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The Impact of NonNormality on the Asymptotic Confidence Interval for an Effect Size Measure in Multiple Regression


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1 THE IMPACT OF NONNORMALITY ON THE ASYMPTOTIC CONFIDENCE INTERVAL FOR AN EFFECT SIZE MEASUR E IN MULTIPLE REGRESSION By LOU ANN MAZULA COOPER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Lou Ann Mazula Cooper

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3 ACKNOWLEDGMENTS I would like to take this opport unity to thank the co-chairs of my dissertation committee, Dr. David Miller and Dr. James Algina. They are both exceptional teac hers and without their guidance, support, and patience, th is work would not have been po ssible. Dr. Miller, through his flexibility, approachable manner, and breadth of knowledge ha s been an invaluable resource throughout the course of my graduate studies. Dr. Algina is perhaps the most generous teacher I have ever known and his door was always ope n to me. His passion for research and his incredible work ethic had a profound influence on me. I would also like to thank the other members of my co mmittee for their help and encouragement. Special thanks go to Dr. Richar d Davidson. I will always be grateful for the opportunity he provided to apply and expand my skills to experime ntal design, data analysis, and psychometric issues in medical education researc h. Thank you for giving me the most rewarding and stimulating job I have ever had. Dr. Walte r Leite, although not ther e at the beginning, has provided a valuable sounding block and I l ook forward to our future collaborations. I would also like to express my gratitude to my family for their love and encouragement during my mid-life career change. I know it has not always been easy to live with me. To my daughters, Abigail and Amanda, my hope is that I have provided a good example for you in pursuing my life-long love of l earning. Finally, and most esp ecially to Brian, my husband and best friend, whose love and unw avering belief in me made achieving this goal possible.

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4 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................3 LIST OF TABLES................................................................................................................. ..........6 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............10 CHAPTER 1 INTRODUCTION..................................................................................................................12 Effect Sizes and Confidence Interval s in Multiple Regression Analysis...............................14 Asymptotic Confidence Intervals for Correlations.................................................................15 The Impact of Nonnormality on Statistical Estimates............................................................24 Statement of the Problem....................................................................................................... .26 Purpose of the Study........................................................................................................... ....26 2 METHODS........................................................................................................................ .....28 Study Design................................................................................................................... ........28 Number of predictors.......................................................................................................28 Squared multiple correlations..........................................................................................28 Sample size.................................................................................................................... ..29 Distributions.................................................................................................................. ..29 Background and Theoretical Justific ation for the Simulation Method...................................33 Data Simulation................................................................................................................ ......37 Data Analysis.................................................................................................................. ........39 3 RESULTS........................................................................................................................ .......47 Replication of Results for Multivariate Normal Data.............................................................47 Simulation Proper.............................................................................................................. .....49 Analysis of Variance and Mean Square Components............................................................54 The Influence of Nonnormality on Coverage Probability......................................................58 Nonnormal predictors......................................................................................................58 Nonnormal error distribution...........................................................................................59 The Impact of Squared Multiple Correlations on Coverage Probability................................59 The Impact of Sample Size on Coverage Probability.............................................................64 Probability Above and Below the Confidence Interval..........................................................65 The Relationship between Estimated Asymptotic Variance, Empirical Sampling Variance of R2, and Coverage Probability........................................................................66

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5 4 DISCUSSION..................................................................................................................... ..124 Limitations.................................................................................................................... ........126 Further Research............................................................................................................... ....128 Conclusion..................................................................................................................... .......130 APPENDIX A PROGRAM FOR COMPUTING MARDIAS MULTIVARIATE MEASURES OF SKEWNESS AND KURTOSIS IN SAS.............................................................................135 B DATA SIMULATION PROGRAM IN SAS......................................................................138 LIST OF REFERENCES.............................................................................................................142 BIOGRAPHICAL SKETCH.......................................................................................................146

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6 LIST OF TABLES Table page 2-1 Study Design............................................................................................................... .......41 2-2 Mardias Multivariate Skewness, b1, k, for the Nonnormal Distributions...........................42 2-3 Mardias Multivariate Kurtosis, b2, k, for the Nonnormal Distributions.............................43 3-1 Replication of Algina and Moulders Results for Multivariate Data and Two Predictors..................................................................................................................... ......72 3-2 Replication of Algina and Moulder s Results for Multivariate Data and Six Predictors..................................................................................................................... ......73 3-3 Replication of Algina and Moulders Results for Multivariate Data and Ten Predictors..................................................................................................................... ......74 3-4 Empirical Coverage Probabilities for Normal Predictors and Normal Errors...................75 3-5 Empirical Coverage Probabilities for Normal Predictors and Nonnormal Errors.............82 3-6 Empirical Coverage Probabilities for Nonnormal Predictors and Normal Errors.............89 3-7 Empirical Coverage Probabilities for Predictors Nonnormal and Errors Nonnormal.......96 3-8 Descriptive Statistics for Coverage Probability by Distributional Condition..................104 3-9 Analysis of Variance, Estimated Mean S quare Components, and Percentage of Total..105 3-10 Descriptive Statistics for Coverage Proba bility by Distribution for the Predictors.........105 3-11 Descriptive Statistics for Coverage Pr obability by Distribution for the Errors...............106 3-12 Coverage Probability by 2 and the Distribution for the Predictors..............................106 3-13 Coverage Probability by 2 and the Distribution for the Errors....................................106 3-14 Coverage Probability by 2 rand the Distribution fo r the Predictors...............................107 3-15 Coverage Probability by 2 rand the Distribution for the Errors......................................107 3-16 Coverage Probability by 2 rand 2 for X Distributed Multivariate Normal.................107 3-17 Coverage Probability by 2r and 2 for X Distributed Pseudot10( g = 0, h = .058).......108 3-18 Coverage Probability by 2r and 2 for X Distributed 2 10Pseudo...............................108

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7 3-19 Coverage Probability by 2rand 2 for X Distributed 2 4Pseudo................................108 3-20 Coverage Probability by 2 rand 2 for X Distributed Pseudo-exponential...................109 3-21 Coverage Probability by Sample Size and 2................................................................109 3-22 Coverage Probability by Sample Size and Number of Predictors...................................110 3-23 Analysis of Variance, Estimated Mean S quare Components, and Percentage of Total..110 4-1 Coverage Probability as a Function of n, Selected Values for 22and ,r and Distribution for the Predictors.........................................................................................132

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8 LIST OF FIGURES Figure page 2-1 Plot of the empirical cumulative dist ribution function for a univariate nonnormal distribution where g = 0, h = .058 overlaid with a normal curve with gh = 0, gh = 1.097....................................................................................................................... ..44 2-2 Plot of the empirical cumulative dist ribution function for a univariate nonnormal distribution where g = .301, h = -.017 overlaid with a normal curve with gh = .150, gh = 1.041....................................................................................................................... ..44 2-3 Plot of the empirical cumulative dist ribution function for a univariate nonnormal distribution where g = .502, h = -.048 overlaid with a normal curve with gh = .249, gh = 1.108....................................................................................................................... ..45 2-4 Plot of the empirical cumulative dist ribution function for a univariate nonnormal distribution where g = .760, h = -.098 overlaid with a normal curve with gh = .378, gh = 1.252....................................................................................................................... ..45 2-5 Comparison of Mardias multivariate skewness for the multivariate normal distribution to that of the distributions investigated..........................................................46 2-6 Mardias multivariate kurtosis for the multivariate normal distribution and the nonnormal distributions investigated.................................................................................46 3-1 Mean estimated coverage probability by normality vs. nonnormality in the predictors, normality vs. nonnormality in the errors, and sample size............................111 3-2 Empirical coverage prob ability as a function of dist ributional condition and sample size........................................................................................................................... ........112 3-3 Box plots of the distributions of covera ge probability estimates by distribution for the predictors (ni = 14,700)..............................................................................................113 3-4 Box plots of the distributions of covera ge probability estimates by distribution for the errors (ni = 14,700).....................................................................................................114 3-5 Main effect of the squared se mipartial correlation coefficient 2, and the effect of the interaction of 2 and X on coverage probability for 2 > 0...................................115 3-6 Main effect of the squared se mipartial correlation coefficient 2, and the effect of the interaction of 2 and e on coverage probability for 2 > 0....................................115 3-7 Effect of the interaction between the size of the square d multiple correlation in the reduced model,2r and the distribution for the predictors, X, on coverage probability.................................................................................................................... ....116

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9 3-8 Interaction between the size of the sq uared multiple correlation in the reduced model,2r, and the distribution for the errors, e, and its relationship to coverage probability.2 r..................................................................................................................116 3-9 Effect of the 2 r 2 interaction on coverage probability for 2 > 0...........................117 3-10 Effect of the X 2r 2interaction on covera ge probability for 2 > 0..................118 3-11 Interaction between sample size, n, and the population squared semipartial correlation, 2, and the impact on coverage probability for 2 > 0.............................119 3-12 Effect of the interact ion between sample size, n and number of predictors, k, on coverage probability.........................................................................................................119 3-13 Ratio of mean estimated asymptotic variance to the variance in R2 (MEAV/Var R2) as a function of the dist ribution for the predictors, 2, and 2 r .............................120 3-14 Relationship between coverage probab ility and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0...................121 3-15 Relationship between coverage probab ility and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 for multivariate normal data (g = 0, h = 0)............................................................................121 3-16 Relationship between coverage probab ility and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed pseudo-t10 (g = 0, h = .058)............................................................................122 3-17 Relationship between coverage probab ility and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed 2 10pseudo(g = .502, h = -.048)....................................................................122 3-18 Relationship between coverage probab ility and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed 2 4pseudo(g = .502, h = -.048).....................................................................123 3-19 Relationship between coverage probab ility and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed pseudo-exponential (g =.760, h = -.098)........................................................123 4-1 Coverage probability as a function of sample size and several combinations of 22 and rfor predictors sampled from a normal distribution and (A) pseudo-t10; (B) pseudo-22 104; (C) pseudo-;and (D) pseudo-exponential distributions...........................133

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10 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE IMPACT OF NONNORMALITY ON TH E ASMPTOTIC CONFIDENCE INTERVAL FOR AN EFFECT SIZE MEASUR E IN MULTIPLE REGRESSION By Lou Ann Mazula Cooper May 2007 Chair: M. David Miller Co chair: James Algina Major: Research and Evaluation Methodology The increase in the squared multiple correlation coefficient, R2, associated with an individual predictor in a regr ession analysis is a measure commonly used to evaluate the importance of that variable in a multiple re gression analysis. Prev ious research using multivariate normal data had shown that relative ly large sample sizes are necessary for an acceptably accurate confidence interval for this regression effect size measure. The coverage probability that an asymptotic confidence in terval contained the population squared semipartial correlation, 2, was investigated by simula ting data from a range of nonnormal distributions such that (a) the predictors we re nonnormal, (b) the error distribution was nonnormal, or (c) both predictors and er rors were nonnormal. Additional factors manipulated included (a) the number of predictor variables, (b) the magni tude of the population squared multiple correlation coefficient in the original model, 2,r (c) the magnitude of the population squared semipartial correlation, 2, and (d) sample size. This study showed that when nonnormality is introduced, empirical coverage probability was always less than the nominal confidence leve l, often dramatically so. The degree of

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11 nonnormality in the predictors was the most im portant factor influe ncing poor coverage probability. Although coverage probability incr eased as a function of sample size, when nonnormality in the predictors was substantial, the confidence interval is likely to be inaccurate no matter how large a sample size is used. With multivariate normal data, coverage probability improved as both 2rand 2 increased. When predictors are sampled from a nonnormal distribution, coverage probability tended to decrease as 2r and 2 increased and became even worse as the degree of nonnormality increased. It was further demonstrated that the asymptotic variance underestimates the sampling variance of R2. This produces standard errors that are too small and results in a confidence in terval that is too narrow. Re liance on this confidence interval as a measure of the strength of the effect size wi ll lead us to underestim ate the importance of an individual predictor to the regression.

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12 CHAPTER 1 INTRODUCTION There is a growing consensus that the tr adition of null hypothesis significance testing (NHST) has led to over-reliance on statistical sign ificance in evaluating re search results in the behavioral and social sciences. According to Co hen (1994), the biggest flaw in NHST is that it does not tell us what we want to know. A statis tical test evaluates the pr obability of the sample results given the size of the sample assuming th at the sample is drawn from a population where the null hypothesis is exactly true. In this fram ework, the outcome of a significance test is a dichotomous decision whether or not to reject the null hypothesis. As noted by Steiger and Fouladi (1997, p. 225), this dichotomy is inherently dissatisfyi ng to psychologists and educators, who frequently use the null hypothesis as a statemen t of no effect, and are more interested in knowing how big an effect is than whether it is (p recisely) zero. Fundamentally, we are interested in determining how accurately the population effect has been estimated from the sample data and whether the observed effect size has practical signi ficance. Statistical significance testing fails to provide the answers. Within the behavioral and social sciences methodological recommendations for reporting research results have increasingly emphasized the importance of reporting confidence intervals (Cumming & Finch, 2001; Smiths on, 2001), effect sizes (Olejnik & Algina, 2002; Vacha-Hasse & Thompson, 2004), and confidence intervals for e ffect sizes (Cohen, 1990; Steiger & Fouladi, 1997; Thompson, 2002) to complement the re sults of hypothesis testing. Among the recommendations of the APAs Task Force on St atistical Inference (Wilkinson & Task Force on Statistical Inference, 1999) wa s a proposal to move away fr om routine reliance on NHST as a primary means of analyzing data to exploring, summarizing and analyzin g data using visual representations, effect-size measures, and confid ence intervals. The most recent edition of The

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13 Publication Manual of the Amer ican Psychological Association (2001, p. 25-26) states, For the reader to fully understand the importance of your findings, it is almost always necessary to include some index of effect si ze or strength of relationship in your Results sectionThe general principle to be followed, howev er, is to provide the reader not only with information about statistical significance but also with enough inform ation to assess the magn itude of the observed effect or relationship. The Manual also states that failure to re port an effect size is a defect (p. 5). In 1996, Thompson recommended that Ameri can Educational Research Association (AERA) journals require that e ffect sizes be reported and interp reted in all studies. Ten years later the AERA Council recommends that statistica l results should include an effect size measure as well as an indication of the un certainty of that index of effect such as a confidence interval. The recently adopted Standards for Reporting on Em pirical Social Scienc e Research in AERA Publications (AERA, 2006) states that when quantita tive methods are employed, It is important to report the results of analyses that are critical for the interpre tation of findings in ways that capture the magnitude as well as the si gnificance of those results (p. 37). Editors of over 20 APA and other social science journals have published guidelines explicitly requiring auth ors to report effect sizes (Ellis, 2000; Harris, 2003; Heldref Foundation, 1997; Hresko, 2000; McLean & Kaufman, 2000; Royer, 2000; Snyder, 2000; Thompson, 1994; Vacha-Haase, Nilsson, Re ntz, Lance, & Thompson, 2000) and the Editor of Journal of Applied Psychology requires an author to provide an explan ation when an effect size is not reported (Murphy, 1997). Although this is evidence that editorial practices have evolved somewhat, effect size reporting is unlikely to become th e norm until we move from recommendation and encouragement to requirement (Thompson, 1996; 1999).

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14 Effect Sizes and Confidence Intervals A confidence interval establishe s a range of parameter values that are reasonably consistent with the data observed from a sample. Because a confidence interval gives a best point estimate of a parameter of interest and an interval about it reflecting an estim ate of likely error, it contains all the information to be found in a significance test and more (Cohen, 1994). The likely range of the parameter values provides researchers with a better understanding of their data. If the parameter estimated has meaningful units, a confidence interval can be used to make statistical inferences that provide information in the same metric. According to Cumming and Finch (2001), there are four main reasons for promoting the use of confidence in tervals: (a) they are readily interpretable, (b) are linked to familiar statistical tests, (c) can encourage replication and meta-analytic thinking, and (d) gi ve information about precision. The term effect size is broadly used to refer to any statistic that provides information that helps us judge the practical significance of the results of a study (K irk, 1996). Cohen (1990) recommends that in addition to reporting an effect size, researchers should provide confidence intervals for effect sizes in orde r to gauge the possible range of va lues an effect size may assume. Absent a confidence interval, it is difficult to evaluate the accuracy of the effect size estimate. This, in turn, has implications for drawing meaningful conclusions. Unfortunately, despite the increasing demand fo r researchers to do so, reporting effect sizes and confidence intervals has yet to beco me commonplace in educational and psychological journals. Vacha-Hasse, Nilsson, Rentz, Lance, and Thompson (2000) reviewed ten studies of effect size reporting in 23 journa ls, and found effect size(s) to be reported in roughly 10 to 50 percent of articles, notwithstanding the encouragem ent to do so from the fourth edition of the APA manual (1994). Empirical studies show th at even when effect sizes are reported, interpretation is often give n short shrift (Finch et al, 2002; Keselman et al., 1998).

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15 It is likely that the emphasis on null hypothesis significance tes ting in graduate courses in statistics and research methodol ogy has contributed to a genera l lack of knowledge concerning confidence intervals. Moreover, techniques for computing confidence intervals are often neglected in popular statis tics textbooks and are not easily ava ilable in the statistical software that is routinely employed by app lied researchers in the social sc iences (Smithson, 2001). Even if these factors were not operating, researchers mi ght be reluctant to report confidence intervals because as Steiger and Fouladi (1997, p. 228) observe, interval estimates are sometimes embarrassing. Reporting confidence intervals can highlight the level of imprecision of statistical estimates and exposes the trivial nature of many publis hed studies. Smithson (2001, p. 614) notes, Almost any literature review or meta-analysis in psychology would give a very different impression from that conveyed by NHST if we routinely reconstructed CIs for multiple R2 and related GLM parameters. Asymptotic Confidence Intervals for Correlations A confidence interval establishes a range of hypothetical parameter va lues that cannot be ruled out given the observed sample data. The pr obability that the random interval includes, or covers, the true value of the parameter is the cove rage probability of the interval. When the exact distribution of a statistic is known, the coverage is equal to th e confidence level and the interval is said to be exact. A confiden ce interval is exact if it can be expected to contain a parameters true value 100(1 )% of the time. Often exact interval s are not available or are difficult to calculate, and approximate intervals are used instead. Confidence intervals are based on the sampling distribution of a statistic. Due to the central limit theorem, when sample size is suffici ently large, the sampling distribution of statistic will become more symmetric and eventually appe ar nearly normal, even when the population itself is not normally distributed. Methods based on asymptotic th eory use approximations to the

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16 sampling variance of a statistic. If only the asym ptotic distribution of the statistic is known, we can obtain an approximate confidence interval, wh ich may or may not be reasonably accurate in finite samples. If the asymptotic confidence inte rval procedure is fully adequate, under repeated random sampling under identical conditions, a 95% c onfidence interval would contain the true population parameter 95% of the time. The accur acy of the approximation depends on whether there is a lack of bias and the degree to whic h the sampling distribution deviates from normality. If a statistic has no bias as an es timator of a parameter, its sampli ng distribution is centered at the true value of a parameter. An unbiased confid ence interval is one where the probability of including any value other than the parameters true value is less than or equal to 100(1 )%. An interval is said to be conservative if the rate of coverage is greater than 100(1 )%, the nominal confidence level. If the coverage probability is less than the nominal, the interval is said to be liberal. In general, c onservative intervals are preferred ov er liberal ones (Smithson, 2003). Whenever a statistic based on asymptotic theory has poor finite sample properties, a confidence interval based on that sta tistic has poor coverage. Multiple regression analysis is a common statistic al application frequently used to predict a dependent variable (outcome) fr om two or more independent variables (predictors). The interpretation of results would be enhanced by the reporting of confidence intervals and effect sizes. The sample statistic, R2, which estimates the proportion of variance in the dependent variable that is explained by the set of predic tors, is commonly used to evaluate a multiple regression model. Published rese arch studies frequently report R2 values without any evidence of the precision with which they have been estimated. It is unfortunate that a confidence interval for the population parameter, 2, is not computed by most popular statistical software packages.

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17 Perhaps more significant, the topic is not even di scussed in many applied or theoretical statistics texts. In addition to the amount of variance explained by a given multiple regression model, researchers are often interested in evaluating th e contribution that one variable makes to the regression, over and above a set of other explanatory variables. The increase in R2, R2, when a variable (Xj) is added to a multiple regression model is a useful measure of the strength of the relationship between Xj and the dependent variable, Y, controlling for all other independent variables in the model. The change in R2 that we observe by including each new Xj in the regression equation is the squa red semipartial correlation corr esponding to a given regression coefficient. Typically, whether Xj has made a statistically significant contribution to predicting Y is tested by conducting a tor F-test on that regression coefficient. But, the squared semipartial correlation itself is a useful measure of effect size and as recommended by Cohen (1990) and Thompson (2002), we should calculate a confidence in terval to evaluate th e precision with which it has been estimated and the range of likely values. Hedges and Olkin (1981) presente d procedures for constructi ng a confidence interval for the squared semipartial correlation based on calc ulating the asymptotic covariance matrix for commonality components. Commonality analys is is a procedure by which the variance accounted for in the criterion is partitioned into two parts, the unique part and the common part. The unique part is attributable to the predictors individually. This is essentially the partial contribution of each predictor to the squared multip le correlation with the criterion. The second part is the common part, attributable to a combin ation of the predictors, which is the contribution to the multiple correlation with the criterion that a ll of the predictors in the combination share.

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18 Thus, commonality analysis is a way to measur e the importance of vari ables through the use of partial correlations. Hedges and Olkins results can be used to construct a confidence interval for R2. Olkin and Finn (1995) derived explicit ex pressions for asymptotic (large -sample) confidence intervals for functions of simple, partial, and multiple correl ations. Since the focus of this study is on the squared semipartial correlation, the following di scussion will be limite d to Olkin and Finns Model A (p. 157-159). Model A is the special case for use in determining whether an additional variable provides an improvement in predicting the criterion. All of the procedures for comparing two samp le correlation coefficients or two sample squared correlation coefficients described by Olkin and Finn have the same general form. Let rA and rB be the two sample correlations to be compared and A and B denote their corresponding population values. The large-sample distributional form for the differe nce in two correlations is 2()()~0,ABABrrN (1.1) where 2var()var()2cov(,)ABABrrrr (1.2) is the asymptotic variance of the differe nce of the two correlation coefficients; 2 is dependent on the population correlations (Olkin & Finn, 1995, p. 156). When squared correlation coefficients are co mpared, the expressions in Equations 1.1 and 1.2 become 22222[()()]~(0,)ABABrrN (1.3) and 2222222var2cov,.ABABABrrrr (1.4)

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19 Olkin and Finn present the general form for the large-sample variance of functions of correlations 2(,,)ijikjkfrrr aa (1.5) specialized to a function of three correlations, rij, rik, and rjk where f( ) is a function of the correlations, is the sampling variance-covariance ma trix of the correlations, and vector a contains a set of coefficients that depend on the function of the correlations to be evaluated. The variance of sample correlation rij is 22var()(1)/ijijrn (1.6) and the covariance of two correlations is 22221 cov(,)() 2 ()/.ijikijklikiljkjlikjliljk ijikiljijkjlkikjklliljlkrr n (1.7) When two correlations have one variable in common, Equation 1.7 simplifies to 22231 2cov(,)(2)(1)/.ijikjkijikijikjkjkrrn (1.8) Large-sample estimates are obtained by replac ing the population parameters with values computed from sample data. Using the delta method, it can be shown that if f(rij, rik. rjk) is a function of the three correlations, then the vector a consists of the partial derivatives a = 121323,,. f ff rrr (1.9) In the simplest case, suppose that two variables X1 and X2 are used to predict a third variable, X0. In order to determine whether X2, makes a significant contributio n to the regression, we are interested in the difference,22 0(12)01 R r Here, we use a capital R to signify a multiple correlation rather than a bivariate correlatio n, denoted by a lower case r. The symbol 2 0(12)R

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20 denotes the squared multiple correlation between X0, X1 and X2, which is a function of the correlations among the variables r01, r02, and r12 given by 22 22 0102010212 0(12)0(12) 2 122 1rrrrr R r (1.10) The squared correlation between X0 and X1 is represented by2 01r. Therefore, a c onfidence interval for22 0(12)01 R r can be computed using Olkin and Finns re sults for comparing two squared multiple correlation coefficients. In order to compare the population squared multiple correlations 2 0(12)and2 01 we use the estimates2 0(12)R, 2 01r, and 2 the estimated variance of the difference 2 0(12)R-2 01rwhere 22 0(12)01var() Rr a a. (1.11) The upper triangular of the symmetri c population correlation matrix is 0102 121 1 1 P (1.12) and the elements of the vector, a, are 12 1011202 2 122 (), 1 a (1.13) 2020112 2 122 (), 1 a (1.14) 222 3120112020102010212 22 122 (). (1) a (1.15) The variance-covariance matrix for the sample correlations is 1112130101020112 2223020212 3312var()cov(,)cov(,) var()cov(,). var() rrrrr rrr r (1.16)

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21 The sample correlation matrix, R, estimates P and the sample values in R can be used to compute the elements of a. Because the calculation of anal ytic derivatives becomes increasingly complicated as the number of variables increases, Olkin and Finn i llustrated their method for a multiple regression model with no more than two predictors. Graf and Alf (1999) e xpanded Olkin and Finns procedures to more general forms. Graf and Alf substituted numerical derivatives and offered two BASIC programs for calculating asymptotic c onfidence limits on the difference between two squared multiple correlations and the differe nce between two partial correlations. These programs, REDUX-AB, to compare two multiple correlations, and REDUX-CD, to compare two partial correlations, compute the matrix, the partial derivatives in vector a, and a 95% confidence interval. Alf and Graf (1999) present a further simp lification that does not employ numerical derivatives, is less computationally demanding, and produces results equi valent to the method described by Olkin and Finn. A ll computations are based on samp le estimates. The problem is approached by representing a multiple correlatio n as a zero-order correlation between the outcome variable and another singl e variable that is a weighted sum of the predictors. Alf and Graf defined 0 0B AB Ar r r (1.17) where the subscripts A and B denote weighted sums of tw o sets of predictors and rAB is the correlation between the two composite variables. The confidence interval for the squared semipart ial correlation coefficient is determined by the special case in which one set of predictors is a proper subset of the predictors in the other correlation. The two squared multiple correlations are computed using the same sample and the

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22 variables in the reduced model are a subset of the variables in the full model. Let 2r and 2 f denote the population squared multiple correlation coefficients corresponding to2 r R and2 f R The subscript, f refers to the full model with all predictors; the subscript, r refers to the reduced model. The reduced model contains all predictors with the exception of the variable of interest. The asymptotic variance of 2 f R is 22 241 ().ff fVarR n (1.18) The asymptotic variance of 2r R is 22 241 ().rr rVarR n (1.19) The asymptotic covariance between 2r R and 2 f R is 222233 224.52/1// (,) f rrffrfrrfrf frCovRR n (1.20) For the squared semipartial correlation, let 2 R 22. f r R R The asymptotic variance of 2 R is 22222()()2(,) f rfrVarRVarRCovRR. (1.21) An asymptotically correct 100(1 )% confidence interval for 222 f r is 2 /2 Rz (1.22) where z /2 is the (1 /2)th percentile of the standard normal distribution and is the estimate of In practice, the large-sample variance is estimated by substituting2 f R for 2 f and 2 r R for2 r in Equations 1.18, 1.19, and 1.20. Equations 1.18 and 1.19 are problematic when the population squared multiple correlations are zero because the implication is that the sampling variance of R2 is also zero (Stuart, Ord, &

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23 Arnold, 1999). Similarly, Equation 1.20 implies th at the sampling covarian ce is zero if either population multiple correlation coefficient, f or r is zero. If it were known that both 2 f and 2 r were zero and these values were used to constr uct a confidence interval, we would incorrectly conclude that the width of the resulting interval is zero. This computa tional problem is unlikely to occur in practice since we substitute samp le multiple correlation coefficients for their population values and it is doubtful that either2 f R or 2 r R will ever be exactly zero. The Alf and Graf formulas rely on asymptotic results. As such, they are only exactly correct for infinitely large samples. Thus, the accuracy of this approximation is heavily dependent on sample size. Alf and Graf (1999, p.74) concluded that the correlation between two multiple correlations will be extremely high when the variables in one multiple correlation are a subset of the variables in another multip le correlation and to ensure that coverage probability is equal to the nominal for the confidence interval on 2, moderately large to large sample sizes are necessary. In the absence of more specific recommendations on sample sizes, Algina and Moulder (2001) conducted a simulation study to evaluate th e empirical probability that the interval in Equation 1.22 includes 2 for 95% confidence interval. Algina and Moulder manipulated 2 f 2 r the number of predictors in the model ( k ), and the sample size ( n ). When the data are distributed multivariate normal, results indicate that when 2 > 0, for sample sizes representative of those used in psychology (i.e., n 600), coverage probab ilities for a nominal 95% confidence interval were less than .95. This tends to be true even with relatively large sample sizes, i.e. between 600 and 1200.

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24 When 22 f r= 0 all coverage probabilities were at l east .999 for all sample sizes studied. That is, when 2 does not increase when a predictor is added to a multiple regression model, the confidence interval is always t oo wide. Algina and Moulder ( 2001) posited two reasons for this defect in the confidence interval: (a) for all conditions in which 22 f r = 0 the asymptotic variance overestimated the sampling variance and (b) the distribution of 22 f r R R is positively skewed with a lower limit of 0. Because the c onfidence interval does not take this lower limit into account, even if the asym ptotic variance was not overestim ated, the lower limit would tend to be smaller than zero. Algina and Moulder (2001) showed that co verage probability tends to increase as 2 r increases and as 2 increases and tends to decrease as the number of predictors increases. Further, when the interval does not contain 2, there is a tendency for th e interval to be entirely below 2. Algina and Moulder conc lude that using the Alf a nd Graf method to compute a confidence interval with an inadequate sample size will underestimate the strength of the relationship between the predic tor and the outcome variable. The Impact of Nonnormality on Statistical Estimates Every procedure used to make statistical infere nces is based on a set of core assumptions. If the assumptions are met, the test will perfor m as theorized. However, the results may be misleading when the assumptions are violat ed. The most common method for estimating regression coefficients is ordinary least squares (OLS). Ordinary least squares yields unbiased, efficient, and normally distributed estimates when the following conditions are met: (a) No measurement error; (2) the mean of the residuals is zero; (3) the residuals have constant variance; (4) the residuals are not inter-correlated; a nd (5) the residuals ar e normally distributed.

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25 In terms of power and accura te probability coverage, stan dard analysis of variance (ANOVA) and regression me thods are affected by arbitrarily small departures from normality. As early as 1960, Tukey found that nonnormality could have a sizeable impact on power and measures of effect size could be misleading wh enever means are being compared. By sampling from a contaminated normal distribution, Tuke y showed that classical estimators are quite sensitive to distributions with h eavy tails. The contaminated nor mal distribution is a mixture of two normal distributions, one of which has a larg e variance; the other di stribution is standard normal. This results in a di stribution with heavier tails th an the Gaussian. Heavy-tailed distributions are characterized by unusually large or small values. Both heavy-tailed and skewed distributions are commonplace in applied work (Micceri, 1989). The presence of these characteristics in the data can diminish the chances of detecting true associations among random variables and obtaining accurate confiden ce intervals for the parameters of interest (Wilcox, 1998). After reviewing over 400 large data sets fr om educational and psychological research, Micceri (1989) found the majo rity did not follow univariate normal distributions. Approximately two-thirds of ability measures and over 80% of the psychometric measures examined exhibited at least moderate asymmetr y. For all data sets studied, 31% of the distributions showed skewness, 1, greater than .70 and 52% of psychometric measures demonstrated extreme to exponential asymmetry, 1 > 2.00. Psychometric measures also exhibited heavier tails than ability measures. Ku rtosis estimates ranged from .70 to 37.37. To put this in some perspective, the kurtosis for the double exponential distribution is 3.0. Breckler (1990) considered 72 articles in pe rsonality and social ps ychology journals and found that in analyses relying on the assumption of multivariate normality, only 19% of authors

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26 acknowledged this assumption and less than 10% considered whet her it had been violated. Keselman and his colleagues (1998) reviewed articles in promin ent educational and behavioral sciences research journals published during 1994 and 1995 and conc luded (a) the majority of researchers conduct statistical anal yses without considering the di stributional assumptions of the tests they are using and therefore use analyses that are not robust; (b) researchers rarely reported effect sizes; and (c) researchers failed to perform power analyses in order to inform sample size decisions. Statement of the Problem Methods for constructing confidence intervals based on asymptotic theory, such as those proposed by Olkin and Finn and Alf and Graf, have the potential to be very attractive to applied researchers. In the case of the equations presen ted by Alf and Graf, a hand calculator can be used to compute a confidence interval using the appropri ate estimates from the results of data analysis obtained using standard statisti cal analysis software. Howe ver, as Algina and Moulder demonstrated, even under the best case scenar io, where data are drawn from a multivariate normal distribution, the coverage probability of the asymptotic confidence interval for 2 is less than optimal, and when sample size is relatively small, e.g., < 200, would be considered unacceptable by most researchers. Since multivariate normal data is rare, the performance of Alf and Grafs procedure under real world cond itions warrants further investigation. Purpose of the Study My dissertation will extend the work of Algina and Moulde r (2001) and investigate the effect of the magnitude of population s quared multiple correlation coefficients,2 r and 2 f as well as the number of predictors, on th e asymptotic confidence interval for 2 under a range of nonnormal conditions. The study will investigate c overage probability when (a) the predictor

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27 variables are not distributed multivariate normal; (b) the residuals are not normal; and (c) both predictors and residuals are nonnor mal. Empirical coverage proba bilities will be compared to nominal coverage probabilities over a wide ra nge of sample sizes. My research will address the following questions: How adequate is Alf and Grafs asymptotic confidence interval procedure for the squared semipartial correlation coefficient when used with sample sizes typically employed in research in education, psychology and the behavioral sciences under conditions of nonnormality? Is there a minimum sample size for which th is method meets established standards for accuracy over a wide range of situations such that recommendations can be made for the use of this procedure in reporting the results of applied research?

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28 CHAPTER 2 METHODS In conducting a simulation study, especially wh en the goal is to inform the practice of researchers, it is important to ensure that the re levant factors are manipulat ed and that the levels of these factors reflect those rout inely observed. To that end, si x factors were manipulated in a factorial design using values ty pical of those observed in app lied research: the number of predictors, the size of the squared multiple corre lation in the reduced model, the size of the squared semipartial correlation, sample size, the distribution for th e predictors, and the distribution for the error. These f actors, and the levels of these f actors, are detailed in Table 2-1. Study Design Number of predictors Algina, Moulder, and Moser (2002) examin ed sample size requirements for accurate estimation of squared semipartial correlation co efficients and found a modest effect on the distribution of R2 due to the number of predictors included in the multiple regression model. Therefore, it follows that the sample size required for the confidence interval on 2 to be robust, i.e. to have the coverage probability equal to the nominal confidence level, will likewise depend on the number of predictors. The number of pr edictors in the initial set of predictors ( k 1) ranged from 2 to 10 in increments of 2. This allowed investigation of the performance of the asymptotic confidence interval for a reasonable range of model sizes. Squared multiple correlations Algina, Moulder, and Moser also show ed that the sampling distribution of R2 strongly depends on the population squared multiple correlations in both the full and reduced models, 2 f and 2 r Based on a survey of all APA journal articles published in 1992 reporting multiple

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29 regression results, Jaccard and Wan (1995) found the median squared multiple correlation in these studies to be .30. The 75th percentile for squared multiple correlations was approximately .50. Based on these results, the values for the squared multiple correlation coefficients for the predictors in the initial set (2 r ) ranged from .00 to .60 in steps of .10 (7 levels of the factor). Cohen (1988) proposed, as a convention, that .02, .13, and .26 represent small, medium, and large effect sizes for squared semipartial correlations. By manipulating the squared multiple correlation coefficient for the entire set of predictors (2 f ) such that it ranged from 2 r to 2 r + .30 in steps of .05, values for 2 that ranged from .00 to .30 in steps of .05 were produced (7 levels of the factor). The values for 2 are reasonably representative of likely effect sizes and the values selected for 2 r and 2 f cover a comprehensive range of population squared multiple correlations for multiple regression models from 2 = .00 to 2 = .90. Sample size Jaccard and Wan also reported typical sample sizes for studies using regression analysis. The median sample size was 175; a sample size of 400 was at the 75th percentile. However, Algina and Moulder found with multivariate normal data empirical estimates of the coverage probability were smaller than .95 even with a samp le size as large as 1200. Since we expected empirical coverage probabilities to be worse for nonnormal data, larger sample sizes than are usually observed in psychological research were included. Sample size ranged from 100 to 1000 in steps of 100 and from 1000 to 2000 in st eps of 250 (14 levels of the factor). Distributions The distributions chosen for study represent varying levels of nonnormality and were selected to: (a) allow examination of the effects of skewness and kurtosis; and (b) be representative of the types of univariate nonnormality commonly en countered in applied research

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30 in education and psychology. The method de scribed in Hoaglin ( 1985) and Martinez and Iglewicz (1984) using the g -andh distributions was used to genera te data that is characterized by varying degrees of skewness ( 1) and kurtosis ( 2). A g -andh distribution is generated by a single transformation of the standard normal distribution and allows for asymmetry and a variety of tail weights. In the case of the standard normal distribution, g = h = 0 and 1 = 2 = 0. When g = 0, a distribution is symmetric. Distri butions with positive skew typically have 1 > 0 and in distributions with negative skew, 1 < 0. The tails of the distribution become heavier as h increases in value. Long-ta iled distributions, such as the t -distribution, are characterized by 2 > 0. Short-tailed distributions, su ch as the uniform distribution, have 2 < 0. The distributions selected for this study a nd their skewness and kurtosis are presented in Table 2-1. Distribution 1 is the multivariate normal case. Distributi on 2 is symmetric and long-tailed and has the same skew and kurtosis as a t -distribution with 10 degrees of freedom. Distribution 3 is both asymmetric and leptokurtotic with the same skew and kurtosis as a 2 distribution with 10 degrees of freedom. Since distributions 2 and 3 have similar kurtosis, but differ with respect to asymmetry, this allowed us to evaluate the relativ e importance of skewness and kurtosis on the coverage probability of the co nfidence interval. Distribution 4 has the same skew and kurtosis as2 4 Distribution 5 is extremely skewed with heavy tails and has skew and kurtosis equal to the exponentia l distribution. Nonnormality was manipulated in either (a) the predictors, (b) the residuals, or (c) in both the predictors and the residuals. The error distribution is a univariate distribution. The empirical cumulative distribution functions for the four nonnormal distributions selected for this study, generated by sampling 1,000,000 random variates from each g -andh distribution, are depicted in Figures 2-1 to 2-4. In addition, the deviation from normality is shown by including the normal curve with mean equal

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31 to gh and standard deviation equal to gh for each distribution. The population mean and standard deviation for each g -andh distribution were calculated using the formulas given by Hoaglin (1985, p. 502-503). In multiple regression, the predictors are mu ltivariate. Multivariate normality, however, is a stronger assumption than univariate normality. Un ivariate normality of each of the variables is necessary, but not sufficient, and a nonnormal multivariate distribution can have normal marginals. Therefore, a preliminary step in evaluating multivariate normality is to study the reasonableness of assuming marginal normality for the observations on each of the variables (Gnanadesikan, 1997). In additi on to graphical approaches, a common method for evaluating the normality of univariate observations is by m eans of skewness and kurtosis coefficients, 1b and b2: 3 1 1 3/2 2 1 n i i n i inxx b xx (2.1) and 4 1 2 2 2 1.n i i n i inxx b xx (2.2) These are sample estimates of the popul ation skewness and kurtosis parameters 1 and 2, respectively. When the population is normal, 10 and 2 = 3. If 2 < 3, there is negative kurtosis; if 2 > 3, there is positive kurtosis. Popul ation skewness and kurtosis are also

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32 commonly described by 1 and 2 (Hoaglin, 1985) where 11 (2.3) and 223. (2.4) Mardia (1970) proposed indices for a ssessing multivariate normality that are generalizations of the univariat e skewness and kurtosis measures 1b and b2. Let X1,,Xn be a random sample from a population with mean vector and covariance matrix The sample mean vector and covariance matrix are denoted by X and S, respectively. The skewness and kurtosis, 1, k and 2, k, for a multivariate population, as defined by Mardia, are 3 1 1, kijExx (2.5) and 2 1 2,.kijExx (2.6) According to Rencher (1995), since third orde r central moments for the multivariate normal distribution are zero, 1, k = 0 when X ~ N ( ). Furthermore, it can be shown that for multivariate normal X 2,(2)kkk (2.7) where k is equal to the number of va riables. Sample estimates of 1, k and 2, k are given by 3 1 1, 2 111 ()nn kij ijbXXSXX n (2.8)

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33 and 2 1 2,1 ()().kij ibXXSXX n (2.9) Multivariate skewness and kurtosis were calculated by simulating 1,000,000 random variates sampled from each g -andh distribution for each level of k under investigation and then applying equations 2.8 and 2.9 to obtain estimates of Mardias multivariate measures, b1, k and b2, k. The SAS program used to estimate these i ndices is included in Appendix A. Mardias multivariate skewness estimates are presented in Table 2-2 and Table 2-3 presents Mardias multivariate kurtosis estimates. Figures 2-5 and 2-6 are graphic presentations that compare the coefficients for the nonnormal distributions to the values expected under multivariate normality for the number of predictors unde r investigation in this study. The design for the study is a 5 (data gene rating distribution fo r the predictors) 5 (data generating distributi on for the errors) 7 (2 r ) 7 ( 2) 5 ( k ) 14 ( n ) fully crossed factorial. This resulted in a total of 85,750 unique conditions. Each combination of factors was replicated 10,000 times and for each replication, a 95% conf idence interval was cons tructed using the Alf and Graf method. Background and Theoretical Justific ation for the Simu lation Method The multiple regression model can be written as 01122.... j jjkkjjYXXX (2.10) In the standardized multiple regres sion model, in the population with k 1 predictors and one criterion, all variables are standardized to m ean zero and unit variance so an intercept is not needed. This model is 1122 1...k j jjkkjjiijj iYXXXX (2.11)

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34 where i is the population standard ized regression coefficient associated with the i th predictor; eij ~ N (0, 2); i = 1, k ; j = 1, n Assuming that we are ope rating on the population and that the model is correct, pr edicted values are given by 1k jiij iYX (2.12) and the squared correlation between the observed ( Y ) and the predicted ( Y) values is denoted as 2 YY In the sample, this is estimated by R2. When the predictors are uncorrelated, the sum of the squared correlations is equa l to the variation accounted for by all the predictors 22 1.ik YX YY i (2.13) A simplifying transformation (Browne, 1975) holds that for any set of predictors that has a squared multiple correlation, 2, with Y it is always possible to transform the predictors so that (a) the transformed predictors are mutually unc orrelated, (b) have un it variance, and (c) the regression coefficients are equal to any set of values such that 222 1.k jy j (2.14) The quantity 2 is a function of the elements of the c ovariance matrix for the predictors and the criterion. In order to illustrate the app lication of Brownes results to the current simulation, let x denote the vector of standardized predictor variables, with kk correlation matrix P and 1 k vector of correlation coefficients between the predictors and the criterion variable, y. The squared multiple correlation coefficient for all k variables is denoted by 2 f and for the first 1k variables is denoted by2 r We seek a transformation of the predictors toxsuch that the new

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35 variables are standardized and uncorrelated, and the regression coefficients relating y to the variables in xare 0ifor the first 2 k variables and 2 1 kr and 22 kfr for the last two variables, respectively. The transformation can be constructed in two step s. It is well known that the variables in the vector ,xAx where A is akk matrix, will be uncorrelated dependent on an appropriate choice of A. For example, A can be selected as the inverse of the left Cholesky factor of R1Ti.e., ,RAA where T Aindicates the inverse of'A. The vector of correlation coefficients between the transfor med predictors and the criterion is A and because the transformed variables are uncorrelated, A is the vector of regression coefficients relating the criterion variable to the variables in x % Because the criterion is a standardized variable andxAx% is a nonsingular transformation,2 f is unchanged by the transformation, and2 f. We next seek a transformation xTx where Tis kk such that the variables in xare standardized and uncorrelated a nd so that the regression coeffi cients for the variables in xare 0i for the first 2 k variables and 2 1 kr and 22 kfr for the last two variables, respectively. We see that 2 f Because the variables in x are standardized and uncorrelated, the matrix T must be orthogonal so that the variables in x will be standardized and uncorrelated. With an orthogonal transformation, .T The matrix T can be constructed as follows (M. W. Browne, personal co mmunication with J. Algina, 1999): Let u Then, 12 TIuuuu is an orthogonal matrix, and T Because 2 1 u uu and T it follows that 2 u T uu Thus, if the variables in

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36 x are transformed toxTAx with Tand A defined as above, the transformed variables will be uncorrelated and standardized and the regression coefficients will be for the first k 2 variables, and 2 1 kr and 22 kfr for the last two variables, respectively. Because the variables are standardized and uncorrelated, the squared multiple correlation coefficient for the first k 1 variables will be 1 2 1 k ir i and the squared multiple correlation coefficient for all k variables will be 22222 1 k irfrf i The implication of Brownes result is that if the predictors are correlated, they can be transformed so that (a) the predictors ar e uncorrelated, (b) the predictive power of k 1 of the predictors is channeled into one of the transfor med predictors, (c) the pr edictive power of the remaining predictor is channeled into anothe r of the transformed pr edictors, and (d) the remaining k 2 predictors have no predictive power (A lgina, Moulder, & Moser, 2002). Rather than simulating various covarian ce structures for the predictors, the application of Brownes results allows us to operate with uncorrelated pred ictors since it is always possible to transform these variables to correlated variables. This dr amatically reduces the num ber of conditions in the simulation to a more manageable number. In ad dition, when the focus of the study is squared multiple correlation coefficients, there is no loss of generality if the means of the predictors and the criterion are rescaled to zero. Therefore, in the simulation, (a) the independe nt variables are mutually uncorrelated with mean zero and variance one; (b) the criterion has mean zero and variance one; and (c) the regression coefficients are 1 = r, 2 == k -1 = 0, k = 2 2r f The squared multiple correlation is 2 f for variables X1 to Xk and 2 r for variables X1 to Xk -1. Given these conditions,

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37 the covariance between Y and X1 is r, the covariances of Y with the remaining independent variables, X2 to Xk -1 are all zero, the covariance between Y and Xk is 2 2 r f and the covariance for any pair of X variables is zero. Data Simulation The data were simulated using the random-nu mber generating function in SAS Version 9.13. Computations were pe rformed using SAS Interactive Matrix Language (PROC IML). Data management and follow up analyses were also conducted using SAS. Normal random deviates were generated for the n x k data matrix of predictors, X, using the SAS RANNOR function. All nk scores were generated to be statistically independent. In order to generate data from a g-and-h distribution, standard unit normal variables, Zij, were transformed via the following equation 2exp1 exp 2ij ij ijgZ hZ X g (2.15) when both g and h were nonzero. When g is zero, equation 2.15 is reduced to 2exp. 2ij ijijhZ XZ (2.16) The g -andh distributed variables were then standard ized by subtracting th e population mean and dividing by the population sta ndard deviation. If g = 0, gh = 0. When g > 0, the population mean is 2exp1 2(1) 1ghg h gh (2.17)

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38 and for h the population standard deviation is 2 222 2 22 exp2exp1exp1 2(1)2(12)2(1) (1) 12ghggg hhh gh gh (2.18) In a similar manner, an n 1 vector of standard normal ra ndom variables was generated. All n scores were generated to be statistically in dependent. The results of this vector were multiplied by 2 1f The result is a vector of residuals, e, with mean zero and variance equal to 1 2 f These steps ensured that the dependent variable, y, has mean of zero and variance equal to 1.0. As detailed above, applyi ng Brownes results, the k 1 vector of regression coefficients was constructed such that elements 1 to k are zero and the next two elements are r and 22 f r respectively. The sample covariance matrix, S, was calculated from the data according to the model y = X + e. Let Rf be the correlation matrix for the full set of k predictor variables, Rf + be the k + 1 correlation matrix for all variab les (including the criterion), Rr be the correlation matrix for the first k -1 predictors, and Rr + be the correlation matrix for the first k -1 predictors and the criterion variable. All four correlation matrices can then be calculated from S. The squared multiple correlation coefficients for the full and reduced models are given by 2det 1 detf f fR R R (2.19) and 2det 1 detr r r R R R (2.20)

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39 where det ( ) represents the de terminant of the matrix (Mulai k, 1972). For each of the 10,000 replications of each distributiona l condition, the asymptotic conf idence interval was calculated using the method describe d by Alf and Graf (1999). Data Analysis Coverage probability, the probabi lity that a confidence interval contains the parameter for which the confidence interval was constructed, was used to evaluate the adequacy of the confidence intervals. Coverage probability was estimated as the proportion of the 10,000 replications in which the confidence interval contained the population squared semipartial correlation, 2. In order to investigate bias, the pr obability that the confidence interval was wholly below 2 and the probability the confiden ce interval was entirely above 2 were also estimated. To evaluate the conditions under which a hypot hesis test is insensitive to assumption violations, Bradley (1978; 1980) pr oposed three criteria. Given th e nominal Type I error rate, a test is robust if the empirical estimate of falls within the interval /s. A liberal criteria is established when s = 2 and the limits are given by .025 = [.025, .075]. Using s = 5, the interval for a moderate criterion is [.04, .06]. To establish a strict criterion, s = 10 and the interval is [.045, .055]. If these recomme ndations are adapted and applied to criteria for a confidence interval with a nominal coverage probab ility of .95, the criterion intervals become (a) [.925, .975]; (b) [.94, .96]; and (c) [.945, .955]. Although there is no universally accepted sta ndard by which procedures are considered robust or not, Lix and Keselman (1998) suggest that applied researcher s should be comfortable working with a procedure that controls Type I error within the bounds established by Bradleys liberal criterion, as long as th e procedure also limits the erro r rate across a wide range of

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40 assumption violations. Applyi ng this recommendation to the procedure for constructing an asymptotic confidence interval means that in or der to be controlled, th e coverage probability should fall within the interval [.925, .975]. We used this interval for judging the adequacy of the confidence intervals. Because there are those who w ould consider this standa rd to be too lenient, confidence intervals were also evaluated according to the more stringent criterion level of .94 to .96.

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41 Table 2-1. Study Design Number of predictors, k (5 levels) 1. k = 2 2. k = 4 3. k = 6 4. k = 8 5. k = 10 Size of the squared multiple correlation coe fficient for the reduced model (7 levels) 1. 2 r = .00 2. 2 r = .10 3. 2 r = .20 4. 2 r = .30 5. 2 r = .40 6. 2 r = .50 7. 2 r = .60 Size of the squared semipartial correlation coefficient (7 levels) 1. 2 = .00 2. 2 = .05 3. 2 = .10 4. 2 = .15 5. 2 = .20 6. 2 = .25 7. 2 = .30 Sample size, n (14 levels) 1. n = 100 2. n = 200 3. n = 300 4. n = 400 5. n = 500 6. n = 600 7. n = 700 8. n = 800 9. n = 900 10. n = 1000 11. n = 1250 12. n = 1500 13. n = 1750 14. n = 2000

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42 Table 2-1 Continued Distribution for the pr edictor variables, X (5 levels) 1. g = 0, h = 0 = 0, = 1, 1 = .00, 2 = .00 2. g = 0, h = .058 = 0, = 1.097, 1 = .00, 2 = 1.00 3. g = .301, h = -.017 = .150, = 1.041, 1 = .89, 2 = 1.20 4. g = .502, h = -.048 = .249, = 1.108, 1 = 1.41, 2 = 3.00 5. g = .760, h = -.098 = .378, = 1.252, 1 = 2.00, 2 = 6.00 Distribution for the residuals, e (5 levels) 1. g = 0, h = 0 = 0, = 1, 1 = .00, 2 = .00 2. g = 0, h = .058 = 0, = 1.097, 1 = .00, 2 = 1.00 3. g = .301, h = -.017 = .150, = 1.041, 1 = .89, 2 = 1.20 4. g = .502, h = -.048 = .249, = 1.108, 1 = 1.41, 2 = 3.00 5. g = .760, h = -.098 = .378, = 1.252, 1 = 2.00, 2 = 6.00 Table 2-2. Mardias Multivariate Skewness, b1, k, for the Nonnormal Distributions. g = .301 h = -.017 k b1, kb1, kb1, kb1, k2.01(-.03,.05)1.55(1.51,1.59)3.90(3.81,3.98)7.87(7.71,8.03) 4.02(-.05,.08)3.15(3.09,3.22)7.65(7.52,7.78)15.80(15.57,16.02) 6.00(-.09,.08)4.71(4.62,4.80)11.50(11.33,11.66)23.74(23.47,24.02) 8-.01(-.12,.10)6.23(6.11,6.35)15.47(15.26,15.68)31.64(31.32,31.96) 10.01(-.13,.14)7.71(7.57,7.86)19.43(19.18,19.68)39.61(39.23,39.98)1This interval represents .025 and .975 percentiles of the 1,000,000 replications.Interval1Interval1Interval1Interval1g = .760 h = -.098 Distribution g = 0 h = .058 g = .502 h = -.048

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43 Table 2-3. Mardias Multivariate Kurtosis, b2, k, for the Nonnormal Distributions. g = .301 h = -.017 k b2, kb2, kb2, kb2, k210.0510.35(10.32,10.38)13.8419.95(19.88,20.02) 428.0728.75(28.70,28.80)35.6948.01(47.90,48.13) 654.0755.09(55.02,55.16)65.5084.08(83.93,84.23) 888.1089.44(89.34,89.53)103.41128.05(127.87,128.24) 10130.12131.79(131.67,131.91)149.21180.03(179.80,180.25)1This interval represents .025 and .975 percentiles of the 1,000,000 replications.Distribution (65.39,64.62) (35.60,35.77) Interval1Interval1Interval1Interval1g = 0g = .502 (149.02,149.40) (13.79,13.89) (10.03,10.08) (28.02,28.11) (54.00,53.13) (88.01,88.19) (130.01,130.23) (103.26,103.57) g = .760 h = .058 h = -.048 h = -.098

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44 Figure 2-1. Plot of the empi rical cumulative distribution func tion for a univariate nonnormal distribution where g = 0, h = .058 overlaid with a normal curve with gh = 0, gh = 1.097. Figure 2-2. Plot of the empi rical cumulative distribution func tion for a univariate nonnormal distribution where g = .301, h = -.017 overlaid with a normal curve with gh = .150, gh = 1.041.

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45 Figure 2-3. Plot of the empi rical cumulative distribution func tion for a univariate nonnormal distribution where g = .502, h = -.048 overlaid with a normal curve with gh = .249, gh = 1.108. Figure 2-4. Plot of the empi rical cumulative distribution func tion for a univariate nonnormal distribution where g = .760, h = -.098 overlaid with a normal curve with gh = .378, gh = 1.252

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46 246810 0 10 20 30 40 50 Multivariate Skewness b1, kNumber of Predictors, k g = 0, h = 0 (Multivariate Normal) g = 0, h = .058 g = .301, h = -.017 g = .502, h = -.048 g = .760, h = -.098 Figure 2-5. Comparison of Mardias multivariate skewness for the multivariate normal distribution to that of the distributions investigated. 246810 0 20 40 60 80 100 120 140 160 180 200 Multivariate Kurtosis b2, kNumber of Predictors, k g = 0, h = 0 (Multivariate Normal) g = 0, h = .058 g = .301, h = -.017 g = .502, h = -.048 g = .760, h = -.098 Figure 2-6. Mardias multivariate kurtosis for the multivariate normal distribution and the nonnormal distributions investigated.

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47 CHAPTER 3 RESULTS Replication of Results for Multivariate Normal Data Prior to conducting the study, data were simula ted for the multivariate normal case in order to replicate key findings repor ted by Algina and Moulder (2001) Replication served two additional purposes. It verifi ed that the simulation program was functioning properly and that reasonably close agreement was achieved betwee n coverage probabilities estimated with 10,000 replications and coverage probabilities es timates reported by Algi na and Moulder based on 50,000 replications. Results are compared for k = 2, 6, and 10 in Tables 3-1, 3-2, and 3-3. The shaded columns are the results from this si mulation; the unshaded columns reproduce tabled results reported by Algina and Moul der (p. 638). In these tables, as well as subsequent tables reporting coverage probabilities, it alics indicate that the estimated coverage probability falls within the interval from .925 to .975. Results in bold represent estimated coverage probabilities between .94 and .96. As Olkin and Finn warned, and Algina and Moul der demonstrated, this procedure does not work at all when the population squared semipart ial correlation is zero. Regardless of sample size, number of predictors, or the value of th e population squared multiple correlation in the reduced model, the coverage probability when 2 is zero is always too large, i.e., p .999. This is because if 2 f =2 r = 0 even though the actu al sampling variance of R2 is not zero. Because of this defect in the asymptotic conf idence interval, Alf and Graf recommended that researchers perform a hypothesi s test of the significance of the corresponding regression coefficient and apply the asymptotic confidence interval procedure only when the null hypothesis is rejected. Given the coverage probability results when 2 = 0, although coverage probabilities are reported in Tables 3-1 to 3-3, they are not included in the assessment of agreement that

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48 follows as doing so would tend to exaggerate th e degree of correspondence between the two sets of estimates. Comparing the coverage probability estim ates generated by the two studies, for 2 > 0, 79% were within .003 and 94% were within .005. Of the 504 comparisons, 73 (15%) showed no difference to 3 decimal places. When coverage probabilities differed, 208 (41%) estimates from the current study were greater and 223 (44%) were smaller than coverage probabilities reported by Al gina and Moulder. For k = 2, reported in Table 3-1, 90% of the estimates from the two simulations were within .003 and only 5 differen ces were greater than .005. For 15 of the 168 cases, estimated coverage probability would have been categorized differently with respect to Bradleys criteria for robustness, [.925,.975] or [.94,.96]. These di screpancies were evenly split with 8 estimates from Algina and Moulders study falling in the more stringent interval, th at is, closer to the nominal level, and 7 values of p estimated in this study satisfied th e more stringent criterion. Both sets of estimates when k = 2 showed that empirical co verage probability approached the nominal as sample size increased and as the magnitude of the squared semipartial correlation increased. The confidence interval was leas t accurate for the smallest sample size, n = 175, for all levels of 2 r when 2 = .05. There was good coverage probability, i.e. at least .94, for n 425 and 2 > .10. Depending on the tolerance one has for the difference between coverage probability and the nominal confidence level, coverage probability could be considered marginally adequate, that is, at l east .925, for all sample sizes and 2 > .10. The agreement between the two replications was somewhat worse as the number of predictors increased. As shown in Tables 3-2 and 3-3, for both k = 6 and k = 10, 127 (76%) comparisons were within .003. There were 8 (5%) differences greater than .005 with 6

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49 predictors and 16 (9%) differences exceeded .005 with 10 predictors. Although for k = 6 the large differences favored the results repor ted by Algina and Moul der (6 vs. 2), for k = 10 a large difference was just as likely to favor the estimat es from the current simulation where favoring is defined as an estimated coverage probability that is closer in value to the nominal. In Algina and Moulders data, there was also a tendency for the estimated coverage probability to meet the more stringent evaluation criterion when th ere was mismatch in categorization. For k = 6 and 2 > .10, all coverage probabilities were greater than .925 for n 425, and all but one were greater than .94 for n = 600. At k = 10, although all coverage probabilities met the liberal criterion at n = 600, there was no level of 2 for which all were greater than .94. Overall, agreement between the two studies was quite good and therefore, the current study was conducted by simulating 10,000 replications of each condition. Simulation Proper In this simulation, 857,500,000 independent co nfidence intervals were calculated. Given there were 10,000 replicati ons of each combination of X, e, n k ,2 r and 2, coverage probability was computed as the proportion of times the constructed confidence interval contained 2, the population squared semipartial correla tion. In this manner, 85,750 coverage probabilities were estimated. Since the distribution from which predictors were sampled and the distribution for the residuals were both manipulated, this allowed us to examine four dist inct situations that might be encountered when analyzing data us ing multiple regression: (a) normal X, normal e, (b) normal X, nonnormal e, (c) nonnormal X, normal e, and (d) nonnormal X, nonnormal e. Average empirical coverage probability estimates for these four scenarios, as a function of sample size, are depicted in Figure 3-1. Results for all values of k ,2 r and 2, for selected sample sizes, are

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50 reported in Tables 3-4 to 3-7. Estimates for conditions where 2 = 0 were omitted since all were either .999 or 1.000, r ounded to three decimal places. Table 3-4 presents results for normal predicto rs with normal errors. If we consider Bradleys liberal interval, .925 to .9 75, as evidence for robustness, for k = 2, 4, 6, 8, and 10, the percentages of nonrobust values at n = 200 were 9%, 12%, 14%, 38% and 71%, respectively. At n = 400, the percentages of empirical values th at were not robust decr eased dramatically to 0%, 0%, 0%, 2%, and 2%. All estimated coverage probabilities were robust at n 600. At the largest sample sizes reported, n = 1500 and n = 2000, all exceeded .94 and met the more stringent standard for robustness When predictors were normal with nonnorma l residuals, reported in Table 3-5, the percentage of nonrobust coverage probabilities increased. For k = 2, 4, 6, 8, and 10 and n = 200, the percentages of nonrobust values were 31%, 38 %, 50%, 76%, and 100%, respectively. As expected, the number of nonrobust cove rage probabilities decreased as n grew larger. This decrease was notable between n = 200 and n = 400 (7%, 7%, 5%, 14%, and 19%) and less so for n = 600 (5%, 2%, 2%, 5%, 7%) and n = 800 (2%, 2%, 2%, 5%, 5%). For n 1000, all coverage probabilities were robust except when 2 r = 0 and 2 = .30. Table 3-6 shows coverage probability estimat es when the predictors were nonnormal and the distribution of the residuals was normal. At n = 200, there were no robust empirical estimates at any level of k For n = 400, the percentages of estim ates outside Bradleys liberal interval were 64%, 62%, 76%, 95%, and 100% for k = 2, 4, 6, 8 and 10, respectively. For n = 600, the percentage of coverage probabilit ies that were nonrobust for these values of k were 50%, 55%, 60%, 64%, and 69%. For sample sizes greater than 600, improvement, as measured by a decrease in nonrobust values, was much more gradual. For k = 2, 4, 6, 8, and 10, and

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51 n = 800, the percentages were 50% 50%, 50%, 57%, and 60%; for n = 1000, 48%, 50%, 50%, 48%, and 55%; and for n = 1500, 45%, 45%, 50%, 48%, and 52%. At the largest sample size, n = 2000, at least 45% of empirical cove rage probabilities at every level of k failed to meet even the liberal standard for robustness. The coverage probabilities contained in Tabl e 3-7 were estimated for the case where both predictors and errors were nonnormal. For n 400, there were only 6 estimates greater than .925. Of these, 5 were observed for n = 400 and k = 2, and 1 at n = 400 and k = 4. For n = 600, the percentages of c overage probabilities that were nonrobust were 74%, 71%, 81%, 86%, and 88% for k = 2, 4, 6, 8, and 10, respectively. Similar to what was observed with nonnormal X and normal e, the improvement in coverage probabilities is minor for n > 600 such that when n = 2000, nonrobust estimates were 71% 71% 74% 74% and 76% for k = 2, 4, 6, 8, and 10, respectively. For all four scenarios, coverage probability te nded to decrease as more predictors were included in the model, particularly with smaller sample sizes. Coverage improved as sample size increased. Figure 3-1 suggests that nonnormality in the predictors was more detrimental to the adequacy of the confidence interval than was a nonnormal error distribution. A modest decline in coverage probability was observed between normal X, normal e and normal X, nonnormal e, but there was a considerable drop off in performance when X was nonnormal even when the errors were normally distributed. In addition, coverage probability was examined by distributional condition. A distributional condition was defined by the combinat ion of the distribution for the predictors and the distribution for the errors. There were 25 di stributional conditions in cluded in this study.

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52 For clarity and ease of presentation, the gandh distributions from which data were generated will be referred to as: (a) pseudot10 for g = 0, h = .058; (b) 2 10pseudofor g = .301, h = -.017; (c) 2 4pseudofor g = .502, h = -.048; and (d) ps eudo-exponential for g = .760, h = -.098. The descriptive statistics reported in Table 3-8 were based on 2940 coverage probability estimates per distributional condition, excluding those cases where 2 = 0. Average coverage probability was closest to the nominal confidence level when both X and e were normally distributed. The average coverage probability was smallest for the most seriously nonnormal case, both X and e sampled from the pseudo-exponential distribution. Within each level of X, mean coverage probability decrease d as the error dist ribution exhibited increasing nonnormality. A similar pattern was observed for the median. In the extremes, the median for multivariate normal data was .944. In contrast, for the condition where both X and e were distributed pseudo-exponential, half the estimated coverage probabilities were less than .868. For all distributional conditions in which X was distributed pseudo-e xponential, at each error distribution, at least 50% of the estimated c overage probabilities were below .90. The variability in coverage probability increa sed with greater skewne ss and kurtosis in the data. When X was distributed pseudo-exponential, regardless of the distribution for e, the standard deviation was over three times that observed for the multivariate normal case. Although the maximum value did not differ a great deal as a function of distributional condition, the minimum was much lower and the range was wider for conditions with greater nonnormality. Also included in Table 3-8 is an examinati on of the robustness of the confidence interval as a function of distributional condition at n = 600 and n = 2000. Applying the liberal criterion,

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53 .925 to .975, all coverage probabilities were robust at n = 600 for multivariate normal data and when predictors were normally distributed a nd the distribution for th e errors was either pseudot10 or 2 10pseudo. There was no other distributio nal condition for which coverage probability was adequate for the entire range of values for k,2 r and 2, even for the largest sample size investigated, n = 2000. For the most extreme distributional condition simulated, both X and e drawn from a pseudo-exponentia l distribution, 100% of th e coverage probabilities were nonrobust at n = 600. There was only slight improvement at n = 2000 where 90% of the estimates were not robust. Althoug h it could be argued that data like this is unlikel y to occur in practice, with an error distribution with seve re nonnormality, i.e. pseudo-exponential, there was poor coverage even when the predictors were multivariate normal. At n = 2000, 25.2% of the estimates were not robust. Furthermore, when using multiple regression, applied researchers are much more likely to be concerne d about the error dist ribution since violati on of this assumption influences the power and accuracy of hypothesis test s. Researchers may not even investigate the multivariate skewness and kurtosis for the predic tors. With a normal error distribution, the percentages of nonrobust estimates at n = 2000 for predictors distributed pseudot10, 2 10pseudo, 2 4pseudo, and pseudo-exponential, were roughly 7%, 10%, 49%, and 96%, respectively. Although results are reported for only two sample sizes, for all distri butional conditions and sample sizes investigated, when an estimated cove rage probability was outside of either criterion interval, it was without exception, too small. Figure 3-2 illustrates the relationship betw een coverage probability, distributional condition, and sample size. The best coverage pr obability, over the full range of sample sizes investigated, was observed for the condition in which both X and e were normal. However, at best, average coverage probability never reached the nominal confidence level, .95. There was a

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54 slight degradation in perfo rmance for conditions where X was normal and the nonnormal errors were distributed pseudot10 or 2 10pseudo. Although it is a bit hard to discern because of the overlap for conditions where X was distributed pseudot10 and 2 10pseudo, results were similar for normal X with e distributed 2 4pseudo, and X sampled from pseudot10 with normal error. That is, coverage probability estimates were similar when the predictors were normal with markedly nonnormal errors and when predictors were sampled from a pseudot10 distribution with a normal error distribution. Similarly, the condition in which X was distributed 2 4pseudowith normal error exhibits coverage probability comparable to the conditions where predictors were moderately nonnormal, sampled from pseudot10 and 2 10pseudo, with errors that were extremely skewed and kurtotic (pseudo-exponent ial). Thereafter, as the distribution for X became increasingly nonnormal, coverage probability decreased and was least adequate when the predictors were sampled from a pseudo-exponential distribution regardless of the distribution for the residuals. Within each condition for X, coverage probability decreased in the same systematic way as a function of the nonnormality in the error distribution such that coverage probability was best with normally distributed erro rs and worst when the errors were distributed pseudo-exponential. Analysis of Variance and Mean Square Components Given the sheer volume of data collected in th is study, analysis of variance (ANOVA) was used to identify the experimental factors that were important in determining the estimated coverage probability, p Factorial ANOVA assumes that mu ltiple factors contribute to the variance in the data. The total variance is pa rtitioned into main effects corresponding to each factor, the interactions among them, and random error. Th e factors manipulated in the study were all treated as between-subjects effects in a fully-crossed ANOVA model that consisted of 6

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55 main effects and 56 interactions. Since the proc edure for calculating the confidence interval is clearly inappropriate when 2 = 0, the 12,250 coverage probabilities calculated for this value were not included in this analysis. It was felt this provided a more accurate reflection of the data. ANOVA analyses and variance parti tioning of coverage probabiliti es were therefore based on N = 73,500. The mean squares, F -statistics, and p -values associated with each effect in the full model were computed using the ANOVA procedure in SAS. These results are reported in Table 3-9. The combination of a large number of effects and a very large sample size ensured that there were many statistically signifi cant effects, including higher-order interactions. In all, 34 of the 62 effects estimated were significant at p < .0001. Because statistical significance is in large part a function of sample size, a statistically significant effect is not very informative when the sample size is very large. To better understand the relative impact of these effects on coverage probability, it was necessary to obtain a measure of influence to determine which effect s were associated with a meaningful proportion of the variance. The term varian ce component is used in the contex t of analysis of variance with random effects and denotes the estimate of the amount of variance that can be attributed to each effect. In the current context, the levels of each factor were purposively selected. Because effects are fixed and not random, the more accura te term is mean square component. The ANOVA method for estimating mean square components equates mean squares to their expected values, EMS, and solves for the mean square component s in those expectations. The estimated mean square component for each main effect a nd interaction was comput ed using the general formula 2()(Residual) MSMS j

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56 where is the effect of interest and j is the product of the number of levels for each factor not involved in (Myers & Well, 2003). In this case, the residual mean square, .0000079, includes the mean square for the six-way interaction and the mean square for error. For example, the mean square component for X is given by 27.5162.00000797.516192 .0005113. (5)(5)(7)(6)(14)14700X Since these are simultaneous linear equati ons with as many unknowns as there are equations, they have unique solutions and mean square components are estimated noniteratively. An unfortunate characteristic of ANOVA estimato rs is that they can yield negative estimates even though, by definition, they are nonnegative. Negative compone nts were set equal to zero before calculating the proportion of variance that could be attrib uted to each effect. The components were then summed and the ratio of each mean square component to the sum was used as a measure of influence. Effects significant at = .0001 that accounted for at leas t .5% of the variance are reported in Table 3-9. The distribution for the predictors, X, was responsible for 44.51% of the total variance in coverage probability. Th e variance component associated with X was nearly four times greater than that of any other main effect. The main effects of 2 and 2 r were comparatively less important factors in determin ing average coverage probability, accounting for 10.12% and 3.26% of the total vari ance, respectively. Effects of 2 and 2 r were moderated by their interaction. This two-way interaction accounted for an additional 1.60% of the variance. The mean square component associated with sample size, n accounted for 9.41% of the total variability in p The main effect of e accounted for only 3.38% of the variance indicating that the error distribution had a much smaller impact on the coverage probability of the confidence

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57 interval than did the distribu tion of the predictors. The num ber of predicto r variables, k had very little impact on p accounting for only .69% of the variability. The critical importance of nonnormality in the predictors was further substantiated by the fact that interactions involving X explained an additional 22.24% of the variance in p The variance components fo r the two-way interactions between X and 2 and X and 2 r were associated with 11.38% and 8.82% of the total va riance, respectively. The three-way interaction of these factors, X 2 r 2, moderated the three main effects and the two-way interactions and accounted for an additional 2.04% of the total variance in coverage probability. The main effects of X, 2, and2 r and the interactions of these three factors explained 81.7% of the total variance in coverage probabilities. The effect of e was also moderated, although to a lesse r extent, by the twoway interactions between e and 2 and e and2 r These interaction effects were responsible for .70% and 1.28%, respectively. The three-way interaction, e 2 r 2, explained .54% of the total variance. The main effects of e, 2, and2 r and their interactions accounted for 5.85% of the variance in p This was further evidence that although a nonnor mal error distribution had some effect on the coverage of the confidence interval it was not nearly as important as nonnormality in the predictors. Sample size interacted with the number of predictors, k and the size of the squared semipartial correlation coefficient, 2. The n k and n 2 interaction effects each explained approximately 1% of the total variance. Importa nt effects involving sample size were associated with 11.35% of the variability in coverage probab ility. Thus, it appears that sample size was also more important than nonnormality in the error di stribution in determining the adequacy of the

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58 confidence interval. The effects reported in Ta ble 3-9 accounted for an estimated 99.6% of the total variance in coverage probability. The fo llowing sections describe the important factors influencing coverage probability as identified by the mean square components analysis. The Influence of Nonnormality on Coverage Probability Nonnormal predictors When coverage probability was averaged ove r all other factors, Table 3-10 shows the adequacy of the confidence interval, as measured by coverage probability, worsened as the distribution for the predictors be came increasingly nonnormal. When X was distributed multivariate normal, average coverage probability was .935 (SD = .014). When the set of predictors was made up of va riables sampled from a pseudot10 distribution, that is symmetric, but more peaked and heavier tailed than the nor mal distribution, average coverage probability dropped to .925 (SD = .015). A similar estimate of average coverage probability, p =.923 (SD = .015), was obtained when the explanator y variables were sampled from a population distributed as2 10pseudo. Because these distributions had similar values for both univariate and multivariate kurtosis, but differed with respect to skewness, this result seems to suggest, at least for moderate nonnormality, that skewness may be less important than kurtosis in determining the adequacy of the confidence interval procedure. When predictors were sampled from a 2 4pseudopopulation distribution, the average coverage probability was .906 (SD = .022). The average coverage probability when predictors were sa mpled from a distribution that has the same skewness and kurtosis as the expone ntial distribution was .877 (SD = .037). The median was also related to the degree of nonnormality present and declined in a manner similar to the mean. In addition, the rang e of coverage probability values estimated in the simulation expanded as the degree of nonnormality became more extreme. Figure 3-3

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59 presents boxplots that describe th e distribution of coverage probabi lity estimates as a function of the distribution for X. We see that all distributions for p are skewed to the right, but the distribution was flatter, more sp read out, and longer-tailed as th e degree of skewness and kurtosis in the distribution for the predictors increased. Nonnormal error distribution Table 3-11 shows descriptive statistics for the main effect of the distribution for error. The means, by error distribution, also declined as a function of the degree of nonnormality present. The range between the largest mean, .919 for norma lly distributed errors, and the smallest, .903 for errors distributed pseudo-e xponential, was much smaller than observed in Table 3-10 for the main effect of the distribution for the predictors There was also less va riability in the median, ranging from .929 for normal errors to .911 fo r pseudo-exponential er rors. The range of coverage probabilities and the standard deviatio ns were essentially equal suggesting that there was little difference in the variab ility of coverage probability estim ates as a function of the error distribution. The boxplots depicted in Figure 3-4 s upported this contention. The Impact of Squared Multiple Correl ations on Coverage Probability Figure 3-5 depicts the relations hip between coverage probabil ity and the magnitude of the population squared semipartial correlation. Averag ed over all other factors, coverage probability tended to decrease as the size of the squared semi partial correlation increa sed. Figure 3-5 also shows that the effect of 2 on coverage probability varied depending on the distribution for the predictors hence the significant interaction between X and 2. Figure 3-5 and Table 3-12 show the relationship between 2 and coverage probability within each distribution for X. Under normality there was actually a slight increase in p from 2 = .05 to 2 = .10, the smallest values investigated. This increase essentially leve led off thereafter. Stable coverage probability

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60 between 2 = .05 and 2 = .10 was observed for pseudot10 and 2 10pseudodistributions. In both distributions, p showed a steady, but modest, decline for 2 > .10. The decline in p when X was distributed 2 4pseudowas modest between 2 = .05 ( p = .924) and 2 = .10 ( p = .919). The rate of change was much steeper for 2 > .10 such that p decreased to .886 at 2 = .30. For X sampled from the pseudo-exponential dist ribution, coverage probability was essentially a linear function of 2 that declined sharply over the range of 2 from p = .915 to p = .840. There was also a significant interacti on, depicted in Figure 3-6, between e and 2. However, as reported in Table 39, this effect while statistically significant, accounted for little of the variance in coverage probability. A co mparison of Figure 3-6 with Figure 3-5 shows a similar pattern for the relationship between the error distribution and 2 with less extreme variation in the rate at which coverage probability declined. When the error distribution was normal, pseudot10, or 2 10pseudo, p declined slightly between 2 = .05 and 2 = .10 with a steady, gradual decrease for 2 > .10. The decline in coverage probability between 2 = .05 and 2 = .60 was more nearly linear, with a steeper slope, when th e error distribution was sampled from either a2 4pseudoor pseudo-exponential distribut ion. The decrease in coverage probability was most dramatic when er rors were distributed pseudo-exponential. At 2 = .05, p = .923 and at 2 = .60, coverage probability dropped to p = .883. Coverage probabilities, as a function of e and 2, are reported in Table 3-13. Figure 3-7 depicts the relationship between2 r and coverage probability. Coverage probability stayed relatively constant between 2 r = .00 and 2 r = .40 and then decreased for

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61 2 r = .50 and 2 r = .60. The interaction between X and 2 r is also demonstrated in Figure 3-7. When the predictors were distributed multivariate normal, coverage probability was a linear function of2 r gradually increasing from .928 at 2 r = .00 to .940 at 2 r = .60. For X distributed pseudot10 and2 10pseudo, there was a minor increase in covera ge probability, roughly .92 to .95, between 2 r = .00 and 2 r = .40. Coverage probability was smaller for 2 r .50. As the distribution for X demonstrated greater skewness and kurto sis, the coverage probability function tended to be more curvilinear. For X distributed 2 4pseudo, coverage probability was relatively consistent between 2 r = .00 and 2 r = .30 and decreased steadily for 2 r .40 to a minimum of .887 at 2 r = .60. When X was sampled from a pseudo-exponential distribution, coverage probability started out at p = .896 at2 r = .00 and decreased between 2 r = .00 and 2 r = .30 to .885. The decline in p was at a much faster rate thereafter such that when2 r = .60, p = .837. As Table 3-14 shows, the differences in coverage probabil ities, as a function of the degree of nonnormality in X, had their smallest range of values at 2 r = .00 (.928 to .896) and the range was maximized at 2 r = .60 (.940 to .837). While the behavior of p over the levels of 2, as a function of the error distribution, was comparable to the relationship between 2 and the distribution fo r the predictors, the e 2 r interaction, presented in Figure 3-8, shows this was not the case for2 r In contrast, the differences in p as a function of nonnormality in the error distribution, were greatest at2 r = .00. Coverage probability, reported in Table 3-15, ranged from .927 for normal errors to .897 when errors were pseudo-exponential. By the time 2 r = .60, coverage probability had essentially

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62 converged and was approximately .90 regardle ss of the degree of nonnormality in the error distribution. Furthermore, for normal errors, maximum coverage probability, .927, occurred for 2 r = .00. For e distributed pseudot10 and2 10pseudo, the largest coverage probability, .922, occurred at 2 r = .10. For 2 10pseudo, the largest average coverage probability, .915, was observed for 2 r = .20 and 2 r = .30. When the error dist ribution was sampled from a pseudo-exponential population distri bution, the largest coverage probability, .907, was observed at 2 r = .30 and 2 r = .40. These results suggest that the X 2 r and e 2 r interactions might have a counterbalancing effect. However, the e 2 r interaction, although statistically significant, explained a modest 1.3% of the to tal variance in coverage probability while the X 2 r interaction accounted for 9.5% of the total variance. The impact of the interaction between 2 and2 r on coverage probability is shown in Figure 3-9. Although there was a te ndency for estimated coverage probability to be further from the nominal as2 r increased, this was not th e case for all values of 2. When 2 = .05, there was an increasing trend in coverage probability over the range of2 r values. For 2 > .05, coverage probability was relatively stable between 2 r = .00 and 2 r =.30, but then decreased substantially from 2 r = .30 to 2 r = .60. However, the relationship of p to2 r and 2 varied depending on the distribution for X. Figure 3-10 shows the effect of th e three-way interaction between X,2 r and 2 on coverage probability. To aid in the description and interpre tation of effects, coverage probabilities, as a function of 2 r and 2, for each level of X are reported in Tables 3-16 through 3-20. For the multivariate normal case, coverage probability tended to be worse when 2 = .05 and for all

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63 levels of 2 coverage probability increased as 2 r increased. The plots of coverage probability as a function of 2 and2 r for pseudot10 and 2 10pseudolook remarkably similar to each other and have the same pattern of results de scribed for the two-way interaction of 2 r and 2, albeit over a narrower range of values. Coverage probability increased over the levels of 2 r when 2 = .05, but for 2 > .05, coverage probabili ty tended to increase from 2 r = .00, reached a maximum at 2 r = .30, and decreased thereafter. Alt hough coverage probability was best for 2 = .05 and 2 r = .60, for all other levels of 2, coverage probability was lowest at 2 r = .60. Coverage probability was cons istent at approximately 925 over the full range for 2 r for 2 = .05 when the predic tors were distributed 2 4pseudo. For 2 > .05, coverage probability was stable between 2 r = .00 and 2 r = .20, but showed a decline between 2 r = .30 and2 r = .60. The rate of decline was faster for larger values of 2. For X sampled from the pseudo-exponential distribution, coverage probability decreased as 2 r increased for all levels of 2. The rate of decline vari ed according to the value of 2 with steeper slopes associated with larger values of 2. The drop in coverage probability was minor for 2 = .05, where p = .917 at 2 r = .00, falling to p =.907 at 2 r = .60. However, when 2 = .30, at 2 r = .00 coverage probability was .870 and decreased markedly to p =.777 at 2 r = .60. Thus, when nonnormality in the predictors was extreme, the importance of the magnitude of the squared multiple correlations, 2 and2 r was critical for determining the adequacy of the confidence interval pro cedure. Although no condition, on average, demonstrated acceptable coverage over the entire range of factors mani pulated in this study,

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64 Figure 3-10 illustrates how inaccurate the asymptot ic confidence interval can be under conditions that could occur in practice. The Impact of Sample Size on Coverage Probability As seen in Figures 3-1 and 3-2, regardless of the distribution for the predictors or the distribution for error, coverage pr obability increase d rapidly between n = 100 and n = 400. The average coverage probability at n = 100 was .882 increasing to .912 at n = 400. The rate of increase, from .914 to .917, wa s considerably slower between n = 500 and n = 800. Furthermore, it appears that there was little to be gained by increas ing the size of the sample beyond n = 1000 with respect to the adequacy of the confidence interval. Coverage probability is increasing so slowly between n = 1000 and n = 2000 (from .918 to .920) that it is likely that sample sizes well in excess of 2000 would be required to ensure th e robustness of the confidence interval over a wide range of nonnormal conditions. Evidence to support this contention was evaluated by estimating coverage probabilities for X and e distributed pseudo-exponential; n = 5000; 2 r = .00, .30, and .60; and 2 = .05, .10, .15, .20, .25, and .30. Results indicated that even with an extremely large sample size, when nonnormality is severe, coverage probability remained inadequate. Only 7 of 54 coverage probability estimates exceeded .925 and consequently, 87% were nonrobust. Six of the r obust estimates were observed for 2 r = .00 or 2 r =.30 and 2 = .05 for all three levels of k The remaining robust estimate occurred for k = 10, 2 r = .00, and 2 = .10. Figure 3-11 shows that the effect of sample size was not the same at every level of 2. The interaction between n and 2 was due to the fact that the effect of 2 was smaller when the sample size was smaller than the effect of 2 when the sample size is larger. In addition, the average values for p are not is the same or der as a function of 2 for smaller sample sizes. For

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65 example, at n = 100, although coverage probability was clearly inadequate for all levels of 2, it was worse for the smallest value, 2 = .05, as well as the largest values, 2 = .25 and 2 =.30. Coverage probability improved noticeably for 2 = .05 at n = 200 although it was still not as large as it was for 2 = .10. By n = 300 coverage probability for 2 = .05 and 2 = .10 were equal. At n 400, coverage probability was a function of 2 growing worse as 2 increased. Coverage probabilities, as a function of sample size and 2, are presented in Table 3-21. As shown in Figure 3-12, the rate of increas e in coverage probabil ity as a function of sample size depended on the number of predictors in the model. For the smaller sample sizes, most notably at n = 100, although average coverage probabi lity was clearly inadequate, it was considerably worse when there were more predictors in the model. As the sample size increased, the difference between coverage probabilities as a function of the number of predictors became progressively smaller. Table 3-22 shows that at sample sizes greater than 1000, the difference in coverage probability was minimal and it appears that the number of predictors exerted very little influence on coverage probability. Probability Above and Below the Confidence Interval When the confidence interval did not c ontain the population squared semipartial correlation coefficient, the probability that the confidence interval was below 2 and the probability that the confidence interval was above 2 were also estimated. When 2 = 0, average coverage probability was .9998. On ly 18,754 of the 122,250,000 confidence intervals constructed did not contain the population paramete r. There were only 4 instances in which the interval was wholly below 2; 18,750 confidence intervals were wholly above 2. When the increase in the squared multiple correlation was zero the confidence interval was too conservative, but for all other values of 2, the confidence intervals te nded to be too liberal.

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66 For the 73,500 conditions where 2 > .05, the probability that the confidence interval was wholly below 2 was twice the probability that the confidence interval was entirely above 2 (.664 vs. .336). The confidence interval is biased in the sense that there is a systematic error that causes the estimated confidence lim its to regularly mi ss the population para meter in the same direction. The tendency to underestimate 2 occurs because the estimated asymptotic standard error declines as R2 declines. As a result, when R2 < 2 there is a tendency for the interval to be completely below 2 (Algina & Moulder, 2001). The Relationship between Estimated Asympt otic Variance, Empirical Sampling Variance of R2, and Coverage Probability As previously noted, all coverage pr obabilities were at least .998 for 2 = 0. This result indicates that when a predictor was added to a multiple regression model and there was no increase in 2, the confidence interval was always too wi de. As previously noted, there were two reasons for this shortcoming in the c onfidence interval. The distribution of R2 is skewed to the right and since the increase in R2 cannot be less than zero it has a lower limit of zero. Because the confidence interval formula does not recogn ize this lower limit, when the population value was 2 = 0, the confidence interval tended to have a lower limit less than zero. The second basis for the problem, identified by Al gina and Moulder, is that the asymptotic variance overestimates the sampling variance of R2. This was verified in the current study by calculating for each combination of X, e, n k 2 r and 2 (a) the mean estimated asymptotic variance over the 10,000 repli cations and (b) the empiri cal sampling variance of R2. For all conditions where 2 = 0, the ratio of the average value of (a) to (b), denoted as MEAV/Var R2, ranged from 1.27 to 2.18 with a mean of 1.95 and a median of 1.96.

PAGE 67

67 The ratio, MEAV/Var R2, was also evaluated for 2 > 0. ANOVA and mean square components analyses were conducted for MEAV/Var R2 as the outcome variable. As was the case with coverage probability, due to the larg e sample size, only 24 of 62 effects failed to demonstrate significance at p < .0001. Effects significant at = .0001 that accounted for at least .5 % of the variance are repor ted in Table 3-23. These eff ects accounted for 97.8% of the variability in the variance ratio, MEAV/Var R2. The distribution for th e predictors explained 51.78% of the variance in the ratio. An additiona l 21.06% was attributable to the size of the squared semipartial correlation coefficient. Less important for accurate estimation of the variance were the main effects of e and2 r These effects explained 6. 26% and 2.67% of the total variance, respectively. As observed for coverage probability, a subs tantial proportion of the variance, 89.5%, was accounted for by the main effects of X, 2, and 2 r and the interaction of these effects: 6.81% was associated with the X 2 interaction, 6.45% was associated with the X 2 r interaction, and the three-way interaction, X 2 r 2, explained a modest .72%. The interaction between the distribution for the errors and 2 r accounted for an additional 2.02%. Although sample size plays a role in determining the c overage probability, it was not important in determining the ratio since the effect of n was included in calculating the variance. Figure 3-13 illustrates how MEAV/Var R2 varies as a function of the distribution for the predictors,2 r and 2. This figure corresponds to Figure 310, describing coverage probability as a function of the X 2 r 2 interaction, and shows a similar pattern. For the multivariate normal case, variance ratios got further from 1.0 as 2 increased for 2 r = 0. As 2 r increased,

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68 variance ratios improve d for all values of 2. This improvement was greater for larger values of 2. By the time 2 r = .60, there was no difference in the MEAV/Var R2 ratio as a function of2 r The behavior of the variance ratio helps to explain the fact that for normal data coverage probability increases with both 2 and 2 r For X distributed pseudot10 and 2 10pseudo, the pattern for MEAV/Var R2 as a function of 2 and 2 r was very similar. This was also observed for coverage probability. At all values of2 r the variance ratio got smaller as 2 increased. For 2 = .05, the MEAV/Var R2 ratio was consistent across the range for2 r There was a slight curvilinear relationship in the 2 2 r plots for 2 > .15 such that variance estimation improved slightly from 2 r = .00 to 2 r = .30 and then declined from 2 r = .30 to 2 r = .60. Therefore, variance estimates were best for all values of 2 at 2 r = .30 and the most serious varian ce underestimation occurred when both 2 r and 2 were largest. When X was distributed 2 4pseudoor pseudo-exponential the difference between the variance ratio at the smallest value of 2 and the largest was greater than for the previous distributions at 2 r = .00 and this difference became progressively larger as2 r increased. For the most extreme degree of nonnormality, although MEAV/Var R2 was never greater than .90, when 2 represents a large effect si ze, the accuracy of the estimat ed variance was particularly poor over the range of 2 r values. The scatterplot in Figure 3-14 is further evid ence of a strong positive association between coverage probability and MEAV/Var R2. The correlation between coverage probability and the

PAGE 69

69 variance ratio was r = .91. As the asymptotic variance more accurately estimated the actual sampling variance of R2, coverage probability approached th e nominal confidence level. When coverage probabilities were poor, the estimated as ymptotic variance could be less than half that of the empirical sampling variance of R2. The strength of the rela tionship between coverage probability and MEAV/Var R2 depends on the distribution for the predictors, as shown in Figures 3-15 to 3-19. For multivariate normal data, presented in Figure 3-15, the mean varian ce ratio was .946 (SD = .06). The median was .963 with a range from .666 to 1.050. Approximate ly 10% of the estimates were greater than 1.0 indicating that the asymptotic variance, albeit rarely, sometimes overestimated the empirical sampling variance. As the plot shows, however a variance ratio near 1.0 was not a guarantee that the coverage probability will necessarily be close to .95 and coverage probability was as low as .85. Not surprisingly, the correlati on between coverage probability and MEAV/Var R2 was lower than that for the full data set, r = .62. Although the correlation between co verage probability and MEAV/Var R2 was similar to that for normal data, r = .63, when the predictors were sampled from the pseudot10 distribution, less than 1% of the variance ratios were above 1 (Figure 3-16). The mean variance ratio was .881 (SD=.065), the median was .886, and the range was .631 to 1.044. The estimates from the 2 10pseudodistribution again demonstrat e close similarity to the pseudot10 distribution. Although the s catterplot in Figure 3-17 is somewhat less dispersed reflected in a slightly higher correlation, r = .68, the descriptive sta tistics show close agreement. The mean variance ratio was .870 (SD=.068), the median was .874, and the range was .626 to 1.044. Again, less than 1% of the ratio estimates were greater than 1.0.

PAGE 70

70 As multivariate skewness and kurtosis incr eased, the correlation between coverage probability and MEAV/Var R2 became much stronger. For the 2 4pseudodistribution r = .86. As Figure 3-18 demonstrates, the scatte rplot was more compact and more spread out. The range of values was wider, 548 to 1.004, due to a lower minimum value. There was only 1 variance ratio greater than 1. The mean wa s .785 (SD = .100) and the median was .788. Figure 3-19 shows the st rongest relationship ( r = .91) between coverage probability and MEAV/Var R2 for the pseudo-exponential distribu tion. With skewness and kurtosis corresponding to the exponential distribution, the s catterplot was tightly concentrated and substantially more elong ated. None of the variance ratios were greater than 1 and over 25% were less than .60. Variance ra tios ranged from a low of .381 to a high of .972. The mean was .881 (SD=.132) and the median was .673. In summary, for multivariate normal data, MEAV/Var R2 was best when 2 was small, but as2 r increased, variance was more accu rately estimated and by the time 2 r = .60, MEAV/Var R2 was not dependent on 2. This pattern of results did not hold when nonnormality was introduced in the predictors. For moderate nonnormality, MEAV/Var R2 tended to be more dependent on the value of 2 than on the magnitude of2 r When nonnormality was more extreme, variance estimation became more inaccurate as both 2 and 2 r increased. Thus, when a variable was adde d to a multiple regression model that already explained a sizeable proportion of the va riation in the outcome, for example, 2 r = .60, the effect size associated with that vari able was large, for example, 2 = .30, and the data were not multivariate normal, using Alf and Grafs formul a underestimated the variance. Furthermore, this study showed that when nonnormality was seve re, the estimated asymptotic variance could

PAGE 71

71 be less than half that indicated by the sampling distribution of R2. In practice, this is likely to produce standard errors that are t oo small resulting in a confidence interval that is too narrow. Reliance on this confidence interval as a measure of the strength of the effect size will lead us to underestimate the importance of an indi vidual predictor to the regression.

PAGE 72

72Table 3-1. Replication of Algi na and Moulders Results for Multivariate Data and Two Predictors. n1750.001.0001.0000.9070.904 0.9250.9250.9310.9300.9360.9370.9380.939 0.940 0.934 0.101.0001.0000.9110.910 0.926 0.922 0.9330.9320.9380.935 0.940 0.9380.9380.938 0.201.0001.0000.9130.912 0.9300.9290.9350.9330.9380.938 0.9420.9400.940 0.939 0.301.0001.0000.9190.919 0.9310.9280.935 0.941 0.9380.9390.939 0.9410.942 0.939 0.401.0001.0000.9220.923 0.9340.934 0.940 0.9390.939 0.9410.9420.9400.9410.944 0.501.0001.0000.923 0.9290.9360.9390.9390.938 0.9410.9420.942 0.939 0.9430.941 0.601.0001.000 0.9310.9280.9390.937 0.9410.9430.9410.9400.942 0.939 0.9430.940 3000.001.0001.0000.9230.923 0.9370.9320.9380.936 0.9430.9430.9430.9450.9440.944 0.101.0001.000 0.9280.9270.9350.939 0.9420.9400.9440.9460.9440.9420.9430.944 0.201.0001.000 0.9290.9310.9380.936 0.9400.9460.9420.9410.9430.9460.9460.946 0.301.0001.000 0.9300.935 0.940 0.939 0.9410.9420.9440.9440.9440.9440.9430.943 0.401.0001.000 0.9340.929 0.9410.9430.9440.9420.9450.9470.9460.9460.9460.946 0.501.0001.000 0.9350.936 0.9410.9430.9430.9440.9460.9480.9450.9440.9460.948 0.601.0001.000 0.9380.937 0.9430.9420.9440.9430.9440.9420.9450.9450.9450.945 4250.001.0001.000 0.9310.9330.938 0.9420.9420.9430.9440.9480.9450.9440.9470.947 0.101.0001.000 0.9330.934 0.9410.9400.9440.9440.9440.9460.9440.9480.9470.948 0.201.0001.000 0.933 0.9400.9420.9420.9450.9460.9450.9440.9470.9420.9460.947 0.301.0001.000 0.9350.937 0.9420.9440.9450.9410.9460.9430.9460.9470.9450.947 0.401.0001.000 0.9360.935 0.9430.9410.9450.9460.9440.9490.9460.9470.9490.948 0.501.0001.000 0.9390.939 0.9440.9430.9460.9430.9470.9460.9480.9450.9450.946 0.601.0001.000 0.9410.9410.9440.9470.9460.9450.9450.9470.9470.9470.9460.945 6000.001.0001.000 0.9350.935 0.9430.9410.9450.9440.9450.9460.9470.9440.9460.950 0.101.0001.000 0.9370.936 0.9450.9420.9450.9440.9480.9440.9470.9460.9490.949 0.201.0001.000 0.939 0.9410.9450.9470.9460.9420.9460.9440.9480.9480.9480.944 0.301.0001.000 0.9390.939 0.9450.9430.9450.9480.9450.9420.9450.9450.9460.948 0.401.0001.000 0.9400.9420.9450.9450.9470.9460.9490.9480.9480.9510.9480.948 0.501.0001.000 0.9430.9410.9460.9480.9460.9430.9490.9420.9470.9480.9490.947 0.601.0001.000 0.9430.9420.9460.9450.9460.9450.9490.9480.9480.9480.9470.9450.000.050.100.150.200.250.30 Note : Bold results are estimated coverage probabilities between .9 4 and .96; italicized results are estimated coverage probabiliti es between .925 and .975. Shaded columns are results from this study; unshaded columns are the results reported by Algina and Moulder (2001, p 638-640).

PAGE 73

73Table 3-2. Replication of Algi na and Moulders Results for Multivariate Data and Six Predictors n1750.001.0001.0000.8970.8960.9180.915 0.9270.9250.9300.9280.9330.9370.9350.936 0.101.0001.0000.9030.9060.9200.920 0.928 0.923 0.9320.9340.9350.9350.9340.932 0.201.0001.0000.9080.9090.922 0.9260.9260.9310.9310.9340.9340.9280.9340.935 0.301.0001.0000.9090.9140.922 0.9250.9300.9280.9300.9340.9330.9320.9330.935 0.401.0001.0000.9120.915 0.925 0.924 0.9300.9320.9320.9250.9320.9320.9340.933 0.501.0001.0000.9180.918 0.9270.9290.9310.9340.9310.9260.9320.9330.9300.939 0.601.0001.0000.9210.919 0.9290.9330.9290.9340.9300.9290.9320.9310.9310.927 3000.001.0001.0000.9200.919 0.9320.9320.9350.9360.9380.9370.939 0.9400.9420.941 0.101.0001.0000.9230.918 0.9320.9270.9370.9370.9390.9390.939 0.9400.9400.944 0.201.0001.000 0.9250.9260.9330.9350.9380.9390.9390.939 0.9400.9410.9400.942 0.301.0001.000 0.926 0.924 0.9350.9380.9390.935 0.9410.9420.940 0.938 0.9400.940 0.401.0001.000 0.9270.9310.9360.9310.936 0.9400.9410.9420.9410.9400.9400.942 0.501.0001.000 0.9330.9310.9350.9350.9380.9360.9380.934 0.9400.942 0.9390.939 0.601.0001.000 0.9330.9330.938 0.943 0.9380.934 0.940 0.9390.939 0.940 0.9390.937 4250.001.0001.000 0.9270.9250.9380.939 0.940 0.937 0.9410.9450.9430.9410.9440.944 0.101.0001.000 0.9300.9270.9370.936 0.941 0.935 0.9410.9440.9430.9450.9430.942 0.201.0001.000 0.9310.9320.939 0.9420.9400.9420.9430.9410.9440.9440.9430.945 0.301.0001.000 0.9340.9350.937 0.9430.9410.9400.9410.9420.945 0.938 0.9430.942 0.401.0001.000 0.9350.937 0.941 0.937 0.9420.9440.943 0.939 0.9410.9440.9430.946 0.501.0001.000 0.9360.934 0.9410.9400.940 0.936 0.9430.9400.9430.9400.9430.941 0.601.0001.000 0.9360.939 0.9420.9440.9420.9430.9410.9440.9410.9440.9410.944 6000.001.0001.000 0.9330.934 0.941 0.938 0.9440.9410.9440.9440.9450.9490.9460.947 0.101.0001.000 0.9370.936 0.9410.9430.9420.9420.9430.9450.9440.9490.9470.945 0.201.0001.000 0.9370.935 0.942 0.936 0.9410.9430.9430.9470.9440.9460.9450.946 0.301.0001.000 0.939 0.9410.943 0.938 0.9440.9410.9460.9410.9450.9490.9470.943 0.401.0001.0000.940 0.939 0.9420.9410.9460.9420.9450.9410.9450.9460.9460.941 0.501.0001.000 0.942 0.935 0.9420.9440.9450.9430.9450.9450.9450.9460.9430.944 0.601.0001.000 0.941 0.942 0.942 0.9420.9450.9450.9450.946 0.943 0.9420.9440.9460.000.050.100.150.200.250.30 Note : Bold results are estimated coverage probabilities between .9 4 and .96; italicized results are estimated coverage probabiliti es between .925 and .975. Shaded columns are results from this study; unshaded columns are the results reported by Algina and Moulder (2001, p 638-640).

PAGE 74

74Table 3-3. Replication of Algi na and Moulders Results for Multivariate Data and Ten Predictors. n1750.001.0001.0000.8900.8930.9100.9130.9170.9190.9210.9200.9210.920 0.9270.925 0.101.0001.0000.8950.8880.9100.9160.9190.9180.9190.9220.923 0.926 0.923 0.926 0.201.0001.0000.8970.8980.9130.9110.9190.9170.9210.9200.923 0.927 0.9210.921 0.301.0001.0000.9010.9040.9160.9180.9200.9200.9200.9210.9230.9210.9220.914 0.401.0001.0000.9040.9070.9180.9140.9200.9240.9210.9220.9190.9170.9180.916 0.501.0001.0000.9090.9100.9200.9160.9180.9240.9180.9140.9170.9220.9160.916 0.601.0001.0000.9110.9120.9190.9170.9190.9220.9150.9160.9110.9120.9100.904 3000.001.0001.0000.9140.922 0.9280.9260.9290.9340.9350.9350.9360.9340.9380.935 0.101.0001.0000.9170.921 0.926 0.923 0.9330.9340.9340.9360.9350.9370.9360.935 0.201.0000.9990.9190.919 0.927 0.924 0.9330.9340.9320.9360.9360.9370.9350.931 0.301.0001.0000.9220.923 0.9310.9310.9320.9340.9340.9360.9340.9360.9340.930 0.401.0001.0000.9240.919 0.9290.9340.9340.9370.9330.9320.9320.9340.9310.931 0.501.0001.0000.9240.922 0.9300.9310.9330.9380.9320.9340.9310.9330.9300.930 0.601.0001.000 0.9260.9270.930 0.924 0.9310.9350.9310.9320.9270.9350.9270.926 4250.001.0001.0000.9230.924 0.9350.9350.936 0.941 0.9380.9370.938 0.9410.9400.940 0.101.0001.000 0.9270.9260.9340.9310.9380.9340.9390.9350.9380.938 0.9400.940 0.201.0001.000 0.9280.9280.9360.9390.9370.938 0.940 0.9380.9390.9390.9390.934 0.301.0001.000 0.9300.9340.9360.9340.9360.9340.9390.9390.9390.9390.9380.936 0.401.0001.000 0.9320.9300.9360.9340.9380.9360.9390.937 0.941 0.9350.9360.937 0.501.0001.000 0.9300.9310.9360.9390.9360.9390.9380.9390.9390.9340.9350.935 0.601.0001.000 0.9350.9340.9360.9350.9370.9370.9350.9380.9370.9350.9360.938 6000.001.0001.000 0.9330.9260.9380.938 0.9410.9400.944 0.938 0.9420.9430.9430.946 0.101.0001.0000.933 0.9390.937 0.9410.9410.9400.9410.9410.9450.9440.943 0.939 0.201.0001.000 0.9350.9290.9390.937 0.940 0.938 0.9410.9490.9420.9400.9440.941 0.301.0001.000 0.9350.9300.939 0.9410.9410.9400.9410.9420.9410.9420.943 0.937 0.401.0001.000 0.9360.937 0.9410.9440.9440.9440.9420.9420.9420.9400.942 0.937 0.501.0001.000 0.934 0.9400.9420.9420.941 0.939 0.9420.9410.9400.942 0.9390.938 0.601.0001.000 0.940 0.939 0.940 0.9400.940 0.9360.939 0.940 0.939 0.9420.940 0.9370.200.250.30 0.000.050.100.15 Note : Bold results are estimated coverage probabilities between .9 4 and .96; italicized results are estimated coverage probabiliti es between .925 and .975. Shaded columns are results from this study; unshaded columns are the results reported by Algina and Moulder (2001, p 638-640).

PAGE 75

75 Table 3-4. Empirical Coverage Probabilities for Normal Predictors and Normal Errors. k n 2246810 2000.000.050.9120.9110.9010.8960.898 0.10 0.9260.9250.926 0.9180.915 0.15 0.9300.9300.9320.928 0.916 0.20 0.9340.9360.9340.928 0.924 0.25 0.941 0.9360.9340.92 7 0.928 0.30 0.9430.9400.940 0.9320.931 0.100.050.9120.9070.9040.9040.906 0.10 0.9320.9320.925 0.9230.918 0.15 0.9340.9350.9290.925 0.922 0.20 0.942 0.9350.9380.9250.92 7 0.25 0.940 0.9380.9360.930 0.923 0.30 0.9420.9400.942 0.93 7 0.930 0.200.050.9160.9170.9160.9160.902 0.10 0.9290.9300.928 0.9210.922 0.15 0.942 0.9330.932 0.924 0.925 0.20 0.9380.9380.9380.9280.928 0.25 0.942 0.93 7 0.9390.9300.928 0.30 0.9400.9420.942 0.9300.925 0.300.05 0.925 0.9130.9150.9110.906 0.10 0.9350.9320.92 7 0.9240.919 0.15 0.9390.9320.9330.9310.926 0.20 0.9380.9360.931 0.923 0.926 0.25 0.9440.940 0.9350.9310.928 0.30 0.9430.940 0.93 7 0.9320.92 7 0.400.050.9240.9190.9190.9170.909 0.10 0.9350.9360.9280.92 7 0.919 0.15 0.941 0.9350.9330.930 0.920 0.20 0.942 0.9380.9380.928 0.922 0.25 0.945 0.9390.9330.930 0.924 0.30 0.949 0.9390.9330.931 0.920 0.500.05 0.9320.92 7 0.925 0.9130.914 0.10 0.9340.9350.931 0.9230.922 0.15 0.9460.941 0.9350.928 0.923 0.20 0.942 0.9380.9300.928 0.924 0.25 0.9400.942 0.93 7 0.930 0.918 0.30 0.9410.942 0.9350.92 7 0.921 0.600.05 0.93 7 0.928 0.9220.9230.918 0.10 0.941 0.9390.931 0.9230.924 0.15 0.943 0.93 7 0.9310.929 0.923 0.20 0.940 0.9380.9350.92 7 0.921 0.25 0.949 0.9380.9350.929 0.916 0.30 0.944 0.9390.934 0.9230.919 2 r Note : Bold results are estimated coverage probabilities be tween .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.

PAGE 76

76 Table 3-4. Continued k n 2246810 4000.000.05 0.9310.92 7 0.929 0.9230.924 0.10 0.935 0.940 0.9340.9340.936 0.15 0.9420.9410.940 0.93 7 0.939 0.20 0.9400.9450.9400.941 0.93 7 0.25 0.9450.9430.9420.947 0.93 7 0.30 0.9460.9470.9430.942 0.938 0.100.05 0.9290.9320.9280.9280.92 7 0.10 0.9400.942 0.9390.9380.936 0.15 0.9430.9420.9410.940 0.93 7 0.20 0.9460.942 0.9380.9390.936 0.25 0.9460.9420.9410.9420.941 0.30 0.9460.9470.9460.945 0.93 7 0.200.05 0.9310.9290.9290.9300.929 0.10 0.939 0.9400.941 0.93 7 0.931 0.15 0.9440.942 0.9360.9360.936 0.20 0.9440.9450.9420.943 0.935 0.25 0.9470.9470.942 0.939 0.941 0.30 0.9420.9460.9400.942 0.936 0.300.05 0.9380.9330.9340.9350.929 0.10 0.9410.944 0.9350.9360.93 7 0.15 0.9420.9400.941 0.9380.939 0.20 0.9450.9440.9420.941 0.938 0.25 0.9460.9440.943 0.9350.938 0.30 0.9460.9450.9430.940 0.939 0.400.05 0.9330.9350.9340.9300.930 0.10 0.9440.941 0.9360.93 7 0.931 0.15 0.9500.9400.9410.941 0.934 0.20 0.9440.9470.944 0.9390.939 0.25 0.9480.9440.9400.944 0.933 0.30 0.9470.9460.9430.944 0.934 0.500.05 0.9350.9350.9360.9320.931 0.10 0.9490.9400.942 0.9360.93 7 0.15 0.9460.9420.945 0.93 7 0.939 0.20 0.9500.942 0.938 0.941 0.93 7 0.25 0.9460.9460.942 0.9380.939 0.30 0.9480.9490.943 0.9390.93 7 0.600.05 0.9430.940 0.9380.9350.932 0.10 0.9410.943 0.9340.9350.938 0.15 0.9430.9440.943 0.9380.935 0.20 0.9450.9430.944 0.9390.935 0.25 0.9430.943 0.9380.9320.935 0.30 0.9500.9470.945 0.9360.92 7 2 r

PAGE 77

77 Table 3-4. Continued k n 2246810 6000.000.05 0.9310.9350.93 7 0.92 7 0.929 0.10 0.9450.9430.9420.941 0.936 0.15 0.9470.9460.9450.942 0.93 7 0.20 0.9450.9490.9440.9410.942 0.25 0.9420.9500.9480.9440.944 0.30 0.9480.9440.9450.9470.941 0.100.05 0.943 0.933 0.940 0.9350.932 0.10 0.939 0.9420.9460.9400.940 0.15 0.9430.9450.9420.9420.941 0.20 0.9450.9500.9400.9430.942 0.25 0.9480.9470.9470.9440.940 0.30 0.9500.9490.9410.9420.943 0.200.05 0.9380.9360.9390.9380.936 0.10 0.9480.9440.9460.9400.941 0.15 0.9490.9460.9450.9400.945 0.20 0.9450.9490.9460.9430.943 0.25 0.9460.9470.9440.9410.944 0.30 0.9520.9460.9460.9470.940 0.300.05 0.939 0.940 0.9330.9380.936 0.10 0.9450.9420.9410.943 0.938 0.15 0.9430.9510.9450.9430.943 0.20 0.9440.9450.9450.941 0.938 0.25 0.9490.9450.9410.9470.941 0.30 0.9480.9450.9480.941 0.938 0.400.05 0.93 7 0.943 0.9360.9340.939 0.10 0.9490.9410.9440.9430.940 0.15 0.9490.9450.9450.9440.944 0.20 0.9490.9450.9440.946 0.939 0.25 0.9490.9490.9480.9440.943 0.30 0.9450.9490.9450.9430.943 0.500.05 0.9410.9410.9400.943 0.936 0.10 0.9460.9490.9410.9460.941 0.15 0.9440.9410.9450.9420.948 0.20 0.9500.9500.9430.9430.944 0.25 0.9460.9460.9410.9460.941 0.30 0.9510.9470.9450.945 0.938 0.600.05 0.9400.9450.9440.9420.944 0.10 0.9450.9460.945 0.939 0.944 0.15 0.9480.9450.9460.9430.942 0.20 0.9520.9420.9410.940 0.93 7 0.25 0.9500.9470.9450.9450.944 0.30 0.9480.9480.9450.9420.944 2 r

PAGE 78

78 Table 3-4. Continued k n 2246810 8000.000.05 0.939 0.940 0.9390.93 7 0.936 0.10 0.9440.9430.9460.9420.945 0.15 0.9470.9410.9460.9450.945 0.20 0.9480.9510.9430.9440.946 0.25 0.9480.9490.9480.9440.946 0.30 0.9490.9510.9460.9450.944 0.100.05 0.938 0.942 0.9390.9380.939 0.10 0.9430.9490.9420.941 0.938 0.15 0.9450.9420.9460.9470.943 0.20 0.9400.9430.9480.9460.943 0.25 0.9420.9450.9460.9470.946 0.30 0.9510.9480.9460.9470.944 0.200.05 0.938 0.940 0.9380.935 0.940 0.10 0.9440.9440.9440.945 0.939 0.15 0.9500.9440.9410.9450.946 0.20 0.9460.9460.9470.9450.946 0.25 0.9490.9470.9440.9480.946 0.30 0.9470.9480.9430.9460.941 0.300.05 0.944 0.938 0.942 0.9390.939 0.10 0.9490.9450.9460.9420.942 0.15 0.9470.9490.9450.9480.943 0.20 0.9430.9490.9460.9490.943 0.25 0.9470.9480.944 0.939 0.944 0.30 0.9510.9450.9480.9480.943 0.400.05 0.946 0.938 0.9410.9440.942 0.10 0.9470.9470.9450.9440.942 0.15 0.9460.9480.9440.9430.941 0.20 0.9470.9460.9470.9440.944 0.25 0.9480.9470.9500.9430.946 0.30 0.9480.9450.9460.9460.941 0.500.05 0.9470.9400.9510.942 0.939 0.10 0.9480.9470.9490.9430.941 0.15 0.9500.9460.9490.944 0.939 0.20 0.9490.9440.9440.9490.944 0.25 0.9510.9460.9460.9470.943 0.30 0.9530.9480.9510.9440.942 0.600.05 0.9450.9440.9450.943 0.93 7 0.10 0.9530.9480.9480.9440.943 0.15 0.9480.9460.9430.9410.942 0.20 0.9430.9480.9450.9460.942 0.25 0.9470.9480.9450.9430.944 0.30 0.9500.9510.9440.9450.943 2 r

PAGE 79

79 Table 3-4. Continued k n 2246810 10000.000.05 0.9400.943 0.9380.9390.93 7 0.10 0.9470.9460.9420.9440.943 0.15 0.9470.9450.9450.9410.946 0.20 0.9470.9470.9440.9480.941 0.25 0.9490.9470.9460.9480.950 0.30 0.9530.9500.9490.9450.948 0.100.05 0.9450.9460.9450.9440.943 0.10 0.9500.9440.9450.9460.942 0.15 0.9450.9480.9430.9450.947 0.20 0.9500.9510.9460.9440.944 0.25 0.9490.9490.9470.9480.944 0.30 0.9490.9510.9460.9450.950 0.200.05 0.9440.9440.9450.9430.940 0.10 0.9490.9410.9460.9440.944 0.15 0.9480.9440.9470.9470.946 0.20 0.9470.9460.9430.9450.945 0.25 0.9500.9480.9490.9480.947 0.30 0.9470.9470.9490.9460.945 0.300.05 0.945 0.93 7 0.9400.9460.942 0.10 0.9460.9490.9430.9430.944 0.15 0.9450.9480.9460.9440.946 0.20 0.9480.9480.9450.9480.945 0.25 0.9470.9490.9470.9430.943 0.30 0.9510.9480.9470.9450.950 0.400.05 0.9430.9440.9480.9450.940 0.10 0.9440.9450.9480.9460.946 0.15 0.9470.9490.9450.9400.952 0.20 0.9480.9470.9440.9470.949 0.25 0.9470.9480.9490.9480.945 0.30 0.9510.9460.9500.9490.947 0.500.05 0.9460.9460.9450.9460.947 0.10 0.9500.9430.9480.9420.943 0.15 0.9490.9490.9450.9440.945 0.20 0.9510.9480.9480.9430.946 0.25 0.9460.9470.9460.9470.948 0.30 0.9480.9450.9510.9470.941 0.600.05 0.9460.9430.9420.9450.943 0.10 0.9510.9450.9460.9450.946 0.15 0.9460.9440.9440.9430.948 0.20 0.9510.9500.9470.9450.942 0.25 0.9440.9490.9440.9460.941 0.30 0.9470.9510.9490.9440.943 2 r

PAGE 80

80 Table 3-4. Continued k n 2246810 15000.000.05 0.9460.9420.9440.9440.944 0.10 0.9490.9470.9490.9430.945 0.15 0.9460.9490.9490.9470.945 0.20 0.9490.9440.9490.9450.947 0.25 0.9470.9460.9470.9450.946 0.30 0.9500.9480.9490.9450.944 0.100.05 0.9450.9450.9490.9420.946 0.10 0.9450.9440.9480.9450.947 0.15 0.9500.9490.9540.9470.952 0.20 0.9450.9510.9530.9460.945 0.25 0.9490.9470.9480.9450.949 0.30 0.9490.9480.9470.9490.944 0.200.05 0.9450.9440.9450.9400.944 0.10 0.9510.9430.9450.9450.948 0.15 0.9490.9500.9470.9520.945 0.20 0.9480.9490.9500.9480.948 0.25 0.9510.9470.9460.9490.947 0.30 0.9470.9460.9480.9520.952 0.300.05 0.9510.9420.9450.9410.943 0.10 0.9500.9500.9470.9470.945 0.15 0.9470.9490.9460.9500.946 0.20 0.9500.9470.9500.9490.948 0.25 0.9490.9520.9540.9500.941 0.30 0.9530.9520.9490.9490.944 0.400.05 0.9440.9440.9420.9420.948 0.10 0.9490.9490.9470.9480.948 0.15 0.9490.9550.9470.9480.944 0.20 0.9470.9500.9470.9460.945 0.25 0.9460.9490.9490.9470.945 0.30 0.9470.9520.9470.9470.949 0.500.05 0.9480.9480.9470.9450.943 0.10 0.9490.9510.9500.9500.942 0.15 0.9470.9510.9500.9460.949 0.20 0.9510.9510.9470.9480.945 0.25 0.9520.9470.9480.9470.947 0.30 0.9470.9520.9450.9500.949 0.600.05 0.9510.9500.9440.9490.945 0.10 0.9480.9480.9480.9470.947 0.15 0.9480.9510.9490.9480.949 0.20 0.9530.9520.9440.9460.948 0.25 0.9490.9510.9480.9460.941 0.30 0.9500.9510.9520.9490.947 2 r

PAGE 81

81 Table 3-4. Continued k n 2246810 20000.000.05 0.9460.9450.9450.9430.940 0.10 0.9500.9480.9450.9480.944 0.15 0.9460.9440.9480.9520.950 0.20 0.9470.9440.9490.9480.949 0.25 0.9450.9450.9490.9480.945 0.30 0.9510.9490.9470.9440.951 0.100.05 0.9490.9480.9430.9460.946 0.10 0.9490.9520.9510.9470.944 0.15 0.9470.9450.9450.9470.948 0.20 0.9460.9540.9490.9490.948 0.25 0.9450.9500.9480.9460.948 0.30 0.9470.9430.9500.9500.950 0.200.05 0.9510.9460.9440.9470.944 0.10 0.9460.9470.9430.9480.943 0.15 0.9490.9520.9500.9510.949 0.20 0.9480.9510.9460.9520.950 0.25 0.9480.9500.9470.9440.949 0.30 0.9450.9450.9510.9490.946 0.300.05 0.9470.9480.9450.9500.944 0.10 0.9450.9510.9450.9480.945 0.15 0.9460.9470.9500.9470.948 0.20 0.9490.9480.9480.9450.945 0.25 0.9450.9510.9500.9470.946 0.30 0.9480.9490.9500.9450.945 0.400.05 0.9480.9510.9480.9470.948 0.10 0.9470.9490.9480.9470.944 0.15 0.9460.9510.9500.9510.949 0.20 0.9550.9510.9490.9480.944 0.25 0.9490.9470.9480.9500.948 0.30 0.9490.9510.9430.9520.948 0.500.05 0.9490.9440.9460.9460.948 0.10 0.9480.9510.9470.9520.945 0.15 0.9490.9460.9500.9470.947 0.20 0.9460.9490.9480.9480.950 0.25 0.9480.9530.9510.9500.947 0.30 0.9480.9490.9490.9520.949 0.600.05 0.9480.9500.9430.9440.945 0.10 0.9470.9460.9490.9540.941 0.15 0.9490.9490.9500.9480.946 0.20 0.9530.9510.9500.9490.948 0.25 0.9460.9510.9470.9510.944 0.30 0.9530.9520.9500.9480.943 2 r

PAGE 82

82 Table 3-5. Empirical Coverage Probabilities for Normal Predictors and Nonnormal Errors. k n 2246810 2000.000.050.9100.9060.9020.9010.898 0.100.9190.9170.9200.9100.910 0.150.9210.9200.9190.9160.909 0.200.9200.9200.9150.9140.912 0.250.9160.9170.9140.9130.907 0.300.9160.9160.9130.9090.906 0.100.050.9130.9080.9060.9020.901 0.100.9240.9210.9180.9150.913 0.15 0.9250.926 0.9210.9180.915 0.20 0.925 0.9230.9200.9140.911 0.250.9230.9240.9180.9180.911 0.300.9220.9210.9160.9140.909 0.200.050.9150.9120.9070.9090.904 0.10 0.9270.926 0.9220.9160.916 0.15 0.928 0.9240.9220.9220.920 0.20 0.9310.9270.925 0.9210.919 0.25 0.9290.927 0.9240.9190.913 0.30 0.9270.925 0.9230.9180.915 0.300.050.9200.9160.9140.9090.906 0.10 0.9300.927 0.9240.9180.917 0.15 0.9340.9290.925 0.9230.918 0.20 0.9320.9290.925 0.9200.920 0.25 0.9320.9300.92 6 0.9230.919 0.30 0.9340.9300.928 0.9220.916 0.400.050.9240.9190.9140.9130.910 0.10 0.9320.9300.92 6 0.9210.918 0.15 0.9350.9310.9300.925 0.919 0.20 0.9350.9320.92 7 0.925 0.919 0.25 0.9370.9330.9290.925 0.917 0.30 0.9380.9350.931 0.9210.917 0.500.05 0.926 0.9240.9200.9150.913 0.10 0.9350.9310.928 0.9240.921 0.15 0.9370.9320.9280.925 0.920 0.20 0.9380.9350.9330.928 0.921 0.25 0.941 0.9340.930 0.9240.917 0.30 0.9390.9360.9310.92 7 0.918 0.600.05 0.9320.9250.92 6 0.9200.914 0.10 0.9380.9370.9300.925 0.921 0.15 0.940 0.9370.9300.925 0.920 0.20 0.942 0.9380.9300.925 0.920 0.25 0.941 0.9370.9300.92 6 0.917 0.30 0.943 0.9370.930 0.9230.915 2 r Note : Bold results are estimated coverage probabilities be tween .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.

PAGE 83

83 Table 3-5. Continued k n 2246810 4000.000.05 0.9270.9250.92 7 0.9220.922 0.10 0.9300.9290.9310.9280.92 7 0.15 0.9310.9320.931 0.9230.924 0.20 0.9270.9290.925 0.9230.922 0.250.9240.9210.9220.9210.921 0.300.9180.9200.9200.9210.916 0.100.05 0.9290.9260.92 6 0.92 6 0.920 0.10 0.9330.9320.9300.9280.92 7 0.15 0.9320.9310.9290.92 7 0.92 6 0.20 0.9300.9300.9290.9280.925 0.25 0.9290.9280.92 6 0.9250.925 0.300.9240.924 0.925 0.9210.919 0.200.05 0.9300.9270.92 6 0.92 6 0.925 0.10 0.9340.9340.9330.9300.929 0.15 0.9320.9320.9320.9290.925 0.20 0.9330.9330.9300.9300.92 7 0.25 0.9350.9340.9300.9300.92 6 0.30 0.9320.9310.9300.92 6 0.925 0.300.05 0.9330.9300.9280.930 0.924 0.10 0.9360.9380.9330.9320.932 0.15 0.9380.9370.9320.9330.931 0.20 0.9370.9370.9330.9320.930 0.25 0.9350.9360.9350.9310.931 0.30 0.9380.9360.9340.9310.930 0.400.05 0.9360.9350.9310.9320.928 0.10 0.9370.9380.9380.9340.934 0.15 0.9390.9380.93 7 0.9340.931 0.20 0.940 0.9390.93 7 0.9340.932 0.25 0.940 0.9350.93 7 0.9340.931 0.30 0.943 0.9370.9380.9330.930 0.500.05 0.9370.9370.9350.9320.931 0.10 0.9390.9390.93 7 0.93 7 0.933 0.15 0.9420.940 0.9380.9350.934 0.20 0.945 0.9390.93 6 0.93 7 0.933 0.25 0.9450.941 0.9380.93 6 0.934 0.30 0.9440.9410.940 0.9390.931 0.600.05 0.9390.9360.93 6 0.9330.931 0.10 0.9440.943 0.9380.93 7 0.934 0.15 0.9430.9400.942 0.9380.933 0.20 0.9460.943 0.9390.9350.934 0.25 0.9470.9440.941 0.9380.932 0.30 0.9440.944 0.9390.9380.932 2 r

PAGE 84

84 Table 3-5. Continued k n 2246810 6000.000.05 0.9320.9330.9310.9320.92 7 0.10 0.9320.9310.9310.9320.930 0.15 0.9330.9300.9310.9300.928 0.20 0.9290.9300.9300.92 6 0.925 0.250.922 0.9260.92 6 0.9230.922 0.300.9240.9220.9210.9180.922 0.100.05 0.9350.9350.9320.9300.930 0.10 0.9350.9360.9340.9330.932 0.15 0.9370.9320.9340.9310.930 0.20 0.9340.9320.9310.9310.928 0.25 0.9290.9300.92 7 0.928 0.922 0.30 0.9270.9270.92 7 0.92 6 0.92 6 0.200.05 0.9360.9350.9330.9320.933 0.10 0.9370.9380.9350.9350.935 0.15 0.940 0.9350.9340.9340.932 0.20 0.9380.9360.9340.9350.932 0.25 0.9350.9310.9310.9320.928 0.30 0.9340.9350.9290.9300.929 0.300.05 0.9380.9360.93 6 0.9340.935 0.10 0.940 0.9380.93 6 0.9350.934 0.15 0.940 0.9370.9380.9350.932 0.20 0.9380.9350.93 7 0.93 6 0.933 0.25 0.9390.9370.93 7 0.93 6 0.932 0.30 0.9380.9380.9330.9340.931 0.400.05 0.9370.9380.93 6 0.9350.935 0.10 0.9420.9400.940 0.93 7 0.93 6 0.15 0.9410.940 0.9390.9380.935 0.20 0.940 0.9390.9380.93 7 0.93 7 0.25 0.9410.941 0.9390.9390.934 0.30 0.9400.940 0.9390.93 6 0.93 6 0.500.05 0.9410.941 0.9380.93 7 0.93 6 0.10 0.9420.943 0.9390.93 7 0.938 0.15 0.9450.9410.941 0.9380.939 0.20 0.9440.9410.943 0.9390.938 0.25 0.9430.9440.942 0.9390.93 7 0.30 0.9450.944 0.939 0.940 0.93 6 0.600.05 0.9440.9410.940 0.9390.93 7 0.10 0.9450.9450.9410.942 0.938 0.15 0.9430.9420.9410.942 0.93 6 0.20 0.9440.9430.9420.943 0.93 7 0.25 0.9470.9430.9420.941 0.938 0.30 0.9460.9460.942 0.9390.938 2 r

PAGE 85

85 Table 3-5. Continued k n 2246810 8000.000.05 0.9360.9360.9330.93 6 0.933 0.10 0.9370.9330.9340.9330.931 0.15 0.9330.9330.9330.9320.930 0.20 0.9320.9290.9280.9280.92 6 0.25 0.9260.9280.92 6 0.9240.924 0.300.9230.9210.9240.9210.920 0.100.05 0.9390.9380.9340.9350.934 0.10 0.9380.9340.9380.9330.934 0.15 0.9340.9350.9340.93 6 0.932 0.20 0.9330.9350.9310.9300.930 0.25 0.9300.9300.9300.9290.928 0.30 0.9290.9260.9280.925 0.923 0.200.05 0.9370.9370.93 7 0.9350.93 7 0.10 0.939 0.940 0.9390.9380.934 0.15 0.9380.9380.9380.9380.935 0.20 0.9370.9350.93 7 0.9340.933 0.25 0.9340.9350.9320.9340.933 0.30 0.9330.9320.9330.9310.930 0.300.05 0.941 0.9360.9380.9380.93 6 0.10 0.9420.943 0.9390.9350.939 0.15 0.9410.940 0.9390.9380.93 7 0.20 0.941 0.9380.9390.93 7 0.934 0.25 0.9380.9370.9380.93 7 0.935 0.30 0.9380.9370.9380.93 6 0.933 0.400.05 0.9400.9400.941 0.9380.938 0.10 0.9430.9410.941 0.9390.93 7 0.15 0.9420.9410.9420.940 0.938 0.20 0.9420.9410.940 0.93 7 0.939 0.25 0.9410.941 0.9380.9380.93 6 0.30 0.943 0.9380.9380.9390.93 7 0.500.05 0.9430.9420.9410.942 0.938 0.10 0.9430.9400.9420.942 0.939 0.15 0.9450.9420.9430.9420.940 0.20 0.9440.9430.942 0.938 0.940 0.25 0.9460.9440.943 0.939 0.941 0.30 0.9450.9430.9440.9420.941 0.600.05 0.9430.9400.9400.9430.940 0.10 0.9460.9440.9450.9420.942 0.15 0.9460.9440.9430.9410.942 0.20 0.9450.9450.9430.9430.941 0.25 0.9430.9460.9450.9420.942 0.30 0.9480.9440.9440.9410.941 2 r

PAGE 86

86 Table 3-5. Continued k n 2246810 10000.000.05 0.9380.9390.93 6 0.9350.935 0.10 0.9380.9370.93 7 0.93 6 0.935 0.15 0.9360.9320.9320.9340.93 7 0.20 0.9330.9300.9310.9300.92 7 0.25 0.9270.9300.92 7 0.923 0.92 7 0.300.9240.9240.9220.9220.923 0.100.05 0.9380.9380.93 7 0.9380.93 7 0.10 0.9390.9360.9380.93 6 0.934 0.15 0.9360.9390.93 7 0.9350.935 0.20 0.9360.9340.9330.9310.935 0.25 0.9290.9300.9310.9280.929 0.30 0.9290.9280.9280.9250.92 7 0.200.05 0.940 0.939 0.940 0.9380.93 7 0.10 0.939 0.941 0.9380.9380.939 0.15 0.9390.9390.93 6 0.9380.93 6 0.20 0.9350.9370.9340.9340.934 0.25 0.9360.9340.9330.9340.933 0.30 0.9340.9340.9320.9340.931 0.300.05 0.940 0.939 0.940 0.939 0.940 0.10 0.9410.9410.9410.940 0.938 0.15 0.942 0.939 0.941 0.9390.93 6 0.20 0.941 0.9390.9380.9380.93 7 0.25 0.941 0.9370.93 6 0.9380.93 6 0.30 0.9390.9390.9380.93 6 0.938 0.400.05 0.9430.9400.9410.940 0.938 0.10 0.9430.9430.9410.9430.942 0.15 0.9430.9410.9430.9400.940 0.20 0.9420.941 0.9390.9380.93 7 0.25 0.941 0.939 0.9400.9400.941 0.30 0.9410.942 0.939 0.9400.941 0.500.05 0.9430.9440.9430.9400.940 0.10 0.9460.9440.9430.9430.944 0.15 0.9420.9450.9420.9410.942 0.20 0.9430.9440.9420.9420.940 0.25 0.9430.9430.9430.9410.940 0.30 0.9460.9450.9440.9410.940 0.600.05 0.9450.9440.9440.9410.941 0.10 0.9460.9420.9440.9430.940 0.15 0.9450.9450.9450.9410.942 0.20 0.9460.9470.9450.9430.941 0.25 0.9470.9470.9440.9430.941 0.30 0.9480.9470.9460.9420.945 2 r

PAGE 87

87 Table 3-5. Continued k n 2246810 15000.000.05 0.9410.940 0.9390.9390.938 0.10 0.9390.937 0.940 0.93 7 0.93 6 0.15 0.9370.9350.9350.9340.935 0.20 0.9320.9330.9300.9310.932 0.25 0.9290.9270.9290.92 6 0.928 0.300.924 0.925 0.9210.9240.922 0.100.05 0.9430.9410.9400.940 0.938 0.10 0.941 0.939 0.9400.940 0.938 0.15 0.9370.9390.9390.9350.93 6 0.20 0.9340.9350.9330.9330.934 0.25 0.9330.9310.9320.9310.930 0.30 0.9310.9280.9300.92 6 0.931 0.200.05 0.9440.9420.9440.9410.942 0.10 0.9400.9430.9400.941 0.939 0.15 0.939 0.9400.940 0.93 7 0.941 0.20 0.9390.9390.9350.93 7 0.93 6 0.25 0.9330.9350.93 7 0.9340.935 0.30 0.9350.9340.93 6 0.9310.934 0.300.05 0.9420.9420.9430.9410.944 0.10 0.9450.9440.9430.9430.942 0.15 0.9430.9400.9410.942 0.939 0.20 0.939 0.9410.942 0.9390.93 7 0.25 0.9390.9390.9380.93 7 0.93 6 0.30 0.9410.940 0.93 7 0.9380.938 0.400.05 0.9450.9440.9430.9430.942 0.10 0.9430.9410.9440.9420.943 0.15 0.9440.9410.9420.9400.943 0.20 0.9420.9410.9430.9400.940 0.25 0.9420.9430.9420.941 0.939 0.30 0.9420.9410.9440.9420.941 0.500.05 0.9440.9460.9450.9430.945 0.10 0.9450.9430.9410.9430.944 0.15 0.9440.9440.9450.9430.942 0.20 0.9440.9460.9430.9420.945 0.25 0.9450.9450.9430.9460.941 0.30 0.9470.9450.9430.9440.944 0.600.05 0.9470.9470.9450.9470.946 0.10 0.9470.9460.9460.9430.943 0.15 0.9440.9460.9450.9440.945 0.20 0.9460.9450.9460.9420.944 0.25 0.9480.9440.9470.9470.944 0.30 0.9500.9480.9480.9460.943 2 r

PAGE 88

88 Table 3-5. Continued k n 2246810 20000.000.05 0.9440.9440.9400.9400.941 0.10 0.940 0.937 0.940 0.9390.939 0.15 0.9360.9380.9350.93 6 0.935 0.20 0.9330.9330.9330.9310.932 0.25 0.9280.9300.9280.9310.92 6 0.300.924 0.925 0.9230.9210.924 0.100.05 0.9410.9410.9420.9410.941 0.10 0.9420.9410.9420.9400.940 0.15 0.940 0.9350.9380.9380.93 7 0.20 0.9360.9330.9340.9340.93 6 0.25 0.9290.9310.9320.9300.930 0.30 0.9310.9300.9290.9300.930 0.200.05 0.9430.9440.9430.9420.943 0.10 0.9410.9400.9400.9420.942 0.15 0.9420.9400.940 0.939 0.940 0.20 0.940 0.9370.9390.9350.93 6 0.25 0.9350.9360.93 6 0.9350.935 0.30 0.9340.9350.9330.9320.933 0.300.05 0.9430.9440.9430.9440.943 0.10 0.9420.9440.9430.9430.942 0.15 0.9420.9400.9400.9400.940 0.20 0.9430.9400.9420.940 0.938 0.25 0.941 0.938 0.941 0.9380.93 6 0.30 0.9390.9380.9380.9390.938 0.400.05 0.9470.9470.9460.9450.945 0.10 0.9450.9430.9420.9430.943 0.15 0.9420.9440.9430.9440.944 0.20 0.9410.9420.9410.9420.940 0.25 0.9430.9440.9420.9420.941 0.30 0.9420.9410.9410.9410.942 0.500.05 0.9450.9480.9460.9470.944 0.10 0.9460.9450.9450.9460.944 0.15 0.9440.9450.9460.9430.942 0.20 0.9440.9440.9440.9440.942 0.25 0.9440.9440.9430.9430.942 0.30 0.9460.9460.9460.9460.943 0.600.05 0.9460.9450.9470.9450.945 0.10 0.9480.9470.9450.9450.944 0.15 0.9460.9490.9480.9430.946 0.20 0.9480.9460.9460.9460.944 0.25 0.9480.9490.9470.9460.946 0.30 0.9490.9490.9480.9460.946 2 r

PAGE 89

89 Table 3-6. Empirical Coverage Probabilities for Nonnormal Predictors and Normal Errors. k n 2246810 2000.000.050.9040.9040.8980.8980.892 0.100.9180.9140.9140.9090.906 0.150.9190.9180.9160.9110.909 0.200.9210.9170.9170.9100.907 0.250.9190.9170.9130.9110.908 0.300.9160.9120.9100.9100.902 0.100.050.9100.9080.9030.8970.898 0.100.9190.9140.9120.9100.905 0.150.9210.9180.9150.9110.906 0.200.9190.9140.9110.9110.905 0.250.9180.9150.9130.9060.902 0.300.9120.9150.9070.9030.899 0.200.050.9100.9110.9070.9020.899 0.100.9180.9160.9140.9080.905 0.150.9200.9170.9130.9080.905 0.200.9190.9150.9110.9070.904 0.250.9170.9100.9090.9030.900 0.300.9140.9070.9040.8990.893 0.300.050.9140.9100.9090.9030.901 0.100.9190.9160.9130.9100.906 0.150.9180.9150.9110.9070.903 0.200.9130.9110.9070.9030.900 0.250.9060.9040.9030.9010.893 0.300.9020.9010.8980.8920.886 0.400.050.9180.9130.9090.9050.902 0.100.9170.9160.9120.9100.904 0.150.9140.9110.9090.9030.897 0.200.9100.9060.9010.8930.888 0.250.9010.8970.8940.8900.885 0.300.8940.8890.8880.8790.873 0.500.050.9160.9130.9090.9060.901 0.100.9150.9110.9070.9050.900 0.150.9070.9060.9000.8980.890 0.200.8990.8940.8910.8860.879 0.250.8900.8870.8810.8760.871 0.300.8810.8780.8750.8650.860 0.600.050.9170.9130.9100.9050.900 0.100.9100.9070.9010.8970.893 0.150.8960.8960.8910.8800.879 0.200.8870.8820.8780.8730.864 0.250.8710.8700.8630.8580.853 0.300.8660.8600.8570.8480.841 2 r Note : Bold results are estimated coverage probabilities be tween .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.

PAGE 90

90 Table 3-6. Continued k n 2246810 4000.000.050.924 0.925 0.9200.9220.917 0.10 0.9300.9280.9280.928 0.922 0.15 0.92 6 0.9290.92 6 0.9240.924 0.20 0.92 7 0.9250.925 0.9240.920 0.250.922 0.925 0.9230.9190.916 0.300.9190.9220.9170.9140.913 0.100.05 0.92 7 0.9240.9220.9230.919 0.10 0.9290.9290.9280.92 6 0.923 0.15 0.92 7 0.92 7 0.92 6 0.9220.920 0.200.924 0.925 0.9220.9230.917 0.250.9220.9210.9170.9160.916 0.300.9160.9190.9170.9110.909 0.200.05 0.92 6 0.928 0.9240.9220.919 0.10 0.9290.92 6 0.92 6 0.9240.922 0.15 0.92 6 0.92 7 0.9230.9210.921 0.20 0.925 0.9220.9190.9190.916 0.250.9180.9160.9170.9140.914 0.300.9150.9140.9120.9120.908 0.300.05 0.9280.9290.92 6 0.9240.919 0.10 0.9290.92 7 0.92 7 0.9240.921 0.150.9240.9230.9230.9200.918 0.200.9170.9180.9160.9120.911 0.250.9120.9120.9110.9050.904 0.300.9060.9040.9040.9020.899 0.400.05 0.9290.92 6 0.925 0.9230.922 0.100.9240.9240.9240.9200.919 0.150.9180.9180.9150.9150.912 0.200.9140.9140.9090.9110.904 0.250.9070.9030.9010.8970.894 0.300.8980.8970.8960.8950.891 0.500.05 0.92 6 0.92 6 0.925 0.9220.922 0.100.9230.9200.9200.9180.914 0.150.9130.9080.9090.9040.903 0.200.9010.9030.9010.8980.892 0.250.8950.8910.8910.8870.884 0.300.8870.8860.8820.8780.873 0.600.05 0.92 6 0.925 0.9240.9200.920 0.100.9150.9160.9100.9090.908 0.150.9010.9010.8980.8950.893 0.200.8910.8870.8880.8810.876 0.250.8770.8750.8730.8700.866 0.30 0.8700.8660.8610.8590.856 2 r

PAGE 91

91 Table 3-6. Continued k n 2246810 6000.000.05 0.9310.9290.9300.9280.92 7 0.10 0.9340.9330.9310.9300.928 0.15 0.9340.9320.9280.9290.928 0.20 0.9320.9280.92 7 0.9280.92 6 0.25 0.92 7 0.9240.9230.9220.922 0.300.9220.9220.9220.9190.918 0.100.05 0.9340.9330.9280.9280.928 0.10 0.9330.9340.9310.9310.929 0.15 0.9310.9290.9280.9280.92 6 0.20 0.92 6 0.9250.92 7 0.9230.923 0.250.9240.9240.9210.9190.919 0.300.9180.9210.9160.9160.917 0.200.05 0.9320.9320.9290.9300.929 0.10 0.9300.9340.9310.9300.930 0.15 0.9300.9300.9250.925 0.923 0.20 0.92 6 0.92 7 0.9230.9200.919 0.250.9210.9190.9160.9170.917 0.300.9170.9150.9140.9110.912 0.300.05 0.9320.9300.9290.9300.929 0.10 0.9310.9310.9300.92 7 0.92 7 0.15 0.92 7 0.92 7 0.9240.9200.920 0.200.9190.9160.9170.9170.912 0.250.9150.9150.9130.9130.910 0.300.9090.9050.9060.9080.904 0.400.05 0.9310.9310.9310.9290.929 0.10 0.9300.92 7 0.92 6 0.9230.922 0.150.9220.9210.9230.9170.915 0.200.9140.9140.9110.9110.907 0.250.9080.9070.9060.9030.902 0.300.8980.9020.8950.8940.892 0.500.05 0.9310.9330.9300.9280.92 6 0.10 0.92 6 0.9210.9190.9180.919 0.150.9140.9140.9120.9100.907 0.200.9050.9040.9040.9010.897 0.250.8920.8930.8920.8870.890 0.300.8860.8850.8840.8800.878 0.600.05 0.9300.9250.9290.92 7 0.924 0.100.9180.9170.9150.9130.910 0.150.9020.9030.9000.9000.896 0.200.8870.8890.8890.8850.881 0.250.8800.8760.8760.8750.869 0.300.8680.8670.8630.8630.863 2 r

PAGE 92

92 Table 3-6. Continued k n 2246810 8000.000.05 0.93 7 0.9350.9310.9340.932 0.10 0.93 6 0.9350.93 6 0.9340.934 0.15 0.9330.9320.9340.9300.931 0.20 0.9300.9290.9290.9290.928 0.25 0.9250.92 6 0.9250.925 0.922 0.300.9240.9220.9220.9230.922 0.100.05 0.9380.9330.9340.9350.932 0.10 0.9340.9340.9340.9350.931 0.15 0.9310.9310.9320.9280.931 0.20 0.9280.9290.92 6 0.92 6 0.924 0.250.923 0.92 6 0.92 6 0.9210.921 0.300.9200.9200.9210.9180.918 0.200.05 0.9350.93 7 0.9330.9350.932 0.10 0.9340.9330.9330.9330.931 0.15 0.9320.9280.9290.9280.92 7 0.20 0.92 6 0.924 0.925 0.9240.922 0.250.9220.9200.9180.9190.916 0.300.9180.9140.9150.9120.914 0.300.05 0.93 6 0.93 7 0.9330.9310.931 0.10 0.9310.9320.9300.9290.929 0.15 0.92 7 0.92 7 0.9230.923 0.92 7 0.200.9200.9190.9210.9170.917 0.250.9160.9140.9130.9130.912 0.300.9090.9080.9070.9070.906 0.400.05 0.93 6 0.9340.9350.9350.934 0.10 0.9330.9300.9300.9280.92 7 0.150.9220.9240.9210.9200.920 0.200.9160.9130.9100.9130.910 0.250.9070.9070.9060.9030.900 0.300.9020.8990.8990.8970.895 0.500.05 0.9330.9340.9310.9320.928 0.10 0.92 6 0.9250.925 0.9230.922 0.150.9160.9120.9130.9120.911 0.200.9070.9060.9080.9020.898 0.250.8960.8930.8950.8910.889 0.300.8850.8850.8840.8830.881 0.600.05 0.9320.9290.9310.92 7 0.928 0.100.9180.9170.9180.9150.914 0.150.9010.9010.9010.9030.899 0.200.8930.8890.8910.8870.887 0.250.8790.8780.8740.8730.872 0.300.8690.8640.8660.8660.861 2 r

PAGE 93

93 Table 3-6. Continued k n 2246810 10000.000.05 0.93 6 0.9380.9340.9350.935 0.10 0.9380.9380.9360.9350.934 0.15 0.9330.9340.9340.9340.931 0.20 0.9310.9310.9290.9290.927 0.25 0.92 7 0.9270.9270.928 0.924 0.30 0.9230.9220.923 0.925 0.922 0.100.05 0.93 7 0.9360.9360.9360.934 0.10 0.93 6 0.9360.9350.9340.932 0.15 0.9320.9330.9310.9290.932 0.20 0.9300.9300.9270.9280.926 0.25 0.92 7 0.924 0.9250.926 0.923 0.30 0.9200.9230.9150.9200.920 0.200.05 0.941 0.9370.9350.9380.935 0.10 0.9350.9350.9330.9330.932 0.15 0.9300.9310.9300.9290.931 0.20 0.92 7 0.9280.925 0.9240.921 0.25 0.9220.9210.9200.9200.919 0.30 0.9170.9160.9160.9130.910 0.300.05 0.9380.9360.9350.9370.935 0.10 0.9330.9360.9340.9320.929 0.15 0.9280.9280.9270.9270.928 0.20 0.9230.9230.9200.9170.920 0.25 0.9170.9150.9150.9140.914 0.30 0.9110.9080.9090.9070.907 0.400.05 0.9330.9380.9340.9320.934 0.10 0.9320.9290.9280.9290.928 0.15 0.9240.9220.9200.9220.922 0.20 0.9150.9160.9130.9120.912 0.25 0.9070.9080.9060.9040.902 0.30 0.9010.8980.9000.8980.898 0.500.05 0.9340.9360.9340.9330.933 0.10 0.9280.927 0.924 0.9260.925 0.15 0.9140.9140.9140.9140.915 0.20 0.9060.9030.9020.9020.902 0.25 0.8970.8940.8930.8970.894 0.30 0.8890.8870.8840.8850.882 0.600.05 0.9320.9320.9300.9310.929 0.10 0.9220.9170.9170.9180.916 0.15 0.9040.9060.9020.9020.900 0.20 0.8920.8880.8890.8850.886 0.25 0.8800.8800.8760.8740.872 0.30 0.8670.8670.8670.8660.863 2 r

PAGE 94

94 Table 3-6. Continued k n 2246810 15000.000.05 0.941 0.9390.939 0.9400.941 0.10 0.9390.9380.9380.9380.934 0.15 0.9340.9320.9370.9350.936 0.20 0.9320.9310.9290.9310.931 0.25 0.9280.9280.9270.9250.928 0.30 0.923 0.925 0.9230.9210.921 0.100.05 0.9400.940 0.9390.9380.938 0.10 0.93 6 0.9370.9390.9380.934 0.15 0.9340.9340.9350.9330.930 0.20 0.9300.9280.9300.9300.929 0.25 0.92 6 0.9250.925 0.9240.924 0.30 0.9220.9210.9230.9160.917 0.200.05 0.9410.941 0.9390.9380.938 0.10 0.93 7 0.9360.9340.9350.935 0.15 0.9320.9300.9310.9300.928 0.20 0.9280.9270.9260.9250.925 0.25 0.9220.9220.9210.9190.920 0.30 0.9180.9150.9170.9180.913 0.300.05 0.9390.9380.9380.937 0.940 0.10 0.9330.9330.9350.9300.932 0.15 0.9300.9300.9290.9290.927 0.20 0.9230.9220.9200.9220.921 0.25 0.9150.9180.9140.9150.913 0.30 0.9100.9120.9100.9080.907 0.400.05 0.93 7 0.9380.9380.9360.934 0.10 0.9320.9320.9320.9310.929 0.15 0.925 0.9230.923 0.925 0.919 0.20 0.9180.9130.9150.9120.913 0.25 0.9090.9080.9050.9090.908 0.30 0.9010.9010.9020.9000.898 0.500.05 0.9350.9340.9380.9350.936 0.10 0.92 7 0.925 0.924 0.925 0.924 0.15 0.9190.9180.9140.9160.911 0.20 0.9060.9040.9070.9030.900 0.25 0.8960.8970.8960.8980.893 0.30 0.8900.8910.8860.8880.883 0.600.05 0.9330.9350.9320.9310.933 0.10 0.9180.9200.9170.9170.915 0.15 0.9060.9030.9030.9050.903 0.20 0.8910.8910.8890.8910.890 0.25 0.8800.8750.8780.8780.878 0.30 0.8680.8670.8670.8670.866 2 r

PAGE 95

95 Table 3-6. Continued k n 2246810 20000.000.050.9410.9410.9400.9410.939 0.10 0.9390.9380.9390.9390.939 0.15 0.93 7 0.9350.93 6 0.9350.935 0.20 0.9310.9290.9300.9330.930 0.25 0.92 6 0.92 7 0.924 0.92 7 0.928 0.30 0.9250.9250.92 6 0.9240.922 0.100.050.9410.9420.9410.9390.9410.100.9420.9390.9380.93 7 0.93 7 0.15 0.93 6 0.9340.9340.93 6 0.933 0.20 0.9310.9320.9280.9310.928 0.25 0.92 6 0.92 6 0.92 7 0.9250.92 6 0.300.9220.9200.9220.9200.924 0.200.050.9430.9420.9400.9400.9410.10 0.9380.93 7 0.93 7 0.9380.93 7 0.15 0.9330.9310.9310.9320.930 0.20 0.9250.9280.92 7 0.9250.928 0.250.9230.9230.9210.9210.921 0.300.9170.9190.9170.9160.917 0.300.05 0.9390.9380.9420.9410.938 0.10 0.93 6 0.9340.93 7 0.9350.934 0.15 0.92 7 0.9280.92 6 0.9280.928 0.200.9210.9200.9230.9210.924 0.250.9140.9170.9150.9160.914 0.300.9100.9070.9110.9120.908 0.400.050.9410.9390.9390.93 7 0.9410.10 0.9310.9330.9340.9290.929 0.150.9240.9240.9230.9230.924 0.200.9150.9150.9160.9160.913 0.250.9080.9080.9040.9050.909 0.300.9040.8990.8970.8990.898 0.500.05 0.9390.93 7 0.9380.9350.935 0.10 0.92 7 0.92 7 0.9250.92 6 0.925 0.150.9180.9150.9150.9140.914 0.200.9070.9060.9050.9070.905 0.250.8940.9000.8970.8960.895 0.300.8890.8890.8880.8850.881 0.600.05 0.9340.9330.9330.9340.930 0.100.9180.9200.9170.9180.916 0.150.9060.9060.9050.9030.903 0.200.8920.8900.8920.8880.890 0.250.8790.8790.8780.8770.875 0.300.8720.8700.8690.8680.866 2 r

PAGE 96

96 Table 3-7. Empirical Coverage Probabilities for Predictors Nonnormal and Errors Nonnormal. k n 2246810 2000.000.050.9030.9000.8970.8940.891 0.100.9110.9080.9050.9020.899 0.150.9090.9070.9040.9000.898 0.200.9040.9010.8980.8950.893 0.250.8970.8960.8930.8900.888 0.300.8910.8900.8880.8850.881 0.100.050.9080.9040.9000.8960.892 0.100.9140.9110.9090.9030.901 0.150.9100.9080.9060.9010.898 0.200.9060.9040.9010.8980.892 0.250.9020.8990.8960.8930.888 0.300.8940.8930.8910.8860.882 0.200.050.9100.9060.9030.9010.896 0.100.9150.9110.9090.9060.901 0.150.9120.9090.9050.9010.899 0.200.9060.9040.9010.8970.893 0.250.9010.8970.8950.8930.886 0.300.8960.8930.8910.8870.881 0.300.050.9130.9090.9060.9020.899 0.100.9160.9130.9080.9050.901 0.150.9120.9080.9050.9020.896 0.200.9060.9020.9000.8960.891 0.250.8990.8970.8930.8890.884 0.300.8940.8920.8880.8830.878 0.400.050.9160.9130.9070.9040.900 0.100.9150.9130.9080.9040.901 0.150.9090.9060.9030.8980.895 0.200.9030.8990.8940.8910.885 0.250.8950.8920.8880.8830.877 0.300.8880.8860.8830.8750.868 0.500.050.9160.9130.9090.9040.902 0.100.9120.9090.9060.9010.897 0.150.9050.9000.8970.8940.888 0.200.8960.8930.8870.8820.876 0.250.8880.8820.8780.8730.867 0.300.8790.8750.8710.8650.857 0.600.050.9160.9130.9100.9060.901 0.100.9080.9050.9020.8950.891 0.150.8960.8920.8890.8830.877 0.200.8850.8810.8750.8700.863 0.250.8740.8700.8640.8570.849 0.300.8630.8600.8560.8470.838 2 r Note : Bold results are estimated coverage probabilities be tween .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.

PAGE 97

97 Table 3-7. Continued k n 2246810 4000.000.050.9200.9190.9190.9160.915 0.100.9220.9210.9200.9170.915 0.150.9160.9160.9150.9120.911 0.200.9100.9070.9070.9060.905 0.250.9030.9020.9010.9020.897 0.300.8960.8950.8940.8930.890 0.100.050.9230.9230.9200.9180.916 0.100.9230.9220.9220.9190.917 0.150.9180.9170.9150.9140.912 0.200.9120.9100.9090.9070.906 0.250.9050.9040.9030.9010.900 0.300.9010.8990.8970.8950.893 0.200.05 0.92 6 0.9230.9220.9200.918 0.100.9240.9200.9220.9190.917 0.150.9190.9160.9160.9130.911 0.200.9110.9100.9090.9070.906 0.250.9050.9030.9050.9010.898 0.300.9000.8980.8970.8960.894 0.300.05 0.92 6 0.9240.9230.9210.918 0.100.9230.9220.9200.9180.916 0.150.9170.9150.9140.9130.912 0.200.9100.9070.9070.9050.902 0.250.9020.9020.9010.8990.895 0.300.8990.8960.8930.8930.891 0.400.05 0.92 6 0.9240.9240.9220.918 0.100.9230.9190.9200.9170.916 0.150.9160.9130.9110.9090.907 0.200.9070.9050.9020.9000.898 0.250.8980.8980.8950.8930.888 0.300.8910.8890.8890.8850.883 0.500.05 0.92 7 0.926 0.9230.9200.920 0.100.9210.9180.9180.9130.912 0.150.9100.9070.9060.9030.900 0.200.9000.8970.8940.8930.890 0.250.8900.8870.8850.8820.879 0.300.8800.8800.8780.8750.872 0.600.05 0.925 0.9230.9210.9200.918 0.100.9130.9120.9100.9070.906 0.150.9000.8980.8950.8930.891 0.200.8870.8860.8830.8810.875 0.250.8760.8730.8710.8690.864 0.30 0.8640.8640.8620.8590.853 2 r

PAGE 98

98 Table 3-7. Continued k n 2246810 6000.000.05 0.9280.92 7 0.92 6 0.9240.923 0.10 0.92 6 0.925 0.9230.9230.921 0.150.9190.9180.9180.9180.916 0.200.9140.9120.9110.9100.908 0.250.9040.9050.9030.9030.903 0.300.8970.8980.8970.8950.895 0.100.05 0.9300.9280.9280.9260.925 0.10 0.92 7 0.926 0.9240.9240.922 0.150.9210.9180.9180.9170.917 0.200.9140.9130.9140.9110.909 0.250.9080.9050.9040.9040.902 0.300.9010.9000.8970.8980.896 0.200.05 0.9300.9290.9290.92 7 0.925 0.10 0.92 7 0.9250.925 0.9240.922 0.150.9210.9190.9190.9170.915 0.200.9130.9140.9120.9110.909 0.250.9060.9070.9030.9030.903 0.300.9010.9000.8990.8980.896 0.300.05 0.9300.9300.9300.92 7 0.92 7 0.10 0.92 6 0.926 0.9240.9240.921 0.150.9190.9190.9180.9150.914 0.200.9110.9100.9090.9090.907 0.250.9050.9040.9030.9010.899 0.300.8980.8970.8970.8950.893 0.400.05 0.9310.9300.9290.9280.92 7 0.100.924 0.925 0.9220.9210.920 0.150.9160.9150.9130.9130.910 0.200.9070.9070.9050.9040.903 0.250.9000.8980.8970.8960.895 0.300.8930.8920.8890.8890.886 0.500.05 0.9310.9290.9290.92 7 0.926 0.100.9230.9200.9180.9180.916 0.150.9100.9090.9080.9070.903 0.200.9010.8990.8980.8970.895 0.250.8890.8910.8880.8870.884 0.300.8840.8820.8810.8770.876 0.600.05 0.9280.9260.92 6 0.925 0.923 0.100.9140.9140.9120.9110.910 0.150.9010.9000.8980.8960.895 0.200.8890.8870.8850.8830.881 0.250.8760.8760.8740.8720.869 0.300.8680.8640.8640.8600.858 2 r

PAGE 99

99 Table 3-7. Continued k n 2246810 8000.000.05 0.9320.9310.9290.9300.928 0.10 0.9280.9260.92 6 0.9260.925 0.150.9220.9190.9200.9190.917 0.200.9130.9130.9140.9110.911 0.250.9060.9050.9060.9050.904 0.300.8970.8980.8970.8970.896 0.100.05 0.9320.9310.9310.9290.930 0.10 0.9280.92 7 0.92 7 0.92 7 0.925 0.150.9210.9200.9220.9200.919 0.200.9160.9150.9140.9120.911 0.250.9070.9080.9050.9050.905 0.300.9020.9010.9000.8990.899 0.200.05 0.9320.9320.9310.9320.930 0.10 0.9290.9280.92 7 0.92 7 0.926 0.150.9230.9210.9200.9180.918 0.200.9150.9130.9120.9120.911 0.250.9070.9080.9070.9060.905 0.300.9010.9010.9010.8990.897 0.300.05 0.9330.9330.9320.9300.929 0.10 0.9280.92 7 0.92 7 0.926 0.924 0.150.9210.9190.9180.9180.916 0.200.9120.9120.9110.9090.910 0.250.9050.9040.9040.9030.902 0.300.8990.8970.8980.8970.894 0.400.05 0.9330.9320.9320.9300.930 0.10 0.9250.926 0.9240.9230.923 0.150.9160.9180.9140.9150.914 0.200.9090.9070.9070.9050.903 0.250.9000.8990.8980.8970.895 0.300.8940.8920.8890.8910.888 0.500.05 0.9320.9310.9300.9290.928 0.100.9230.9220.9190.9190.919 0.150.9110.9110.9100.9070.907 0.200.9010.9010.8990.8980.897 0.250.8910.8900.8900.8880.887 0.300.8850.8840.8800.8800.878 0.600.05 0.9300.9300.92 7 0.92 7 0.925 0.100.9160.9150.9140.9140.911 0.150.9010.9010.8990.8980.896 0.200.8890.8880.8870.8850.883 0.250.8780.8760.8750.8730.871 0.300.8690.8660.8660.8630.862 2 r

PAGE 100

100 Table 3-7. Continued k n 2246810 8000.000.05 0.9320.9310.9290.9300.928 0.10 0.9280.9260.9260.9260.925 0.15 0.9220.9190.9200.9190.917 0.20 0.9130.9130.9140.9110.911 0.25 0.9060.9050.9060.9050.904 0.30 0.8970.8980.8970.8970.896 0.100.05 0.9320.9310.9310.9290.930 0.10 0.9280.9270.9270.9270.925 0.15 0.9210.9200.9220.9200.919 0.20 0.9160.9150.9140.9120.911 0.25 0.9070.9080.9050.9050.905 0.30 0.9020.9010.9000.8990.899 0.200.05 0.9320.9320.9310.9320.930 0.10 0.9290.9280.9270.9270.926 0.15 0.9230.9210.9200.9180.918 0.20 0.9150.9130.9120.9120.911 0.25 0.9070.9080.9070.9060.905 0.30 0.9010.9010.9010.8990.897 0.300.05 0.9330.9330.9320.9300.929 0.10 0.9280.9270.9270.926 0.924 0.15 0.9210.9190.9180.9180.916 0.20 0.9120.9120.9110.9090.910 0.25 0.9050.9040.9040.9030.902 0.30 0.8990.8970.8980.8970.894 0.400.05 0.9330.9320.9320.9300.930 0.10 0.9250.926 0.9240.9230.923 0.15 0.9160.9180.9140.9150.914 0.20 0.9090.9070.9070.9050.903 0.25 0.9000.8990.8980.8970.895 0.30 0.8940.8920.8890.8910.888 0.500.05 0.9320.9310.9300.9290.928 0.10 0.9230.9220.9190.9190.919 0.15 0.9110.9110.9100.9070.907 0.20 0.9010.9010.8990.8980.897 0.25 0.8910.8900.8900.8880.887 0.30 0.8850.8840.8800.8800.878 0.600.05 0.9300.9300.9270.9270.925 0.10 0.9160.9150.9140.9140.911 0.15 0.9010.9010.8990.8980.896 0.20 0.8890.8880.8870.8850.883 0.25 0.8780.8760.8750.8730.871 0.30 0.8690.8660.8660.8630.862 2 r

PAGE 101

101 Table 3-7. Continued k n 2246810 10000.000.05 0.9330.9320.9310.9320.931 0.10 0.9290.9300.9270.9270.926 0.15 0.9220.9220.9210.9210.919 0.20 0.9150.9140.9130.9130.913 0.25 0.9050.9060.9060.9050.904 0.30 0.8990.8990.9000.8970.898 0.100.05 0.9340.9350.9340.9320.931 0.10 0.9290.9300.9290.9280.926 0.15 0.9230.9220.9210.9210.920 0.20 0.9170.9150.9140.9120.912 0.25 0.9080.9080.9070.9060.906 0.30 0.9000.9010.9000.9010.901 0.200.05 0.9350.9350.9340.9330.933 0.10 0.9290.9290.9270.9280.926 0.15 0.9220.9220.9210.9200.920 0.20 0.9150.9140.9140.9130.913 0.25 0.9080.9070.9060.9060.906 0.30 0.9010.9020.9010.9010.899 0.300.05 0.9340.9340.9330.9340.933 0.10 0.9290.9280.9280.9260.926 0.15 0.9220.9210.9190.9210.918 0.20 0.9140.9130.9120.9100.912 0.25 0.9060.9050.9050.9050.903 0.30 0.8990.8990.8970.8970.897 0.400.05 0.9350.9340.9320.9310.931 0.10 0.92 7 0.927 0.9240.9230.924 0.15 0.9170.9160.9150.9150.915 0.20 0.9110.9070.9080.9070.906 0.25 0.9000.9010.9000.8990.896 0.30 0.8940.8930.8910.8910.891 0.500.05 0.9330.9320.9330.9310.930 0.10 0.9240.9230.9210.9210.921 0.15 0.9110.9110.9100.9100.909 0.20 0.9000.9000.9000.9000.899 0.25 0.8930.8900.8890.8890.888 0.30 0.8830.8830.8830.8810.881 0.600.05 0.9310.9300.9290.9290.928 0.10 0.9150.9160.9150.9140.914 0.15 0.9010.9020.9000.9000.896 0.20 0.8890.8890.8870.8870.885 0.25 0.8790.8770.8750.8740.873 0.30 0.8690.8680.8650.8650.864 2 r

PAGE 102

102 Table 3-7. Continued k n 2246810 15000.000.05 0.93 7 0.9350.9350.9350.935 0.10 0.9310.9310.9290.9290.928 0.15 0.9240.9220.9220.9220.921 0.20 0.9150.9140.9160.9150.914 0.25 0.9070.9080.9070.9060.907 0.30 0.9000.8980.8980.8980.898 0.100.05 0.9380.9350.9360.9360.934 0.10 0.9320.9310.9300.9300.929 0.15 0.9230.9220.9230.9220.922 0.20 0.9150.9160.9160.9150.914 0.25 0.9090.9090.9070.9080.907 0.30 0.9020.9020.9010.9010.899 0.200.05 0.9380.9370.9360.9350.936 0.10 0.9300.9300.9290.9290.929 0.15 0.9240.9240.9230.9220.921 0.20 0.9160.9150.9150.9150.912 0.25 0.9080.9080.9090.9060.905 0.30 0.9030.9020.9010.9010.900 0.300.05 0.93 7 0.9360.9350.9350.934 0.10 0.9300.9290.9290.9290.928 0.15 0.9210.9210.9210.9200.919 0.20 0.9120.9140.9130.9110.913 0.25 0.9070.9070.9060.9050.904 0.30 0.9010.8990.9000.8990.898 0.400.05 0.93 7 0.9360.9370.9350.935 0.10 0.92 7 0.9270.9270.9260.925 0.15 0.9190.9170.9180.9170.916 0.20 0.9120.9090.9080.9080.907 0.25 0.9020.9000.9010.9010.898 0.30 0.8960.8930.8930.8910.892 0.500.05 0.9340.9340.9340.9340.933 0.10 0.9240.9240.9230.9230.923 0.15 0.9130.9120.9110.9110.911 0.20 0.9030.9020.9020.9000.899 0.25 0.8910.8910.8910.8900.890 0.30 0.8840.8850.8830.8810.880 0.600.05 0.9310.9310.9300.9300.931 0.10 0.9180.9170.9170.9150.914 0.15 0.9030.9010.9010.9020.902 0.20 0.8890.8880.8880.8880.885 0.25 0.8780.8770.8770.8760.873 0.30 0.8670.8660.8660.8660.864 2 r

PAGE 103

103 Table 3-7. Continued k n 2246810 20000.000.05 0.9380.9380.9370.9380.936 0.10 0.9310.9310.9310.9300.931 0.15 0.9240.9230.9230.9230.922 0.20 0.9160.9150.9150.9140.915 0.25 0.9080.9080.9070.9080.905 0.30 0.9000.8990.8980.9000.898 0.100.05 0.9390.9380.9370.9370.936 0.10 0.9320.9320.9310.9310.932 0.15 0.9240.9240.922 0.925 0.923 0.20 0.9170.9170.9150.9160.916 0.25 0.9090.9080.9080.9090.907 0.30 0.9030.9010.9020.9000.902 0.200.05 0.9390.9390.9380.9370.936 0.10 0.9310.9310.9310.9300.930 0.15 0.9230.9230.9220.9230.921 0.20 0.9160.9150.9140.9150.916 0.25 0.9090.9090.9070.9090.909 0.30 0.9030.9020.9020.9030.902 0.300.05 0.93 6 0.9380.9370.9360.936 0.10 0.9300.9300.9300.9290.928 0.15 0.9230.9210.9210.9200.920 0.20 0.9130.9130.9130.9140.912 0.25 0.9060.9050.9060.9050.907 0.30 0.9010.9000.8990.8990.898 0.400.05 0.93 7 0.9370.9370.9360.937 0.10 0.9290.9280.9270.9250.927 0.15 0.9190.9180.9170.9180.917 0.20 0.9090.9110.9080.9090.908 0.25 0.9010.9000.9010.9000.900 0.30 0.8950.8940.8940.8930.894 0.500.05 0.93 6 0.9360.9340.9340.934 0.10 0.9250.925 0.9240.9230.923 0.15 0.9140.9130.9110.9110.911 0.20 0.9000.9020.9020.9010.900 0.25 0.8930.8920.8900.8910.891 0.30 0.8830.8850.8850.8840.883 0.600.05 0.9330.9330.9310.9310.931 0.10 0.9200.9180.9160.9150.916 0.15 0.9010.9030.9020.9020.901 0.20 0.8900.8900.8890.8870.887 0.25 0.8770.8770.8770.8760.875 0.30 0.8680.8680.8680.8680.866 2 r

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104Table 3-8. Descriptive Statistics for Cove rage Probability by Distributional Condition. SDMedian MinimumMaximum n = 600 n = 2000 0.9400.01230.9440.8560.9560.00.0 0.9380.01200.9410.8510.9530.00.0 0.9370.01190.9410.8530.9530.00.0 0.9330.01240.9360.8550.9536.71.4 0.9250.01550.9280.8560.95134.325.2 0.9310.01310.9340.8550.95011.07.1 0.9280.01280.9310.8450.95012.46.2 0.9280.01270.9300.8420.94915.27.6 0.9230.01290.9250.8450.94546.226.2 0.9150.01540.9170.8500.94672.955.7 0.9290.01380.9330.8440.94915.210.0 0.9260.01320.9290.8410.94821.412.4 0.9260.01320.9290.8430.94723.814.3 0.9210.01330.9230.8380.94852.933.8 0.9130.01560.9150.8440.94474.361.0 0.9120.02130.9170.7960.94662.449.1 0.9090.02080.9130.8040.94570.558.1 0.9090.02090.9130.8000.94571.963.3 0.9040.02050.9060.8040.94081.970.0 0.8960.02150.8960.8020.93991.483.3 0.8830.03730.8910.7330.94081.496.2 0.8800.03670.8870.7290.93999.586.2 0.8800.03690.8860.7410.93895.785.7 0.8750.03630.8790.7450.93699.087.1 0.8670.03630.8680.7380.933100.090.0 Range Percent Nonrobust Distributional Condition Normal Normal Distribution for X Distribution for e Pseudo t (10) Normal Normal Normal Normal Pseudo t (10) Pseudo t (10) Pseudo t (10) Normal Normal Pseudo t (10) Pseudo t (10) Pseudo t (10) Pseudo t (10) Pseudo (10) Pseudo (10) Pseudo (4) Pseudo exponential Pseudo exponential Pseudo t (10) Pseudo (10) Pseudo (10) Normal Normal Pseudo (4) Pseudo (4) Pseudo (4) Pseudo (4) Pseudo (10) Pseudo (10) Pseudo (10) Pseudo (10) Pseudo (10) Pseudo t (10) Pseudo exponential Pseudo exponential Pseudo exponential Pseudo (10) Pseudo exponentialPseudo exponential Pseudo exponential Pseudo exponential Pseudo (4) Pseudo (4) Pseudo (4) Pseudo (4) Pseudo (4) Pseudo exponential p M

PAGE 105

105 Table 3-9. Analysis of Variance, Estimated Mean Square Components, and Percentage of Total Variance in Cove rage Probability Explaine d by the Study Variables. Source of Variance df SS Mean Square F p Mean Square Component Percentage of Total Variance X 4 30.065 7.5162 951467.0< .0001 .0005113 44.51 X 2 20 6.408 .3204 40557.4< .0001.0001308 11.38 2 5 7.117 .1423 180191.0< .0001.0001162 10.12 n 13 7.378 .5675 71840.3< .0001.0001081 9.41 X 2 r 24 5.107 .2128 26938.3< .0001.0001013 8.82 e 4 2.251 .5629 71251.3< .0001.0000383 3.33 2 r 6 2.357 .3929 49734.9< .0001.0000374 3.26 X 2 r 2 120 .984 .0082 1038.4< .0001.0000234 2.04 2 r 2 30 .963 .0321 4062.8< .0001.0000183 1.60 e 2 r 24 .738 .0308 3895.6< .0001.0000147 1.28 n 2 65 .649 .0099 1263.4< .0001.0000114 .99 n k 52 .599 .0115 1459.0< .0001.0000110 .95 e 2 20 .396 .0198 2505.4< .0001.0000081 .70 k 4 .465 .1162 14712.1< .0001.0000079 .69 e 2 r 2 120 .259 .0022 273.5< .0001.0000062 .54 Table 3-10. Descriptive Statistics for Coverage Probability by Distribution for the Predictors. M SD MedianMinimumMaximum 0.9350.0140.9390.8510.956 0.9250.0150.9280.8420.950 0.9230.0150.9260.8380.949 0.9060.0220.9100.7920.946 0.8770.0370.8830.7290.940 Pseudo (4) Pseudo exponential Distribution Range Normal Pseudo t (10) Pseudo (10)

PAGE 106

106 Table 3-11. Descriptive Statistics for Covera ge Probability by Distribution for the Errors. M SD MedianMinimumMaximum 0.9190.02970.9290.7330.956 0.9160.02930.9260.7290.953 0.9160.02940.9260.7410.953 0.9110.02920.9200.7450.953 0.9030.03020.9110.7380.951Pseudo (10) Pseudo (4)Pseudo exponential Range Distribution Normal Pseudo t (10) Table 3-12. Coverage Probability by 2 and the Distribution for the Predictors. Normal Pseudot (10) Pseudo(10)Pseudo(4)Pseudo-exponential 0.050.9320.9290.9290.9240.915 0.100.9360.9300.9290.9190.901 0.150.9360.9270.9260.9110.885 0.200.9350.9240.9220.9020.868 0.250.9350.9210.9180.8940.853 0.300.9340.9170.9150.8860.840 Distribution for the Predictors ( X ) Table 3-13. Coverage Probability by 2 and the Distributi on for the Errors. Normal Pseudot (10) Pseudo(10)Pseudo(4)Pseudo-exponential 0.050.9280.9270.9270.9250.923 0.100.9270.9250.9250.9220.917 0.150.9220.9200.9200.9150.907 0.200.9170.9140.9140.9080.899 0.250.9120.9080.9080.9010.890 0.300.9070.9030.9020.8950.883 Distribution for the Errors ( e )

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107 Table 3-14. Coverage Probability by 2 r and the Distribution for the Predictors. r 2Normal Pseudot (10) Pseudo(10)Pseudo(4)Pseudo-exponential 0.000.9280.9220.9220.9120.896 0.100.9300.9250.9240.9130.895 0.200.9330.9260.9250.9130.891 0.300.9350.9270.9250.9110.885 0.400.9370.9270.9250.9070.875 0.500.9390.9250.9230.8990.860 0.600.9400.9220.9180.8870.837 Distribution for the Predictors ( X ) Table 3-15. Coverage Probability by 2 r and the Distribution for the Errors. r 2Normal Pseudot (10) Pseudo(10)Pseudo(4)Pseudo-exponential 0.000.9270.9220.9210.9130.897 0.100.9260.9220.9220.9140.902 0.200.9250.9210.9210.9150.905 0.300.9220.9200.9190.9150.907 0.400.9180.9160.9160.9130.907 0.500.9120.9100.9100.9080.905 0.600.9020.9020.9010.9000.899 Distribution for the Errors ( e ) Table 3-16. Coverage Probability by 2 r and 2 for X Distributed Multivariate Normal. r 20.050.100.150.200.250.30 0.000.9230.9240.9200.9150.9100.904 0.100.9250.9250.9210.9160.9110.906 0.200.9260.9250.9210.9160.9110.906 0.300.9270.9250.9200.9140.9090.904 0.400.9270.9240.9180.9110.9050.899 0.500.9270.9210.9130.9050.8970.891 0.600.9260.9170.9050.8950.8850.877 2

PAGE 108

108 Table 3-17. Coverage Probability by 2 r and 2 for X Distributed Pseudot10( g = 0, h = .058). r 20.050.100.150.200.250.30 0.000.92480.92770.92590.92260.91900.9151 0.100.92660.92920.92740.92450.92130.9183 0.200.92820.93030.92820.92580.92320.9201 0.300.92930.93080.92890.92590.92330.9211 0.400.93110.93130.92860.92570.92260.9201 0.500.93170.93100.92760.92360.92010.9168 0.600.93240.92950.92410.91940.91450.9102 2 Table 3-18. Coverage Probability by 2 r and 2 for X Distributed 2 10Pseudo( g = .301, h = -.017). r 20.050.100.150.200.250.30 0.000.92470.92750.92520.92160.91810.9138 0.100.92650.92850.92650.92320.92020.9168 0.200.92800.92940.92730.92420.92110.9182 0.300.92910.93020.92770.92440.92160.9184 0.400.93030.93020.92720.92370.91990.9169 0.500.93120.92990.92530.92090.91620.9128 0.600.93130.92750.92130.91550.90980.9053 2 Table 3-19. Coverage Probability by 2 r and 2 for X Distributed 2 4Pseudo( g = .502, h = -.048). r 20.050.100.150.200.250.30 0.000.92190.92190.91700.91090.90460.8980 0.100.92310.92240.91740.91130.90490.8987 0.200.92440.92240.91680.91040.90370.8976 0.300.92480.92170.91500.90750.90050.8939 0.400.92530.91980.91120.90260.89380.8862 0.500.92470.91580.90460.89350.88290.8737 0.600.92250.90890.89390.87880.86550.8542 2

PAGE 109

109 Table 3-20. Coverage Probability by 2 r and 2 for X Distributed Pseudo-exponential ( g = .760, h = -.098). r 20.050.100.150.200.250.30 0.000.91690.91200.90260.89220.88110.8703 0.100.91790.91170.90130.89010.87950.8685 0.200.91810.90990.89810.88580.87370.8632 0.300.91720.90660.89250.87830.86440.8518 0.400.91570.90110.88360.86610.84960.8349 0.500.91250.89180.86890.84750.82780.8111 0.600.90630.87680.84690.81950.79590.7771 2 Table 3-21. Coverage Proba bility by Sample Size and 2. n 0.050.100.150.200.250.30 1000.8770.8900.8890.8870.8830.878 2000.9070.9120.9090.9040.8990.894 3000.9180.9190.9150.9090.9040.899 4000.9240.9230.9180.9120.9070.902 5000.9280.9250.9200.9140.9080.903 6000.9300.9270.9210.9150.9090.904 7000.9320.9280.9220.9160.9100.905 8000.9330.9290.9230.9170.9100.905 9000.9340.9300.9230.9170.9110.906 10000.9350.9300.9240.9170.9110.906 12500.9360.9300.9230.9160.9090.904 15000.9370.9310.9230.9160.9100.904 17500.9380.9310.9240.9170.9100.904 20000.9380.9310.9240.9170.9100.905 2

PAGE 110

110 Table 3-22. Coverage Probability by Sa mple Size and Number of Predictors. kn 246810 1000.8970.8900.8830.8740.864 2000.9090.9060.9020.8980.893 3000.9130.9110.9090.9060.903 4000.9160.9140.9120.9100.908 5000.9170.9160.9140.9130.911 6000.9180.9170.9160.9140.913 7000.9180.9180.9170.9150.914 8000.9190.9180.9170.9160.915 9000.9190.9190.9180.9170.916 10000.9200.9190.9180.9180.917 12500.9200.9200.9190.9190.918 15000.9210.9200.9200.9190.918 17500.9210.9210.9200.9200.919 20000.9210.9210.9200.9200.920 Table 3-23. Analysis of Variance, Estimated Mean Square Components, and Percentage of Total Variance Explained in the Ratio of Mean Estimated Asymptotic Variance to the Empirical Sampling Variance of R2. Source of Variance df SS Mean Square F p Mean Square Component Percentage of Total Variance X 4 640.90 160.22 1094917.0< .0001 .0108996 51.78 2 5 271.56 54.31 371145.0< .0001.0044336 21.06 X 2 20 70.21 3.51 23988.1< .0001.0014327 6.81 X 2 r 24 68.45 2.85 19489.7< .0001.0013580 6.45 e 4 77.48 19.37 132371.0< .0001.0013177 6.26 2 r 6 35.40 5.90 40318.4< .0001.0005619 2.67 e 2 r 24 21.42 0.89 6098.1< .0001.0004249 2.02 X 2 r 2 120 6.42 0.05 365.8< .0001.0001525 0.72

PAGE 111

111 0500100015002000 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage ProbabilitySample Size Normal X Normal e Normal X Nonnormal e Nonnormal X Normal e Nonnormal X Nonnormal e Figure 3-1. Mean estimated coverage probability by normality vs. nonnormality in the predictors, normality vs. nonnormality in the errors, and sample size.

PAGE 112

1120500100015002000 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage ProbabilitySample Size Predictors Distribution: Error Distribution: Black g=0, h=0 g=0, h=0 Red g=0, h=.058 g=.760, h=-.098 Green g=.301, h=-.017 g=0, h=.058 Blue g=.502, h=-.048 g=.301, h=-.017 Purple g=.760, h=-.098 g=.502, h=-.048 Figure 3-2. Empirical coverage probability as a function of di stributional condition and sample size.

PAGE 113

113 0.975 + | | | | 0.950 + | | | | +-----+ | | | | *--+--* +-----+ | | | | +-----+ | | +-----+ | | 0.925 + | *--+--* *--+--* +-----+ | | | +-----+ +-----+ | | | | | | | *-----* | | 0 | | | + | +-----+ 0.900 + 0 | | | | | | | 0 0 | +-----+ | | | 0 0 0 | | | | 0 0 | *-----* 0.875 + 0 0 | | + | | 0 0 | | | | * 0 | | | | * | +-----+ 0.850 + * | | | * 0 | | 0 | | 0 | 0.825 + 0 | | 0 | | 0 | | 0 | 0.800 + 0 | | | | | | | 0.775 + 0 | 0 | 0 | 0 0.750 + 0 | 0 | 0 | 0 0.725 + ------------+-----------+-----------+-----------+-----------+----------g 0 0 0.301 0.502 0.760 h 0 0.058 -0.017 -0.048 -0.098 Figure 3-3. Box plots of the distributions of coverage probability estimates by distribution for the predictors ( ni = 14,700).

PAGE 114

114 | 0.975 + | | | | | 0.950 + | | | | | | | | | | | | +-----+ +-----+ +-----+ | | | *-----* | | | | +-----+ | 0.925 + | | *-----* *-----* | | +-----+ | | + | | + | | + | *-----* | | | | | | | | | | + | *-----* | +-----+ +-----+ +-----+ | | | + | 0.900 + | | | +-----+ | | | | | | | | | | | | | | +-----+ | | | | | | 0.875 + | | | | | | | | | | | | 0 0 0 | | | 0 0 0 0 | 0.850 + 0 0 0 0 | | 0 0 0 0 | | 0 0 0 0 0 | 0 0 0 0 0 0.825 + 0 0 0 0 0 | 0 0 0 0 0 | * 0 0 | * 0 0 0.800 + * * 0 | * * 0 | * * 0 | * * 0 0.775 + * * | * * | * * | * 0.750 + * | * | * | * 0.725 + ------------+-----------+-----------+-----------+-----------+----------ge 0 0 0.301 0.502 0.760 he 0 0.058 -0.017 -0.048 -0.098 Figure 3-4. Box plots of the distributions of coverage probability estimates by distribution for the errors ( ni = 14,700).

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1150.050.100.150.200.250.30 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage Probability Averaged over all conditions Main Effect of 2Effect of 2 by distribution for X: g = 0, h = 0 g = 0, h = .058 g = .301, h = -.017 g = .502, h = -.048 g = .760, h = -.098 Figure 3-5. Main effect of the squared semipartial correlation coefficient 2, and the effect of the interaction of 2 and X on coverage probability for 2 > 0. 0.050.100.150.200.250.30 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage Probability2 Distribution for e : g = 0, h = 0 g = 0, h = .058 g = .301, h = -.017 g = .502, h = -.048 g = .760, h = -.098 Figure 3-6. Main effect of the squared semipartial correlation coefficient 2, and the effect of the interaction of 2 and e on coverage probability for 2 > 0.

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1160.000.100.200.300.400.500.60 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage Probabilityr 2 Main effect of r 2Effect of r 2 by distribution for X: g = 0, h = 0 g = 0, h = .058 g = .301, h = -.017 g = .502, h = .048 g = .760, h = -.098 Figure 3-7. Effect of the inte raction between the size of the s quared multiple correlation in the reduced model,2 r and the distribution for the predictors, X, on coverage probability. 0.000.100.200.300.400.500.60 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage Probabilityr 2 Distribution for e : g = 0, h = 0 g = 0, h = .058 g = .301, h = -.017 g = .502, h = -.048 g = .760, h = -.098 Figure 3-8. Interaction between the size of th e squared multiple correlation in the reduced model,2 r and the distribution for the errors, e, and its relationship to coverage probability.2 r

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1170.000.100.200.300.400.500.60 0.87 0.88 0.89 0.90 0.91 0.92 0.93 Coverage Probabilityr 2 2=.05 2=.10 2=.15 2=.20 2=.25 2=.30 Figure 3-9. Effect of the 2 r 2 interaction on coverage probability for 2 > 0.

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1180.000.100.200.300.400.500.60 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 .05 .10 .15 .20 .25 .30Coverage Probability2 rA 0.000.100.200.300.400.500.60 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 .05 .10 .15 .20 .25 .30Coverage Probability2 rB 0.000.100.200.300.400.500.60 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 .05 .10 .15 .20 .25 .30Coverage Probability2 rC 0.000.100.200.300.400.500.60 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 .05 .10 .15 .20 .25 .30Coverage Probability2 rD 0.000.100.200.300.400.500.60 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 .05 .10 .15 .20 .25 .30Coverage Probability2 rE Figure 3-10. Effect of the X 2 r 2 interaction on coverage probability for 2 > 0. A) X sampled from a multivariate normal population. B) X sampled from a pseudot (10) distribution ( g = 0, h = .058). C) X sampled from a pseudo2(10) distribution ( g = .301, h = -.017). D) X sampled from a pseudo2(10) distribution ( g = .502, h = -.048). E) X sampled from a pseudo-exponential distribution ( g = .760, h = -.098).

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1190200400600800100012001400160018002000 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 2=.05 2=.10 2=.15 2=.20 2=.25 2=.30Coverage ProbabilitySample Size ( N ) Figure 3-11. Interacti on between sample size, n and the population squared semipartial correlation, 2, and the impact on coverage probability for 2 > 0. 0200400600800100012001400160018002000 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Coverage ProbabilitySample Size ( N ) k =2 k =4 k =6 k =8 k =10 Figure 3-12. Effect of the interaction between sample size, n and number of predictors, k, on coverage probability.

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1200.000.100.200.300.400.500.60 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 MEAV/R2r 2A .05 .10 .15 .20 .25 .300.000.100.200.300.400.500.60 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 .05 .10 .15 .20 .25 .30MEAV/R2r 2B 0.000.100.200.300.400.500.60 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 .05 .10 .15 .20 .25 .30MEAV/R2r 2C 0.000.100.200.300.400.500.60 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 .05 .10 .15 .20 .25 .30MEAV/R2r 2D 0.000.100.200.300.400.500.60 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 .05 .10 .15 .20 .25 .30MEAV/R2r 2E Figure 3-13. Ratio of mean estimated as ymptotic variance to the variance in R2 (MEAV/Var R2) as a function of the distribution for the predictors, 2, and 2 r .A) X sampled from a multivariate normal population. B) X sampled from a pseudot (10) distribution ( g = 0, h = .058). C) X sampled from a pseudo2(10) distribution ( g = .301, h = -.017). D) X sampled from a pseudo2(10) distribution ( g = .502, h = -.048). E) X sampled from a pseudo-exponential distribution ( g = .760, h = -.098).

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121 Figure 3-14. Relationship between coverage pr obability and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 Figure 3-15. Relationship between coverage pr obability and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 for multivariate normal data ( g = 0, h = 0). r = .91 r = .62

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122 Figure 3-16. Relationship between coverage pr obability and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed pseudot10 ( g = 0, h = .058). Figure 3-17. Relationship between coverage pr obability and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed 2 10pseudo( g = .502, h = -.048). r = .63 r = .68

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123 Figure 3-18. Relationship between coverage pr obability and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed 2 4pseudo( g = .502, h = -.048). Figure 3-19. Relationship between coverage pr obability and the ratio of mean estimated asymptotic variance to the empirical sampling variance of R2 for 2 > 0 and X distributed pseudo-exponential ( g =.760, h = -.098).. r = .91 r = .86

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124 CHAPTER 4 DISCUSSION Given the current emphasis on re porting effect sizes and confidence intervals, Alf and Grafs approach to constructi ng confidence intervals for the s quared semipartial correlation is appealing in its simplicity. Confidence limits can be computed using only a hand calculator and the computer output from a multiple regression analysis. Results of this study would caution against widespread use because the procedure dem onstrated poor control of coverage probability. Even when the distributional assumption of multivariate normality holds for the data, the asymptotic confidence interval procedure is bias ed and with sample sizes typically used in psychology, the coverage probability for a nominal 95% confidence interval will tend to be less than .95. As shown in this study, when nonnor mality is introduced, depending on the degree, coverage probability can be drama tically less than .95 even with samples as large as 2000. This is especially true when the predictors included in a multiple regression model do not follow a multivariate normal distribution. When the predictors are nonnormal, the Alf and Graf procedure produces a confidence interval that tends to be too liberal. With extreme nonnormality, the interval will be much too narrow and the contribution of a single variable to the regression will be minimized. Since multivariate normality is rarely observed in practice, the poor performance of this procedur e for use as a measure of effect size accuracy is particularly disappointing. It appears that accuracy is sacr ificed for the sake of computational facility. One of the goals of this study was to determ ine if we could identify a minimum, fail safe, sample size for which the confidence interv al offers adequate coverage probability over a wide range of distributional conditions and regression model characteristics. Despite the sizeable amount of data simulated and analyzed, we must conclude that a sample size well in excess of 2000 would be necessary to demonstrat e the robustness of this procedure against

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125 nonnormality even if we were willing to adopt a more liberal standard, i.e. > .925. Increasing the sample size to a level that is atypical for resear ch in the behavioral and social sciences (e.g., 1000) in an attempt to ensure adequate covera ge probability when the distributional assumption is violated will not yield the desired result. In the case of extreme nonnormality in the predictors, the extremely slow incremental improve ment in coverage probability, as a function of sample size, suggests that the pro cedure is likely to be inaccurate no matter how large a sample is used. For both normal and nonnormal data, coverage probability is also dependent on the population effect sizes for both th e squared semipartia l correlation coeffici ent and the squared multiple correlation coefficient for the model to which the variable of interest has been added; however, the pattern of empirical coverage probabilities over the ra nge of factors manipulated is completely different. In order to illustrate this Figure 4-1 presents thre e hypothetical situations: (a) a variable with a small population effect size is added to a model that explains none of the variance in the criterion, 22.05,.00;r (b) a variable with a me dium population effect size is added to a model in which the effect size associated with the population squared multiple correlation coefficient is already relatively large, 22.15,.30;r and (c) a variable with a large population effect size is added to a model for which the effect size associated with the population squared multiple correlation coefficient is very large. For each of these situations, average coverage probability for each nonnormal di stribution is compared to the case where X is multivariate normal as a function of sample size. Coverage probabilities are averaged over the distribution for e and the number of predictors, k Although estimates are based on only 25 cases, there is a clear trend. Not only is c overage probability worse for nonnormal data, but the behavior of the confidence interval shows a pattern that is the dire ct opposite of that observed for

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126 normal predictors regardless of the degree of nonnormality. For normal predictors, coverage probability improves as both 2 r and 2 become larger. This suggests that coverage probability is closer to the nominal confidence level for larger effect sizes than it is when the effect size associated with an added variable is small. Th e reverse is true when there is nonnormality in the predictors. Here, the confidence interval is more accurate for smaller effect sizes and coverage probability decreases as the effect size gets larg er. Table 4-1 shows coverage probability is clearly unacceptable for a large effect under the most nonnormal conditions investigated. Due to this inadequacy, this procedure for estimating co nfidence intervals falls fa r short as a reliable measure of importance in reporting research re sults and its use defeat s the purpose of those interested in reforming statistical practice. Limitations It is obviously not possible to investigate ev ery imaginable type of nonnormality that might occur in applied research situa tions. This study has investigat ed a reasonably broad range of nonnormal distributions wit hout claiming to have exhausted th e possibilities. Some might argue that the method of simulating nonnormality is not representative of real world data. For example, it is unlikely that all of the variables in a multiple regression analysis are samples from a population that is distributed with the same skewness a nd kurtosis as the exponential distribution and this is where estimated cove rage probability deviates markedly from the nominal. However, the premise in simulation studies investigating r obustness is that if a procedure performs well under extreme conditions it can be expected to work under most conditions likely to be encountered by researchers. Therefore, adequate coverage probability across a wide range of possible conditions should be demonstrat ed in order to recommend a statistical procedure.

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127 In this study, coverage probability characterist ics were very similar when the data were distributed either pseudot10 or 2 10pseudo. Both univariate and multivariate kurtosis measures are nearly equal for the tw o distributions, but pseudot10 is symmetrical while2 10pseudois moderately skewed. Although this would suggest that skewness ha s less influence on coverage probability than kurtosis, the evidence is base d on only one comparison. In this study, the distribution for the predictors, in a sense, se rves as a proxy for the degree of multivariate nonnormality and the distribution for the errors is a reflection of the additive influence of univariate nonnormality. In the ANOVA and varian ce components analyses reported, these variables were treated as categor ical, as were the levels of n k 2 and 2 r Since multivariate skewness and kurtosis can be quantified as cont inuous variables using Mardias measures of multivariate skewness, b1, k, and multivariate kurtosis, b2, k, and univariate skewness can be measured by 1 and 2, the impact of skewness and kurtosis on coverage probability could be modeled using multiple regression. The size of the squared semipartial correlati on may have examined too wide a range and the values studied were somewhat coarse. The la rge values studied, i.e. .25 and .30, where the procedure tends to perform es pecially poorly, correspond to a large effect size and may be relatively rare in practice. At the other extreme, Algina a nd Moulder reported results for 2 < .05 that were similar to the coverage probabilities observed for 2 =.05 with multivariate normal data. Since small to medium effect sizes are much more common than large effect sizes in research in the social and behavioral sciences, values for 2 less than .05 and in smaller steps between .05 and .15 would have given a more comp lete picture. Nonetheless, this would not have changed the findings or conclusions of this study.

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128Further Research The asymptotic method of constructing confidence intervals assumes symmetry by referring to a single normal distri bution with two-equal sized tails sliced off. Our concern in using an approximation based on asymptotic theory is whether the distribution of the difference in squared correlations approaches normality for a given sample size. This approach assumes that the sampling distribution re tains the same shape regardless of the value of the population parameter. However, confidence intervals are not always symmetrical around sample statistics and their sampling distributions do not always have the same shap e for different values. If the distribution that generates the da ta is not symmetric, using the asymptotic variance derived under the assumption of symmetry typically underes timates the actual asymptotic variance. When one set of predictors is a subset of the other, as is true for the case where, 222,fr R RR 2 r R will never be larger than2 f R and as a result, the sampling distribution of R2 is truncated at zero. Therefore, if the difference between 22and f r R Ris not significant, the approximation will be inappropriate regardless of sample size (Graf & Alf, 1999). Research directed at understand ing the distribution of R2, with the goal of deve loping more appropriate methods of approximation, is needed. The poor coverage of the asym ptotic confidence interval de scribed in this study suggests that developing an alternative method fo r constructing confidence intervals for 2 that has adequate probability coverage for practical sample sizes is an important goal. The effectiveness of asymptotically distribution-free and nonpara metric methods employing the bootstrap for use in constructing a confidence interval for 2 should be explored. It may be possible to obtain much more accurate confidence intervals than were demonstrated in this study, with smaller, more realistic sample sizes, using bootstrap methods. These studies will be demanding given

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129 the computational demands of simulations employi ng resampling techniques, the wide variety of nonnormal distributions that coul d be studied, and the number of potential combinations that could be generated for the pr edictors and the residuals. In addition, when one set of va riables is not a subset of the other, there are no small sample procedures to test the significan ce of the difference in correlations Thus, when we are interested in whether a set of predictors performs equa lly well in two populations, i.e. the difference between two independent samples,22 ab R R asymptotic confidence inte rval methods are the only procedures available (Graf & Alf, 1999, p. 119). For the comparison of squared multiple correlations from independent samples, Algina and Keselman (1999) found that asymptotic confidence intervals were inaccurate when the populat ion multiple correlations were zero or very small and in some conditions, were inaccurate when sample sizes and coefficients were unequal. The procedure worked reasonably well w ith equal sample sizes as small as n = 40 and equal population multiple correlations that were suffi ciently different from zero. When multiple correlation coefficients and sample sizes were un equal, the sample sizes required for control of coverage probabilities ranged from 40 to 960 fo r the smaller sample. These results, however, were obtained by simulating multivariate normal data. Since the present study showed unequivocally that asymptotic confidence interval procedures are inaccurate with nonnormal data when the two multiple correlation coefficients comp ared are estimated from the same sample, an important next step is to evaluate coverage pr obability for the comparison of multiple correlation coefficients from independent samples under conditions of nonnormality. This will be a huge undertaking since the range of nonnormal distributions is cons iderable. For independent samples, this problem is compounded by the nece ssity of manipulating not only the degree of nonnormality in the predictor and error distributi ons, but simulating a broa d range of conditions

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130 for two populations instead of one. The number of possible combinations is extraordinary. Further complexity is introduced in that the ma nipulation of sample size must include conditions in which the two samples are equal and unequal. Conclusion The purpose of this dissertation was to eval uate the accuracy of confidence intervals around an effect size measure in multiple regression analysis ( R2), based on an asymptotic approach to the problem as out lined by Hedges and Olkin (198 1), Olkin and Finn (1995), and Graf and Alf (1999) when the distributional assu mption of multivariate normality does not hold. Algina and Moulder (2001) found, and this study confirms, that even with multivariate normal data asymptotic confidence intervals were generally inaccurate except when sample sizes were large, i.e. n 600. There are many considerations that influence sample size decisions, including power and accuracy. With the current emphasis on reporting effect sizes, it is recommended that researchers plan studies with sufficient sample size so that effect sizes are estimated with adequate accuracy and hypotheses are tested with sufficient power. A researcher interested in setting confidence intervals for 2 to estimate the importance of individual variables to the regression will find it challenging to design a study with a large enough sample to ensure accuracy even in the unlikely event that multivariate normal data is anticipated. It might be tempting for researchers to c ontinue to use the asymptotic method for constructing confidence intervals in situations wh ere there is the expectation that the data is approximately multivariate normal. This s hould be discouraged. Although it is highly recommended that one should always carefully in spect ones data, there are very few tests for examining multivariate normality. Graphical met hods are not reliable for evaluating the degree of nonnormality present in multivariate data The graphical test, similar to the Q-Q plot

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131 discussed for the univariate case is a plot of the ordered squa red Mahalanobis distance against the 2 distribution with k degrees of freedom. The analytical tests simply assess the multivariate measures of skewness and kurtosis and the distribution of these test statistics is not known. In addition, indices of skew and kurtosis can be dece ptive because these estimates are likely to have very large standard errors unless the sample si ze is very large (Algina, Keselman, & Penfield, 2005). Our conclusion is that use of Alf and Grafs method for constructing confidence intervals for the squared semipartial correlation coefficien t should be abandoned. Rather, we should turn our efforts toward the search for a confidence interval that has good coverage probability for sample sizes typical of research in the behavioral and social sciences, over a wide range of distributions and values for2 r 2 f and k

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132Table 4-1. Coverage Proba bility as a Function of n Selected Values for 22and ,r and Distribution for the Predictors. Predictor Distribution 10020030040050060070080090010001250150017502000 Normal .00.05.864.904.916 .927.932.932.937.938.938.939 .940.944.945.944 .30.15.913 .932.937 .940.941.945.944.946.944.945.948.948.949.948 .60.30.915 .932.937 .941 .939 .945.945.947.946.947.947.950.948.949 Pseudot (10) .00.05.866.899.915.922 .926.929.931.934.934.934.937 .940 .939 .941 .30.15.900.919 .926.928.931.931.932.933.934.934.934.934.935.936 .60.30.882.900.906.908.911.912.913.914.915.914.917.915.917.918 Pseudo2(10) .00.05.865.900.914.921 .926.929.932.934.934.935.938.938 .940.941 .30.15.899.918.924 .927.929.931.931.932.931.933.934.934.934.933 .60.30.875.896.902.905.905.908.909.909.909.911.911.911.911.912 Pseudo2(4) .00.05.862.897.911.919.923 .926.928.930.934.933.934.936.937.938 .30.15.889.904.912.915.916.917.919.919.920.921.921.921.920.920 .60.30.829.845.851.853.855.857.858.858.859.858.858.858.860.860 Pseudo-exponential .00.05.854.893.907.913.918.921.924 .925.927.928.930.931.932.933 .30.15.862.882.888.893.893.894.896.896.896.899.899.899.900.900 .60.30.761773.776.777.778.777.779.779.779.780.781.778.780.782Sample Size ( n ) 2 r2 Note : Bold results are estimated coverage probabilities between .9 4 and .96; italicized results are estimated coverage probabiliti es between .925 and .975.

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1330500100015002000 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 Coverage ProbabilitySample Size X Multivariate Normal: X Pseudot (10): r 2 = .00, 2 = .05 r 2 = .00, 2 = .05 r 2 = .30, 2 = .15 r 2 = .30, 2 = .15 r 2 = .60, 2 = .30 r 2 = .60, 2 = .30 0500100015002000 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 X Multivariate Normal: X Pseudo-(10): r 2 = .00, 2 = .05 r 2 = .00, 2 = .05 r 2 = .30, 2 = .15 r 2 = .30, 2 = .15 r 2 = .60, 2 = .30 r 2 = .60, 2 = .30Coverage ProbabilitySample Size A B Figure 4-1. Coverage probability as a function of sample size and several combinations of 22and r for predictors sampled from a normal distribution and (A) pseudot10; (B) pseudo-22 104; (C) pseudo-; and (D) pseudo-exponential distributions.

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134 0500100015002000 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 Figure 4-1. Continued X Multivariate Normal: X Pseudo-(4): r 2 = .00, 2 = .05 r 2 = .00, 2 = .05 r 2 = .30, 2 = .15 r 2 = .30, 2 = .15 r 2 = .60, 2 = .30 r 2 = .60, 2 = .30Coverage ProbabilitySample SizeC 0500100015002000 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 X Multivariate Normal: X Pseudo-exponential: r 2 = .00, 2 = .05 r 2 = .00, 2 = .05 r 2 = .30, 2 = .15 r 2 = .30, 2 = .15 r 2 = .60, 2 = .30 r 2 = .60, 2 = .30Coverage ProbabilitySample SizeD

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135 APPENDIX A PROGRAM FOR COMPUTING MARDIAS MU LTIVARIATE MEASURES OF SKEWNESS AND KURTOSIS IN SAS /*This program was used to compute values for Mardia's multivariate indices of skewness and kurtosis for the four nonnormal distributions investigated in this study.*/ data ; g= .00 ; h= .058 ; *g=.301; *h=-.017; *g=.502; *h=-.048; *g=.760; *h=-.098; if g^= 0 then do ; popmu=(exp(g** 2 /( 2 *( 1 -h)))1 )/(g*(sqrt( 1 -h))); num1=exp( 2 *g** 2 /( 1 2 *h)); num2= 2 *exp(g** 2 /( 2 *( 1 2 *h))); den=g** 2 *(sqrt( 1 2 *h)); popvar=(1 )*popmu** 2 + (num1-num2+ 1 )/den;; end ; if g= 0 then do ; popmu= 0 ; popvar= 1 /(( 1 2 *h)**( 3 / 2 )); end ; do i = 1 to 1000000 ; tempx1=rannor( 0 ); tempy1=rannor( 0 ); tempx2=rannor( 0 ); tempy2=rannor( 0 ); tempx3=rannor( 0 ); tempy3=rannor( 0 ); tempx4=rannor( 0 ); tempy4=rannor( 0 ); tempx5=rannor( 0 ); tempy5=rannor( 0 ); tempx6=rannor( 0 ); tempy6=rannor( 0 ); tempx7=rannor( 0 ); tempy7=rannor( 0 ); tempx8=rannor( 0 ); tempy8=rannor( 0 ); tempx9=rannor( 0 ); tempy9=rannor( 0 ); tempx10=rannor( 0 ); tempy10=rannor( 0 );

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136 if g= 0 then do ; zx1=(tempx1*exp(h*tempx1** 2 / 2 )-popmu)/sqrt(popvar); zy1=(tempy1*exp(h*tempy1** 2 / 2 )-popmu)/sqrt(popvar); zx2=(tempx2*exp(h*tempx2** 2 / 2 )-popmu)/sqrt(popvar); zy2=(tempy2*exp(h*tempy2** 2 / 2 )-popmu)/sqrt(popvar); zx3=(tempx3*exp(h*tempx3** 2 / 2 )-popmu)/sqrt(popvar); zy3=(tempy3*exp(h*tempy3** 2 / 2 )-popmu)/sqrt(popvar); zx4=(tempx4*exp(h*tempx4** 2 / 2 )-popmu)/sqrt(popvar); zy4=(tempy4*exp(h*tempy4** 2 / 2 )-popmu)/sqrt(popvar); zx5=(tempx5*exp(h*tempx5** 2 / 2 )-popmu)/sqrt(popvar); zy5=(tempy5*exp(h*tempy5** 2 / 2 )-popmu)/sqrt(popvar); zx6=(tempx6*exp(h*tempx6** 2 / 2 )-popmu)/sqrt(popvar); zy6=(tempy6*exp(h*tempy6** 2 / 2 )-popmu)/sqrt(popvar); zx7=(tempx7*exp(h*tempx7** 2 / 2 )-popmu)/sqrt(popvar); zy7=(tempy7*exp(h*tempy7** 2 / 2 )-popmu)/sqrt(popvar); zx8=(tempx8*exp(h*tempx8** 2 / 2 )-popmu)/sqrt(popvar); zy8=(tempy8*exp(h*tempy8** 2 / 2 )-popmu)/sqrt(popvar); zx9=(tempx9*exp(h*tempx9** 2 / 2 )-popmu)/sqrt(popvar); zy9=(tempy9*exp(h*tempy9** 2 / 2 )-popmu)/sqrt(popvar); zx10=(tempx10*exp(h*tempx10** 2 / 2 )-popmu)/sqrt(popvar); zy10=(tempy10*exp(h*tempy10** 2 / 2 )-popmu)/sqrt(popvar); end ; if g^= 0 then do ; zx1=(( 1 /g)*(exp(g*tempx1)1 )*exp(h*tempx1** 2 / 2 )-popmu)/sqrt(popvar); zy1=(( 1 /g)*(exp(g*tempy1)1 )*exp(h*tempy1** 2 / 2 )-popmu)/sqrt(popvar); zx2=(( 1 /g)*(exp(g*tempx2)1 )*exp(h*tempx2** 2 / 2 )-popmu)/sqrt(popvar); zy2=(( 1 /g)*(exp(g*tempy2)1 )*exp(h*tempy2** 2 / 2 )-popmu)/sqrt(popvar); zx3=(( 1 /g)*(exp(g*tempx3)1 )*exp(h*tempx3** 2 / 2 )-popmu)/sqrt(popvar); zy3=(( 1 /g)*(exp(g*tempy3)1 )*exp(h*tempy3** 2 / 2 )-popmu)/sqrt(popvar); zx4=(( 1 /g)*(exp(g*tempx4)1 )*exp(h*tempx4** 2 / 2 )-popmu)/sqrt(popvar); zy4=(( 1 /g)*(exp(g*tempy4)1 )*exp(h*tempy4** 2 / 2 )-popmu)/sqrt(popvar); zx5=(( 1 /g)*(exp(g*tempx5)1 )*exp(h*tempx5** 2 / 2 )-popmu)/sqrt(popvar); zy5=(( 1 /g)*(exp(g*tempy5)1 )*exp(h*tempy5** 2 / 2 )-popmu)/sqrt(popvar); zx6=(( 1 /g)*(exp(g*tempx6)1 )*exp(h*tempx6** 2 / 2 )-popmu)/sqrt(popvar); zy6=(( 1 /g)*(exp(g*tempy6)1 )*exp(h*tempy6** 2 / 2 )-popmu)/sqrt(popvar); zx7=(( 1 /g)*(exp(g*tempx7)1 )*exp(h*tempx7** 2 / 2 )-popmu)/sqrt(popvar); zy7=(( 1 /g)*(exp(g*tempy7)1 )*exp(h*tempy7** 2 / 2 )-popmu)/sqrt(popvar); zx8=(( 1 /g)*(exp(g*tempx8)1 )*exp(h*tempx8** 2 / 2 )-popmu)/sqrt(popvar); zy8=(( 1 /g)*(exp(g*tempy8)1 )*exp(h*tempy8** 2 / 2 )-popmu)/sqrt(popvar); zx9=(( 1 /g)*(exp(g*tempx9)1 )*exp(h*tempx9** 2 / 2 )-popmu)/sqrt(popvar); zy9=(( 1 /g)*(exp(g*tempy9)1 )*exp(h*tempy9** 2 / 2 )-popmu)/sqrt(popvar); zx10=(( 1 /g)*(exp(g*tempx10)1 )*exp(h*tempx10** 2 / 2 )-popmu)/sqrt(popvar); zy10=(( 1 /g)*(exp(g*tempy10)1 )*exp(h*tempy10** 2 / 2 )-popmu)/sqrt(popvar); end ; b12=(zx1*zy1+zx2*zy2)** 3 ; b22=(zx1** 2 +zx2** 2 )** 2 ; b14=(zx1*zy1+zx2*zy2+zx3*zy3+zx4*zy4)** 3 ; b24=(zx1** 2 +zx2** 2 +zx3** 2 +zx4** 2 )** 2 ; b16=(zx1*zy1+zx2*zy2+zx3*zy3+zx4*zy4+zx5*zy5+zx6*zy6)** 3 ; b26=(zx1** 2 +zx2** 2 +zx3** 2 +zx4** 2 +zx5** 2 +zx6** 2 )** 2 ; b18=(zx1*zy1+zx2*zy2+zx3*zy3+zx4*zy4+zx5*zy5+zx6*zy6+zx7*zy7+zx8*zy8)** 3 ; b28=(zx1** 2 +zx2** 2 +zx3** 2 +zx4** 2 +zx5** 2 +zx6** 2 +zx7** 2 +zx8** 2 )** 2 ; b110=(zx1*zy1+zx2*zy2+zx3*zy3+zx4*zy4+zx5*zy5+zx6*zy6+zx7*zy7 +zx8*zy8+zx9*zy9+zx10*zy10)** 3 ;

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137 b210=(zx1** 2 +zx2** 2 +zx3** 2 +zx4** 2 +zx5** 2 +zx6** 2 +zx7** 2 +zx8** 2 +zx9** 2 +zx10** 2 )** 2 ; uniskew=(zx1*zx2)** 3 ; unikurt=zx1** 4 ; output ; end ; keep b12 b22 b14 b24 b16 b26 b18 b28 b110 b210 uniskew unikurt; proc means ; var b12 b22 b14 b24 b16 b26 b18 b28 b110 b210 uniskew unikurt; run ; quit ;

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138 APPENDIX B DATA SIMULATION SAS PROGRAM /*This program was used to simulate data for the conditions under study, construct a 95% confidence interval for the squared semipartial correlation according to the formulas presented by Alf and Graf, and compute the coverage probability.*/ proc iml; filename output 'C:\Dissertation\nx1nne4.dat'; start1=0; reps = 10000; n=nrow(x); p=ncol(x); do i=1 to 5; /*Predictor distributions*/ if i=1 then do; /*Normal distribution*/ g=0; h=0; end; if i=2 then do; /*t-distribution with 10 df*/ g = 0; h = .058; end; if i=3 then do; /*Chi-sq with 10 df*/ g=.301; h=-.017; end; if i=4 then do; /*Chi-sq with 4 df*/ g=.502; h=-.048; end; if i=5 then do; /*Exponential distribution*/ g =.760; h=-.098; end; do j=1 to 5; /*Distributions for errors*/ if j=1 then do; /*Normal distribution*/ ge=0; he=0; end; if j=2 then do; /*t-distribution with 10 df*/ ge = 0; he = .058; end; if j=3 then do; /*Chi-sq with 10 df*/ ge=.301; he=-.017; end; if j=4 then do; /*Chi-sq with 4 df*/ ge=.502; he=-.048; end; if j=5 then do; /*Exponential*/ ge =.760;

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139 he=-.098; end; do n = 100 to 1000 by 100; /*Sample size*/ do k = 2 to 10 by 2; /*Predictors*/ do rho2r = .00 to .60 by .10; /*R-sq for reduced model*/ do rho2f = rho2r to (rho2r + .30) by .05; /*R-sq for full model*/ do rep = 1 to reps; /*Replications*/ start1=start1+1; counter1=0; counter2=0; counter3=0; rhor = root(rho2r); rhof = root(rho2f); rho2inc = rho2f-rho2r; /*Calculate vector of regression coefficients transform to uncorrelated Predictors*/ B=j(k,1,0); B[k-1,1]=rhor; B[k,1]=rho2inc##.5; /*Generates a matrix of predictor variables with mean=0 and variance = 1*/ z=rannor(repeat(0,n,k)); if g^=0 then do; /*uses g and h generator to transform x*/ x=((exp(g#z)-j(n,p,1))/g)#exp(h#z##2/2); popmu=(exp(g##2/(2#(1-h)))-1)/(g#(sqrt(1-h))); num1=exp(2#g##2/(1-2#h)); num2=2#exp(g##2/(2#(1-2#h))); den=g##2#(sqrt(1-2#h)); popvar=(-1)#popmu##2+(num1-num2+1)/den; xx=(x-popmu)/popvar##.5; end; if g=0 then do; popmu=0; popvar=1/((1-2#h)##(3/2)); x=z#exp(h#z##2/2); xx=(x-popmu)/popvar##.5; end; /*Generate a vector of errors with mean = 0 and variance = 1-rho2f*/ temp=j(n,1,0); ze = rannor(temp); if ge^=0 then do; e = ((exp(ge#ze)-j(n,1,1))/ge)#exp(he#ze##2/2); popmue =(exp(ge##2/(2#(1-he)))-1)/(ge#(sqrt (1-he))); num1e=exp(2#ge##2/(1-2#he)); num2e=2#exp(ge##2/(2#(1-2#he))); dene=ge##2#(sqrt(1-2#he)); popvare=(-1)#popmue##2+(num1e-num2e+1)/dene; ee = ((e-popmue)/popvare##.5)#((1-rho2f)##.5); end;

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140 if ge=0 then do; popmue=0; popvare=1/((1-2#he)##(3/2)); e=ze#exp(he#ze##2/2); ee = ((e-popmue)/popvare##.5)#((1-rho2f)##.5); end; /*Calculate y according to the model*/ y = (xx*B) + ee; YX=y||xx; /*YX is the matrix with all variables*/ sum=YX[+,]; /*Calculate covariance and correlation matrices based on YX*/ YXPYX=YX`*YX-SUM`*SUM/N; S=DIAG(1/SQRT(VECDIAG(YXPYX))); R=S*YXPYX*S; R2 = R[2:p+1,2:p+1]; /*Calculate the remaining R matrices*/ R3 = R[1:k,1:k]; R4 = R[k:p,2:k]; R2F=1-((det(R))/(det(R2))); R2R=1-((det(R3))/(det(R4))); R2INC=R2F-R2R; RR = R2R##.5; RF = R2F##.5; RFR = RR/RF; if RF = 0 then do; print RF; RFR=0; end; /*Alf and Graf variance formula for case 2 where one set of predictors is a subset of another*/ V=(4#R2F#(1-R2F)##2/n) + (4#R2R#(1-R2R)##2/n) (8#RF#RR#(.5#(2#RFR-(RF#RR))*(1-R2F-R2R-(RFR##2)) + (RFR##3))/n); if V < 0 then do; print R2F R2R; V = 0; end; LCL=R2INC-(1.96*(V##.5)); UCL=R2INC+(1.96*(V##.5)); if LCL<=rho2inc & UCL>=rho2inc then counter1=counter1+1; if LCL>rho2inc then counter2=counter2+1; if UCL
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141 end; *end for rho2r; end; *end for k; end; *end for n; end; *for k; end; *for i; closefile output; quit; /*Calculate coverage probability*/ data ; infile 'C:\Dissertation\nx1nne4.dat' ; input n 4.0 + 2 k 2.0 + 2 rho2r 4.2 + 2 rho2f 4.2 + 2 rho2inc 4.2 + 2 R2R 6.4 + 2 R2F 6.4 + 2 R2INC 6.4 + 2 LCL 8.6 + 2 UCL 8.6 + 2 counter1 5.0 + 2 counter2 5.0 + 2 counter3; proc sort ; by k n rho2r rho2inc; proc means noprint n mean std maxdec = 4 ; by k n rho2r rho2inc; var counter1 counter2 counter3; output mean =covprob below above out = res14; proc print noobs data = res14; var k n rho2r rho2inc covprob below above; run ; /*Calculate mean estimated asympto tic variance and sampling variance of R2*/ data varratio; infile 'c:\Dissertation\nx1nne4.dat' ; input n k rho2r rho2f rho2inc R2R R2F R2INC; RF=sqrt(R2F); RR=sqrt(R2R); RFR=RR/RF; if RF = 0 then RFR = 0 ; EAV=( 4 *R2F*( 1 -R2F)** 2 /N) + ( 4 *R2R*( 1 -R2R)** 2 /N) ( 8 *RF*RR*( .5 *( 2 *RFR-(RF*RR))*( 1 -R2F-R2R-(RFR** 2 )) + (RFR** 3 ))/N); proc means noprint n mean var maxdec = 4 ; by n k rho2r rho2inc; var R2INC EAV; output mean =MR2INC MEAV var =VR2INC VEAV out =res14; run ; data varratio2; set res14; ratio = MEAV/VR2INC; proc print noobs data =varratio2; var n k rho2r rho2inc MR2INC VR2INC MEAV RATIO; run ;

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142 REFERENCES Alf, E. F., Jr. & Graf, R. G. (1999). Asymptotic confidence limits for the difference between two squared multiple correlations: A simplified approach. Psychological Methods, 4 (1), 70-75. Algina, J., & Keselman, H. J. (1999). Compar ing squared multiple correlation coefficients: Examination of a confidence interval and a test of significance. Psychological Methods, 4 (1), 76-83. Algina, J., & Olejnik, S. (2000). Determining sa mple size for accurate estimation of the squared multiple correlation. Multivariate Behavioral Research 35 (1), 119-137. Algina, J. & Moulder, B. C. (2001). Sample sizes for confidence intervals on the increase in the squared multiple correlation coefficient. Educational and Psychological Measurement 61 (4), 633-649. Algina, J., Moulder, B. C., & Moser, B. K. (2002). Sample size requirements for accurate estimation of squared semi-partial correlation coefficients. Multivariate Behavioral Research 37 (1), 37-57. Algina, J., Keselman, H. J., & Penfield, R. D. (2005). An alternative to Cohens standardized mean difference effect size: A robust para meter and confidence interval in the two independent groups case. Psychological Methods 10 (3), 317-328. American Educational Research Association. (2006). Standards for reporting on empirical social science research in AERA publications. Educational Researcher 35 (6), 33-40. American Psychological Association. (2001). Publication Manual of the American Psychological Association (5th Ed.). Washington, DC: Author. Bradley, J. V. (1978). Robustness? British Journal of Mathematical and Statistical Psychology 31 144-152. Bradley, J. V. (1980). Nonrobustness in classi cal tests on means and variances: A large-scale sampling study. Bulletin of the Psychonomic Society 15 275-278. Breckler, S. J. (1990). Application of covari ance structure modeling in psychology: Cause for concern? Psychological Bulletin, 107 260-273. Browne, M. W. (1969). Precision of prediction (Research Bulletin No. 69-69). Princeton, NJ: Educational Testing Service. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum. Cohen, J. (1990). Things I have learned (so far). American Psychologist 45 1304-1312.

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143 Cohen, J. (1994). The earth is round ( p < .05). American Psychologist 49 997-1003. Cumming, G. & Finch, S. ( 2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement 61(4), 532-574. Ellis, N. (2000). Editors statement. Language Learning 50 xi-xiii. Finch, S., Thomason, N., & Cumming, G. (2002) Past and future American psychological association guidelines fo r statistical practice. Theory & Psychology 12 (6), 825-853. Gnanadesikan, R. (1977). Methods for statistical data analysis of multivariate observations New York: John Wiley & Sons. Graf, R. G. & Alf, E. F., Jr. (1999). Asymptotic confidence limits for par tial and squared partial correlations. Applied Psychological Methods, 23 116-119. Harris, K. (2003). Journal of Educational Psychology: Instruc tions for authors. Journal of Educational Psychology 95 201. Hedges, L. V. & Olkin, I. (1981). The asympt otic distribution of commonality components. Psychometrika 46 331-336. Heldref Foundation (1997). Guid elines for contributors. Journal of Experimental Education 65 95-96. Hoaglin, D. C. (1985). Using quantiles to study shape. In Exploring data tables, trends, and shapes. p.417-460. D. C. Hoaglin, F. Mosteller & J. W. Tukey (Eds.), New York: John Wiley & Sons. Hresko, W. (2000). Editorial policy. Journal of Learning Disabilities 33 214-215. Jaccard, J. & Wan, C. K. (1995). Measurement e rror in the analysis of interaction effect between continuous predictors using multiple re gression: Multiple indicator and structural equation approaches. Psychological Bulletin 117 348-357. Keselman, H.J., Huberty, C.J., Lix, L.M., Olej nik, S., Cribbie, R.A., Donahue, B., Kowalchuk, R.K., Lowman, L.L., Petoskey, M.D., Keselman, J. C., & Levin, J.R. Statistical practices of educational researchers: An analys is of their ANOVA, MANOVA, and ANCOVA analyses. Review of Educational Research 68 350-386. Kirk, R. E. (1996). Practi cal significance: A concep t whose time has come. Educational and Psychological Measurement 56 746-759. Lix, L. M. & Keselman, H. J. (1998). To trim or not to trim: tests of location equality under heteroscedastcity and nonnormality. Educational and Psychological Measurement 58 (3), 409-430.

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144 McLean, J. & Kaufman, A. (2000). Editori al: Statistical sign ificance testing and Research in the Schools Research in the Schools 7 (2), 1-2. Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika 57 (3), 519-530. Martinez, J. & Iglewicz, B. (1984) Some properties of the Tukey g and h family of distributions. Communications in Statistics: Theory and Methods 13 (3), 353-369. Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin 105 156-166. Mulaik, S. A. (1972). The foundations of factor analysis New York: McGraw-Hill. Murphy, K. R. (1997). Editorial. Journal of Applied Psychology 82 3-5. Myers, J. L., & Well, A. D. (2003). Research design and statistical analysis (2nd Ed.). Mahwah, NJ: Lawrence Erlbaum Associates. Olejnik, S. & Algina, J. (2002). Measures of effect size fo r comparative studies: Applications, interpretations, and limitations. Contemporary Educational Psychology 25 241-286. Olkin, I. & Finn, J. D. (1995). Correlations redux. Psychological Bulletin 118 155-164. Rencher, A. C. (1995). Methods of multivariate analysis New York: John Wiley & Sons. Royer, J. M. (2000). A policy on reporting of effect sizes. Contemporary Educational Psychology 25 239. SAS Institute. (1999). SAS/IML users guide, Version 8. Cary, NC: Author. Smithson, M. (2001). Correct confidence interv als for various regression effect sizes and parameters: The importance of noncentral distributions in computing intervals. Educational and Psychological Measurement 61 (4), 605-632. Smithson, M. (2003). Confidence intervals Thousand Oaks, CA: Sage. Snyder, P. (2000). Guidelines for reporting re sults of group quantitative investigations. Journal of Early Intervention 23 145-150. Steiger, J. H. & Fouladi, R. T. (1997). Noncen trality interval estimati on and the evaluation of statistical models, In L.L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 222-257). Hillsdale, NJ: Lawrence Erlbaum. Stuart, A., Ord, J. K., & Arnold, S. (1999). The advanced theory of statistics (6th ed., Vol. 2a). New York: Oxford University Press.

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145 Thompson, B. (1994). Guidelines for authors. Educational and Psyc hological Measurement 54 837-847. Thompson, B. (1996). AERA editori al policies regarding statistic al significance testing: Three suggested reforms. Educational Researcher 25 (2), 26-30. Thompson, B. (1999). Journal editori al policies regarding st atistical significance tests: Heat is to fire as p is to importance. Educational Psychology Review 11 157-169. Thompson, B. (2002). What future quantitativ e social science rese arch could look like: confidence intervals for effect sizes. Educational Researcher 31 (3), 24-31. Tukey, J. W. (1960). A survey of sampling from contaminated normal distributions. In I. Olkin et al. (Eds.), Contributions in Proba bility and Statistics. Stanford, CA: Stanford University Press. Vacha-Haase, T., Nilsson, J. E ., Rentz, D. R., Lance, T. S ., & Thompson, B. (2000). Reporting practices and APA editorial policies regardi ng statistical significance and effect size. Theory and Psychology 10 413-425. Vacha-Haase, T. & Thompson, B. (2004). How to estimate and interpret various effect sizes. Journal of Counseling Psychology 51 (4), 73-81. Wilcox, R. R. (1998). How many di scoveries have been lost by ignoring modern statistical methods? American Psychologist 53 (3), 300-314. Wilkinson, L. & APA Task Force on Statisti cal Inference. (1999). Statistical methods in psychology journals: Guidel ines and explanations. American Psychologist 54 (8), 594-604.

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146 BIOGRAPHICAL SKETCH Lou Ann Mazula Cooper, the daughter of Willia m and Priscilla Mazula, was born in Cedar Falls, Iowa. She graduated from Parkersburg High School and majored in psychology at the University of Iowa where she earned a Bachelor of Science degree with h onors in 1977. While a student at the University of Iowa, she met and married Brian Cooper. When her husband accepted a postdoctoral fellowship at the Univ ersity of Florida in 1979, she moved to Gainesville, Florida. She worked for the Flor ida Department of Labor coordinating youth and employment and training programs. She was later employed at Santa Fe Community College where she held several positions: work evaluator, career counselor, and adjunct faculty. In 1992, she earned a Master of Education degree in educational psychology from the University of Florida in 1992. She developed and wrote su ccessful grant proposals to fund educational opportunity programs for local hi gh school students. She was th e director of SFCCs Upward Bound Program from 1995 to 2000. In 2000, she be gan her doctoral studies in research and evaluation methods in the Department of Educati onal Psychology at the Univ ersity of Florida. She earned a Master of Arts in Education in 2004 and will graduate with the Ph.D. degree in May 2007.


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Permanent Link: http://ufdc.ufl.edu/UFE0019420/00001

Material Information

Title: The Impact of NonNormality on the Asymptotic Confidence Interval for an Effect Size Measure in Multiple Regression
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0019420:00001

Permanent Link: http://ufdc.ufl.edu/UFE0019420/00001

Material Information

Title: The Impact of NonNormality on the Asymptotic Confidence Interval for an Effect Size Measure in Multiple Regression
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0019420:00001


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THE IMPACT OF NONNORMALITY ON THE ASYMPTOTIC CONFIDENCE INTERVAL
FOR AN EFFECT SIZE MEASURE IN MULTIPLE REGRESSION



















By

LOU ANN MAZULA COOPER


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

UNIVERSITY OF FLORIDA

2007

































O 2007 Lou Ann Mazula Cooper









ACKNOWLEDGMENTS

I would like to take this opportunity to thank the co-chairs of my dissertation committee,

Dr. David Miller and Dr. James Algina. They are both exceptional teachers and without their

guidance, support, and patience, this work would not have been possible. Dr. Miller, through his

flexibility, approachable manner, and breadth of knowledge has been an invaluable resource

throughout the course of my graduate studies. Dr. Algina is perhaps the most generous teacher I

have ever known and his door was always open to me. His passion for research and his

incredible work ethic had a profound influence on me.

I would also like to thank the other members of my committee for their help and

encouragement. Special thanks go to Dr. Richard Davidson. I will always be grateful for the

opportunity he provided to apply and expand my skills to experimental design, data analysis, and

psychometric issues in medical education research. Thank you for giving me the most rewarding

and stimulating j ob I have ever had. Dr. Walter Leite, although not there at the beginning, has

provided a valuable sounding block and I look forward to our future collaborations.

I would also like to express my gratitude to my family for their love and encouragement

during my mid-life career change. I know it has not always been easy to live with me. To my

daughters, Abigail and Amanda, my hope is that I have provided a good example for you in

pursuing my life-long love of learning. Finally, and most especially to Brian, my husband and

best friend, whose love and unwavering belief in me made achieving this goal possible.











TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............3.....


LIST OF TABLES ........._..... ...............6.._.._ ......


LIST OF FIGURES .............. ...............8.....


AB S TRAC T ........._. ............ ..............._ 10...


CHAPTER


1 INTRODUCTION ................. ...............12.......... ......


Effect Sizes and Confidence Intervals in Multiple Regression Analysis .............. ................14
Asymptotic Confidence Intervals for Correlations ................. ...............15........... ...
The Impact of Nonnormality on Statistical Estimates ................. ............... ......... ...24
Statement of the Problem ................. ...............26................
Purpose of the Study ................. ...............26.......... .....

2 M ETHODS .............. ...............28....


Study Design............... ...............28.
Number of predictors ................. ...............28........... ....
Squared multiple correlations ................. ...............28........... ....
Sample size ................. ...............29.................
D istributions ................. ......... ...... .... ... .. .. ..... .........2

Background and Theoretical Justification for the Simulation Method............... .................3
Data Simulation .............. ...............37....
Data Analysis............... ...............39

3 RE SULT S .............. ...............47....


Replication of Results for Multivariate Normal Data ................. ............... ......... ...47
Simulation Proper ................. .. ........... ........... .............4
Analysis of Variance and Mean Square Components ................ ..............................54
The Influence of Nonnormality on Coverage Probability ................. .......... ...............58
Nonnormal predictors ................. ...............58.................
Nonnormal error distribution............... ... .. .......... ........5
The Impact of Squared Multiple Correlations on Coverage Probability ............... .... ........._..59
The Impact of Sample Size on Coverage Probability............... ..............6
Probability Above and Below the Confidence Interval .............. .....___ .............. .65
The Relationship between Estimated Asymptotic Variance, Empirical Sampling
Variance of AR2, and Coverage Probability ................. ...............66...............












4 DI SCUS SSION ................. ................. 124........ ....


Limitations ................. ...............126................
Further Research ................. ...............128................
Conclusion ................ ...............130................


APPENDIX


A PROGRAM FOR COMPUTING MARDIA' S MULTIVARIATE MEASURES OF
SKEWNESS AND KURTOSIS INT SAS .............. ...............135....


B DATA SIMULATION PROGRAM INT SAS ................ ...............138..............


LIST OF REFERENCES ................ ...............142................


BIOGRAPHICAL SKETCH ................. ...............146......... ......










LIST OF TABLES
Table page

2-1 Study Design ................. ...............41........... ....

2-2 Mardia' s Multivariate Skewness, bl~k, for the Nonnormal Distributions ................... ........42

2-3 Mardia' s Multivariate Kurtosis, b2,k, for the Nonnormal Distributions. ................... .........43

3-1 Replication of Algina and Moulder' s Results for Multivariate Data and Two
Predictors. ............. ...............72.....

3-2 Replication of Algina and Moulder' s Results for Multivariate Data and Six
Predictors .............. ...............73....

3-3 Replication of Algina and Moulder' s Results for Multivariate Data and Ten
Predictors. ............. ...............74.....

3-4 Empirical Coverage Probabilities for Normal Predictors and Normal Errors. ..................75

3-5 Empirical Coverage Probabilities for Normal Predictors and Nonnormal Errors. ............82

3-6 Empirical Coverage Probabilities for Nonnormal Predictors and Normal Errors. ...........89

3-7 Empirical Coverage Probabilities for Predictors Nonnormal and Errors Nonnormal. ......96

3-8 Descriptive Statistics for Coverage Probability by Distributional Condition. .................1 04

3-9 Analysis of Variance, Estimated Mean Square Components, and Percentage of Total ..105

3-10 Descriptive Statistics for Coverage Probability by Distribution for the Predictors......... 105

3-11 Descriptive Statistics for Coverage Probability by Distribution for the Errors. ..............106

3-12 Coverage Probability by Ap2 and the Distribution for the Predictors .............. ...............106

3-13 Coverage Probability by Ap2 and the Distribution for the Errors. ............. ..................106

3-14 Coverage Probability by p~ and the Distribution for the Predictors. .............. ...............107

3-15 CoeaePoaiiyb and the Distribution for the Errors. ............. ...................107


3-6Coverage Probability by p and Ap2 for X Distributed Multivariate Normal. ................107


3-17 Coverage Probability by p~ and Ap2 for X Distributed Pseudo-tlo(g = 0, h = .058). ......108


3-8 Coverage Probability by p and Ap2 for X Distributed Pseudo-g ................0










3-9Coverage Probability by p2 and Ap2 for X Distributed Pseudo-( .10

3-0Coverage Probability by p2 and Ap2 for X Distributed Pseudo-exponential. ..................1 09

3-21 Coverage Probability by Sample Size and Ap2 ................ ...............109.............

3 -22 Coverage Probability by Sample Size and Number of Predictors. .........._... ............... 110

3-23 Analysis of Variance, Estimated Mean Square Components, and Percentage of Total ..1 10

4-1 Coverage Probability as a Function of n, Selected Values for p2 and Ap2, and
Distribution for the Predictors. ............. ...............132....










LIST OF FIGURES


Figure page

2-1 Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = 0, h = .058 overlaid with a normal curve with Cash= 0,
ags = 1.097. ............. ...............44.....

2-2 Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = .301, h = -.017 overlaid with a normal curve with Cash= .150,
ags = 1.041. ............. ...............44.....

2-3 Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = .502, h = -.048 overlaid with a normal curve with Cas = .249,
gsh = 1.108. ............. ...............45.....

2-4 Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = .760, h = -.098 overlaid with a normal curve with Cash= .378,
gsh = 1.252 .............. ...............45....

2-5 Comparison of Mardia' s multivariate skewness for the multivariate normal
distribution to that of the distributions investigated. ................ ................. ..........46

2-6 Mardia's multivariate kurtosis for the multivariate normal distribution and the
nonnormal distributions investigated. .............. ...............46....

3-1 Mean estimated coverage probability by normality vs. nonnormality in the
predictors, normality vs. nonnormality in the errors, and sample size. ................... ........111

3-2 Empirical coverage probability as a function of distributional condition and sample
size. .............. .. ...............112......... ......

3-3 Box plots of the distributions of coverage probability estimates by distribution for
the predictors (n, = 14,700) ................. ...............113........... ...

3-4 Box plots of the distributions of coverage probability estimates by distribution for
the errors (n, = 14,700)............... ...............114

3-5 Main effect of the squared semipartial correlation coefficient Ap2, and the effect of
the interaction of Ap2 and X on coverage probability for Ap2 > 0 ................. ...............1 15

3-6 Main effect of the squared semipartial correlation coefficient Ap2, and the effect of
the interaction of Ap2 and e on coverage probability for Ap2 > 0. .........._... ................115

3-7 Effect of the interaction between the size of the squared multiple correlation in the
reduced model, pi, and the distribution for the predictors, X, on coverage
probability. ................ ...............116................









3-8 Interaction between the size of the squared multiple correlation in the reduced
model, p and the distribution for the errors, e, and its relationship to coverage
probability. p~ ................ ...............116......... ......

3-9 Effect of the p~ x Ap2 interaction on coverage probability for Ap2 > 0. ................... ....... 117

3-10 Effect of the X x p~ x Ap2 interaction on coverage probability for Ap2 > 0 ..................1 18

3-11 Interaction between sample size, n, and the population squared semipartial
correlation, Ap2, and the impact on coverage probability for Ap2 > 0. ................... .........119

3-12 Effect of the interaction between sample size, n and number of predictors, k, on
coverage probability ................. ...............119................

3-13 Ratio of mean estimated asymptotic variance to the variance in AR2 (MEAVn/Var
AR2) as a function of the distribution for the predictors, Ap2, and p, ...............120

3-14 Relationship between coverage probability and the ratio of mean estimated
asymptotic variance to the empirical sampling variance of AR2 for Ap2 > 0 ................... 121

3-15 Relationship between coverage probability and the ratio of mean estimated
asymptotic variance to the empirical sampling variance of AR2 for Ap2 > 0 for
multivariate normal data (g = 0, h = 0). ................ ...............121........... .

3-16 Relationship between coverage probability and the ratio of mean estimated
asymptotic variance to the empirical sampling variance of AR2 for Ap2 > 0 and X
distributed pseudo-tlo (g = 0, h = .058)............... ...............122.

3-17 Relationship between coverage probability and the ratio of mean estimated
asymptotic variance to the empirical sampling variance of AR2 for Ap2 > 0 and X
distributed pseudo-X, (g = .502, h = -.048) ................. ...............122............

3-18 Relationship between coverage probability and the ratio of mean estimated
asymptotic variance to the empirical sampling variance of AR2 for Ap2 > 0 and X
distributed pseudo-X! (g = .502, h = -.048)............... ...............123

3-19 Relationship between coverage probability and the ratio of mean estimated
asymptotic variance to the empirical sampling variance of AR2 for Ap2 > 0 and X
distributed pseudo-exponential (g =.760, h = -.098)............... ...............123

4-1 Coverage probability as a function of sample size and several combinations of
p~ and Ap for predictors sampled from a normal distribution and (A) pseudo-tlo; (B)
pseudVno-7 \, (C)psudo-(;, and (D) pseudo-exponential distributions. ................... .......133











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 IMPACT OF NONNORMALITY ON THE ASMPTOTIC CONFIDENCE INTERVAL
FOR AN EFFECT SIZE MEASURE IN MULTIPLE REGRESSION

By

Lou Ann Mazula Cooper

May 2007

Chair: M. David Miller
Co chair: James Algina
Major: Research and Evaluation Methodology

The increase in the squared multiple correlation coefficient, AR2, associated with an

individual predictor in a regression analysis is a measure commonly used to evaluate the

importance of that variable in a multiple regression analysis. Previous research using

multivariate normal data had shown that relatively large sample sizes are necessary for an

acceptably accurate confidence interval for this regression effect size measure.

The coverage probability that an asymptotic confidence interval contained the population

squared semipartial correlation, Ap2, was investigated by simulating data from a range of

nonnormal distributions such that (a) the predictors were nonnormal, (b) the error distribution

was nonnormal, or (c) both predictors and errors were nonnormal. Additional factors

manipulated included (a) the number of predictor variables, (b) the magnitude of the population

squared multiple correlation coefficient in the original model, pi, (c) the magnitude of the

population squared semipartial correlation, Ap2, and (d) sample size.

This study showed that when nonnormality is introduced, empirical coverage probability

was always less than the nominal confidence level, often dramatically so. The degree of









nonnormality in the predictors was the most important factor influencing poor coverage

probability. Although coverage probability increased as a function of sample size, when

nonnormality in the predictors was substantial, the confidence interval is likely to be inaccurate

no matter how large a sample size is used. With multivariate normal data, coverage probability

improved as both p~ and Ap2 increased. When predictors are sampled from a nonnormal

distribution, coverage probability tended to decrease as p~ and Ap2 increased and became even

worse as the degree of nonnormality increased. It was further demonstrated that the asymptotic

variance underestimates the sampling variance of AR2. This produces standard errors that are too

small and results in a confidence interval that is too narrow. Reliance on this confidence interval

as a measure of the strength of the effect size will lead us to underestimate the importance of an

individual predictor to the regression.









CHAPTER 1
INTRODUCTION

There is a growing consensus that the tradition of null hypothesis significance testing

(NHST) has led to over-reliance on statistical significance in evaluating research results in the

behavioral and social sciences. According to Cohen (1994), the biggest flaw in NHST is that it

does not tell us what we want to know. A statistical test evaluates the probability of the sample

results given the size of the sample assuming that the sample is drawn from a population where

the null hypothesis is exactly true. In this framework, the outcome of a significance test is a

dichotomous decision whether or not to reject the null hypothesis. As noted by Steiger and

Fouladi (1997, p. 225), "this dichotomy is inherently dissatisfying to psychologists and

educators, who frequently use the null hypothesis as a statement of no effect, and are more

interested in knowing how big an effect is than whether it is (precisely) zero." Fundamentally,

we are interested in determining how accurately the population effect has been estimated from

the sample data and whether the observed effect size has practical significance. Statistical

significance testing fails to provide the answers.

Within the behavioral and social sciences, methodological recommendations for reporting

research results have increasingly emphasized the importance of reporting confidence intervals

(Cumming & Finch, 2001; Smithson, 2001), effect sizes (Olejnik & Algina, 2002; Vacha-Hasse

& Thompson, 2004), and confidence intervals for effect sizes (Cohen, 1990; Steiger & Fouladi,

1997; Thompson, 2002) to complement the results of hypothesis testing. Among the

recommendations of the APA' s Task Force on Statistical Inference (Wilkinson & Task Force on

Statistical Inference, 1999) was a proposal to move away from routine reliance on NHST as a

primary means of analyzing data to exploring, summarizing and analyzing data using visual

representations, effect-size measures, and confidence intervals. The most recent edition of The









Publication Manual of the American Psychological Association (200 1, p. 25-26) states, "For the

reader to fully understand the importance of your findings, it is almost always necessary to

include some index of effect size or strength of relationship in your Results section... The general

principle to be followed, however, is to provide the reader not only with information about

statistical significance but also with enough information to assess the magnitude of the observed

effect or relationship." The Manual also states that failure to report an effect size is a "defect"

(p. 5).

In 1996, Thompson recommended that American Educational Research Association

(AERA) journals require that effect sizes be reported and interpreted in all studies. Ten years

later the AERA Council recommends that statistical results should include an effect size measure

as well as an indication of the uncertainty of that index of effect such as a confidence interval.

The recently adopted Standards for Reporting on Empirical Social Science Research in AERA

Publications (AERA, 2006) states that when quantitative methods are employed, "It is important

to report the results of analyses that are critical for the interpretation of findings in ways that

capture the magnitude as well as the significance of those results" (p. 37).

Editors of over 20 APA and other social science journals have published guidelines

explicitly requiring authors to report effect sizes (Ellis, 2000; Harris, 2003; Heldref Foundation,

1997; Hresko, 2000; McLean & Kaufman, 2000; Royer, 2000; Snyder, 2000; Thompson, 1994;

Vacha-Haase, Nilsson, Rentz, Lance, & Thompson, 2000) and the Editor of Journal of Applied

Psychology requires an author to provide an explanation when an effect size is not reported

(Murphy, 1997). Although this is evidence that editorial practices have evolved somewhat,

effect size reporting is unlikely to become the norm until we move from recommendation and

encouragement to requirement (Thompson, 1996; 1999).









Effect Sizes and Confidence Intervals

A confidence interval establishes a range of parameter values that are reasonably consistent

with the data observed from a sample. Because a confidence interval gives a best point estimate

of a parameter of interest and an interval about it reflecting an estimate of likely error, it contains

all the information to be found in a significance test and more (Cohen, 1994). The likely range

of the parameter values provides researchers with a better understanding of their data. If the

parameter estimated has meaningful units, a confidence interval can be used to make statistical

inferences that provide information in the same metric. According to Cumming and Finch

(2001), there are four main reasons for promoting the use of confidence intervals: (a) they are

readily interpretable, (b) are linked to familiar statistical tests, (c) can encourage replication and

meta-analytic thinking, and (d) give information about precision.

The term effect size is broadly used to refer to any statistic that provides information that

helps us judge the "practical significance" of the results of a study (Kirk, 1996). Cohen (1990)

recommends that in addition to reporting an effect size, researchers should provide confidence

intervals for effect sizes in order to gauge the possible range of values an effect size may assume.

Absent a confidence interval, it is difficult to evaluate the accuracy of the effect size estimate.

This, in turn, has implications for drawing meaningful conclusions.

Unfortunately, despite the increasing demand for researchers to do so, reporting effect

sizes and confidence intervals has yet to become commonplace in educational and psychological

journals. Vacha-Hasse, Nilsson, Rentz, Lance, and Thompson (2000) reviewed ten studies of

effect size reporting in 23 journals, and found effect size(s) to be reported in roughly 10 to 50

percent of articles, notwithstanding the encouragement to do so from the fourth edition of the

APA manual (1994). Empirical studies show that even when effect sizes are reported,

interpretation is often given short shrift (Finch et al, 2002; Keselman et al., 1998).









It is likely that the emphasis on null hypothesis significance testing in graduate courses in

statistics and research methodology has contributed to a general lack of knowledge concerning

confidence intervals. Moreover, techniques for computing confidence intervals are often

neglected in popular statistics textbooks and are not easily available in the statistical software

that is routinely employed by applied researchers in the social sciences (Smithson, 2001). Even

if these factors were not operating, researchers might be reluctant to report confidence intervals

because as Steiger and Fouladi (1997, p. 228) observe, "interval estimates are sometimes

embarrassing." Reporting confidence intervals can highlight the level of imprecision of

statistical estimates and exposes the trivial nature of many published studies. Smithson (2001,

p. 614) notes, "Almost any literature review or meta-analysis in psychology would give a very

different impression from that conveyed by NHST if we routinely 'reconstructed' CIs for

multiple R2 and related GLM parameters."

Asymptotic Confidence Intervals for Correlations

A confidence interval establishes a range of hypothetical parameter values that cannot be

ruled out given the observed sample data. The probability that the random interval includes, or

covers, the true value of the parameter is the coverage probability of the interval. When the exact

distribution of a statistic is known, the coverage is equal to the confidence level and the interval

is said to be exact. A confidence interval is exact if it can be expected to contain a parameter' s

true value 100(1 a)% of the time. Often exact intervals are not available or are difficult to

calculate, and approximate intervals are used instead.

Confidence intervals are based on the sampling distribution of a statistic. Due to the

central limit theorem, when sample size is sufficiently large, the sampling distribution of statistic

will become more symmetric and eventually appear nearly normal, even when the population

itself is not normally distributed. Methods based on asymptotic theory use approximations to the









sampling variance of a statistic. If only the asymptotic distribution of the statistic is known, we

can obtain an approximate confidence interval, which may or may not be reasonably accurate in

Einite samples. If the asymptotic confidence interval procedure is fully adequate, under repeated

random sampling under identical conditions, a 95% confidence interval would contain the true

population parameter 95% of the time. The accuracy of the approximation depends on whether

there is a lack of bias and the degree to which the sampling distribution deviates from normality.

If a statistic has no bias as an estimator of a parameter, its sampling distribution is centered at the

true value of a parameter. An unbiased confidence interval is one where the probability of

including any value other than the parameter's true value is less than or equal to 100(1 a)%.

An interval is said to be conservative if the rate of coverage is greater than 100(1 a)%, the

nominal confidence level. If the coverage probability is less than the nominal, the interval is said

to be liberal. In general, conservative intervals are preferred over liberal ones (Smithson, 2003).

Whenever a statistic based on asymptotic theory has poor finite sample properties, a confidence

interval based on that statistic has poor coverage.

Multiple regression analysis is a common statistical application frequently used to predict a

dependent variable (outcome) from two or more independent variables (predictors). The

interpretation of results would be enhanced by the reporting of confidence intervals and effect

sizes. The sample statistic, R2, which estimates the proportion of variance in the dependent

variable that is explained by the set of predictors, is commonly used to evaluate a multiple

regression model. Published research studies frequently report R2 ValUeS without any evidence

of the precision with which they have been estimated. It is unfortunate that a confidence interval

for the population parameter, p2, iS not computed by most popular statistical software packages.









Perhaps more significant, the topic is not even discussed in many applied or theoretical statistics

texts.

In addition to the amount of variance explained by a given multiple regression model,

researchers are often interested in evaluating the contribution that one variable makes to the

regression, over and above a set of other explanatory variables. The increase in R2, R2, when a

variable (4) is added to a multiple regression model is a useful measure of the strength of the

relationship between 4 and the dependent variable, Y, controlling for all other independent

variables in the model. The change in R2 that we observe by including each new 4 in the

regression equation is the squared semipartial correlation corresponding to a given regression

coefficient. Typically, whether 4 has made a statistically significant contribution to predicting Y

is tested by conducting a t- or F-test on that regression coefficient. But, the squared semipartial

correlation itself is a useful measure of effect size and as recommended by Cohen (1990) and

Thompson (2002), we should calculate a confidence interval to evaluate the precision with which

it has been estimated and the range of likely values.

Hedges and Olkin (1981) presented procedures for constructing a confidence interval for

the squared semipartial correlation based on calculating the asymptotic covariance matrix for

commonality components. Commonality analysis is a procedure by which the variance

accounted for in the criterion is partitioned into two parts, the unique part and the common part.

The unique part is attributable to the predictors individually. This is essentially the partial

contribution of each predictor to the squared multiple correlation with the criterion. The second

part is the common part, attributable to a combination of the predictors, which is the contribution

to the multiple correlation with the criterion that all of the predictors in the combination share.









Thus, commonality analysis is a way to measure the importance of variables through the use of

partial correlations.

Hedges and Olkin's results can be used to construct a confidence interval for AR2. Olkin

and Finn (1995) derived explicit expressions for asymptotic (large-sample) confidence intervals

for functions of simple, partial, and multiple correlations. Since the focus of this study is on the

squared semipartial correlation, the following discussion will be limited to Olkin and Finn's

Model A (p. 157-159). Model A is the special case for use in determining whether an additional

variable provides an improvement in predicting the criterion.

All of the procedures for comparing two sample correlation coefficients or two sample

squared correlation coefficients described by Olkin and Finn have the same general form. Let rA

and rB be the two sample correlations to be compared and pA and pB denote their corresponding

population values. The large-sample distributional form for the difference in two correlations is

[(r~-A B A B)]~ N 0,01) (1.1)

where

01 = var(rA )+ Var(rg) -2 cov(rA B) (1.2)

is the asymptotic variance of the difference of the two correlation coefficients; ol is dependent

on the population correlations (Olkin & Finn, 1995, p. 156).

When squared correlation coefficients are compared, the expressions in Equations 1.1 and

1.2 become

[(]- j -( -\ p )]_ ~~C N(, (1.3)

and

02 = var r- rj-p p ) 2cov( r ,rj). (1.4)










Olkin and Finn present the general form for the large-sample variance of functions of

correlations


o f (q, rk ',k, = B08' (1.5)

specialized to a function of three correlations, rzi, rGk, and ryk where f( ) is a function of the

correlations,
contains a set of coefficients that depend on the function of the correlations to be evaluated. The

variance of sample correlation rzi is

var(rl) = (1- p )) / It (1.6)

and the covariance of two correlations is


cor rik zy k1 7k I k + 1 T 7k ~1 T 71 ]~k
2 (1.7)



When two correlations have one variable in common, Equation 1.7 simplifies to


cov(rry, rik) = (2p~ pyk ( zy Pk zy Pk Pk k 7 77.i (1.8)


Large-sample estimates are obtained by replacing the population parameters with values

computed from sample data. Using the delta method, it can be shown that if f(r,,, rzk r7k) is a

function of the three correlations, then the vector a consists of the partial derivatives


a = d -3d (1 .9 )


In the simplest case, suppose that two variables X1 and X2 are USed to predict a third variable, Xo.

In order to determine whether X2, makes a significant contribution to the regression, we are

interested in the difference, 42 4 Here, we use a capital "R" to signify a multiple


correlation rather than a bivariate correlation, denoted by a lower case "r"'. The symbol R412









denotes the squared multiple correlation between Xo, X1 and X2, which is a function of the

correlations among the variables rol, r02, and rl2 given by


xiaZ) = Pla2 1_ (1.10)
1-r1

The squared correlation between Xo and X1 is represented by r,, Therefore, a confidence interval

for IC, -4,] can be computed using Olkin and Finn's results for comparing two squared multiple

correlation coefficients. In order to compare the population squared multiple correlations

p and z w use, the estimates@ ,2 4, and 61, the estimated variance of the difference


R(1Z) -;,where

var(R4 \) = a~a'. (1.11I)

The upper triangular of the symmetric population correlation matrix is

P=1 p, 1 p,,1 (1.12)



and the elements of the vector, a, are
2p,
az = (poiP12 902), (1.13)
I-J2
as = (zpoz P0112), (1.14)

2p
az= (p1 p + p1 p2 -DPO poPo 01902912?). (1.15)
(1 p 2)2
The variance-covariance matrix for the sample correlations is

i1 ~12 ~13233 vr ovq, ,)co:,,r
22 2 = vr~q ) cv~q r) .(1.16)










The sample correlation matrix, R, estimates P and the sample values in R can be used to compute

the elements of a.

Because the calculation of analytic derivatives becomes increasingly complicated as the

number of variables increases, Olkin and Finn illustrated their method for a multiple regression

model with no more than two predictors. Graf and Alf (1999) expanded Olkin and Finn' s

procedures to more general forms. Graf and Alf substituted numerical derivatives and offered

two BASIC programs for calculating asymptotic confidence limits on the difference between two

squared multiple correlations and the difference between two partial correlations. These

programs, REDUX-AB, to compare two multiple correlations, and REDUX-CD, to compare two

partial correlations, compute the # matrix, the partial derivatives in vector a, and a 95%

confidence interval.

Alf and Graf (1999) present a further simplification that does not employ numerical

derivatives, is less computationally demanding, and produces results equivalent to the method

described by Olkin and Finn. All computations are based on sample estimates. The problem is

approached by representing a multiple correlation as a zero-order correlation between the

outcome variable and another single variable that is a weighted sum of the predictors. Alf and

Graf defined


rAB OB(1.17)
r0A

where the subscripts A and B denote weighted sums of two sets of predictors and reBis the

correlation between the two composite variables.

The confidence interval for the squared semipartial correlation coefficient is determined by

the special case in which one set of predictors is a proper subset of the predictors in the other

correlation. The two squared multiple correlations are computed using the same sample and the









variables in the reduced model are a subset of the variables in the full model. Let pS and p'

denote the population squared multiple correlation coefficients corresponding to Rf- and R The

subscript,J; refers to the "full" model with all predictors; the subscript, r, refers to the "reduced"

model. The reduced model contains all predictors with the exception of the variable of interest.

The asymptotic variance of R is


VarR ) (1.18)

The asymptotic variance of RY is

4p' (1-p
Var (R )= (1.19)


The asymptotic covariance between Rf and R~ is


2 4p,p, [.5 2p,/ip,-p,p,( 1- p -p -p /p >+p /p3
Cov(R,, Rf ) = (1.20)


For the squared semnipartial correlation, let AR2 = R~ -R. The asymptotic variance of AR2 is

,o = Var(R: )+Var(R))- 2Cov(R ,Rf- ). (1.21)

An asymptotically correct 100(1 a)% confidence interval for Ap2 = p pS is

AR2 fZn/2 m (1.22)

where zu/2 is the (1 a/2)th percentile of the standard normal distribution and ois the estimate of

am. Inm+,~ prctc,the lagesmple variance is estimatedl byr substituti;ng R for p and Ry2 for p in


Equations 1.18, 1.19, and 1.20.

Equations 1.18 and 1.19 are problematic when the population squared multiple correlations

are zero because the implication is that the sampling variance of R2 is also zero (Stuart, Ord, &

22









Arnold, 1999). Similarly, Equation 1.20 implies that the sampling covariance is zero if either

population multiple correlation coefficient, p, or py is zero. If it were known that both p: and

p, were zero and these values were used to construct a confidence interval, we would incorrectly

conclude that the width of the resulting interval is zero. This computational problem is unlikely

to occur in practice since we substitute sample multiple correlation coefficients for their

population values and it is doubtful that either Rp or Rf- will ever be exactly zero.

The Alf and Graf formulas rely on asymptotic results. As such, they are only exactly

correct for infinitely large samples. Thus, the accuracy of this approximation is heavily

dependent on sample size. Alf and Graf (1999, p.74) concluded that "the correlation between

two multiple correlations will be extremely high when the variables in one multiple correlation

are a subset of the variables in another multiple correlation" and to ensure that coverage

probability is equal to the nominal for the confidence interval on Ap2, "mOderately large to large"

sample sizes are necessary.

In the absence of more specific recommendations on sample sizes, Algina and Moulder

(2001) conducted a simulation study to evaluate the empirical probability that the interval in

Equation 1.22 includes Ap2 for 95% confidence interval. Algina and Moulder manipulated p ,

pi, the number of predictors in the model (k), and the sample size (n). When the data are

distributed multivariate normal, results indicate that when Ap2 > 0, for sample sizes

representative of those used in psychology (i.e., n < 600), coverage probabilities for a nominal

95% confidence interval were less than .95. This tends to be true even with relatively large

sample sizes, i.e. between 600 and 1200.









When p. p~ = 0 all coverage probabilities were at least .999 for all sample sizes studied.

That is, when p2 does not increase when a predictor is added to a multiple regression model, the

confidence interval is always too wide. Algina and Moulder (2001) posited two reasons for this

defect in the confidence interval: (a) for all conditions in which p~ p~ = 0 the asymptotic

variance overestimated the sampling variance and (b) the distribution of R Rf is positively

skewed with a lower limit of 0. Because the confidence interval does not take this lower limit

into account, even if the asymptotic variance was not overestimated, the lower limit would tend

to be smaller than zero.

Algina and Moulder (2001) showed that coverage probability tends to increase as p~

increases and as Ap2 increases and tends to decrease as the number of predictors increases.

Further, when the interval does not contain Ap2, there is a tendency for the interval to be entirely

below Ap2. Algina and Moulder conclude that using the Alf and Graf method to compute a

confidence interval with an inadequate sample size will underestimate the strength of the

relationship between the predictor and the outcome variable.

The Impact of Nonnormality on Statistical Estimates

Every procedure used to make statistical inferences is based on a set of core assumptions.

If the assumptions are met, the test will perform as theorized. However, the results may be

misleading when the assumptions are violated. The most common method for estimating

regression coefficients is ordinary least squares (OLS). Ordinary least squares yields unbiased,

efficient, and normally distributed estimates when the following conditions are met: (a) No

measurement error; (2) the mean of the residuals is zero; (3) the residuals have constant variance;

(4) the residuals are not inter-correlated; and (5) the residuals are normally distributed.









In terms of power and accurate probability coverage, standard analysis of variance

(ANOVA) and regression methods are affected by arbitrarily small departures from normality.

As early as 1960, Tukey found that nonnormality could have a sizeable impact on power and

measures of effect size could be misleading whenever means are being compared. By sampling

from a contaminated normal distribution, Tukey showed that classical estimators are quite

sensitive to distributions with heavy tails. The contaminated normal distribution is a mixture of

two normal distributions, one of which has a large variance; the other distribution is standard

normal. This results in a distribution with heavier tails than the Gaussian. Heavy-tailed

distributions are characterized by unusually large or small values. Both heavy-tailed and skewed

distributions are commonplace in applied work (Micceri, 1989). The presence of these

characteristics in the data can "diminish the chances of detecting true associations among

random variables and obtaining accurate confidence intervals for the parameters of interest"

(Wilcox, 1998).

After reviewing over 400 large data sets from educational and psychological research,

Micceri (1989) found the maj ority did not follow univariate normal distributions.

Approximately two-thirds of ability measures and over 80% of the psychometric measures

examined exhibited at least moderate asymmetry. For all data sets studied, 31% of the

distributions showed skewness, yl, greater than .70 and 52% of psychometric measures

demonstrated extreme to exponential asymmetry, yl > 2.00. Psychometric measures also

exhibited heavier tails than ability measures. Kurtosis estimates ranged from -1.70 to 37.37. To

put this in some perspective, the kurtosis for the double exponential distribution is 3.0.

Breckler (1990) considered 72 articles in personality and social psychology journals and

found that in analyses relying on the assumption of multivariate normality, only 19% of authors









acknowledged this assumption and less than 10% considered whether it had been violated.

Keselman and his colleagues (1998) reviewed articles in prominent educational and behavioral

sciences research j ournals published during 1994 and 1995 and concluded (a) the maj ority of

researchers conduct statistical analyses without considering the distributional assumptions of the

tests they are using and therefore use analyses that are not robust; (b) researchers rarely reported

effect sizes; and (c) researchers failed to perform power analyses in order to inform sample size

deci sions.

Statement of the Problem

Methods for constructing confidence intervals based on asymptotic theory, such as those

proposed by Olkin and Finn and Alf and Graf, have the potential to be very attractive to applied

researchers. In the case of the equations presented by Alf and Graf, a hand calculator can be used

to compute a confidence interval using the appropriate estimates from the results of data analysis

obtained using standard statistical analysis software. However, as Algina and Moulder

demonstrated, even under the best case scenario, where data are drawn from a multivariate

normal distribution, the coverage probability of the asymptotic confidence interval for Ap2 is leSS

than optimal, and when sample size is relatively small, e.g., < 200, would be considered

unacceptable by most researchers. Since multivariate normal data is rare, the performance of Alf

and Graf s procedure under "real world" conditions warrants further investigation.

Purpose of the Study

My dissertation will extend the work of Algina and Moulder (2001) and investigate the

effect of the magnitude of population squared multiple correlation coefficients, p~ and p as

well as the number of predictors, on the asymptotic confidence interval for Ap2 under a range of

nonnormal conditions. The study will investigate coverage probability when (a) the predictor









variables are not distributed multivariate normal; (b) the residuals are not normal; and (c) both

predictors and residuals are nonnormal. Empirical coverage probabilities will be compared to

nominal coverage probabilities over a wide range of sample sizes. My research will

address the following questions:

* How adequate is Alf and Graf s asymptotic confidence interval procedure for the squared
semipartial correlation coefficient when used with sample sizes typically employed in
research in education, psychology and the behavioral sciences under conditions of
nonnormality?

* Is there a minimum sample size for which this method meets established standards for
accuracy over a wide range of situations such that recommendations can be made for the
use of this procedure in reporting the results of applied research?










CHAPTER 2
METHOD S

In conducting a simulation study, especially when the goal is to inform the practice of

researchers, it is important to ensure that the relevant factors are manipulated and that the levels

of these factors reflect those routinely observed. To that end, six factors were manipulated in a

factorial design using values typical of those observed in applied research: the number of

predictors, the size of the squared multiple correlation in the reduced model, the size of the

squared semipartial correlation, sample size, the distribution for the predictors, and the

distribution for the error. These factors, and the levels of these factors, are detailed in Table 2-1.

Study Design

Number of predictors

Algina, Moulder, and Moser (2002) examined sample size requirements for accurate

estimation of squared semipartial correlation coefficients and found a modest effect on the

distribution of AR2 due to the number of predictors included in the multiple regression model.

Therefore, it follows that the sample size required for the confidence interval on Ap2 to be robust,

i.e. to have the coverage probability equal to the nominal confidence level, will likewise depend

on the number of predictors. The number of predictors in the initial set of predictors (k 1)

ranged from 2 to 10 in increments of 2. This allowed investigation of the performance of the

asymptotic confidence interval for a reasonable range of model sizes.

Squared multiple correlations

Algina, Moulder, and Moser also showed that the sampling distribution of AR2 Strongly

depends on the population squared multiple correlations in both the full and reduced models,

pf and p~ Based on a survey of all APA journal articles published in 1992 reporting multiple









regression results, Jaccard and Wan (1995) found the median squared multiple correlation in

these studies to be .30. The 75th percentile for squared multiple correlations was approximately

.50. Based on these results, the values for the squared multiple correlation coefficients for the

predictors in the initial set (p~ ) ranged from .00 to .60 in steps of .10 (7 levels of the factor).

Cohen (1988) proposed, as a convention, that .02, .13, and .26 represent small, medium, and

large effect sizes for squared semipartial correlations. By manipulating the squared multiple




p~ + .30 in steps of .05, values for Ap2 that ranged from .00 to .30 in steps of .05 were produced

(7 levels of the factor). The values for Ap2 are reasonably representative of likely effect sizes

andthevalesselcte fo pandu p~ cover a comprehensive range of population squared

multiple correlations for multiple regression models from p2 = .00 to p2 = .90.

Sample size

Jaccard and Wan also reported typical sample sizes for studies using regression analysis.

The median sample size was 175; a sample size of 400 was at the 75th percentile. However,

Algina and Moulder found with multivariate normal data empirical estimates of the coverage

probability were smaller than .95 even with a sample size as large as 1200. Since we expected

empirical coverage probabilities to be worse for nonnormal data, larger sample sizes than are

usually observed in psychological research were included. Sample size ranged from 100 to 1000

in steps of 100 and from 1000 to 2000 in steps of 250 (14 levels of the factor).

Distributions

The distributions chosen for study represent varying levels of nonnormality and were

selected to: (a) allow examination of the effects of skewness and kurtosis; and (b) be

representative of the types of univariate nonnormality commonly encountered in applied research









in education and psychology. The method described in Hoaglin (1985) and Martinez and

Iglewicz (1984) using the g-and-h distributions was used to generate data that is characterized by

varying degrees of skewness (yl) and kurtosis (y2). A g-and-h distribution is generated by a

single transformation of the standard normal distribution and allows for asymmetry and a variety

of tail weights. In the case of the standard normal distribution, g = h = 0 and yl = Y2 = 0. When

g = 0, a distribution is symmetric. Distributions with positive skew typically have vi > 0 and in

distributions with negative skew, yl < 0. The tails of the distribution become heavier as h

increases in value. Long-tailed distributions, such as the t-distribution, are characterized by

Y2 > 0. Short-tailed distributions, such as the uniform distribution, have y2 < 0.

The distributions selected for this study and their skewness and kurtosis are presented in

Table 2-1. Distribution 1 is the multivariate normal case. Distribution 2 is symmetric and

long-tailed and has the same skew and kurtosis as a t-distribution with 10 degrees of freedom.

Distribution 3 is both asymmetric and leptokurtotic with the same skew and kurtosis as a X2

distribution with 10 degrees of freedom. Since distributions 2 and 3 have similar kurtosis, but

differ with respect to asymmetry, this allowed us to evaluate the relative importance of skewness

and kurtosis on the coverage probability of the confidence interval. Distribution 4 has the same

skew and kurtosis as X Distribution 5 is extremely skewed with heavy tails and has skew and

kurtosis equal to the exponential distribution. Nonnormality was manipulated in either (a) the

predictors, (b) the residuals, or (c) in both the predictors and the residuals.

The error distribution is a univariate distribution. The empirical cumulative distribution

functions for the four nonnormal distributions selected for this study, generated by sampling

1,000,000 random variates from each g-and-h distribution, are depicted in Figures 2-1 to 2-4. In

addition, the deviation from normality is shown by including the normal curve with mean equal









tO CEgh and standard deviation equal to agh for each distribution. The population mean and

standard deviation for each g-and-h distribution were calculated using the formulas given by

Hoaglin (1985, p. 502-503).

In multiple regression, the predictors are multivariate. Multivariate normality, however, is

a stronger assumption than univariate normality. Univariate normality of each of the variables is

necessary, but not sufficient, and a nonnormal multivariate distribution can have normal

marginals. Therefore, a preliminary step in evaluating multivariate normality is to study the

reasonableness of assuming marginal normality for the observations on each of the variables

(Gnanadesikan, 1997). In addition to graphical approaches, a common method for evaluating the

normality of univariate observations is by means of skewness and kurtosis coefficients, J and

b2.



= => 3/2 (2.1)



and


x, -x_4

b2 n =1 2 (2.2)


These are sample estimates of the population skewness and kurtosis parameters JSand p2,


respectively. When the population is normal, Jp= 0 and p2 = 3. If P2 < 3, there is negative

kurtosis; if p2 > 3, there is positive kurtosis. Population skewness and kurtosis are also









commonly described by yl and y2 (Hoaglin, 1985) where

Yi = J~(2.3)

and

Yz = Pt -3. (2.4)

Mardia (1970) proposed indices for assessing multivariate normality that are

generalizations of the univariate skewness and kurtosis measures and b2. Let X1,...,X, be a

random sample from a population with mean vector Ct and covariance matrix C. The sample

mean vector and covariance matrix are denoted by X and S, respectively. The skewness and

kurtosis, Bl,k and B2,k, for ai multivariate population, as defined by Mardia, are


1,k = Eit (x, p) Ex, @ (2.5)

and


2,k = E (x, -' p)c x E x, (2.6)

According to Rencher (1995), since third order central moments for the multivariate normal

distribution are zero, Bl,k = 0 when X ~ N(CL,1). Furthermore, it can be shown that for

multivariate normal X

2,k, = k(k + 2) (2.7)

where k is equal to the number of variables. Sample estimates of f 1k and B2,k arT giVen by


b1,k 2 X -)S X -X(28









and


b2,k [ (X~, X'S '(X, X) (2.9)


Multivariate skewness and kurtosis were calculated by simulating 1,000,000 random

variates sampled from each g-and-h distribution for each level of k under investigation and then

applying equations 2.8 and 2.9 to obtain estimates of Mardia's multivariate measures, bl~k and

b2,k. The SAS program used to estimate these indices is included in Appendix A. Mardia's

multivariate skewness estimates are presented in Table 2-2 and Table 2-3 presents Mardia's

multivariate kurtosis estimates. Figures 2-5 and 2-6 are graphic presentations that compare the

coefficients for the nonnormal distributions to the values expected under multivariate normality

for the number of predictors under investigation in this study.

The design for the study is a 5 (data generating distribution for the predictors) x 5 (data

generating distribution for the errors) x 7 (pi) x 7 (Ap2) x 5 (k) x 14 (n) fully crossed factorial.

This resulted in a total of 85,750 unique conditions. Each combination of factors was replicated

10,000 times and for each replication, a 95% confidence interval was constructed using the Alf

and Graf method.

Background and Theoretical Justification for the Simulation Method

The multiple regression model can be written as

Y, = Po + PX,, + P2X2, +...+ PkX,, + E,. (2.10)

In the standardized multiple regression model, in the population with k > 1 predictors and one

criterion, all variables are standardized to mean zero and unit variance so an intercept is not

needed. This model is


Y, = P,X,I + P7XZI +...+ BkX,, + s, = C P,X, + El (2.11)









where p, is the population standardized regression coefficient associated with the ith predictor;

er; ~N(0,G2); i = 1, ... k; j = 1, ... n. Assuming that we are operating on the population and

that the model is correct, predicted values are given by


2, = [,X,, (2.12)


and the squared correlation between the observed (Y) and the predicted (Y) values is denoted as

p~ In the sample, this is estimated by R2. When the predictors are uncorrelated, the sum of the

squared correlations is equal to the variation accounted for by all the predictors


i7=1 =PI (.3

A simplifying transformation (Browne, 1975) holds that for any set of predictors that has a

squared multiple correlation, p2, with Y, it is always possible to transform the predictors so that

(a) the transformed predictors are mutually uncorrelated, (b) have unit variance, and (c) the

regression coefficients are equal to any set of values such that


:0 = oP (2.14)


The quantity Ap2 is a function of the elements of the covariance matrix for the predictors and the

criterion.

In order to illustrate the application of Browne's results to the current simulation, let xt

denote the vector of standardized predictor variables, with k x k correlation matrix P and k x 1

vector of correlation coefficients p between the predictors and the criterion variable, y. The

squared multiple correlation coefficient for all k variables is denoted by p and for the first k -1

variables is denoted by pl We seek a transformation of the predictors to x such that the new









variables are standardized and uncorrelated, and the regression coefficients relating y to the

variables in xt are P, = 0 for the first k -2 variables and Bk-1 = and Bk = ,p-I for the

last two variables, respectively.

The transformation can be constructed in two steps. It is well known that the variables in

the vector i = Axt, where A is ak x k matrix, will be uncorrelated dependent on an appropriate

choice of A. For example, A can be selected as the inverse of the left Cholesky factor of

R (i.e., R = A-'A wNhere AT indicates the inverse of A') The vector of correlation

coefficients between the transformed predictors and the criterion is Ap and because the

transformed variables are uncorrelated, P = Ap is the vector of regression coefficients relating

the criterion variable to the variables in 8E Because the criterion is a standardized variable

and %- Ax; is no~nsingular transforrmatio~n, is; u~nchngedl by the transformation, andl~ p = p


We next seek a transformation x = T'i, where T' is k x k, such that the variables in x are

standardized and uncorrelated and so that the regression coefficients for the variables in x are


p, = 0 for the first k -2 variables and Pk-1= Jand Pk = for the last two variables,


respectively. Wer,, see+ tht 'R = p. Because the variables in ii are standardized and

uncorrelated, the matrix T' must be orthogonal so that the variables in x will be standardized

and uncorrelated. With an orthogonal transformation, P = TP. The matrix T can be constructed

as follows (M. W. Browne, personal communication with J. Algina, 1999):

Let u = p Then, T = I -2u (u'u) u' is an orthogonal matrix, and P = Tp. Because


2u'P 2u'P ~B)Tui hevralsi
P'P = P'P = 1, and P TP it follows that TP = P .Thsiftevralsn
uu uu









xt are transformed to x = T'Axt, with T' and A defined as above, the transformed variables will

be uncorrelated and standardized and the regression coefficients will be P, = 0 for the first k 2


variables, and Sk-1 = and Bk = J for the last two variables, respectively. Because the

variables are standardized and uncorrelated, the squared multiple correlation coefficient for the


first k -1 variables will be 10, = p: and the squared multiple correlation coefficient for all k


variables will be 1972 __ ?, 2 _P ? __ 2


The implication of Browne' s result is that if the predictors are correlated, they can be

transformed so that (a) the predictors are uncorrelated, (b) the predictive power of k 1 of the

predictors is channeled into one of the transformed predictors, (c) the predictive power of the

remaining predictor is channeled into another of the transformed predictors, and (d) the

remaining k 2 predictors have no predictive power (Algina, Moulder, & Moser, 2002). Rather

than simulating various covariance structures for the predictors, the application of Browne's

results allows us to operate with uncorrelated predictors since it is always possible to transform

these variables to correlated variables. This dramatically reduces the number of conditions in the

simulation to a more manageable number. In addition, when the focus of the study is squared

multiple correlation coefficients, there is no loss of generality if the means of the predictors and

the criterion are rescaled to zero.

Therefore, in the simulation, (a) the independent variables are mutually uncorrelated with

mean zero and variance one; (b) the criterion has mean zero and variance one; and (c) the

regression coefficients are pt = pr, P2 P- k- = 0, Bk = p p The squared multiple

corre~lationl is p for variables X1 to Xk and p, for variables Xi to Xk-1. Given these conditions,









the covariance between Y and X1 is pr, the covariances of Y with the remaining independent


variables, X2 to Xk-1 arT all ZeoO, the covariance between Y and Xk is jp p and the

covariance for any pair ofXvariables is zero.

Data Simulation

The data were simulated using the random-number generating function in SAS Version

9.13. Computations were performed using SAS Interactive Matrix Language (PROC IML).

Data management and follow up analyses were also conducted using SAS. Normal random

deviates were generated for the n x k data matrix of predictors, X, using the SAS RANNOR

function. All nk scores were generated to be statistically independent. In order to generate data

from a g-and-h distribution, standard unit normal variables, Z,, were transformed via the

following equation


exp (gZ -1g hZ (.5
X = ex 2.5
g 2

when both g and h were nonzero. When g is zero, equation 2. 15 is reduced to


X = Z, exp 2 (2.16)


The g-and-h distributed variables were then standardized by subtracting the population mean and

dividing by the population standard deviation. If g = 0, Clgh = 0. When g > 0, the population

mean is

82
exp -1
2(1 h)
Rgbh =~ i (2.17)









and for h I V/2 the population standard deviation is

2 ex 2(- 2h ~ [ 2(- h '2
2g2 2 2
[~2(1 h) 2( h (1-h
0 gh2 2 1-h (2.1 8)


In a similar manner, an n x 1 vector of standard normal random variables was generated.

All n scores were generated to be statistically independent. The results of this vector were

multiplied by -p The result is a vector of residuals, e, with mean zero and variance equal

to 1 p2, These steps ensured that the dependent variable, y, has mean of zero and variance

equal to 1.0.

As detailed above, applying Browne' s results, the k x 1 vector of regression coefficients

was constructed such that elements 1 to k -2 are zero and the next two elements are pr and

J -p, respectively. The sample covariance matrix, S, was calculated from the data

according to the model y = XP + e.

Let Rf be the correlation matrix for the full set of k predictor variables, Rf+ be the k + 1

correlation matrix for all variables (including the criterion), R, be the correlation matrix for the

first k -1 predictors, and Rr + be the correlation matrix for the first k -1 predictors and the

criterion variable. All four correlation matrices can then be calculated from S. The squared

multiple correlation coefficients for the full and reduced models are given by

2det (Rf,
R, = 1 (2.19)
det (Rf)

and

2'= det (Ri,)
R, = 1-(2.20)
det (R,)









where det () represents the determinant of the matrix (Mulaik, 1972). For each of the 10,000

replications of each distributional condition, the asymptotic confidence interval was calculated

using the method described by Alf and Graf (1999).

Data Analysis

Coverage probability, the probability that a confidence interval contains the parameter for

which the confidence interval was constructed, was used to evaluate the adequacy of the

confidence intervals. Coverage probability was estimated as the proportion of the 10,000

replications in which the confidence interval contained the population squared semipartial

correlation, Ap2. In Order to investigate bias, the probability that the confidence interval was

wholly below Ap2 and the probability the confidence interval was entirely above Ap2 were also

estimated.

To evaluate the conditions under which a hypothesis test is insensitive to assumption

violations, Bradley (1978; 1980) proposed three criteria. Given the nominal Type I error rate, a,

a test is robust if the empirical estimate of a falls within the interval at f /s. A liberal criteria is

established when s = 2 and the limits are given by a f .025 = [.025, .075]. Using s = 5, the

interval for a moderate criterion is [.04, .06]. To establish a strict criterion, s = 10 and the

interval is [.045, .055]. If these recommendations are adapted and applied to criteria for a

confidence interval with a nominal coverage probability of .95, the criterion intervals become (a)

[.925, .975]; (b) [.94, .96]; and (c) [.945, .955].

Although there is no universally accepted standard by which procedures are considered

robust or not, Lix and Keselman (1998) suggest that applied researchers should be comfortable

working with a procedure that controls Type I error within the bounds established by Bradley's

liberal criterion, as long as the procedure also limits the error rate across a wide range of










assumption violations. Applying this recommendation to the procedure for constructing an

asymptotic confidence interval means that in order to be controlled, the coverage probability

should fall within the interval [.925, .975]. We used this interval for judging the adequacy of the

confidence intervals. Because there are those who would consider this standard to be too lenient,

confidence intervals were also evaluated according to the more stringent criterion level of .94 to

.96.











Table 2-1. Study Design
Number of predictors, k (5 levels)
1. k= 2
2. k= 4
3. k= 6
4. k= 8
5. k= 10

Size of the squared multiple correlation coefficient for the reduced model (7 levels)
1. p = .00

2. p = .10

3. p = .20

4. p = .30

5. p = .40

6. p = .50

7. p = .60

Size of the squared semipartial correlation coefficient (7 levels)
1. Ap2 =.00
2. Ap2 =.05
3. Ap2= .10
4. Ap2= .15
5. Ap2 =.20
6. Ap2 =.25
7. Ap2= .30

Sample size, n (14 levels)
1. n =100
2. n =200
3. n =300
4. n =400
5. n =500
6. n =600
7. n =700
8. n =800
9. n =900
10. n =1000
11. n =1250
12. n =1500
13. n =1750
14. n = 2000































Table 2-2. Mardia's Multivariate Skewness, bl~k, for the Nonnormal Distributions.
Distribution

g =0 g =.301 g =.502 g =.760
h= .058 h= -.017 h= -.048 h= -.098


k bl~k Interval bl,k Interval bl,k Interval bl,k Intervall

2 .01 (-.03, .05) 1.55 (15,15) 3.90 (3.81, 3.98) 7.87 (7.71, 8.03)
4 .02 (-.05, .08) 3.15 (3.09, 3.22) 7.65 (7.52, 7.78) 15.80 (15.57, 16.02)
6 .00 (-.09, .08) 4.71 (4.62, 4.80) 11.50 (11.33,11.66) 23.74 (23.47, 24.02)
8 -.01(.2,.0 6.23 (6. 11, 6.35) 15.47 (15.26,15.68) 31.64 (31.32, 31.96)
10 .01(.3,.4 7.71 (7.57, 7.86) 19.43 (19.18,19.68) 39.61 (39.23, 39.98)


Table 2-1 Continued
Distribution for the predictor variables, X (5 levels)


1. g= 0, h = 0
2. g=0, h =.058
3. g= .301, h= -.017
4. g =.502, h =-.048
5. g =.760, h =-.098


CL= 0, a
CL= 0, a:
L= .150,
CL = .249,
CL = .378,


= 1, yi = .00, y2 = .00
= 1.097, yi= .00, y2= 1.00
o = 1.041, yi = .89, y2= 1.20
a = 1.108, yi = 1.41, y2 = 3.00


S= 1.252, yr


2.00, y2 = 6.00


Distribution for the residuals,
1. g=0, h =0
2. g=0, h =.058
3. g =.301, h =-.017
4. g =.502, h =-.048
5. g =.760, h =-.098


e (5 levels)
CL= 0, a = 1, yi=.00, y2= .00
CL= 0, = 1.097, yl= .00, y2= 1.00


C = .150,
CL = .249,
CL = .378,


0 =1.041, yr
S= 1.108, yr
0 =1.252, yr


.89, 72
1.41, 72
2.00, 72


1.20
= 3.00
= 6.00


1This interval represents .025 and .975 percentiles of the 1,000,000 replications.














g= .502
h= -.048
Interval'
(13.79,13.89)
(35.60,35.77)
(65.39,64.62)
(103.26,103.57)
(149.02,149.40)


g= .760
h= -.098
Interval
(19.88, 20.02)
(47.90, 48.13)
(83.93, 84.23)
(127.87, 128.24)
(179.80, 180.25)


Interval'
(10.32, 10.38)
(28.70, 28.80)
(55.02, 55.16)
(89.34, 89.53)
(131.67, 131.91)


b2,k
10.05
28.07
54.07
88.10
130.12


b2,k
10.35
28.75
55.09
89.44
131.79


b2,k
13.84
35.69
65.50
103.41
149.21


b2,k
19.95
48.01
84.08
128.05
180.03


1This interval represents .025 and .975 percentiles of the 1,000,000 replications.


Table 2-3. Mardia' s Multivariate Kurtosis, b2,k, for the Nonnormal Distributions.
Di stributi on


I


g= .301
h= -.017


g= 0
h= .058
Interval'
(10.03,10.08)
(28.02,28.11)
(54.00,53.13)
(88.01,88.19)
(130.01,130.23)







































































OC ',
-6 -5 -4 -3 -2


-9 -7875 -675 -5625 -45 -3375 -225 -1125 0 1 125 2 25 3 375 4 5 5 625 675 7 875 9






Figure 2-1. Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = 0, h = .058 overlaid with a normal curve with CEgh = 0,
Ggh = 1.097.





O 451


4 5 6


Figure 2-2. Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = .301, h = -.017 overlaid with a normal curve with CEgh = .150,
Ggh = 1.041.

































































-6 -5 -4 -3 -2-


1 0 1 2 3 4 5 67


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6





Figure 2-3. Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = .502, h = -.048 overlaid with a normal curve with Egh = .249,
Ggh = 1.108.


Figure 2-4


.Plot of the empirical cumulative distribution function for a univariate nonnormal
distribution where g = .760, h = -.098 overlaid with a normal curve with CEgh = .378,
agh = 1.252















-0-g9 = 0, h = 0 (Multivariate Normal)
-*- g =0, h =.058
~-- g = .301, h = -.017
,40 -- g = .502, h = -.048
-9-g =.760, h = -098



~30-



e 20-









2 4 6 8 10

Number of Predictors, k



Figure 2-5. Comparison of Mardia's multivariate skewness for the multivariate normal
distribution to that of the distributions investigated.



200-
-m-g- = 0, h = 0 (Multivariate Normal)
18--o-g9=09,h =.058
-A~- g = .301, h = -.017
160 -1 -r-g =.502, h =-.048
-4-g =.760, h =-.098








120-

10-
2 81

NubrofPeicos

Figre26Madasmliaitkutssfrtemliaitnomldsrbtoanth







Fiue2-.Mriasmltvrae utsnon hemlivraenormal distributions invetigted









CHAPTER 3
RESULTS

Replication of Results for Multivariate Normal Data

Prior to conducting the study, data were simulated for the multivariate normal case in order

to replicate key findings reported by Algina and Moulder (2001). Replication served two

additional purposes. It verified that the simulation program was functioning properly and that

reasonably close agreement was achieved between coverage probabilities estimated with 10,000

replications and coverage probabilities estimates reported by Algina and Moulder based on

50,000 replications. Results are compared for k = 2, 6, and 10 in Tables 3-1, 3-2, and 3-3. The

shaded columns are the results from this simulation; the unshaded columns reproduce tabled

results reported by Algina and Moulder (p. 638). In these tables, as well as subsequent tables

reporting coverage probabilities, italics indicate that the estimated coverage probability falls

within the interval from .925 to .975. Results in bold represent estimated coverage probabilities

between .94 and .96.

As Olkin and Finn warned, and Algina and Moulder demonstrated, this procedure does not

work at all when the population squared semipartial correlation is zero. Regardless of sample

size, number of predictors, or the value of the population squared multiple correlation in the

reduced model, the coverage probability when Ap2 is zero is always too large, i.e.,~ j)>.999.


This is because if p: = pi om = 0 even though the actual sampling variance of R2 is not zero.

Because of this defect in the asymptotic confidence interval, Alf and Graf recommended that

researchers perform a hypothesis test of the significance of the corresponding regression

coefficient and apply the asymptotic confidence interval procedure only when the null hypothesis

is rejected. Given the coverage probability results when Ap2 = 0, although coverage probabilities

are reported in Tables 3-1 to 3-3, they are not included in the assessment of agreement that









follows as doing so would tend to exaggerate the degree of correspondence between the two sets

of estimates.

Comparing the coverage probability estimates generated by the two studies, for Ap2 > 0,

79% were within f .003 and 94% were within f .005. Of the 504 comparisons, 73 (15%)

showed no difference to 3 decimal places. When coverage probabilities differed, 208 (41%)

estimates from the current study were greater and 223 (44%) were smaller than coverage

probabilities reported by Algina and Moulder.

For k = 2, reported in Table 3-1, 90% of the estimates from the two simulations were

within f .003 and only 5 differences were greater than f .005. For 15 of the 168 cases, estimated

coverage probability would have been categorized differently with respect to Bradley's criteria

for robustness, [.925,.975] or [.94,.96]. These discrepancies were evenly split with 8 estimates

from Algina and Moulder' s study falling in the more stringent interval, that is, closer to the

nominal level, and 7 values of p estimated in this study satisfied the more stringent criterion.

Both sets of estimates when k = 2 showed that empirical coverage probability approached

the nominal as sample size increased and as the magnitude of the squared semipartial correlation

increased. The confidence interval was least accurate for the smallest sample size, n = 175, for

all levels of pl when Ap2 = .05. There was good coverage probability, i.e. at least .94, for

n > 425 and Ap2 > .10. Depending on the tolerance one has for the difference between coverage

probability and the nominal confidence level, coverage probability could be considered

marginally adequate, that is, at least .925, for all sample sizes and Ap2 > .10.

The agreement between the two replications was somewhat worse as the number of

predictors increased. As shown in Tables 3-2 and 3-3, for both k = 6 and k = 10, 127 (76%)

comparisons were within f .003. There were 8 (5%) differences greater than f .005 with 6









predictors and 16 (9%) differences exceeded + .005 with 10 predictors. Although for k = 6 the

large differences favored the results reported by Algina and Moulder (6 vs. 2), for k = 10 a large

difference was just as likely to favor the estimates from the current simulation where "favoring"

is defined as an estimated coverage probability that is closer in value to the nominal. In Algina

and Moulder' s data, there was also a tendency for the estimated coverage probability to meet the

more stringent evaluation criterion when there was mismatch in categorization. For k = 6 and

Ap2 > .10, all coverage probabilities were greater than .925 for n > 425, and all but one were

greater than .94 for n = 600. At k = 10, although all coverage probabilities met the liberal

criterion at n = 600, there was no level of Ap2 for which all were greater than .94. Overall,

agreement between the two studies was quite good and therefore, the current study was

conducted by simulating 10,000 replications of each condition.

Simulation Proper

In this simulation, 857,500,000 independent confidence intervals were calculated. Given

there were 10,000 replications of each combination of X, e, n, k, p and Ap2, COVerage

probability was computed as the proportion of times the constructed confidence interval

contained Ap2, the population squared semipartial correlation. In this manner, 85,750 coverage

probabilities were estimated.

Since the distribution from which predictors were sampled and the distribution for the

residuals were both manipulated, this allowed us to examine four distinct situations that might be

encountered when analyzing data using multiple regression: (a) normal X, normal e, (b) normal

X, nonnormal e, (c) nonnormal X, normal e, and (d) nonnormal X, nonnormal e. Average

empirical coverage probability estimates for these four scenarios, as a function of sample size,

are depicted in Figure 3-1. Results for all values of k, p and Ap2, for selected sample sizes, are










reported in Tables 3-4 to 3-7. Estimates for conditions where Ap2 = 0 were omitted since all

were either .999 or 1.000, rounded to three decimal places.

Table 3-4 presents results for normal predictors with normal errors. If we consider

Bradley's liberal interval, .925 to .975, as evidence for robustness, for k = 2, 4, 6, 8, and 10, the

percentages of nonrobust values at n = 200 were 9%, 12%, 14%, 38%, and 71%, respectively.

At n = 400, the percentages of empirical values that were not robust decreased dramatically to

0%, 0%, 0%, 2%, and 2%. All estimated coverage probabilities were robust at n > 600. At the

largest sample sizes reported, n = 1500 and n = 2000, all exceeded .94 and met the more

stringent standard for robustness

When predictors were normal with nonnormal residuals, reported in Table 3-5, the

percentage of nonrobust coverage probabilities increased. For k = 2, 4, 6, 8, and 10 and n = 200,

the percentages of nonrobust values were 3 1%, 38%, 50%, 76%, and 100%, respectively. As

expected, the number of nonrobust coverage probabilities decreased as n grew larger. This

decrease was notable between n = 200 and n = 400 (7%, 7%, 5%, 14%, and 19%) and less so for

n = 600 (5%, 2%, 2%, 5%, 7%) and n = 800 (2%, 2%, 2%, 5%, 5%). For n > 1000, all coverage

probabilities were robust except when pi = 0 and Ap2 = .30.

Table 3-6 shows coverage probability estimates when the predictors were nonnormal and

the distribution of the residuals was normal. At n = 200, there were no robust empirical

estimates at any level of k. For n = 400, the percentages of estimates outside Bradley's liberal

interval were 64%, 62%, 76%, 95%, and 100% for k = 2, 4, 6, 8 and 10, respectively. For

n = 600, the percentage of coverage probabilities that were nonrobust for these values of k were

50%, 55%, 60%, 64%, and 69%. For sample sizes greater than 600, improvement, as measured

by a decrease in nonrobust values, was much more gradual. For k = 2, 4, 6, 8, and 10, and










n = 800, the percentages were 50%, 50%, 50%, 57%, and 60%; for n = 1000, 48%, 50%, 50%,

48%, and 55%; and for n = 1500, 45%, 45%, 50%, 48%, and 52%. At the largest sample size,

n = 2000, at least 45% of empirical coverage probabilities at every level of k failed to meet even

the liberal standard for robustness.

The coverage probabilities contained in Table 3-7 were estimated for the case where both

predictors and errors were nonnormal. For n I 400, there were only 6 estimates greater than

.925. Of these, 5 were observed for n = 400 and k = 2, and 1 at n = 400 and k = 4. For

n = 600, the percentages of coverage probabilities that were nonrobust were 74%, 71%, 81%,

86%, and 88% for k = 2, 4, 6, 8, and 10, respectively. Similar to what was observed with

nonnormal X and normal e, the improvement in coverage probabilities is minor for n > 600 such

that when n = 2000, nonrobust estimates were 71% 71% 74% 74% and 76% for k = 2, 4, 6,

8, and 10, respectively.

For all four scenarios, coverage probability tended to decrease as more predictors were

included in the model, particularly with smaller sample sizes. Coverage improved as sample size

increased. Figure 3-1 suggests that nonnormality in the predictors was more detrimental to the

adequacy of the confidence interval than was a nonnormal error distribution. A modest decline

in coverage probability was observed between normal X, normal e and normal X, nonnormal e,

but there was a considerable drop off in performance when X was nonnormal even when the

errors were normally distributed.

In addition, coverage probability was examined by distributional condition. A

distributional condition was defined by the combination of the distribution for the predictors and

the distribution for the errors. There were 25 distributional conditions included in this study.










For clarity and ease of presentation, the g-and-h distributions from which data were generated

willl be referred to as:. (a)j pseuudo-to for g= h= .058; (b) pseudvno-I fo g= .301, h= -.017;)


(c) pseudo-X! for g = .502, h = -.048; and (d) pseudo-exponential for g = .760, h = -.098. The

descriptive statistics reported in Table 3-8 were based on 2940 coverage probability estimates

per distributional condition, excluding those cases where Ap2 = 0.

Average coverage probability was closest to the nominal confidence level when both X

and e were normally distributed. The average coverage probability was smallest for the most

seriously nonnormal case, both X and e sampled from the pseudo-exponential distribution.

Within each level of X, mean coverage probability decreased as the error distribution exhibited

increasing nonnormality.

A similar pattern was observed for the median. In the extremes, the median for

multivariate normal data was .944. In contrast, for the condition where both X and e were

distributed pseudo-exponential, half the estimated coverage probabilities were less than .868.

For all distributional conditions in which X was distributed pseudo-exponential, at each error

distribution, at least 50% of the estimated coverage probabilities were below .90.

The variability in coverage probability increased with greater skewness and kurtosis in the

data. When X was distributed pseudo-exponential, regardless of the distribution for e, the

standard deviation was over three times that observed for the multivariate normal case. Although

the maximum value did not differ a great deal as a function of distributional condition, the

minimum was much lower and the range was wider for conditions with greater nonnormality.

Also included in Table 3-8 is an examination of the robustness of the confidence interval

as a function of distributional condition at n = 600 and n = 2000. Applying the liberal criterion,









.925 to .975, all coverage probabilities were robust at n = 600 for multivariate normal data and

when predictors were normally distributed and the distribution for the errors was either

pseudo-tlo or pseudo-X, ,. There was no other distributional condition for which coverage

probability was adequate for the entire range of values for k, p~ and Ap2, eVen for the largest

sample size investigated, n = 2000. For the most extreme distributional condition simulated,

both X and e drawn from a pseudo-exponential distribution, 100% of the coverage probabilities

were nonrobust at n = 600. There was only slight improvement at n = 2000 where 90% of the

estimates were not robust. Although it could be argued that data like this is unlikely to occur in

practice, with an error distribution with severe nonnormality, i.e. pseudo-exponential, there was

poor coverage even when the predictors were multivariate normal. At n = 2000, 25.2% of the

estimates were not robust. Furthermore, when using multiple regression, applied researchers are

much more likely to be concerned about the error distribution since violation of this assumption

influences the power and accuracy of hypothesis tests. Researchers may not even investigate the

multivariate skewness and kurtosis for the predictors. With a normal error distribution, the

percentages of nonrobust estimates at n = 2000 for predictors distributed pseudo-tlo, pseudo-7 ,,

pseudo-X), and pseudo-exponential, were roughly 7%, 10%, 49%, and 96%, respectively.

Although results are reported for only two sample sizes, for all distributional conditions and

sample sizes investigated, when an estimated coverage probability was outside of either criterion

interval, it was without exception, too small.

Figure 3-2 illustrates the relationship between coverage probability, distributional

condition, and sample size. The best coverage probability, over the full range of sample sizes

investigated, was observed for the condition in which both X and e were normal. However, at

best, average coverage probability never reached the nominal confidence level, .95. There was a









slight degradation in performance for conditions where X was normal and the nonnormal errors

were distributed pseudo-tlo or pseudo-X 0 Although it is a bit hard to discern because of the

overlap for conditions where X was distributed pseudo-tlo and pseudo-X 0,, results were similar

for normal X with e distributed pseudo-X0, and X sampled from pseudo-tlo with normal error.

That is, coverage probability estimates were similar when the predictors were normal with

markedly nonnormal errors and when predictors were sampled from a pseudo-tlo distribution

wit a orml ero ditriutin.Similarly, the condition in which X was distributed pseudo-X,

with normal error exhibits coverage probability comparable to the conditions where predictors

were moderately nonnormal, sampled from pseudo-tlo and pseudo-X ,with errors that were

extremely skewed and kurtoti c (p seudo-exponenti al). Thereafter, as the distribution for X

became increasingly nonnormal, coverage probability decreased and was least adequate when the

predictors were sampled from a pseudo-exponential distribution regardless of the distribution for

the residuals. Within each condition for X, coverage probability decreased in the same

systematic way as a function of the nonnormality in the error distribution such that coverage

probability was best with normally distributed errors and worst when the errors were distributed

pseudo-exponential.

Analysis of Variance and Mean Square Components

Given the sheer volume of data collected in this study, analysis of variance (ANOVA) was

used to identify the experimental factors that were important in determining the estimated

coverage probability, p. Factorial ANOVA assumes that multiple factors contribute to the

variance in the data. The total variance is partitioned into main effects corresponding to each

factor, the interactions among them, and random error. The factors manipulated in the study

were all treated as between-subj ects effects in a fully-crossed ANOVA model that consisted of 6









main effects and 56 interactions. Since the procedure for calculating the confidence interval is

clearly inappropriate when Ap2 = 0, the 12,250 coverage probabilities calculated for this value

were not included in this analysis. It was felt this provided a more accurate reflection of the data.

ANOVA analyses and variance partitioning of coverage probabilities were therefore based on

N= 73,500. The mean squares, F-statistics, and p-values associated with each effect in the full

model were computed using the ANOVA procedure in SAS. These results are reported in Table

3-9. The combination of a large number of effects and a very large sample size ensured that

there were many statistically significant effects, including higher-order interactions. In all, 34 of

the 62 effects estimated were significant at p < .0001.

Because statistical significance is in large part a function of sample size, a statistically

significant effect is not very informative when the sample size is very large. To better

understand the relative impact of these effects on coverage probability, it was necessary to obtain

a measure of influence to determine which effects were associated with a meaningful proportion

of the variance. The term variance component is used in the context of analysis of variance with

random effects and denotes the estimate of the amount of variance that can be attributed to each

effect. In the current context, the levels of each factor were purposively selected. Because

effects are fixed and not random, the more accurate term is mean square component. The

ANOVA method for estimating mean square components equates mean squares to their expected

values, EM~S, and solves for the mean square components in those expectations. The estimated

mean square component for each main effect and interaction was computed using the general

formula

M~S~(o) MS(Residual)
62=









where a is the effect of interest and j is the product of the number of levels for each factor not

involved in a (Myers & Well, 2003). In this case, the residual mean square, .0000079, includes

the mean square for the six-way interaction and the mean square for error. For example, the

mean square component for X is given by

S7.5162 -.0000079 7.516192
6 .0005113.
r(5)(5)(7)(6)(14) 14700

Since these are simultaneous linear equations with as many unknowns as there are

equations, they have unique solutions and mean square components are estimated noniteratively.

An unfortunate characteristic of ANOVA estimators is that they can yield negative estimates

even though, by definition, they are nonnegative. Negative components were set equal to zero

before calculating the proportion of variance that could be attributed to each effect. The

components were then summed and the ratio of each mean square component to the sum was

used as a measure of influence.

Effects significant at a = .0001 that accounted for at least .5% of the variance are reported

in Table 3-9. The distribution for the predictors, X, was responsible for 44.5 1% of the total

variance in coverage probability. The variance component associated with X was nearly four

times greater than that of any other main effect. The main effects of Ap2 and pl were

comparatively less important factors in determining average coverage probability, accounting for

10. 12% and 3.26% of the total variance, respectively. Effects of Ap2 and pl were moderated by

their interaction. This two-way interaction accounted for an additional 1.60% of the variance.

The mean square component associated with sample size, n, accounted for 9.41% of the total

variability in p The main effect of e accounted for only 3.3 8% of the variance indicating that

the error distribution had a much smaller impact on the coverage probability of the confidence









interval than did the distribution of the predictors. The number of predictor variables, k, had

very little impact on p, accounting for only .69% of the variability.

The critical importance of nonnormality in the predictors was further substantiated by the

fact that interactions involving X explained an additional 22.24% of the variance in p The

variance components for the two-way interactions between X and Ap2 and X and p~ were

associated with 1 1.3 8% and 8.82% of the total variance, respectively. The three-way interaction

of these factors, X x p~ x Ap2, mOderated the three main effects and the two-way interactions

and accounted for an additional 2.04% of the total variance in coverage probability. The main

effectsall ofX p adp, and the interactions of these three factors explained 81.7% of the total

variance in coverage probabilities.

The effect of e was also moderated, although to a lesser extent, by the two-way interactions

between e and Ap2 and e and p~ These interaction effects were responsible for .70% and 1.28%,


respectively. The three-way interaction, e x p~ x Ap2, explained .54% of the total variance. The

main effects of e, Ap2, and p~ and their interactions accounted for 5.85% of the variance in p .

This was further evidence that although a nonnormal error distribution had some effect on the

coverage of the confidence interval it was not nearly as important as nonnormality in the

predictors.

Sample size interacted with the number of predictors, k, and the size of the squared

semipartial correlation coefficient, Ap2. The n x k and n x Ap2 interaction effects each explained

approximately 1% of the total variance. Important effects involving sample size were associated

with 1 1.3 5% of the variability in coverage probability. Thus, it appears that sample size was also

more important than nonnormality in the error distribution in determining the adequacy of the









confidence interval. The effects reported in Table 3-9 accounted for an estimated 99.6% of the

total variance in coverage probability. The following sections describe the important factors

influencing coverage probability as identified by the mean square components analysis.

The Influence of Nonnormality on Coverage Probability

Nonnormal predictors

When coverage probability was averaged over all other factors, Table 3-10 shows the

adequacy of the confidence interval, as measured by coverage probability, worsened as the

distribution for the predictors became increasingly nonnormal. When X was distributed

multivariate normal, avera e coverage probability was .935 SD = .014 When the set of

predictors was made up of variables sampled from a pseudo-tlo distribution, that is symmetric,

but more peaked and heavier tailed than the normal distribution, average coverage probability

dropped to .925 (S = .015). A similar estimate of average coverage probability, p =.923

(S = .015), was obtained when the ex lanator variables were sam led from a poulation

distributed as pseudo-X,2. Because these distributions had similar values for both univariate and

multivariate kurtosis, but differed with respect to skewness, this result seems to suggest, at least

for moderate nonnormality, that skewness may be less important than kurtosis in determining the

adequacy of the confidence interval procedure. When predictors were sampled from a pseudo-X;

population distribution, the average coverage probability was .906 (SD = .022). The average

coverage probability when predictors were sampled from a distribution that has the same

skewness and kurtosis as the ex onential distribution was .877 (S = .037).

The median was also related to the degree of nonnormality present and declined in a

manner similar to the mean. In addition, the range of coverage probability values estimated in

the simulation expanded as the degree of nonnormality became more extreme. Figure 3-3









presents boxplots that describe the distribution of coverage probability estimates as a function of

the distribution for X. We see that all distributions for p are skewed to the right, but the

distribution was flatter, more spread out, and longer-tailed as the degree of skewness and kurtosis

in the distribution for the predictors increased.

Nonnormal error distribution

Table 3-11 shows descriptive statistics for the main effect of the distribution for error. The

means, by error distribution, also declined as a function of the degree of nonnormality present.

The range between the largest mean, .919 for normally distributed errors, and the smallest, .903

for errors distributed pseudo-exponential, was much smaller than observed in Table 3-10 for the

main effect of the distribution for the predictors. There was also less variability in the median,

ranging from .929 for normal errors to .911 for pseudo-exponential errors. The range of

coverage probabilities and the standard deviations were essentially equal suggesting that there

was little difference in the variability of coverage probability estimates as a function of the error

distribution. The boxplots depicted in Figure 3-4 supported this contention.

The Impact of Squared Multiple Correlations on Coverage Probability

Figure 3-5 depicts the relationship between coverage probability and the magnitude of the

population squared semipartial correlation. Averaged over all other factors, coverage probability

tended to decrease as the size of the squared semipartial correlation increased. Figure 3-5 also

shows that the effect of Ap2 OH COVerage probability varied depending on the distribution for the

predictors hence the significant interaction between X and Ap2. Figure 3-5 and Table 3-12 show

the relationship between Ap2 and coverage probability within each distribution for X. Under

normality there was actually a slight increase in p from Ap2 = .05 to Ap2 = .10, the smallest

values investigated. This increase essentially leveled off thereafter. Stable coverage probability









between Ap2 = .05 and Ap2 = .10 was observed for pseudo-tlo and pseudo-X,, distributions. In

both distributions, p showed a steady, but modest, decline for Ap2 > .10. The decline in p when

X was distributed pseudo-X! was modest between Ap2 = .05 ( p = .924) and Ap2 = .10

( p = .919). The rate of change was much steeper for Ap2 > .10 such that p decreased to .886 at

Ap2 = .30. For X sampled from the pseudo-exponential distribution, coverage probability was

essentially a linear function of Ap2 that declined sharply over the range of Ap2 fTOm p = .915

to p = .840.

There was also a significant interaction, depicted in Figure 3-6, between e and Ap2

However, as reported in Table 3-9, this effect while statistically significant, accounted for little

of the variance in coverage probability. A comparison of Figure 3-6 with Figure 3-5 shows a

similar pattern for the relationship between the error distribution and Ap2 with less extreme

variation in the rate at which coverage probability declined. When the error distribution was

normal, pseudo-tlo, or pseudo-X,,, p declined slightly between Ap2 = .05 and Ap2 = .10 with a

steady, gradual decrease for Ap2 > .10. The decline in coverage probability between

Ap2 = .05 and Ap2 = .60 was more nearly linear, with a steeper slope, when the error distribution

was sampled from either a pseudo-X! or pseudo-exponential distribution. The decrease in

coverage probability was most dramatic when errors were distributed pseudo-exponential. At

Ap2 = .05, p = .923 and at Ap2 = .60, coverage probability dropped to p = .883. Coverage

probabilities, as a function of e and Ap2, are reported in Table 3-13.

Figure 3-7 depicts the relationship between p~ and coverage probability. Coverage

probability stayed relatively constant between p = .00 and p~ = .40 and then decreased for









pi = .50 and pi = .60. The interaction between X and pl is also demonstrated in Figure 3-7.

When the predictors were distributed multivariate normal, coverage probability was a linear

function of pi, gradually increasing from .928 at p~ = .00 to .940 at p~ = .60. For X distributed

pseudo-tlo and pseudo-X,2,, there was a minor increase in coverage probability, roughly .92 to .95,

between pi = .00 and pi = .40. Coverage probability was smaller for pl > .50. As the

distribution for X demonstrated greater skewness and kurtosis, the coverage probability function

tended to be more curvilinear. For X distributed pseudo-X), coverage probability was relatively

consistent between pi = .00 and pi = .30 and decreased steadily for p~ .40 to a minimum of

.887 at p = .60. When X was sampled from a pseudo-exponential distribution, coverage

probability started out at p= .896 atpi = .00 and decreased between p = .00 and pi = .30 to .885.

The decline in p was at a much faster rate thereafter such that when p~ = .60, p = .837. As Table

3-14 shows, the differences in coverage probabilities, as a function of the degree of nonnormality

in X, had their smallest range of values at p~ = .00 (.928 to .896) and the range was maximized at

p = .60 (.940 to .837).

While the behavior of p over the levels of Ap2, as a function of the error distribution, was

comparable to the relationship between Ap2 and the distribution for the predictors, the e x p~

interaction, presented in Figure 3-8, shows this was not the case for pl In contrast, the

differences in j>, as a function of nonnormality in the error distribution, were greatest at p = .00.

Coverage probability, reported in Table 3-15, ranged from .927 for normal errors to .897 when

errors were pseudo-exponential. By the time p~ = .60, coverage probability had essentially









converged and was approximately .90 regardless of the degree of nonnormality in the error

distribution. Furthermore, for normal errors, maximum coverage probability, .927, occurred for

p~ = .00. For e distributed pseudo-tlo and pseudo-X,2,, the largest coverage probability, .922,

occurred at p~ = .10. For pseudo-X, ,, the largest average coverage probability, .915, was

observed for p~ = .20 and p~ = .30. When the error distribution was sampled from a

pseudo-exponential population distribution, the largest coverage probability, .907, was observed

at p = .30 and p~ = .40. These results suggest that the X x p~ and e x p~ interactions might

have a counterbalancing effect. However, the e x p~ interaction, although statistically

significant, explained a modest 1.3% of the total variance in coverage probability while the

X x p~ interaction accounted for 9.5% of the total variance.

The impact of the interaction between Ap2 and p~ on coverage probability is shown in

Figure 3-9. Although there was a tendency for estimated coverage probability to be further from

the nominal as p~ increased, this was not the case for all values of Ap2. When Ap2 = .05, there

was an increasing trend in coverage probability over the range of p values. For Ap2 > .05,

coverage probability was relatively stable between p~ = .00 and p~ =.30, but then decreased

substantially from p~ =.30 to p~ =.60.

However, the relationship of p to p~ and Ap2 varied depending on the distribution for X.

Figure 3-10 shows the effect of the three-way interaction between X, p~ and Ap2 OH COVerage

probability. To aid in the description and interpretation of effects, coverage probabilities, as a

function of p~ and Ap2, for each level of X are reported in Tables 3-16 through 3 -20. For the

multivariate normal case, coverage probability tended to be worse when Ap2 = .05 and for all









levels of Ap2 COVerage probability increased as p~ increased. The plots of coverage probability

as a function of Ap2 and p~ for pseudo-tlo and pseudo'-7 lokrmrabysmlrtoecte


and have the same pattern of results described for the two-way interaction of p~ and Ap2, albeit

over a narrower range of values. Coverage probability increased over the levels of p~ when

Ap2 = .05, but for Ap2 > .05, coverage probability tended to increase from p~ = .00, reached a

maximum at p~ = .30, and decreased thereafter. Although coverage probability was best for

Ap2 = .05 and p~ = .60, for all other levels of Ap2, COVerage probability was lowest at p~ = .60.

Coverage probability was consistent at approximately .925 over the full range for p ~for

Ap2 = .05 when the predictors were distributed pseudo-X For Ap2 > .05, coverage probability

was stable between p~ = .00 and p~ = .20, but showed a decline between p~ = .30 and p = .60.

The rate of decline was faster for larger values of Ap2

For X sampled from the pseudo-exponential distribution, coverage probability decreased as

p~ increased for all levels of Ap2. The rate of decline varied according to the value of Ap2 with

steeper slopes associated with larger values of Ap2. The drop in coverage probability was minor

for Ap2 = .05, where p = .917 at p~ = .00, falling to p =.907 at p~ = .60. However, when

Ap2 = .30, at p = .00 coverage probability was .870 and decreased markedly to~ p .777 at

p~ = .60. Thus, when nonnormality in the predictors was extreme, the importance of the

magnitude of the squared multiple correlations, Ap2 and p~ was critical for determining the

adequacy of the confidence interval procedure. Although no condition, on average,

demonstrated acceptable coverage over the entire range of factors manipulated in this study,









Figure 3-10 illustrates how inaccurate the asymptotic confidence interval can be under conditions

that could occur in practice.

The Impact of Sample Size on Coverage Probability

As seen in Figures 3-1 and 3-2, regardless of the distribution for the predictors or the

distribution for error, coverage probability increased rapidly between n = 100 and n = 400. The

average coverage probability at n = 100 was .882 increasing to .912 at n = 400. The rate of

increase, from .914 to .917, was considerably slower between n = 500 and n = 800. Furthermore,

it appears that there was little to be gained by increasing the size of the sample beyond n = 1000

with respect to the adequacy of the confidence interval. Coverage probability is increasing so

slowly between n = 1000 and n = 2000 (from .918 to .920) that it is likely that sample sizes well

in excess of 2000 would be required to ensure the robustness of the confidence interval over a

wide range of nonnormal conditions. Evidence to support this contention was evaluated by

estimating coverage probabilities for X and e distributed pseudo-exponential; n = 5000; p~ = .00,

.30, and .60; and Ap2 = .05, .10, .15, .20, .25, and .30. Results indicated that even with an

extremely large sample size, when nonnormality is severe, coverage probability remained

inadequate. Only 7 of 54 coverage probability estimates exceeded .925 and consequently, 87%

were nonrobust. Six of the robust estimates were observed for p~ = .00 or p~ =.30 and


Ap2 = .05 for all three levels of k. The remaining robust estimate occurred for k = 10, p~ = .00,

and Ap2 = .10.

Figure 3-11 shows that the effect of sample size was not the same at every level of Ap2

The interaction between n and Ap2 was due to the fact that the effect of Ap2 was smaller when the

sample size was smaller than the effect of Ap2 when the sample size is larger. In addition, the

average values for p are not is the same order as a function of Ap2 foT Smaller sample sizes. For









example, at n = 100, although coverage probability was clearly inadequate for all levels of Ap2, it

was worse for the smallest value, Ap2 = .05, as well as the largest values, Ap2 = .25 and Ap2=-.30.

Coverage probability improved noticeably for Ap2 = .05 at n = 200 although it was still not as

large as it was for Ap2 = .10. By n = 300 coverage probability for Ap2 = .05 and Ap2 = .10 were

equal. At n > 400, coverage probability was a function of Ap2 grOwing worse as Ap2 increased.

Coverage probabilities, as a function of sample size and Ap2, are presented in Table 3-21.

As shown in Figure 3-12, the rate of increase in coverage probability as a function of

sample size depended on the number of predictors in the model. For the smaller sample sizes,

most notably at n = 100, although average coverage probability was clearly inadequate, it was

considerably worse when there were more predictors in the model. As the sample size increased,

the difference between coverage probabilities as a function of the number of predictors became

progressively smaller. Table 3-22 shows that at sample sizes greater than 1000, the difference in

coverage probability was minimal and it appears that the number of predictors exerted very little

influence on coverage probability.

Probability Above and Below the Confidence Interval

When the confidence interval did not contain the population squared semipartial

correlation coefficient, the probability that the confidence interval was below Ap2 and the

probability that the confidence interval was above Ap2 were also estimated. When Ap2 = 0,

average coverage probability was .9998. Only 18,754 of the 122,250,000 confidence intervals

constructed did not contain the population parameter. There were only 4 instances in which the

interval was wholly below Ap2; 18,750 confidence intervals were wholly above Ap2

When the increase in the squared multiple correlation was zero the confidence interval was

too conservative, but for all other values of Ap2, the confidence intervals tended to be too liberal.









For the 73,500 conditions where Ap2 > .05, the probability that the confidence interval was

wholly below Ap2 was twice the probability that the confidence interval was entirely above Ap2

(.664 vs. .336). The confidence interval is biased in the sense that there is a systematic error that

causes the estimated confidence limits to regularly miss the population parameter in the same

direction. The tendency to underestimate Ap2 Occurs because the estimated asymptotic standard

error declines as AR2 declines. As a result, when AR2 B2 there is a tendency for the interval to

be completely below Ap2 (Algina & Moulder, 2001).

The Relationship between Estimated Asymptotic Variance, Empirical Sampling Variance
of AR2, and Coverage Probability

As previously noted, all coverage probabilities were at least .998 for Ap2 = 0. This result

indicates that when a predictor was added to a multiple regression model and there was no

increase in p2, the confidence interval was always too wide. As previously noted, there were two

reasons for this shortcoming in the confidence interval. The distribution of AR2 is skewed to the

right and since the increase in R2 cannot be less than zero it has a lower limit of zero. Because

the confidence interval formula does not recognize this lower limit, when the population value

was Ap2 = 0, the confidence interval tended to have a lower limit less than zero.

The second basis for the problem, identified by Algina and Moulder, is that the asymptotic

variance overestimates the sampling variance ofA~R2. This was verified in the current study by

calculating for each combination of X, e, n, k, pi and Ap2 (a) the mean estimated asymptotic

variance over the 10,000 replications and (b) the empirical sampling variance of AR2. For all

conditions where Ap2 = 0, the ratio of the average value of (a) to (b), denoted as MEAV/VarAR2,

ranged from 1.27 to 2.18 with a mean of 1.95 and a median of 1.96.









The ratio, MEAV/VarAR2, was also evaluated for Ap2 > 0. ANOVA and mean square

components analyses were conducted for MEAV/VarAR2 as the outcome variable. As was the

case with coverage probability, due to the large sample size, only 24 of 62 effects failed to

demonstrate significance at p < .0001. Effects significant at a = .0001 that accounted for at least

.5 % of the variance are reported in Table 3-23. These effects accounted for 97.8% of the

variability in the variance ratio, MEAV/VarAR2. The distribution for the predictors explained

51.78% of the variance in the ratio. An additional 21.06% was attributable to the size of the

squared semipartial correlation coefficient. Less important for accurate estimation of the

variance were the main effects of e and p~ These effects explained 6.26% and 2.67% of the total

variance, respectively.

As observed for coverage probability, a substantial proportion of the variance, 89.5%, was

accounted for by the main effects of X, Ap2, and p and the interaction of these effects: 6.81%

was associated with the X x Ap2 interaction, 6.45% was associated with the X x p~ interaction,

and the three-way interaction, X x py x Ap2, explained a modest .72%. The interaction between

the distribution for the errors and p~ accounted for an additional 2.02%. Although sample size

plays a role in determining the coverage probability, it was not important in determining the ratio

since the effect of n was included in calculating the variance.

Figure 3-13 illustrates how MEAV/VarAR2 VarieS as a function of the distribution for the

predictors, p and Ap2. This figure corresponds to Figure 3-10, describing coverage probability

as a function of the X x p~ x Ap2 interaction, and shows a similar pattern. For the multivariate

normal case, variance ratios got further from 1.0 as Ap2 increased for p~ = 0. As p~ increased,









variance ratios improved for all values of Ap2. This improvement was greater for larger values

of Ap2. By the time p~ = .60, there was no difference in the MEAV/VarAR2 ratio as a function

ofp~ The behavior of the variance ratio helps to explain the fact that for normal data coverage

probability increases with both Ap2 and p .

Fr Xl dJistibuted pseudo-tio andC pseudo-nlO, the pattern for MEAV/VarAR2 as a function

of Ap2 and p~ was very similar. This was also observed for coverage probability. At all values

of p the variance ratio got smaller as Ap2 increased. For Ap2 = .05, the MEAV/VarAR2 ratio

was consistent across the range for p~ There was a slight curvilinear relationship in the

Ap2 X p~ plOts for Ap2 > .15 such that variance estimation improved slightly from p~ = .00 to

p~ = .30 and then declined from p~ = .30 to p~ = .60. Therefore, variance estimates were best for

all values of Ap2 at p~ = .30 and the most serious variance underestimation occurred when both

p~ and Ap2 were largest.

When X was distributed pseudo-X! or pseudo-exponential the difference between the

variance ratio at the smallest value of Ap2 and the largest was greater than for the previous

distibuion atp =.00 and this difference became progressively larger as p increased. For the

most extreme degree of nonnormality, although MEAV/VarAR2 was never greater than .90,

when Ap2 represents a large effect size, the accuracy of the estimated variance was particularly

poor over the range of pl values.

The scatterplot in Figure 3-14 is further evidence of a strong positive association between

coverage probability and MEAV/VarAR2. The correlation between coverage probability and the









variance ratio was r = .91. As the asymptotic variance more accurately estimated the actual

sampling variance of M2, COVerage probability approached the nominal confidence level. When

coverage probabilities were poor, the estimated asymptotic variance could be less than half that

of the empirical sampling variance of M2.

The strength of the relationship between coverage probability and MEAV/VarMR2 depends

on the distribution for the predictors, as shown in Figures 3-15 to 3-19. For multivariate normal

data, presented in Figure 3-15, the mean variance ratio was .946 (SD = .06). The median was

.963 with a range from .666 to 1.050. Approximately 10% of the estimates were greater than 1.0

indicating that the asymptotic variance, albeit rarely, sometimes overestimated the empirical

sampling variance. As the plot shows, however, a variance ratio near 1.0 was not a guarantee

that the coverage probability will necessarily be close to .95 and coverage probability was as low

as .85. Not surprisingly, the correlation between coverage probability and MEAV/VarMR2 was

lower than that for the full data set, r = .62.

Although the correlation between coverage probability and MEAV/VarMR2 was similar to

that for normal data, r = .63, when the predictors were sampled from the pseudo-tlo distribution,

less than 1% of the variance ratios were above 1 (Figure 3-16). The mean variance ratio was

.881 (SD=.065), the median was .886, and the range was .631 to 1.044.

The estimates from the pseudo-X,2 distribution again demonstrate close similarity to the

pseudo-tlo distribution. Although the scatterplot in Figure 3-17 is somewhat less dispersed

reflected in a slightly higher correlation, r = .68, the descriptive statistics show close agreement.

The mean variance ratio was .870 (SD=.068), the median was .874, and the range was .626 to

1.044. Again, less than 1% of the ratio estimates were greater than 1.0.









As multivariate skewness and kurtosis increased, the correlation between coverage

probability and MEAV/VarAR2 became much stronger. For the pseudo-X0 distribution

r = .86. As Figure 3-18 demonstrates, the scatterplot was more compact and more spread out.

The range of values was wider, 548 to 1.004, due to a lower minimum value. There was only 1

variance ratio greater than 1. The mean was .785 (SD = .100) and the median was .788.

Figure 3-19 shows the strongest relationship (r = .91) between coverage probability and

MEAV/VarAR2 for the pseudo-exponential distribution. With skewness and kurtosis

corresponding to the exponential distribution, the scatterplot was tightly concentrated and

substantially more elongated. None of the variance ratios were greater than 1 and over 25%

were less than .60. Variance ratios ranged from a low of .381 to a high of .972. The mean was

.881 (SD=. 132) and the median was .673.

In summary, for multivariate normal data, MEAV/VarAR2 was best when Ap2 was small,

but as p~ increased, variance was more accurately estimated and by the time p~ = .60,

MEAV/VarAR2 was not dependent on Ap2. This pattern of results did not hold when

nonnormality was introduced in the predictors. For moderate nonnormality, MEAV/VarAR2

tended to be more dependent on the value of Ap2 than on the magnitude ofpi When

nonnormality was more extreme, variance estimation became more inaccurate as both Ap2 and

p~ increased. Thus, when a variable was added to a multiple regression model that already

explained a sizeable proportion of the variation in the outcome, for example, p = .60, the effect

size associated with that variable was large, for example, Ap2 = .30, and the data were not

multivariate normal, using Alf and Graf s formula underestimated the variance. Furthermore,

this study showed that when nonnormality was severe, the estimated asymptotic variance could









be less than half that indicated by the sampling distribution of AR2. In practice, this is likely to

produce standard errors that are too small resulting in a confidence interval that is too narrow.

Reliance on this confidence interval as a measure of the strength of the effect size will lead us to

underestimate the importance of an individual predictor to the regression.











Table 3-1. Replication of Algina and Moulder' s Results for Multivariate Data and Two Predictors.



n pi0.00 0.05 0.10 0.15 0.20 0.25 0.30
175 0.00 1.000 1.000 0.907 0.904 0.925 0.925 0.931 0.930 0.936 0.937 0.938 0.939 0.940 0.934


0.10
0.20
0.30
0.40
0.50
0.60
300 0.00
0.10
0.20
0.30
0.40
0.50
0.60
425 0.00
0.10
0.20
0.30
0.40
0.50
0.60
600 0.00
0.10
0.20
0.30
0.40
0.50
0 60


1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1 000


1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1 000


0.911
0.913
0.919
0.922
0.923
0. 931
0.923
0. 928
0. 929
0. 930
0. 934
0. 935
0. 938
0. 931
0. 933
0. 933
0. 935
0. 936
0. 939
0.941
0. 935
0. 937
0. 939
0. 939
0.940
0.943
0.943


0.910
0.912
0.919
0.923
0. 929
0. 928
0.923
0. 927
0. 931
0. 935
0. 929
0. 936
0. 937
0. 933
0. 934
0.940
0. 937
0. 935
0. 939
0.941
0. 935
0. 936
0.941
0. 939
0.942
0.941
0.942


0. 926
0. 930
0. 931
0. 934
0. 936
0. 939
0. 937
0. 935
0. 938
0.940
0.941
0.941
0.943
0. 938
0.941
0.942
0.942
0.943
0.944
0.944
0.943
0.945
0.945
0.945
0.945
0.946
0.946


0.922
0. 929
0. 928
0. 934
0. 939
0. 937
0. 932
0. 939
0. 936
0. 939
0.943
0.943
0.942
0.942
0.940
0.942
0.944
0.941
0.943
0.947
0.941
0.942
0.947
0.943
0.945
0.948
0.945


0. 933
0. 935
0. 935
0.940
0. 939
0.941
0. 938
0.942
0.940
0.941
0.944
0.943
0.944
0.942
0.944
0.945
0.945
0.945
0.946
0.946
0.945
0.945
0.946
0.945
0.947
0.946
0.946


0. 932
0. 933
0.941
0. 939
0. 938
0.943
0. 936
0.940
0.946
0.942
0.942
0.944
0.943
0.943
0.944
0.946
0.941
0.946
0.943
0.945
0.944
0.944
0.942
0.948
0.946
0.943
0.945


0. 938
0. 938
0. 938
0. 939
0.941
0.941
0.943
0.944
0.942
0.944
0.945
0.946
0.944
0.944
0.944
0.945
0.946
0.944
0.947
0.945
0.945
0.948
0.946
0.945
0.949
0.949
0.949


0. 935
0. 938
0. 939
0.941
0.942
0.940
0.943
0.946
0.941
0.944
0.947
0.948
0.942
0.948
0.946
0.944
0.943
0.949
0.946
0.947
0.946
0.944
0.944
0.942
0.948
0.942
0.948


0.940
0.942
0. 939
0.942
0.942
0.942
0.943
0.944
0.943
0.944
0.946
0.945
0.945
0.945
0.944
0.947
0.946
0.946
0.948
0.947
0.947
0.947
0.948
0.945
0.948
0.947
0.948


0. 938
0.940
0.941
0.940
0. 939
0. 939
0.945
0.942
0.946
0.944
0.946
0.944
0.945
0.944
0.948
0.942
0.947
0.947
0.945
0.947
0.944
0.946
0.948
0.945
0.951
0.948
0.948


0. 938
0.940
0.942
0.941
0.943
0.943
0.944
0.943
0.946
0.943
0.946
0.946
0.945
0.947
0.947
0.946
0.945
0.949
0.945
0.946
0.946
0.949
0.948
0.946
0.948
0.949
0.947


0. 938
0. 939
0. 939
0.944
0.941
0.940
0.944
0.944
0.946
0.943
0.946
0.948
0.945
0.947
0.948
0.947
0.947
0.948
0.946
0.945
0.950
0.949
0.944
0.948
0.948
0.947
0.945


Note: Bold results are estimated coverage probabilities between .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.
Shaded columns are results from this study; unshaded columns are the results reported by Algina and Moulder (2001, p. 638-640).










Table 3-2. Replication of Algina and Moulder' s Results for Multivariate Data and Six Predictors



n p 0.00 0.05 0.10 0.15 0.20 0.25 0.30
175 0.00 1.000 1.000 0.897 0.896 0.918 0.915 0. 927 0. 925 0. 930 0. 928 0. 933 0. 937 0. 935 0. 936
0. 10 1.000 1.000 0.903 0.906 0.920 0.920 0. 928 0.923 0. 932 0. 934 0. 935 0. 935 0. 934 0. 932
0.20 1.000 1.000 0.908 0.909 0.922 0. 926 0. 926 0. 931 0. 931 0. 934 0. 934 0. 928 0. 934 0. 935
0.30 1.000 1.000 0.909 0.914 0.922 0. 925 0. 930 0. 928 0. 930 0. 934 0. 933 0. 932 0. 933 0. 935
0.40 1.000 1.000 0.912 0.915 0. 925 0.924 0. 930 0. 932 0. 932 0. 925 0. 932 0. 932 0. 934 0. 933
0.50 1.000 1.000 0.918 0.918 0. 927 0. 929 0. 931 0. 934 0. 931 0. 926 0. 932 0. 933 0. 930 0. 939
0.60 1.000 1.000 0.921 0.919 0. 929 0. 933 0. 929 0. 934 0. 930 0. 929 0. 932 0. 931 0. 931 0. 927
300 0.00 1.000 1.000 0.920 0.919 0. 932 0. 932 0. 935 0. 936 0. 938 0. 937 0. 939 0.940 0.942 0.941
0. 10 1.000 1.000 0.923 0.918 0. 932 0. 927 0. 937 0. 937 0. 939 0. 939 0. 939 0.940 0.940 0.944
0.20 1.000 1.000 0. 925 0. 926 0. 933 0. 935 0. 938 0. 939 0. 939 0. 939 0.940 0.941 0.940 0.942
0.30 1.000 1.000 0. 926 0.924 0. 935 0. 938 0. 939 0. 935 0.941 0.942 0.940 0. 938 0.940 0.940
0.40 1.000 1.000 0. 927 0. 931 0. 936 0. 931 0. 936 0.940 0.941 0.942 0.941 0.940 0.940 0.942
0.50 1.000 1.000 0. 933 0. 931 0. 935 0. 935 0. 938 0. 936 0. 938 0. 934 0.940 0.942 0. 939 0. 939
0.60 1.000 1.000 0. 933 0. 933 0. 938 0.943 0. 938 0. 934 0.940 0. 939 0. 939 0.940 0. 939 0. 937
425 0.00 1.000 1.000 0. 927 0. 925 0. 938 0. 939 0.940 0. 937 0.941 0.945 0.943 0.941 0.944 0.944
0. 10 1.000 1.000 0. 930 0. 927 0. 937 0. 936 0.941 0. 935 0.941 0.944 0.943 0.945 0.943 0.942
0.20 1.000 1.000 0. 931 0. 932 0. 939 0.942 0.940 0.942 0.943 0.941 0.944 0.944 0.943 0.945
0.30 1.000 1.000 0. 934 0. 935 0. 937 0.943 0.941 0.940 0.941 0.942 0.945 0. 938 0.943 0.942
0.40 1.000 1.000 0. 935 0. 937 0.941 0. 937 0.942 0.944 0.943 0. 939 0.941 0.944 0.943 0.946
0.50 1.000 1.000 0. 936 0. 934 0.941 0.940 0.940 0. 936 0.943 0.940 0.943 0.940 0.943 0.941
0.60 1.000 1.000 0. 936 0. 939 0.942 0.944 0.942 0.943 0.941 0.944 0.941 0.944 0.941 0.944
600 0.00 1.000 1.000 0. 933 0. 934 0.941 0. 938 0.944 0.941 0.944 0.944 0.945 0.949 0.946 0.947
0. 10 1.000 1.000 0. 937 0. 936 0.941 0.943 0.942 0.942 0.943 0.945 0.944 0.949 0.947 0.945
0.20 1.000 1.000 0. 937 0. 935 0.942 0. 936 0.941 0.943 0.943 0.947 0.944 0.946 0.945 0.946
0.30 1.000 1.000 0. 939 0.941 0.943 0. 938 0.944 0.941 0.946 0.941 0.945 0.949 0.947 0.943
0.40 1.000 1.000 0.940 0. 939 0.942 0.941 0.946 0.942 0.945 0.941 0.945 0.946 0.946 0.941
0.50 1.000 1.000 0.942 0. 935 0.942 0.944 0.945 0.943 0.945 0.945 0.945 0.946 0.943 0.944
0.60 1.000 1.000 0.941 0.942 0.942 0.942 0.945 0.945 0.945 0.946 0.943 0.942 0.944 0.946
Note: Bold results are estimated coverage probabilities between .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.
Shaded columns are results from this study; unshaded columns are the results reported by Algina and Moulder (2001, p. 638-640).

















0.920
0.922
0.920
0.921
0.922
0.914
0.916
0. 935
0. 936
0. 936
0. 936
0. 932
0. 934
0. 932
0. 937
0. 935
0. 938
0. 939
0. 937
0. 939
0. 938
0. 938
0.941
0.949
0.942
0.942
0.941
0.940


0.921
0.923
0.923
0.923
0.919
0.917
0.911
0. 936
0. 935
0. 936
0. 934
0. 932
0.931
0. 927
0. 938
0. 938
0. 939
0. 939
0.941
0. 939
0. 937
0.942
0.945
0.942
0.941
0.942
0.940
0. 939


0.920 0. 927 0. 925
0. 926 0.923 0. 926
0. 927 0.921 0.921
0.921 0.922 0.914
0.917 0.918 0.916
0.922 0.916 0.916
0.912 0.910 0.904
0. 934 0. 938 0. 935
0. 937 0. 936 0. 935
0. 937 0. 935 0. 931
0. 936 0. 934 0. 930
0. 934 0. 931 0. 931
0. 933 0. 930 0. 930
0. 935 0. 927 0. 926
0.941 0.940 0.940
0. 938 0.940 0.940
0. 939 0. 939 0. 934
0. 939 0. 938 0. 936
0. 935 0. 936 0. 937
0. 934 0. 935 0. 935
0. 935 0. 936 0. 938
0.943 0.943 0.946
0.944 0.943 0. 939
0.940 0.944 0.941
0.942 0.943 0. 937
0.940 0.942 0. 937
0.942 0. 939 0. 938
0.942 0.940 0. 937


Note: Bold results are estimated coverage probabilities between .94 and .96; italicized results are estimated coverage probabilities between .925 and .975.
Shaded columns are results from this study; unshaded columns are the results reported by Algina and Moulder (2001, p. 638-640).


Table 3-3. Replication of Algina and Moulder' s Results for Multivariate Data and Ten Predictors.



n pr 0.00 0.05 0.10 0.15 0.20 0.25 0.30


0.917
0.919
0.919
0.920
0.920
0.918
0.919
0. 929
0. 933
0. 933
0. 932
0. 934
0. 933
0.931
0. 936
0. 938
0. 937
0. 936
0. 938
0. 936
0. 937
0.941
0.941
0.940
0.941
0.944
0.941
0.940


175 0.00 1.000 1.000 0.890


0.10
0.20
0.30
0.40
0.50
0.60
300 0.00
0.10
0.20
0.30
0.40
0.50
0.60
425 0.00
0.10
0.20
0.30
0.40
0.50
0.60
600 0.00
0.10
0.20
0.30
0.40
0.50
0 60


1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1 000


1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1 000


0.895
0.897
0.901
0.904
0.909
0.911
0.914
0.917
0.919
0.922
0.924
0.924
0. 926
0.923
0. 927
0. 928
0. 930
0. 932
0. 930
0. 935
0. 933
0.933
0. 935
0. 935
0. 936
0. 934
0.940


0.893 0.910 0.913
0.888 0.910 0.916
0.898 0.913 0.911
0.904 0.916 0.918
0.907 0.918 0.914
0.910 0.920 0.916
0.912 0.919 0.917
0.922 0. 928 0. 926
0.921 0. 926 0.923
0.919 0. 927 0.924
0.923 0. 931 0. 931
0.919 0. 929 0. 934
0.922 0. 930 0. 931
0. 927 0. 930 0.924
0.924 0. 935 0. 935
0. 926 0. 934 0. 931
0. 928 0. 936 0. 939
0. 934 0. 936 0. 934
0. 930 0. 936 0. 934
0. 931 0. 936 0. 939
0. 934 0. 936 0. 935
0. 926 0. 938 0. 938
0. 939 0. 937 0.941
0. 929 0. 939 0. 937
0. 930 0. 939 0.941
0. 937 0.941 0.944
0.940 0.942 0.942
0. 939 0.940 0.940


0.919 0.921
0.918 0.919
0.917 0.921
0.920 0.920
0.924 0.921
0.924 0.918
0.922 0.915
0. 934 0. 935
0. 934 0. 934
0. 934 0. 932
0. 934 0. 934
0. 937 0. 933
0. 938 0. 932
0. 935 0. 931
0.941 0.938
0. 934 0. 939
0.938 0.940
0. 934 0. 939
0. 936 0. 939
0. 939 0. 938
0. 937 0. 935
0.940 0.944
0.940 0.941
0.938 0.941
0.940 0.941
0.944 0.942
0.939 0.942
0. 936 0. 939











Table 3-4. Empirical Coverage Probabilities for Normal Predictors and Normal Errors.


n p~ AP2 2 4 6 8 10
200 0.00 0.05 0.912 0.911 0.901 0.896 0.898


Note: Bold results are estimated coverage probabilities between .94 and .96; italicized results are estimated
coverage probabilities between .925 and .975.


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0. 926
0.930
0.934
0.941
0.943
0.912
0.932
0.934
0.942
0.940
0.942
0.916
0. 929
0.942
0.938
0.942
0.940
0. 925
0.935
0.939
0.938
0.944
0.943
0.924
0.935
0.941
0.942
0.945
0.949
0.932
0.934
0.946
0.942
0.940
0.941
0.937
0.941
0.943
0.940
0.949
0.944


0. 925
0.930
0.936
0.936
0.940
0.907
0. 932
0. 935
0. 935
0. 938
0.940
0.917
0. 930
0. 933
0. 938
0. 93 7
0.942
0.913
0. 932
0. 932
0. 936
0.940
0.940
0.919
0.936
0. 935
0. 938
0. 939
0. 939
0. 92 7
0. 935
0.941
0. 938
0.942
0.942
0. 928
0.939
0. 93 7
0. 938
0. 938
0. 939


0. 926
0.932
0.934
0.934
0.940
0.904
0. 925
0. 929
0. 938
0. 936
0.942
0.916
0. 928
0. 932
0. 938
0. 939
0.942
0.915
0. 92 7
0. 933
0. 931
0. 935
0. 937
0.919
0. 928
0. 933
0. 938
0. 933
0. 933
0. 925
0. 931
0. 935
0. 930
0. 937
0.935
0.922
0.931
0. 931
0. 935
0. 935
0. 934


0.918
0. 928
0. 928
0. 92 7
0. 932
0.904
0.923
0. 925
0. 925
0. 930
0. 937
0.916
0.921
0.924
0. 928
0. 930
0. 930
0.911
0.924
0. 931
0.923
0. 931
0. 932
0.917
0. 92 7
0. 930
0. 928
0. 930
0. 931
0.913
0.923
0. 928
0. 928
0. 930
0. 92 7
0.923
0.923
0. 929
0. 927
0. 929
0.923


0.915
0.916
0.924
0. 928
0. 931
0.906
0.918
0.922
0. 92 7
0.923
0. 930
0.902
0.922
0. 925
0. 928
0. 928
0. 925
0.906
0.919
0. 926
0. 926
0. 928
0. 92 7
0.909
0.919
0.920
0.922
0.924
0.920
0.914
0.922
0.923
0.924
0.918
0.921
0.918
0.924
0.923
0.921
0.916
0.919


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-4. Continued



n pi Ap2 2 4 6 8 10
400 0.00 0.05 0.931 0. 92 7 0. 929 0.923 0.924


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.935
0.942
0.940
0.945
0.946
0. 929
0.940
0.943
0.946
0.946
0.946
0.931
0.939
0.944
0.944
0.947
0.942
0.938
0.941
0.942
0.945
0.946
0.946
0.933
0.944
0.950
0.944
0.948
0.947
0.935
0.949
0.946
0.950
0.946
0.948
0.943
0.941
0.943
0.945
0.943
0.950


0.940
0.941
0.945
0.943
0.947
0. 932
0.942
0.942
0.942
0.942
0.947
0. 929
0.940
0.942
0.945
0.947
0.946
0. 933
0.944
0.940
0.944
0.944
0.945
0. 935
0.941
0.940
0.947
0.944
0.946
0. 935
0.940
0.942
0.942
0.946
0.949
0.940
0.943
0.944
0.943
0.943
0.947


0. 934
0.940
0.940
0.942
0.943
0. 928
0.939
0.941
0.938
0.941
0.946
0. 929
0.941
0. 936
0.942
0.942
0.940
0. 934
0. 935
0.941
0.942
0.943
0.943
0. 934
0. 936
0.941
0.944
0.940
0.943
0. 936
0.942
0.945
0. 938
0.942
0.943
0.938
0.934
0.943
0.944
0. 938
0.945


0. 934
0. 937
0.941
0.947
0.942
0. 928
0. 938
0.940
0. 939
0.942
0.945
0. 930
0. 93 7
0. 936
0.943
0. 939
0.942
0. 935
0. 936
0. 938
0.941
0. 935
0.940
0. 930
0. 937
0.941
0. 939
0.944
0.944
0. 932
0. 936
0. 937
0.941
0. 938
0. 939
0. 935
0. 935
0. 938
0. 939
0. 932
0. 936


0. 936
0. 939
0. 937
0. 937
0. 938
0. 92 7
0. 936
0. 937
0. 936
0.941
0. 937
0. 929
0. 931
0. 936
0. 935
0.941
0. 936
0. 929
0. 937
0. 939
0. 938
0. 938
0. 939
0. 930
0. 931
0. 934
0. 939
0. 933
0. 934
0. 931
0. 937
0. 939
0. 937
0. 939
0. 937
0. 932
0. 938
0. 935
0. 935
0. 935
0. 92 7


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-4. Continued


k
6
0.937
0.942
0.945
0.944
0.948
0.945
0.940
0.946
0.942
0.940
0.947
0.941
0.939
0.946
0.945
0.946
0.944
0.946
0.933
0.941
0.945
0.945
0.941
0.948
0. 936
0.944
0.945
0.944
0.948
0.945
0.940
0.941
0.945
0.943
0.941
0.945
0.944
0.945
0.946
0.941
0.945
0.945


0.00


AP2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


2
0.931
0.945
0.947
0.945
0.942
0.948
0.943
0.939
0.943
0.945
0.948
0.950
0.938
0.948
0.949
0.945
0.946
0.952
0.939
0.945
0.943
0.944
0.949
0.948
0.937
0.949
0.949
0.949
0.949
0.945
0.941
0.946
0.944
0.950
0.946
0.951
0.940
0.945
0.948
0.952
0.950
0.948


4
0.935
0.943
0.946
0.949
0.950
0.944
0. 933
0.942
0.945
0.950
0.947
0.949
0.936
0.944
0.946
0.949
0.947
0.946
0.940
0.942
0.951
0.945
0.945
0.945
0.943
0.941
0.945
0.945
0.949
0.949
0.941
0.949
0.941
0.950
0.946
0.947
0.945
0.946
0.945
0.942
0.947
0.948


8
0. 92 7
0.941
0.942
0.941
0.944
0.947
0. 935
0.940
0.942
0.943
0.944
0.942
0. 938
0.940
0.940
0.943
0.941
0.947
0. 938
0.943
0.943
0.941
0.947
0.941
0. 934
0.943
0.944
0.946
0.944
0.943
0.943
0.946
0.942
0.943
0.946
0.945
0.942
0. 939
0.943
0.940
0.945
0.942


10
0. 929
0. 936
0. 937
0.942
0.944
0.941
0. 932
0.940
0.941
0.942
0.940
0.943
0. 936
0.941
0.945
0.943
0.944
0.940
0. 936
0. 938
0.943
0. 938
0.941
0. 938
0. 939
0.940
0.944
0. 939
0.943
0.943
0. 936
0.941
0.948
0.944
0.941
0. 938
0.944
0.944
0.942
0. 937
0.944
0.944


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-4. Continued



n pf Ap2 2 4 6 8 10


0.00


0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.939
0.944
0.947
0.948
0.948
0.949
0.938
0.943
0.945
0.940
0.942
0.951
0.938
0.944
0.950
0.946
0.949
0.947
0.944
0.949
0.947
0.943
0.947
0.951
0.946
0.947
0.946
0.947
0.948
0.948
0.947
0.948
0.950
0.949
0.951
0.953
0.945
0.953
0.948
0.943
0.947
0.950


0.940
0.943
0.941
0.951
0.949
0.951
0.942
0.949
0.942
0.943
0.945
0.948
0.940
0.944
0.944
0.946
0.947
0.948
0. 938
0.945
0.949
0.949
0.948
0.945
0. 938
0.947
0.948
0.946
0.947
0.945
0.940
0.947
0.946
0.944
0.946
0.948
0.944
0.948
0.946
0.948
0.948
0.951


0.939
0.946
0.946
0.943
0.948
0.946
0. 939
0.942
0.946
0.948
0.946
0.946
0. 938
0.944
0.941
0.947
0.944
0.943
0.942
0.946
0.945
0.946
0.944
0.948
0.941
0.945
0.944
0.947
0.950
0.946
0.951
0.949
0.949
0.944
0.946
0.951
0.945
0.948
0.943
0.945
0.945
0.944


0. 93 7
0.942
0.945
0.944
0.944
0.945
0. 938
0.941
0.947
0.946
0.947
0.947
0. 935
0.945
0.945
0.945
0.948
0.946
0. 939
0.942
0.948
0.949
0. 939
0.948
0.944
0.944
0.943
0.944
0.943
0.946
0.942
0.943
0.944
0.949
0.947
0.944
0.943
0.944
0.941
0.946
0.943
0.945


0. 936
0.945
0.945
0.946
0.946
0.944
0. 939
0. 938
0.943
0.943
0.946
0.944
0.940
0. 939
0.946
0.946
0.946
0.941
0. 939
0.942
0.943
0.943
0.944
0.943
0.942
0.942
0.941
0.944
0.946
0.941
0. 939
0.941
0. 939
0.944
0.943
0.942
0. 937
0.943
0.942
0.942
0.944
0.943


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-4. Continued



n p~ Ap2 2 4 6 8 10
1000 0.00 0.05 0.940 0.943 0. 938 0. 939 0. 937


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.947
0.947
0.947
0.949
0.953
0.945
0.950
0.945
0.950
0.949
0.949
0.944
0.949
0.948
0.947
0.950
0.947
0.945
0.946
0.945
0.948
0.947
0.951
0.943
0.944
0.947
0.948
0.947
0.951
0.946
0.950
0.949
0.951
0.946
0.948
0.946
0.951
0.946
0.951
0.944
0.947


0.946
0.945
0.947
0.947
0.950
0.946
0.944
0.948
0.951
0.949
0.951
0.944
0.941
0.944
0.946
0.948
0.947
0. 93 7
0.949
0.948
0.948
0.949
0.948
0.944
0.945
0.949
0.947
0.948
0.946
0.946
0.943
0.949
0.948
0.947
0.945
0.943
0.945
0.944
0.950
0.949
0.951


0.942
0.945
0.944
0.946
0.949
0.945
0.945
0.943
0.946
0.947
0.946
0.945
0.946
0.947
0.943
0.949
0.949
0.940
0.943
0.946
0.945
0.947
0.947
0.948
0.948
0.945
0.944
0.949
0.950
0.945
0.948
0.945
0.948
0.946
0.951
0.942
0.946
0.944
0.947
0.944
0.949


0.944
0.941
0.948
0.948
0.945
0.944
0.946
0.945
0.944
0.948
0.945
0.943
0.944
0.947
0.945
0.948
0.946
0.946
0.943
0.944
0.948
0.943
0.945
0.945
0.946
0.940
0.947
0.948
0.949
0.946
0.942
0.944
0.943
0.947
0.947
0.945
0.945
0.943
0.945
0.946
0.944


0.943
0.946
0.941
0.950
0.948
0.943
0.942
0.947
0.944
0.944
0.950
0.940
0.944
0.946
0.945
0.947
0.945
0.942
0.944
0.946
0.945
0.943
0.950
0.940
0.946
0.952
0.949
0.945
0.947
0.947
0.943
0.945
0.946
0.948
0.941
0.943
0.946
0.948
0.942
0.941
0.943


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-4. Continued



n p~ Ap2 2 4 6 8 10
1500 0.00 0.05 0.946 0.942 0.944 0.944 0.944


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.949
0.946
0.949
0.947
0.950
0.945
0.945
0.950
0.945
0.949
0.949
0.945
0.951
0.949
0.948
0.951
0.947
0.951
0.950
0.947
0.950
0.949
0.953
0.944
0.949
0.949
0.947
0.946
0.947
0.948
0.949
0.947
0.951
0.952
0.947
0.951
0.948
0.948
0.953
0.949
0.950


0.947
0.949
0.944
0.946
0.948
0.945
0.944
0.949
0.951
0.947
0.948
0.944
0.943
0.950
0.949
0.947
0.946
0.942
0.950
0.949
0.947
0.952
0.952
0.944
0.949
0.955
0.950
0.949
0.952
0.948
0.951
0.951
0.951
0.947
0.952
0.950
0.948
0.951
0.952
0.951
0.951


0.949
0.949
0.949
0.947
0.949
0.949
0.948
0.954
0.953
0.948
0.947
0.945
0.945
0.947
0.950
0.946
0.948
0.945
0.947
0.946
0.950
0.954
0.949
0.942
0.947
0.947
0.947
0.949
0.947
0.947
0.950
0.950
0.947
0.948
0.945
0.944
0.948
0.949
0.944
0.948
0.952


0.943
0.947
0.945
0.945
0.945
0.942
0.945
0.947
0.946
0.945
0.949
0.940
0.945
0.952
0.948
0.949
0.952
0.941
0.947
0.950
0.949
0.950
0.949
0.942
0.948
0.948
0.946
0.947
0.947
0.945
0.950
0.946
0.948
0.947
0.950
0.949
0.947
0.948
0.946
0.946
0.949


0.945
0.945
0.947
0.946
0.944
0.946
0.947
0.952
0.945
0.949
0.944
0.944
0.948
0.945
0.948
0.947
0.952
0.943
0.945
0.946
0.948
0.941
0.944
0.948
0.948
0.944
0.945
0.945
0.949
0.943
0.942
0.949
0.945
0.947
0.949
0.945
0.947
0.949
0.948
0.941
0.947


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-4. Continued



n p~ Ap2 2 4 6 8 10
2000 0.00 0.05 0.946 0.945 0.945 0.943 0.940


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.950
0.946
0.947
0.945
0.951
0.949
0.949
0.947
0.946
0.945
0.947
0.951
0.946
0.949
0.948
0.948
0.945
0.947
0.945
0.946
0.949
0.945
0.948
0.948
0.947
0.946
0.955
0.949
0.949
0.949
0.948
0.949
0.946
0.948
0.948
0.948
0.947
0.949
0.953
0.946
0.953


0.948
0.944
0.944
0.945
0.949
0.948
0.952
0.945
0.954
0.950
0.943
0.946
0.947
0.952
0.951
0.950
0.945
0.948
0.951
0.947
0.948
0.951
0.949
0.951
0.949
0.951
0.951
0.947
0.951
0.944
0.951
0.946
0.949
0.953
0.949
0.950
0.946
0.949
0.951
0.951
0.952


0.945
0.948
0.949
0.949
0.947
0.943
0.951
0.945
0.949
0.948
0.950
0.944
0.943
0.950
0.946
0.947
0.951
0.945
0.945
0.950
0.948
0.950
0.950
0.948
0.948
0.950
0.949
0.948
0.943
0.946
0.947
0.950
0.948
0.951
0.949
0.943
0.949
0.950
0.950
0.947
0.950


0.948
0.952
0.948
0.948
0.944
0.946
0.947
0.947
0.949
0.946
0.950
0.947
0.948
0.951
0.952
0.944
0.949
0.950
0.948
0.947
0.945
0.947
0.945
0.947
0.947
0.951
0.948
0.950
0.952
0.946
0.952
0.947
0.948
0.950
0.952
0.944
0.954
0.948
0.949
0.951
0.948


0.944
0.950
0.949
0.945
0.951
0.946
0.944
0.948
0.948
0.948
0.950
0.944
0.943
0.949
0.950
0.949
0.946
0.944
0.945
0.948
0.945
0.946
0.945
0.948
0.944
0.949
0.944
0.948
0.948
0.948
0.945
0.947
0.950
0.947
0.949
0.945
0.941
0.946
0.948
0.944
0.943


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-5. Empirical Coverage Probabilities for Normal Predictors and Nonnormal Errors.


n pZ Ap2 2 4 6 8 10
200 0.00 0.05 0.910 0.906 0.902 0.901 0.898


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.919
0.921
0.920
0.916
0.916
0.913
0.924
0. 925
0. 925
0.923
0.922
0.915
0. 927
0. 928
0. 931
0. 929
0. 927
0.920
0. 930
0. 934
0. 932
0. 932
0. 934
0.924
0. 932
0. 935
0. 935
0. 93 7
0. 938
0. 926
0. 935
0. 93 7
0. 938
0.941
0. 939
0. 932
0. 938
0.940
0.942
0.941
0.943


0.917
0.920
0.920
0.917
0.916
0.908
0.921
0. 926
0.923
0.924
0.921
0.912
0. 926
0.924
0. 927
0. 927
0. 925
0.916
0. 927
0. 929
0. 929
0. 930
0. 930
0.919
0. 930
0. 931
0. 932
0. 933
0. 935
0.924
0. 931
0. 932
0. 935
0. 934
0. 936
0. 925
0. 93 7
0. 93 7
0. 938
0. 93 7
0. 93 7


0.920
0.919
0.915
0.914
0.913
0.906
0.918
0.921
0.920
0.918
0.916
0.907
0.922
0.922
0. 925
0.924
0.923
0.914
0.924
0. 925
0. 925
0. 926
0. 928
0.914
0. 926
0. 930
0. 92 7
0. 929
0. 931
0.920
0. 928
0. 928
0. 933
0. 930
0. 931
0. 926
0. 930
0. 930
0. 930
0. 930
0. 930


0.910
0.916
0.914
0.913
0.909
0.902
0.915
0.918
0.914
0.918
0.914
0.909
0.916
0.922
0.921
0.919
0.918
0.909
0.918
0.923
0.920
0.923
0.922
0.913
0.921
0. 925
0. 925
0. 925
0.921
0.915
0.924
0. 925
0. 928
0.924
0. 92 7
0.920
0. 925
0. 925
0. 925
0. 926
0.923


0.910
0.909
0.912
0.907
0.906
0.901
0.913
0.915
0.911
0.911
0.909
0.904
0.916
0.920
0.919
0.913
0.915
0.906
0.917
0.918
0.920
0.919
0.916
0.910
0.918
0.919
0.919
0.917
0.917
0.913
0.921
0.920
0.921
0.917
0.918
0.914
0.921
0.920
0.920
0.917
0.915


0.10






0.20






0.30






0.40






0.50






0.60


Note: Bold results are estimated coverage probabilities between .94 and .96: italicized results are estimated
coverage probabilities between .925 and .975.











Table 3-5. Continued


k
6
0. 92 7
0. 931
0. 931
0. 925
0.922
0.920
0. 926
0. 930
0. 929
0. 929
0. 926
0. 925
0. 926
0. 933
0. 932
0. 930
0. 930
0. 930
0. 928
0. 933
0. 932
0. 933
0. 935
0. 934
0. 931
0. 938
0. 93 7
0. 93 7
0. 93 7
0. 938
0. 935
0. 93 7
0. 938
0. 936
0. 938
0.940
0. 936
0. 938
0.942
0. 939
0.941
0. 939


A 2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


2
0. 927
0. 930
0. 931
0. 927
0.924
0.918
0. 929
0. 933
0. 932
0. 930
0. 929
0.924
0. 930
0. 934
0. 932
0. 933
0. 935
0. 932
0. 933
0. 936
0. 938
0. 937
0. 935
0. 938
0. 936
0. 937
0. 939
0.940
0.940
0.943
0. 93 7
0. 939
0.942
0.945
0.945
0.944
0. 939
0.944
0.943
0.946
0.947
0.944


4
0. 925
0. 929
0. 932
0. 929
0.921
0.920
0. 926
0. 932
0. 931
0. 930
0. 928
0.924
0. 927
0. 934
0. 932
0. 933
0. 934
0. 931
0. 930
0. 938
0. 93 7
0. 93 7
0. 936
0. 936
0. 935
0. 938
0. 938
0. 939
0. 935
0. 93 7
0. 93 7
0. 939
0.940
0. 939
0.941
0.941
0. 936
0.943
0.940
0.943
0.944
0.944


8
0.922
0. 928
0.923
0.923
0.921
0.921
0. 926
0. 928
0. 92 7
0. 928
0. 925
0.921
0. 926
0. 930
0. 929
0. 930
0. 930
0. 926
0. 930
0. 932
0. 933
0. 932
0. 931
0. 931
0. 932
0. 934
0.934
0.934
0.934
0.933
0. 932
0.937
0. 935
0.937
0. 936
0. 939
0.933
0.937
0. 938
0. 935
0. 938
0. 938


10
0.922
0. 927
0.924
0.922
0.921
0.916
0.920
0. 927
0. 926
0. 925
0. 925
0.919
0. 925
0. 929
0. 925
0. 927
0. 926
0. 925
0.924
0. 932
0. 931
0. 930
0. 931
0. 930
0. 928
0. 934
0. 931
0. 932
0. 931
0. 930
0. 931
0. 933
0. 934
0. 933
0. 934
0. 931
0. 931
0. 934
0. 933
0. 934
0. 932
0. 932


0.10







0.20






0.30






0.40







0.50






0.60











Table 3-5. Continued


k

6
0. 931
0. 931
0. 931
0. 930
0. 926
0.921
0. 932
0. 934
0. 934
0. 931
0. 92 7
0. 92 7
0. 933
0. 935
0. 934
0. 934
0. 931
0. 929
0. 936
0. 936
0. 938
0. 93 7
0. 93 7
0. 933
0. 936
0.940
0. 939
0. 938
0. 939
0. 939
0. 938
0. 939
0.941
0.943
0.942
0. 939
0.940
0.941
0.941
0.942
0.942
0.942


Ap2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


2
0. 932
0. 932
0. 933
0. 929
0.922
0.924
0. 935
0. 935
0. 93 7
0. 934
0. 929
0. 927
0. 936
0. 93 7
0.940
0. 938
0. 935
0. 934
0. 938
0.940
0.940
0. 938
0. 939
0. 938
0. 93 7
0.942
0.941
0.940
0.941
0.940
0.941
0.942
0.945
0.944
0.943
0.945
0.944
0.945
0.943
0.944
0.947
0.946


4
0. 933
0. 931
0. 930
0. 930
0. 926
0.922
0. 935
0. 936
0. 932
0. 932
0. 930
0. 927
0. 935
0. 938
0. 935
0. 936
0. 931
0. 935
0. 936
0. 938
0. 93 7
0. 935
0. 93 7
0. 938
0. 938
0.940
0.940
0. 939
0.941
0.940
0.941
0.943
0.941
0.941
0.944
0.944
0.941
0.945
0.942
0.943
0.943
0.946


8
0. 932
0. 932
0. 930
0. 926
0.923
0.918
0. 930
0. 933
0. 931
0. 931
0. 928
0. 926
0. 932
0. 935
0.934
0. 935
0. 932
0. 930
0.934
0.935
0.935
0.936
0.936
0. 934
0.935
0.937
0. 938
0. 93 7
0. 939
0. 936
0.937
0. 93 7
0. 938
0. 939
0. 939
0.940
0.939
0.942
0.942
0.943
0.941
0. 939


10
0. 927
0. 930
0. 928
0. 925
0.922
0.922
0. 930
0. 932
0. 930
0. 928
0.922
0. 926
0. 933
0. 935
0. 932
0. 932
0. 928
0. 929
0. 935
0. 934
0. 932
0. 933
0. 932
0. 931
0. 935
0. 936
0. 935
0. 93 7
0. 934
0. 936
0. 936
0. 938
0. 939
0. 938
0. 93 7
0. 936
0. 93 7
0. 938
0. 936
0. 93 7
0. 938
0. 938


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-5. Continued


k

6
0. 933
0. 934
0. 933
0. 928
0. 926
0.924
0. 934
0. 938
0. 934
0. 931
0. 930
0. 928
0. 93 7
0. 939
0. 938
0. 93 7
0. 932
0. 933
0. 938
0. 939
0. 939
0. 939
0. 938
0. 938
0.941
0.941
0.942
0.940
0. 938
0. 938
0.941
0.942
0.943
0.942
0.943
0.944
0.940
0.945
0.943
0.943
0.945
0.944


Ap2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


2
0. 936
0. 93 7
0. 933
0. 932
0. 926
0.923
0. 939
0. 938
0. 934
0. 933
0. 930
0. 929
0. 93 7
0. 939
0. 938
0. 93 7
0. 934
0. 933
0.941
0.942
0.941
0.941
0. 938
0. 938
0.940
0.943
0.942
0.942
0.941
0.943
0.943
0.943
0.945
0.944
0.946
0.945
0.943
0.946
0.946
0.945
0.943
0.948


4
0. 936
0. 933
0. 933
0. 929
0. 928
0.921
0. 938
0. 934
0. 935
0. 935
0. 930
0. 926
0. 93 7
0.940
0. 938
0. 935
0. 935
0. 932
0. 936
0.943
0.940
0. 938
0. 93 7
0. 93 7
0.940
0.941
0.941
0.941
0.941
0. 938
0.942
0.940
0.942
0.943
0.944
0.943
0.940
0.944
0.944
0.945
0.946
0.944


8
0. 936
0. 933
0. 932
0. 928
0.924
0.921
0. 935
0. 933
0. 936
0. 930
0. 929
0. 925
0.935
0. 938
0. 938
0.934
0. 934
0. 931
0. 938
0. 935
0. 938
0. 93 7
0. 93 7
0. 936
0. 938
0. 939
0.940
0. 93 7
0. 938
0. 939
0.942
0.942
0.942
0.938
0.939
0.942
0.943
0.942
0.941
0.943
0.942
0.941


10
0. 933
0. 931
0. 930
0. 926
0.924
0.920
0. 934
0. 934
0. 932
0. 930
0. 928
0.923
0. 93 7
0. 934
0. 935
0. 933
0. 933
0. 930
0. 936
0. 939
0. 93 7
0. 934
0. 935
0. 933
0. 938
0. 93 7
0. 938
0. 939
0. 936
0. 93 7
0. 938
0. 939
0.940
0.940
0.941
0.941
0.940
0.942
0.942
0.941
0.942
0.941


0.10






0.20






0.30






0.40






0.50






0.60














Ap2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


p
0.00


2
0. 938
0. 938
0. 936
0. 933
0. 927
0.924
0. 938
0. 939
0. 936
0. 936
0. 929
0. 929
0.940
0. 939
0. 939
0. 935
0. 936
0. 934
0.940
0.941
0.942
0.941
0.941
0. 939
0.943
0.943
0.943
0.942
0.941
0.941
0.943
0.946
0.942
0.943
0.943
0.946
0.945
0.946
0.945
0.946
0.947
0.948


4
0. 939
0. 93 7
0. 932
0. 930
0. 930
0.924
0. 938
0. 936
0. 939
0. 934
0. 930
0. 928
0. 939
0.941
0. 939
0. 93 7
0. 934
0. 934
0. 939
0.941
0. 939
0. 939
0. 93 7
0. 939
0.940
0.943
0.941
0.941
0. 939
0.942
0.944
0.944
0.945
0.944
0.943
0.945
0.944
0.942
0.945
0.947
0.947
0.947


8
0.935
0.936
0. 934
0. 930
0.923
0.922
0.938
0. 936
0.935
0. 931
0. 928
0. 925
0.938
0. 938
0.938
0. 934
0. 934
0. 934
0. 939
0.940
0. 939
0. 938
0.938
0. 936
0.940
0.943
0.940
0. 938
0.940
0.940
0.940
0.943
0.941
0.942
0.941
0.941
0.941
0.943
0.941
0.943
0.943
0.942


10
0. 935
0. 935
0. 93 7
0. 927
0. 927
0.923
0. 93 7
0. 934
0. 935
0. 935
0. 929
0. 927
0. 93 7
0. 939
0. 936
0. 934
0. 933
0. 931
0.940
0. 938
0. 936
0. 93 7
0. 936
0. 938
0. 938
0.942
0.940
0. 93 7
0.941
0.941
0.940
0.944
0.942
0.940
0.940
0.940
0.941
0.940
0.942
0.941
0.941
0.945


1000


0.10






0.20






0.30






0.40






0.50






0.60


Table 3-5. Continued


k
6
0. 936
0. 93 7
0. 932
0. 931
0. 92 7
0.922
0. 93 7
0. 938
0. 93 7
0. 933
0. 931
0. 928
0.940
0. 938
0. 936
0. 934
0. 933
0. 932
0.940
0.941
0.941
0. 938
0. 936
0. 938
0.941
0.941
0.943
0. 939
0.940
0. 939
0.943
0.943
0.942
0.942
0.943
0.944
0.944
0.944
0.945
0.945
0.944
0.946










Table 3-5. Continued


k
6
0. 939
0.940
0. 935
0. 930
0. 929
0.921
0.940
0.940
0. 939
0. 933
0. 932
0. 930
0.944
0.940
0.940
0. 935
0. 93 7
0. 936
0.943
0.943
0.941
0.942
0. 938
0. 93 7
0.943
0.944
0.942
0.943
0.942
0.944
0.945
0.941
0.945
0.943
0.943
0.943
0.945
0.946
0.945
0.946
0.947
0.948


A 2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


2
0.941
0. 939
0. 93 7
0. 932
0. 929
0.924
0.943
0.941
0. 93 7
0. 934
0. 933
0. 931
0.944
0.940
0. 939
0. 939
0. 933
0. 935
0.942
0.945
0.943
0. 939
0. 939
0.941
0.945
0.943
0.944
0.942
0.942
0.942
0.944
0.945
0.944
0.944
0.945
0.947
0.947
0.947
0.944
0.946
0.948
0.950


4
0.940
0. 93 7
0. 935
0. 933
0. 927
0. 925
0.941
0. 939
0. 939
0. 935
0. 931
0. 928
0.942
0.943
0.940
0. 939
0. 935
0. 934
0.942
0.944
0.940
0.941
0. 939
0.940
0.944
0.941
0.941
0.941
0.943
0.941
0.946
0.943
0.944
0.946
0.945
0.945
0.947
0.946
0.946
0.945
0.944
0.948


8
0. 939
0. 93 7
0. 934
0. 931
0. 926
0.924
0.940
0.940
0. 935
0. 933
0. 931
0. 926
0.941
0.941
0. 93 7
0. 93 7
0.934
0.931
0.941
0.943
0.942
0.939
0. 93 7
0.938
0.943
0.942
0.940
0.940
0.941
0.942
0.943
0.943
0.943
0.942
0.946
0.944
0.947
0.943
0.944
0.942
0.947
0.946


10
0. 938
0. 936
0. 935
0. 932
0. 928
0.922
0. 938
0. 938
0. 936
0. 934
0. 930
0. 931
0.942
0. 939
0.941
0. 936
0. 935
0. 934
0.944
0.942
0. 939
0. 93 7
0. 936
0. 938
0.942
0.943
0.943
0.940
0. 939
0.941
0.945
0.944
0.942
0.945
0.941
0.944
0.946
0.943
0.945
0.944
0.944
0.943


1500


0.10






0.20






0.30






0.40






0.50






0.60














Ap2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


p
0.00






0.10






0.20






0.30






0.40






0.50






0.60


2
0.944
0.940
0. 936
0. 933
0. 928
0.924
0.941
0.942
0.940
0. 936
0. 929
0. 931
0.943
0.941
0.942
0.940
0. 935
0. 934
0.943
0.942
0.942
0.943
0.941
0. 939
0.947
0.945
0.942
0.941
0.943
0.942
0.945
0.946
0.944
0.944
0.944
0.946
0.946
0.948
0.946
0.948
0.948
0.949


4
0.944
0. 93 7
0. 938
0. 933
0. 930
0. 925
0.941
0.941
0. 935
0. 933
0. 931
0. 930
0.944
0.940
0.940
0. 93 7
0. 936
0. 935
0.944
0.944
0.940
0.940
0. 938
0. 938
0.947
0.943
0.944
0.942
0.944
0.941
0.948
0.945
0.945
0.944
0.944
0.946
0.945
0.947
0.949
0.946
0.949
0.949


8
0.940
0. 939
0. 936
0. 931
0. 931
0.921
0.941
0.940
0. 938
0. 934
0. 930
0. 930
0.942
0.942
0. 939
0. 935
0.935
0. 932
0.944
0.943
0.940
0.940
0. 938
0. 939
0.945
0.943
0.944
0.942
0.942
0.941
0.947
0.946
0.943
0.944
0.943
0.946
0.945
0.945
0.943
0.946
0.946
0.946


10
0.941
0. 939
0. 935
0. 932
0. 926
0.924
0.941
0.940
0. 93 7
0. 936
0. 930
0. 930
0.943
0.942
0.940
0. 936
0. 935
0. 933
0.943
0.942
0.940
0. 938
0. 936
0. 938
0.945
0.943
0.944
0.940
0.941
0.942
0.944
0.944
0.942
0.942
0.942
0.943
0.945
0.944
0.946
0.944
0.946
0.946


2000


Table 3-5. Continued


k
6
0.940
0.940
0. 935
0. 933
0. 928
0.923
0.942
0.942
0. 938
0. 934
0. 932
0. 929
0.943
0.940
0.940
0. 939
0. 936
0. 933
0.943
0.943
0.940
0.942
0.941
0. 938
0.946
0.942
0.943
0.941
0.942
0.941
0.946
0.945
0.946
0.944
0.943
0.946
0.947
0.945
0.948
0.946
0.947
0.948











Table 3-6. Empirical Coverage Probabilities for Nonnormal Predictors and Normal Errors.
k

n P: Ap2 2 4 6 8 10


0.00


0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.904
0.918
0.919
0.921
0.919
0.916
0.910
0.919
0.921
0.919
0.918
0.912
0.910
0.918
0.920
0.919
0.917
0.914
0.914
0.919
0.918
0.913
0.906
0.902
0.918
0.917
0.914
0.910
0.901
0.894
0.916
0.915
0.907
0.899
0.890
0.881
0.917
0.910
0.896
0.887
0.871
0.866


0.904
0.914
0.918
0.917
0.917
0.912
0.908
0.914
0.918
0.914
0.915
0.915
0.911
0.916
0.917
0.915
0.910
0.907
0.910
0.916
0.915
0.911
0.904
0.901
0.913
0.916
0.911
0.906
0.897
0.889
0.913
0.911
0.906
0.894
0.887
0.878
0.913
0.907
0.896
0.882
0.870
0.860


0.898
0.914
0.916
0.917
0.913
0.910
0.903
0.912
0.915
0.911
0.913
0.907
0.907
0.914
0.913
0.911
0.909
0.904
0.909
0.913
0.911
0.907
0.903
0.898
0.909
0.912
0.909
0.901
0.894
0.888
0.909
0.907
0.900
0.891
0.881
0.875
0.910
0.901
0.891
0.878
0.863
0.857


0.898
0.909
0.911
0.910
0.911
0.910
0.897
0.910
0.911
0.911
0.906
0.903
0.902
0.908
0.908
0.907
0.903
0.899
0.903
0.910
0.907
0.903
0.901
0.892
0.905
0.910
0.903
0.893
0.890
0.879
0.906
0.905
0.898
0.886
0.876
0.865
0.905
0.897
0.880
0.873
0.858
0.848


0.892
0.906
0.909
0.907
0.908
0.902
0.898
0.905
0.906
0.905
0.902
0.899
0.899
0.905
0.905
0.904
0.900
0.893
0.901
0.906
0.903
0.900
0.893
0.886
0.902
0.904
0.897
0.888
0.885
0.873
0.901
0.900
0.890
0.879
0.871
0.860
0.900
0.893
0.879
0.864
0.853
0.841


0.10






0.20






0.30






0.40






0.50






0.60


Note: Bold results are estimated coverage probabilities between .94 and .96: italicized results are estimated
coverage probabilities between .925 and .975.














2
0.924
0. 930
0. 926
0. 927
0.922
0.919
0. 927
0. 929
0. 927
0.924
0.922
0.916
0. 926
0. 929
0. 926
0. 925
0.918
0.915
0. 928
0. 929
0.924
0.917
0.912
0.906
0. 929
0.924
0.918
0.914
0.907
0.898
0. 926
0.923
0.913
0.901
0.895
0.887
0. 926
0.915
0.901
0.891
0.877
0.870


4
0. 925
0. 928
0. 929
0. 925
0. 925
0.922
0.924
0. 929
0. 927
0. 925
0.921
0.919
0. 928
0. 926
0. 927
0.922
0.916
0.914
0. 929
0. 927
0.923
0.918
0.912
0.904
0. 926
0.924
0.918
0.914
0.903
0.897
0. 926
0.920
0.908
0.903
0.891
0.886
0. 925
0.916
0.901
0.887
0.875
0.866


8
0.922
0.928
0.924
0.924
0.919
0.914
0.923
0. 926
0.922
0.923
0.916
0.911
0.922
0.924
0.921
0.919
0.914
0.912
0.924
0.924
0.920
0.912
0.905
0.902
0.923
0.920
0.915
0.911
0.897
0.895
0.922
0.918
0.904
0.898
0.887
0.878
0.920
0.909
0.895
0.881
0.870
0.859


10
0.917
0.922
0.924
0.920
0.916
0.913
0.919
0.923
0.920
0.917
0.916
0.909
0.919
0.922
0.921
0.916
0.914
0.908
0.919
0.921
0.918
0.911
0.904
0.899
0.922
0.919
0.912
0.904
0.894
0.891
0.922
0.914
0.903
0.892
0.884
0.873
0.920
0.908
0.893
0.876
0.866
0.856


0.10






0.20






0.30






0.40






0.50






0.60


Table 3-6. Continued



n pl
400 0.00


k
6
0.920
0. 928
0. 926
0. 925
0.923
0.917
0.922
0. 928
0. 926
0.922
0.917
0.917
0.924
0. 926
0.923
0.919
0.917
0.912
0. 926
0. 927
0.923
0.916
0.911
0.904
0. 925
0.924
0.915
0.909
0.901
0.896
0. 925
0.920
0.909
0.901
0.891
0.882
0.924
0.910
0.898
0.888
0.873
0.861


Ap2
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30











Table 3-6. Continued



n P: Ap2 2 4 6 8 10
600 0.00 0.05 0. 931 0. 929 0. 930 0. 928 0. 927


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.934
0.934
0.932
0. 927
0.922
0. 934
0. 933
0. 931
0. 926
0.924
0.918
0. 932
0. 930
0. 930
0. 926
0.921
0.917
0. 932
0. 931
0. 927
0.919
0.915
0.909
0. 931
0. 930
0.922
0.914
0.908
0.898
0. 931
0. 926
0.914
0.905
0.892
0.886
0. 930
0.918
0.902
0.887
0.880
0.868


0.933
0.932
0. 928
0.924
0.922
0.933
0.934
0. 929
0. 925
0.924
0.921
0.932
0.934
0.930
0. 927
0.919
0.915
0.930
0.931
0. 927
0.916
0.915
0.905
0.931
0. 927
0.921
0.914
0.907
0.902
0.933
0.921
0.914
0.904
0.893
0.885
0. 925
0.917
0.903
0.889
0.876
0.867


0. 931
0. 928
0. 927
0.923
0.922
0. 928
0. 931
0. 928
0. 927
0.921
0.916
0. 929
0. 931
0. 925
0.923
0.916
0.914
0. 929
0. 930
0.924
0.917
0.913
0.906
0. 931
0. 926
0.923
0.911
0.906
0.895
0. 930
0.919
0.912
0.904
0.892
0.884
0. 929
0.915
0.900
0.889
0.876
0.863


0. 930
0.929
0.928
0.922
0.919
0.928
0.931
0.928
0.923
0.919
0.916
0. 930
0. 930
0.925
0.920
0.917
0.911
0. 930
0. 927
0.920
0.917
0.913
0.908
0.929
0.923
0.917
0.911
0.903
0.894
0.928
0.918
0.910
0.901
0.887
0.880
0. 927
0.913
0.900
0.885
0.875
0.863


0. 928
0. 928
0. 926
0.922
0.918
0. 928
0. 929
0. 926
0.923
0.919
0.917
0. 929
0.930
0.923
0.919
0.917
0.912
0. 929
0. 927
0.920
0.912
0.910
0.904
0. 929
0.922
0.915
0.907
0.902
0.892
0. 926
0.919
0.907
0.897
0.890
0.878
0.924
0.910
0.896
0.881
0.869
0.863


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-6. Continued



n pi Ap2 2 4 6 8 10
800 0.00 0.05 0. 937 0.935 0. 931 0. 934 0.932


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0. 936
0.933
0.930
0. 925
0.924
0. 938
0. 934
0. 931
0. 928
0.923
0.920
0. 935
0.934
0.932
0. 926
0.922
0.918
0. 936
0.931
0. 927
0.920
0.916
0.909
0. 936
0. 933
0.922
0.916
0.907
0.902
0. 933
0. 926
0.916
0.907
0.896
0.885
0. 932
0.918
0.901
0.893
0.879
0.869


0.935
0.932
0. 929
0. 926
0.922
0.933
0.934
0.931
0. 929
0. 926
0.920
0.937
0.933
0. 928
0.924
0.920
0.914
0.937
0.932
0. 927
0.919
0.914
0.908
0.934
0.930
0.924
0.913
0.907
0.899
0.934
0. 925
0.912
0.906
0.893
0.885
0. 929
0.917
0.901
0.889
0.878
0.864


0. 936
0. 934
0. 929
0. 925
0.922
0. 934
0. 934
0. 932
0. 926
0. 926
0.921
0. 933
0. 933
0. 929
0. 925
0.918
0.915
0. 933
0. 930
0.923
0.921
0.913
0.907
0. 935
0. 930
0.921
0.910
0.906
0.899
0. 931
0. 925
0.913
0.908
0.895
0.884
0. 931
0.918
0.901
0.891
0.874
0.866


0. 934
0. 930
0.929
0.925
0.923
0.935
0.935
0.928
0. 926
0.921
0.918
0.935
0.933
0.928
0.924
0.919
0.912
0.931
0.929
0.923
0.917
0.913
0.907
0.935
0.928
0.920
0.913
0.903
0.897
0.932
0.923
0.912
0.902
0.891
0.883
0. 927
0.915
0.903
0.887
0.873
0.866


0.934
0.931
0. 928
0.922
0.922
0.932
0.931
0.931
0.924
0.921
0.918
0.932
0.931
0. 927
0.922
0.916
0.914
0.931
0. 929
0. 92 7
0.917
0.912
0.906
0.934
0. 927
0.920
0.910
0.900
0.895
0. 928
0.922
0.911
0.898
0.889
0.881
0. 928
0.914
0.899
0.887
0.872
0.861


0.10






0.20







0.30







0.40







0.50






0.60










Table 3-6. Continued



n pi Ap2 2 4 6 8 10


0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


0. 936
0. 938
0. 933
0. 931
0. 92 7
0.923
0. 93 7
0. 936
0. 932
0. 930
0. 92 7
0.920
0.941
0. 935
0. 930
0. 92 7
0.922
0.917
0. 938
0. 933
0. 928
0.923
0.917
0.911
0. 933
0. 932
0.924
0.915
0.907
0.901
0. 934
0. 928
0.914
0.906
0.897
0.889
0. 932
0.922
0.904
0.892
0.880
0.867


0. 938
0. 938
0. 934
0. 931
0. 92 7
0.922
0. 936
0. 936
0. 933
0. 930
0.924
0.923
0. 937
0. 935
0. 931
0. 928
0.921
0.916
0. 936
0. 936
0. 928
0.923
0.915
0.908
0. 938
0. 929
0.922
0.916
0.908
0.898
0. 936
0. 92 7
0.914
0.903
0.894
0.887
0. 932
0.917
0.906
0.888
0.880
0.867


0. 934
0. 936
0. 934
0. 929
0. 92 7
0.923
0. 936
0. 935
0. 931
0. 927
0. 925
0.915
0. 935
0. 933
0. 930
0. 925
0.920
0.916
0. 935
0. 934
0. 92 7
0.920
0.915
0.909
0. 934
0. 928
0.920
0.913
0.906
0.900
0. 934
0.924
0.914
0.902
0.893
0.884
0. 930
0.917
0.902
0.889
0.876
0.867


0. 935
0. 935
0. 934
0. 929
0. 928
0. 925
0. 936
0. 934
0. 929
0. 928
0. 926
0.920
0. 938
0. 933
0. 929
0.924
0.920
0.913
0. 93 7
0. 932
0. 92 7
0.917
0.914
0.907
0. 932
0. 929
0.922
0.912
0.904
0.898
0. 933
0. 926
0.914
0.902
0.897
0.885
0. 931
0.918
0.902
0.885
0.874
0.866


0. 935
0. 934
0. 931
0. 92 7
0.924
0.922
0. 934
0. 932
0. 932
0. 926
0.923
0.920
0. 935
0. 932
0. 931
0.921
0.919
0.910
0. 935
0. 929
0. 928
0.920
0.914
0.907
0. 934
0. 928
0.922
0.912
0.902
0.898
0. 933
0. 925
0.915
0.902
0.894
0.882
0. 929
0.916
0.900
0.886
0.872
0.863


1000


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-6. Continued

k

n pi Ap2 2 4 6 8 10


0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


0.941
0. 939
0. 934
0. 932
0. 928
0.923
0.940
0. 936
0. 934
0. 930
0. 926
0.922
0.941
0. 93 7
0. 932
0. 928
0.922
0.918
0. 939
0. 933
0. 930
0.923
0.915
0.910
0. 93 7
0. 932
0. 925
0.918
0.909
0.901
0. 935
0. 92 7
0.919
0.906
0.896
0.890
0. 933
0.918
0.906
0.891
0.880
0.868


0. 939
0. 938
0. 932
0. 931
0. 928
0. 925
0.940
0. 93 7
0. 934
0. 928
0. 925
0.921
0.941
0. 936
0. 930
0. 92 7
0.922
0.915
0. 938
0. 933
0. 930
0.922
0.918
0.912
0. 938
0. 932
0.923
0.913
0.908
0.901
0. 934
0. 925
0.918
0.904
0.897
0.891
0. 935
0.920
0.903
0.891
0.875
0.867


0. 939
0. 938
0. 937
0. 929
0. 927
0.923
0. 939
0. 939
0. 935
0. 930
0. 925
0.923
0. 939
0. 934
0. 931
0. 926
0.921
0.917
0. 938
0. 935
0. 929
0.920
0.914
0.910
0. 938
0. 932
0.923
0.915
0.905
0.902
0. 938
0.924
0.914
0.907
0.896
0.886
0. 932
0.917
0.903
0.889
0.878
0.867


0.940
0. 938
0. 935
0. 931
0. 925
0.921
0. 938
0. 938
0. 933
0. 930
0.924
0.916
0. 938
0. 935
0. 930
0. 925
0.919
0.918
0. 93 7
0. 930
0. 929
0.922
0.915
0.908
0. 936
0. 931
0. 925
0.912
0.909
0.900
0. 935
0. 925
0.916
0.903
0.898
0.888
0. 931
0.917
0.905
0.891
0.878
0.867


0.941
0. 934
0. 936
0. 931
0. 928
0.921
0. 938
0. 934
0. 930
0. 929
0.924
0.917
0. 938
0. 935
0. 928
0. 925
0.920
0.913
0.940
0. 932
0. 92 7
0.921
0.913
0.907
0. 934
0. 929
0.919
0.913
0.908
0.898
0. 936
0.924
0.911
0.900
0.893
0.883
0. 933
0.915
0.903
0.890
0.878
0.866


1500


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-6. Continued



n pi Ap2 2 4 6 8 10
2000 0.00 0.05 0.941 0.941 0.940 0.941 0.939


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0. 939
0. 937
0.931
0. 926
0. 925
0.941
0.942
0.936
0. 931
0. 926
0.922
0.943
0.938
0.933
0. 925
0.923
0.917
0. 939
0. 936
0. 927
0.921
0.914
0.910
0.941
0.931
0.924
0.915
0.908
0.904
0. 939
0. 927
0.918
0.907
0.894
0.889
0. 934
0.918
0.906
0.892
0.879
0.872


0.938
0.935
0. 929
0. 927
0. 925
0.942
0.939
0.934
0.932
0. 926
0.920
0.942
0.937
0.931
0. 928
0.923
0.919
0.938
0.934
0. 928
0.920
0.917
0.907
0.939
0.933
0.924
0.915
0.908
0.899
0.937
0. 927
0.915
0.906
0.900
0.889
0.933
0.920
0.906
0.890
0.879
0.870


0. 939
0. 936
0. 930
0.924
0. 926
0.941
0. 938
0. 934
0. 928
0. 927
0.922
0.940
0. 93 7
0. 931
0. 927
0.921
0.917
0.942
0. 93 7
0. 926
0.923
0.915
0.911
0. 939
0. 934
0.923
0.916
0.904
0.897
0. 938
0. 925
0.915
0.905
0.897
0.888
0. 933
0.917
0.905
0.892
0.878
0.869


0.939
0.935
0.933
0. 927
0.924
0.939
0. 93 7
0. 936
0.931
0.925
0.920
0.940
0.938
0.932
0.925
0.921
0.916
0.941
0.935
0.928
0.921
0.916
0.912
0. 93 7
0.929
0.923
0.916
0.905
0.899
0.935
0. 926
0.914
0.907
0.896
0.885
0. 934
0.918
0.903
0.888
0.877
0.868


0.939
0.935
0.930
0. 928
0.922
0.941
0.937
0.933
0. 928
0. 926
0.924
0.941
0.937
0.930
0. 928
0.921
0.917
0.938
0.934
0. 928
0.924
0.914
0.908
0.941
0. 929
0.924
0.913
0.909
0.898
0.935
0. 925
0.914
0.905
0.895
0.881
0.930
0.916
0.903
0.890
0.875
0.866


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-7. Empirical Coverage Probabilities for Predictors Nonnormal and Errors Nonnormal.


n p~ AP2 2 4 6 8 10
200 0.00 0.05 0.903 0.900 0.897 0.894 0.891


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.911
0.909
0.904
0.897
0.891
0.908
0.914
0.910
0.906
0.902
0.894
0.910
0.915
0.912
0.906
0.901
0.896
0.913
0.916
0.912
0.906
0.899
0.894
0.916
0.915
0.909
0.903
0.895
0.888
0.916
0.912
0.905
0.896
0.888
0.879
0.916
0.908
0.896
0.885
0.874
0.863


0.908
0.907
0.901
0.896
0.890
0.904
0.911
0.908
0.904
0.899
0.893
0.906
0.911
0.909
0.904
0.897
0.893
0.909
0.913
0.908
0.902
0.897
0.892
0.913
0.913
0.906
0.899
0.892
0.886
0.913
0.909
0.900
0.893
0.882
0.875
0.913
0.905
0.892
0.881
0.870
0.860


0.905
0.904
0.898
0.893
0.888
0.900
0.909
0.906
0.901
0.896
0.891
0.903
0.909
0.905
0.901
0.895
0.891
0.906
0.908
0.905
0.900
0.893
0.888
0.907
0.908
0.903
0.894
0.888
0.883
0.909
0.906
0.897
0.887
0.878
0.871
0.910
0.902
0.889
0.875
0.864
0.856


0.902
0.900
0.895
0.890
0.885
0.896
0.903
0.901
0.898
0.893
0.886
0.901
0.906
0.901
0.897
0.893
0.887
0.902
0.905
0.902
0.896
0.889
0.883
0.904
0.904
0.898
0.891
0.883
0.875
0.904
0.901
0.894
0.882
0.873
0.865
0.906
0.895
0.883
0.870
0.857
0.847


0.899
0.898
0.893
0.888
0.881
0.892
0.901
0.898
0.892
0.888
0.882
0.896
0.901
0.899
0.893
0.886
0.881
0.899
0.901
0.896
0.891
0.884
0.878
0.900
0.901
0.895
0.885
0.877
0.868
0.902
0.897
0.888
0.876
0.867
0.857
0.901
0.891
0.877
0.863
0.849
0.838


0.10






0.20






0.30






0.40






0.50






0.60


Note: Bold results are estimated coverage probabilities between .94 and .96; italicized results are estimated
coverage probabilities between .925 and .975.











Table 3-7. Continued



n p~ Ap2 2 4 6 8 10
400 0.00 0.05 0.920 0.919 0.919 0.916 0.915


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.922
0.916
0.910
0.903
0.896
0.923
0.923
0.918
0.912
0.905
0.901
0. 926
0.924
0.919
0.911
0.905
0.900
0. 926
0.923
0.917
0.910
0.902
0.899
0. 926
0.923
0.916
0.907
0.898
0.891
0. 927
0.921
0.910
0.900
0.890
0.880
0. 925
0.913
0.900
0.887
0.876
0.864


0.921
0.916
0.907
0.902
0.895
0.923
0.922
0.917
0.910
0.904
0.899
0.923
0.920
0.916
0.910
0.903
0.898
0.924
0.922
0.915
0.907
0.902
0.896
0.924
0.919
0.913
0.905
0.898
0.889
0. 926
0.918
0.907
0.897
0.887
0.880
0.923
0.912
0.898
0.886
0.873
0.864


0.920
0.915
0.907
0.901
0.894
0.920
0.922
0.915
0.909
0.903
0.897
0.922
0.922
0.916
0.909
0.905
0.897
0.923
0.920
0.914
0.907
0.901
0.893
0.924
0.920
0.911
0.902
0.895
0.889
0.923
0.918
0.906
0.894
0.885
0.878
0.921
0.910
0.895
0.883
0.871
0.862


0.917
0.912
0.906
0.902
0.893
0.918
0.919
0.914
0.907
0.901
0.895
0.920
0.919
0.913
0.907
0.901
0.896
0.921
0.918
0.913
0.905
0.899
0.893
0.922
0.917
0.909
0.900
0.893
0.885
0.920
0.913
0.903
0.893
0.882
0.875
0.920
0.907
0.893
0.881
0.869
0.859


0.915
0.911
0.905
0.897
0.890
0.916
0.917
0.912
0.906
0.900
0.893
0.918
0.917
0.911
0.906
0.898
0.894
0.918
0.916
0.912
0.902
0.895
0.891
0.918
0.916
0.907
0.898
0.888
0.883
0.920
0.912
0.900
0.890
0.879
0.872
0.918
0.906
0.891
0.875
0.864
0.853


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-7. Continued

k

n pZ Ap2 2 4 6 8 10


0.00


0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0. 928
0. 926
0.919
0.914
0.904
0.897
0. 930
0. 927
0.921
0.914
0.908
0.901
0. 930
0. 92 7
0.921
0.913
0.906
0.901
0. 930
0. 926
0.919
0.911
0.905
0.898
0. 931
0.924
0.916
0.907
0.900
0.893
0. 931
0.923
0.910
0.901
0.889
0.884
0. 928
0.914
0.901
0.889
0.876
0.868


0. 92 7
0. 925
0.918
0.912
0.905
0.898
0. 928
0. 926
0.918
0.913
0.905
0.900
0. 929
0. 925
0.919
0.914
0.907
0.900
0.930
0. 926
0.919
0.910
0.904
0.897
0.930
0. 925
0.915
0.907
0.898
0.892
0. 929
0.920
0.909
0.899
0.891
0.882
0. 926
0.914
0.900
0.887
0.876
0.864


0. 926
0.923
0.918
0.911
0.903
0.897
0. 928
0.924
0.918
0.914
0.904
0.897
0. 929
0. 925
0.919
0.912
0.903
0.899
0.930
0.924
0.918
0.909
0.903
0.897
0. 929
0.922
0.913
0.905
0.897
0.889
0. 929
0.918
0.908
0.898
0.888
0.881
0. 926
0.912
0.898
0.885
0.874
0.864


0.924
0.923
0.918
0.910
0.903
0.895
0. 926
0.924
0.917
0.911
0.904
0.898
0. 927
0.924
0.917
0.911
0.903
0.898
0. 927
0.924
0.915
0.909
0.901
0.895
0. 928
0.921
0.913
0.904
0.896
0.889
0. 927
0.918
0.907
0.897
0.887
0.877
0. 925
0.911
0.896
0.883
0.872
0.860


0.923
0.921
0.916
0.908
0.903
0.895
0. 925
0.922
0.917
0.909
0.902
0.896
0. 925
0.922
0.915
0.909
0.903
0.896
0. 927
0.921
0.914
0.907
0.899
0.893
0. 927
0.920
0.910
0.903
0.895
0.886
0. 926
0.916
0.903
0.895
0.884
0.876
0.923
0.910
0.895
0.881
0.869
0.858


0.10






0.20






0.30






0.40






0.50






0.60











Table 3-7. Continued



n pi Ap2 2 4 6 8 10
800 0.00 0.05 0. 932 0.931 0. 929 0. 930 0. 928


0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0. 928
0.922
0.913
0.906
0.897
0. 932
0. 928
0.921
0.916
0.907
0.902
0. 932
0. 929
0.923
0.915
0.907
0.901
0. 933
0. 928
0.921
0.912
0.905
0.899
0. 933
0. 925
0.916
0.909
0.900
0.894
0. 932
0.923
0.911
0.901
0.891
0.885
0. 930
0.916
0.901
0.889
0.878
0.869


0. 926
0.919
0.913
0.905
0.898
0.931
0. 92 7
0.920
0.915
0.908
0.901
0.932
0. 928
0.921
0.913
0.908
0.901
0.933
0. 92 7
0.919
0.912
0.904
0.897
0.932
0. 926
0.918
0.907
0.899
0.892
0.931
0.922
0.911
0.901
0.890
0.884
0.930
0.915
0.901
0.888
0.876
0.866


0. 926
0.920
0.914
0.906
0.897
0.931
0. 92 7
0.922
0.914
0.905
0.900
0.931
0. 92 7
0.920
0.912
0.907
0.901
0.932
0. 92 7
0.918
0.911
0.904
0.898
0.932
0.924
0.914
0.907
0.898
0.889
0.930
0.919
0.910
0.899
0.890
0.880
0. 92 7
0.914
0.899
0.887
0.875
0.866


0. 926
0.919
0.911
0.905
0.897
0. 929
0. 927
0.920
0.912
0.905
0.899
0. 932
0. 927
0.918
0.912
0.906
0.899
0. 930
0. 926
0.918
0.909
0.903
0.897
0. 930
0.923
0.915
0.905
0.897
0.891
0. 929
0.919
0.907
0.898
0.888
0.880
0. 927
0.914
0.898
0.885
0.873
0.863


0. 925
0.917
0.911
0.904
0.896
0. 930
0. 925
0.919
0.911
0.905
0.899
0. 930
0. 926
0.918
0.911
0.905
0.897
0. 929
0.924
0.916
0.910
0.902
0.894
0. 930
0.923
0.914
0.903
0.895
0.888
0. 928
0.919
0.907
0.897
0.887
0.878
0. 925
0.911
0.896
0.883
0.871
0.862


0.10






0.20






0.30






0.40






0.50






0.60










Table 3-7. Continued

k

n pi Ap2 2 4 6 8 10


0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30


0.00


0. 932
0. 928
0.922
0.913
0.906
0.897
0. 932
0. 928
0.921
0.916
0.907
0.902
0. 932
0. 929
0.923
0.915
0.907
0.901
0. 933
0. 928
0.921
0.912
0.905
0.899
0. 933
0. 925
0.916
0.909
0.900
0.894
0. 932
0.923
0.911
0.901
0.891
0.885
0. 930
0.916
0.901
0.889
0.878
0.869


0. 931
0. 926
0.919
0.913
0.905
0.898
0. 931
0. 92 7
0.920
0.915
0.908
0.901
0. 932
0. 928
0.921
0.913
0.908
0.901
0. 933
0. 92 7
0.919
0.912
0.904
0.897
0. 932
0. 926
0.918
0.907
0.899
0.892
0. 931
0.922
0.911
0.901
0.890
0.884
0. 930
0.915
0.901
0.888
0.876
0.866


0. 929
0. 926
0.920
0.914
0.906
0.897
0. 931
0. 927
0.922
0.914
0.905
0.900
0. 931
0. 927
0.920
0.912
0.907
0.901
0. 932
0. 927
0.918
0.911
0.904
0.898
0. 932
0.924
0.914
0.907
0.898
0.889
0. 930
0.919
0.910
0.899
0.890
0.880
0. 92 7
0.914
0.899
0.887
0.875
0.866


0. 930
0. 926
0.919
0.911
0.905
0.897
0. 929
0. 92 7
0.920
0.912
0.905
0.899
0. 932
0. 92 7
0.918
0.912
0.906
0.899
0. 930
0. 926
0.918
0.909
0.903
0.897
0. 930
0.923
0.915
0.905
0.897
0.891
0. 929
0.919
0.907
0.898
0.888
0.880
0. 92 7
0.914
0.898
0.885
0.873
0.863


0. 928
0. 925
0.917
0.911
0.904
0.896
0. 930
0. 925
0.919
0.911
0.905
0.899
0. 930
0. 926
0.918
0.911
0.905
0.897
0. 929
0.924
0.916
0.910
0.902
0.894
0. 930
0.923
0.914
0.903
0.895
0.888
0. 928
0.919
0.907
0.897
0.887
0.878
0. 925
0.911
0.896
0.883
0.871
0.862


800


0.10






0.20






0.30






0.40






0.50






0.60