0.806
-1.060
0.930
1.140
33
A
20
1 442
0.525
-1.014
1.535
0.661
34
A
30
1.031
0.595
-0.284
1.191
0.238
35
A
45
0.879
0.879
1.320
1.244
-1.061
36
A
60
0.431
0.747
-0.965
0.862
1.119
37
A
20
0.589
0.214
0.533
0.627
-0.850
38
A
30
1.144
0.661
2.296
1.321
-1.738
39
A
45
0.810
0.810
1.050
1.145
-0.917
40
A
60
0.147
0.254
-0.135
0.293
0.461