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D degree in Counseling , Educational Psychology, and Special Education 7 \ . 7 ,1) ..'_._ _ I X? L (4,.) ,fiLN 636 A; . J Major professor mew MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 i- ‘h‘ LIBRARY Michigan State University PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. "\ DATE DUE I DATE DUE DATE DUE [1.226% filial 1M WWW-p.14 HYPOTHESES AND LIMITING DISTRIBUTIONS FOR P-VALUE SUMMARIES TO COMPARE STUDIES IN META-ANALYSIS By Ji Min Cho A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Counseling, Educational Psychology, and Special Education 1999 ABSTRACT HYPOTHESES AND LIMITING DISTRIBUTIONS FOR P-VALUE SUMMARIES TO COMPARE STUDIES IN META-ANALYSIS By J i Min Cho One of the most widely applicable quantitative methods in research synthesis is the analysis of p values. The purpose of this dissertation is to examine the properties of functions of the significance value to develop a better understanding of several quantitative analyses based on p values in meta-analysis. I examine two of the summaries, called diffuse tests and focused tests (proposed and promoted by Rosenthal and Rubin). These are based on inverse-nonnal-transformed significance values from the sample studies. I described the hypotheses comparing the results of studies tested by those summaries, and derived asymptotic sampling distributions of these tests. Since the small sample behavior of the tests depends on the accuracy of asymptotic distributions of the tests, I did a simulation study to see how the tests behave in finite samples, comparing their derived theoretical moments and distributions to empirical (simulated) moments and distributions. Less than 5% of the results of the Kolmogorov- Smimov fit tests between the asymptotic and the simulated distributions of the focused test statistic were statistically significant. The Pearson tests of association showed that the obtained numbers of significant results of Kolmogorov-Smirnov tests were similar for differing weight values and sampling fractions, but differed for different types of statistics, numbers of studies, sample sizes, and sets of effect-size parameters. fl (It A Pearson chi-square statistic was applied to test the goodness of fit between the asymptotic distributions and the simulated distributions of the diffuse test statistic. The distribution of the diffuse test statistic was well approximated by a modified chi-square distribution (the theoretical distribution) when one-sample t statistics led to the diffuse test statistics. When balanced or unbalanced two-sample t statistics led to the diffuse test statistics, the distribution of the diffuse test statistics was generally well approximated by this same distribution, except for cases with larger numbers of studies and small sample sizes. The asymptotic distributions for the focused and diffuse tests I studied in this dissertation were mostly accurate. The differences in the power values between the simulated and theoretical distributions were mostly small for the focused test, and also for the diffuse tests with few exceptions. Therefore the power values obtained from these asymptotic distributions should be useful for the focused and diffuse tests. The power study for the focused and diffuse tests showed that when the number of studies and sample sizes increased, the power for both tests increased under the alternative hypotheses regardless of other configurations of population parameters. The power study for the focused test supported that the null hypotheses for the effect sizes should be weighted by the square root of their own sample sizes, as expected. However, the power study for the diffuse test did not fully support using the null hypothesis of equality of effect sizes weighted by the square roots of their own sample sizes. To my parents Song Pong Cho and Kyeong Soon Lee iv ACKNOWLEDGEMENTS I would like to express my deepest appreciation to my advisor Dr. Betsy Becker for her academic assistance, guidance and support throughout my graduate study. Her time and effort to encourage me persistently makes me complete this dissertation. She has been just great and I really do owe much more than thanks to her. I also thank my committee members, Dr. James Stapleton, Dr. Deborah Feltz, and Dr. Mark Reckase for their suggestions. I am most thankful to my parents for their endless love, support, and faith. Without my parent’s confidence in me and all their financial support, I even could not start a doctoral program. I really do love them. I also thank my mother-in-law, Ok-Sun Lee who has prayed for me always and have taken best care of my children. Without her help and encouragement I was not able to continue this study. I really do respect her. I have saved my deepest love and thanks for my two lovely children, Namgyu and Yaeyoung for suffering in silence while I was busy finishing this study. Last, but certainly not least, I give my love and thanks to my very special friend and husband, Eekhoon Jho. Without his endless and unconditional love and belief in me, I would not have accomplish anything. I wish to extend thanks to all the nice people I met at Michigan State University, who made my life happier here. TABLE OF CONTENTS LIST OF TABLES ....................... ix LIST OF FIGURES ....................... xv CHAPTER I. INTRODUCTION ................. 1 Introduction ............. 1 Purpose of the Study ............. 2 Justification for Study ............. 2 Goals of the Research ............. 4 Overview of the Dissertation ............. 5 II. LITERATURE REVIEW ................. 7 Meta-Analysis in General ................. 7 Notation ................. 9 The p Value as a Statistic ................. 10 The Transformation Z(p) ................. 12 The p-Value Summaries ................. 13 The Summaries in the Literature ................ 13 The Hypotheses for Focused and Diffuse Tests ...... 17 The Hypotheses for Related Summaries ...... 21 III. DISTRIBUTION OF Z(p) ................... 22 Under the Null Hypothesis ............... 22 Under the Alternative Hypothesis .............. 23 Distribution Based on Rosenthal and Rubin’s Result ........ 24 Distribution Based on Lambert’s Result ............ 25 IV. DISTRIBUTIONS OF THE SUMMARIES ........... 28 Under the Null Hypothesis ........... 28 Under the Alternative Hypothesis ........... 28 Theoretical Distribution for Focused Test when k=2 . . . . 30 Theoretical Distribution for Focused Test when k=3 or more . . . . 31 Theoretical Distribution for Diffuse Test .......... 34 Central Case ................... 35 vi Noncentral Case ................. 37 The three-moment chi-square fit .......... 38 V. SMALL SAMPLE BEHAVIOR OF THE SUMMARIES ...... 42 Parameters in the Simulation Study ............... 42 Values of the Parameters used in the Simulation Study 42 Number of Studies ................. 43 Sample Sizes ................. 43 Sampling Fractions ................. 44 Effect-Size Parameters ............. . . . . 45 Within-Study Sampling Ratio ............... 47 Weight Values ................. 48 Simulation Procedures ................. 49 Accuracy of the Asymptotic Distributions for the Focused Test Statistic 51 Tests for Normality ................. 51 Empirical Proportions ................. 53 The Kolmogorov-Smimov Tests ............... 59 Different Types of Statistics ............... 62 Number of Studies 62 Weight Values ................. 63 Sample Sizes ........... . . . . 63 Effect-Size Patterns ............... 64 Sampling Fractions ............... 65 Within-Study Sampling Ratio ............... 66 Comparisons of Moments ............... 66 Z Test for Mean Differences ............... 70 Power Analysis for the Focused Test Statistic ............. 74 Power Comparisons ............... 74 Power Results for the Focused Test Statistic ........ 75 Under the Null Hypothesis ............... 81 Accuracy of the Asymptotic Distributions for the Diffuse Test Statistic 83 Empirical Proportions ................. 83 Goodness-of-Fit Tests ................. 91 Different Types of Statistics ............... 93 Number of Studies ................. 94 Sample Sizes ................ 95 Sampling Fractions ................ 96 Effect-Size Patterns ............ 96 Within-Study Sampling Ratio ............ 97 Combination of Three Parameters ...... 97 Power Analysis for the Diffuse Test Statistic ...... 99 Power Comparisons . . . . . . 99 Power Results for the Diffuse Test Statistic ...... 100 Under the Null Hypothesis ...... 107 vii VI. CONCLUSIONS AND IMPLICATIONS ...... 111 Summary . . . . 1 11 Practical Implications . . . . 113 Suggestions for Further Research ................. 115 APPENDIX A: SAMPLE SIZES USED IN SIMULATION STUDY . . . . 117 APPENDIX B: MOMENTS AND EMPIRICAL PROPORTIONS FOR THE FOCUSED TEST ................... 124 APPENDIX C: RESULTS OF THE KOLMOGOROV-SMIRNOV TESTS . . 149 APPENDIX D: POWER COMPARISIONS FOR THE FOCUSED TEST . . 152 APPENDIX E: EMPIRICAL PROPORTIONS FOR THE DIFFUSE TEST . . 165 APPENDIX F: RESULTS OF CHI-SQUARE TESTS FOR THE DIFFUSE TEST 178 APPENDIX G: POWER COMPARISIONS FOR THE DIFFUSE TEST . . . . 181 REFERENCES ....................... 186 viii LIST OF TABLES Table Page 1. Meta-analytic Procedures based on Normal-transformed Probability Values for Comparing and Contrasting Studies ...... 14 2. Meta-analytic Procedures for Combining Studies based on Normal-transformed Probability Values ............ 17 3. Rosenthal and Rubin’s Null Models for the Tests in Table 1 ...... 19 4. Null Models involving Sample Sizes for the Tests in Table 1 . . . . 21 5. Theoretical Moments of the Distribution for Z(p) ...... 27 6. Theoretical Means of the Distribution for Focused Test . . . . 32 7. Theoretical Variances of the Distribution for Focused Test . . . . 33 8. Sampling Fractions for Simulation Study . . . . 44 9. Different Patterns of Population Effect Sizes ...... 46 10. Weight Values for Focused Test ...... 48 11. Total Number of Configurations ...... 49 12. Number of Significant Normal Tests for the Focused Test . . . . 53 13. 95% Confidence Intervals for Simultaneous Comparisons of Empirical Proportions for the Focused Test ................. 58 14. Number of Significant Results for Empirical Proportions at Five Cut Points . . 58 15. Results of Kolmogorov-Smimov Tests for Different Statistics . . 62 16. Results of Kolmogorov-Smimov Tests by k . . . . 63 17. Results of Kolmogorov-Smimov Tests by N . . . . 64 18. Results of Kolmogorov-Smimov Tests by Effect-Size Pattern . . . . 65 ix 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. Results of Kolmogorov-Smirnov Tests by Sampling Fractions . . . . 66 Results of Z Tests by k .......... 70 Comparisons for the Theoretical Means with Rosenthal and Robin’s Means for k=5 .......... 71 Comparisons for the Theoretical Variances with Rosenthal and Robin’s Variances for k=5 .......... 72 Analysis of Variance for Power Values of the Focused Test . . . . 75 Analysis of Variance for Power Values for One-Sample t . . . . 77 Analysis of Variance for Power Values for Balanced and Unbalanced Two-Sample t ................... 77 Means of Simulated Power Values for One-Sample 1 Statistics by k x N x 6 78 Means of Simulated Power Values for Balanced and Unbalanced Two-Sample t Statistics by kx N x 6 ................... 79 Power Values under Possible Null Hypotheses by N and 6 for k=5 from one- sample t ..................... 82 95% Confidence Intervals for Comparisons of Empirical Proportions for the Diffuse Test ...................... . 88 Number of Significant Results for Empirical Proportions at eleven Cut Points 88 Accuracy of the Asymptotic Distribution of the Diffuse Test for k=5 . . 90 Results of x2 Goodness-of-Fit Test for Different Statistics ...... 94 Number of Significant x2 Goodness-of—Fit Tests by Number of Studies (k) 95 Number of Significant x2 Goodness-of—F it Tests by N ...... 95 Number of Significant x2 Goodness-of—Fit Tests by Sampling Fractions . . 96 Number of Significant x2 Goodness-of—F it Tests by Effect-Size Pattern for the Diffuse-Test Distributions ................. 97 37. Analysis of Variance for Power Values for the Diffuse Test . . . . 100 38. Analysis of Variance for Power Values for One-Sample t Statistics . . . . 101 39. Analysis of Variance for Power Values for Balanced and Unbalanced . . 101 40. Means of Power Values of the Diffuse Test Statistic for One-Sample t Statistics by k x N x 6 ................... 103 41. Means of Power Values of the Diffuse Test Statistic for Two-Sample t Statistics by k x N x 6 ................... 104 42. Empirical Power Values under Several Null-Hypothesis Scenarios by N and 6 for k=5 from One-Sample t ..................... 107 43. Empirical Power Values under Several Null-Hypothesis Scenarios by N and 6 for k=5 from Balanced Two-Sample t ................ 110 44. Sample Sizes used in Simulation Study ............ 118 45. Moments and Empirical Proportions for k=2 from One-Sample t Statistics for Linear Effect .................. 125 46. Moments and Empirical Pr0portions for k=2 from One-Sample t Statistics for Group Effect .................. 126 47. Moments and Empirical Proportions for k=2 from Balanced Two-Sample t Statistics for Linear Effect .................. 127 48. Moments and Empirical Proportions for k=2 from Balanced Two-Sample t Statistics for Group Effect .................. 128 49. 50. 51. 52. Moments and Empirical Proportions for k=2 from Unbalanced Two-Sample t Statistics for Linear Effect .................. 129 Moments and Empirical Pr0portions for k=2 from Unbalanced Two-Sample t Statistics for Group Effect .................. 130 Moments and Empirical Proportions for k=5 from One-Sample t Statistics for Linear Effect .................. 131 Moments and Empirical Proportions for k=5 from One-Sample t Statistics for Group Effect .................. 132 xi 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 65. 66. 67. Moments and Empirical Proportions for k=5 from Balanced Two-Sample t Statistics for Linear Effect .................. 133 Moments and Empirical Proportions for k=5 from Balanced Two-Sample t Statistics for Group Effect .................. 134 Moments and Empirical Proportions for k=5 from Unbalanced Two-Sample t Statistics for Linear Effect .................. 135 Moments and Empirical Proportions for k=5 from Unbalanced Two-Sample t Statistics for Group Effect .................. 136 Moments and Empirical Proportions for k=10 from One-Sample t Statistics for Linear Effect .................. 137 Moments and Empirical Proportions for k= 10 from One-Sample t Statistics for Group Effect .................. 138 Moments and Empirical Proportions for k=10 from Balanced Two-Sample t Statistics for Linear Effect .................. 139 Moments and Empirical Proportions for k=10 from Balanced Two-Sample t Statistics for Group Effect .................. 140 Moments and Empirical Proportions for k=10 from Unbalanced Two-Sample t Statistics for Linear Effect .................. 141 Moments and Empirical Pr0portions for k=10 from Unbalanced Two-Sample t Statistics for Group Effect ................ 142 Moments and Empirical Proportions for k=50 from One-Sample 1 Statistics for Linear Effect .................. 143 . Moments and Empirical Proportions for k=50 from One-Sample t Statistics for Group Effect .................. 144 Moments and Empirical Proportions for k=50 from Balanced Two-Sample t Statistics for Linear Effect .................... 145 Moments and Empirical Proportions for k=50 from Balanced Two-Sample t Statistics for Group Effect .................... 146 Moments and Empirical Proportions for k=50 from Unbalanced Two-Sample t Statistics for Linear Effect ................ 147 xii . Moments and Empirical Pr0portions for k=50 from Unbalanced Two-Sample t 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. Power Values for k=5 from Balanced Two-Sample t Statistics Power Values for k=5 from Unbalanced Two-Sample t Statistics Power Values for k=10 from One-Sample t Statistics ....... Power Values for k=10 from Balanced Two-Sample t Statistics Power Values for k=10 from Unbalanced Two-Sample t Statistics Power Values for k=50 from One-Sample t Statistics ....... Power Values for k=50 from Balanced Two-Sample t Statistics Power Values for k=50 from Unbalanced Two-Sample t Statistics Empirical Proportions for k=2 from One-Sample t Statistics Empirical Proportions for k=2 from Balanced Two-Sample t Statistics . . . Empirical Proportions for k=2 from Unbalanced Two-Sample t Statistics Empirical Proportions for k=5 from One-Sample t Statistics Empirical Proportions for k=5 from Balanced Two-Sample t Statistics Empirical Proportions for k=5 from Unbalanced Two-Sample t Statistics Empirical Proportions for k=10 from One-Sample t Statistics xiii Statistics for Group Effect ................ 148 . Results of Kolmogorov-Smimov Tests for Linear Effect ...... 150 Results of Kolmogorov-Smirnov Tests for Group Effect ..... 151 Power Values for k=2 from One-Sample t Statistics ....... 153 Power Values for k=2 from Balanced Two-Sample t Statistics 154 Power Values for k=2 from Unbalanced Two-Sample t Statistics 155 Power Values for k=5 from One-Sample t Statistics ....... 156 157 158 160 161 . 163 . 164 166 167 168 169 170 171 .172 90. Empirical Proportions for k=10 from Balanced Two-Sample t Statistics 173 91. Empirical Proportions for k=10 from Unbalanced Two-Sample t Statistics 174 92. Empirical Pr0portions for k=50 from One-Sample t Statistics for Linear Effect 175 93. Empirical Proportions for k=50 from Balanced Two-Sample t Statistics 176 94. Empirical Proportions for k=50 from Unbalanced Two-Sample t Statistics 177 95. Results of Chi-Squared Tests for Equal Sampling Fraction ...... 179 96. Results of Chi-Squared Tests for Unequal Sampling Fraction . . . . 180 97. Power Values for the Diffuse Test for k=2 .......... 182 98. Power Values for the Diffuse Test for k=5 .......... 183 99. Power Values for the Diffuse Test for k=10 ...... . . . . 184 100.Power Values for the Diffuse Test for k=50 .......... 185 xiv LIST OF FIGURES Figure Page 5.1.1 The distribution of the Focused Test Statistic for k=10 from Balanced Two-Sample t Statistics ......................... 55 5.1.2 The Distribution of the Focused Test Statistic for k=50 from Balanced Two-Sample t Statistics ....................... 56 5.1.3 Mean Differences for Linear by k and N ............. 68 5.1.4 Mean Differences for Group by k and N ............. 68 5.1.5 Mean Differences for Linear by k and 6 ............. 69 5.1.6 Mean Differences for Group by k and 6 ............. 69 5.1.7 Power Values for the Focused Test based on Two-Sample t Statistics by N x k 80 5.1.8 Power Values for the Focused Test based on Two-Sample t Statistics by 6 x k 80 5.1.9 Power Values for the Focused Test based on Two-Sample t Statistics by 6 x N 81 5.2.1 The distribution of the Diffuse Test Statistic for k=5 from One-Sample t Statistics ........................ 85 5.2.2 The distribution of the Diffuse Test Statistic for k=5 from Balanced Two-Sample t Statistics ........................ 86 5.2.3 Power Values for the Diffuse Test based on Two-Sample t Statistics by N x k 105 5.2.4 Power Values for the Diffuse Test based on Two-Sample t Statistics by 6 x k 105 5.2.5 Power Values for the Diffuse Test based on Two-Sample t Statistics by 6 x n 106 XV CHAPTER I INTRODUCTION Since Glass (1976) first introduced the term meta-analysis, many researchers in education and related fields have been interested in quantitative methods for the integrative review. In this dissertation I will study two summaries of observed significance values used in meta-analysis. Unlike the traditional narrative review, the quantitative review and its accompanying statistical procedures appear to be more rigorous and objective to the researcher. Glass, McGaw, and Smith (1981) argued that “by recording the properties of studies and their findings in quantitative terms, the meta-analysis of research invites one who would integrate numerous and diverse findings to apply the full power of statistical methods to the task” (p. 21). Johnson, Mullen, and Salas (1995) wrote that meta-analysis generally refers to the statistical integration of the results of independent studies. Meta- analysis can draw stronger and more general conclusions than the traditional narrative review by analyzing the studies with statistical methods. In an experiment which compared both approaches, Cooper and Rosenthal (1980) showed that the conclusions drawn using traditional procedures and using statistical combination procedures differed from one another. Certainly, the use of statistical combination procedures in literature reviews has increased in the social sciences since meta-analysis was introduced. With the growth of meta-analysis in educational and psychological research, meta-analytic software packages also have been introduced. Si. 50 3T. frc It‘ (in Va SIL Two of the main quantitative approaches in research synthesis are to combine or compare significance values (p values) across two or more individual studies, and to analyze and estimate the magnitude of the effect size. When a researcher can get effect- size data from individual studies, analyses of significance values are not recommended, because p values do not provide as much information as the effect size. In addition, the use of analyses of significance values with effect-size analysis is not recommended (Becker, 1987). A reason for doing analyses of significance values, however, is that sometimes researchers cannot get enough information from primary studies for effect-size analysis and significance values are the only statistics reported. Most individual studies do report p values, even if they do not report effect magnitudes. Also p-value analyses need fewer statistical assumptions than effect-size analysis, and p values can be drawn from any test statistic representing a substantive hypothesis of research interest. As a result, analyses of significance values can be applied more broadly than effect-size analyses. Purpose of the Study Justification for the Study My research purpose is to examine the properties of functions of the significance value and to develop a better understanding of several quantitative analyses based on p values in meta-analysis. Rosenthal and Rubin (1979) first suggested the methods I will study as methods for comparing two or more statistical significance values. These tests, also called “diffuse” and “focused” tests (Rosenthal, 1984, p. 60), have been used as a it. It [0 3f. ac: an. diN dic Cor. [he means of comparing the results of studies. Strube (1985) presented similar methods for combining and comparing multiple significance levels of multiple hypothesis tests when the outcomes of the tests are not statistically independent. Hayes (1998) also proposed a related method that adjusts for nonindependence in the dependent variable set, based on Strube’s method. Hayes’s method showed how pooling the significance of nonoverlapping, or doubly nonindependent correlations within a single study was. Rosenthal (1984) wrote that the diffuse test should be applied to reveal whether study results differ significantly and the focused test should be used to reveal whether the results differ in a “theoretically predictable or meaningful way” (p. 61). Strube (1985) also explained that the two procedures (diffuse and focused tests) are for identifying variability in significance levels. However, these descriptions are vague, and do not lead to clearcut hypotheses for these tests. Reviewers have been urged to apply these analyses, in spite of a lack of research on their pr0perties, and minimal clarity about their hypotheses. Rosenthal has consistently promoted the use of these two methods in books such as Meta-analLtic procedures for social research (1984 and 1991), in Judgement studies: Design, analysis, and meta-analysis (1987), and in his chapter in A handbook for data analysis in behavioral sciences: Methodological issues (Rosenthal, 1993). In papers that describe what should typically be included in the introduction, method, results, and discussion sections of a meta-analytic review, Rosenthal (1991, 1995) also has consistently recommended the use of diffuse and focused comparisons of significance levels. In addition, Fisher (1992) proposed using p-value summaries in his study to find the impact of play on development. Mullen (1989) also has recommended use of diffuse ill of (‘0 IE. Ie< EX for Ca? art \ and focused tests in his book Advanced BASIC meta-analysis, and the tests appear in the accompanying software. Since many journals now require authors to report test statistics with degrees of freedom, researchers are more likely to be able to compare effect sizes than before. However, significance levels still have been widely used v_vi_tl_1 effect sizes in meta- analysis. For instance, Mullen and Johnson (1990) used diffuse and focused comparisons of significance levels and analyses of effect sizes for a meta-analytic review of previous research on illusory correlation in stereotyping effects. Burt, Zembar, and Niederehe (1995) used the method presented by Strube to compute a comparison of significance levels to investigate whether moderator variables determine the extent of the association between depression and memory impairment. Mullen (1990) used diffuse and focused comparisons of significance levels and effect sizes in a meta-analytic integration of research that had examined the links between status, expectations, and behavior. In spite of these and other uses of the tests, no one has actually studied how these tests behave. Even Rosenthal and Rubin, who initially suggested the tests, have not explored their properties, while continuing to encourage their use in meta-analytic work in educational and related research. Goals of the Research Continued recommendations for the use of these tests, their presence in software for meta-analysis, and their application in published research syntheses suggest that careful examination of their properties and use is needed. I will describe what hypotheses are tested by the summaries when they are used to compare the results of studies. Rosenthal and Rubin did not write about the hypotheses tested by their summaries, and even stated that diffuse and focused tests are meant to evaluate whether the significance values across the studies differ significantly from one another (Rosenthal & Rubin, 1979). This cannot be the hypothesis to be tested because hypotheses must refer to population parameters. Studying the statistical properties of the significance-value summaries will allow me to determine whether these tests provide reasonable information about the hypotheses that a researcher considers in his or her synthesis. Therefore, both the hypotheses for significance testing and the statistical properties of the tests need to be clearly understood. Overview of the Dissertation In the dissertation, I examine both diffuse and focused tests of the significance levels, clarifying their statistical hypotheses and deriving their asymptotic sampling distributions, which are used to obtain the large sample-approximations to the distributions of those tests. Diffuse and focused test statistics are based on inverse- normal-transformed significance values from the studies, and their distributions are based on certain assumptions under the null hypothesis. Rosenthal and Rubin (1979) discussed the sample significance values only from one sample t test statistics. In this research, I study cases in which the sample significance values result from one-sample and two- sample I statistics, the latter being more common in some areas of educational and psychological research. nc .' sir I review the research on the nonnull asymptotic sampling distributions of inverse- norrnal-transformed significance values, which I then use to obtain the asymptotic sampling distributions for diffuse and focused test statistics. Finally I investigate the accuracy of the large-sample approximations to the distributions of diffuse and focused test statistics. Since the small-sample behavior of the test statistics depends on goodness of the approximation of the distribution of test statistics by asymptotic distribution, I do a simulation study to see how diffuse and focused test statistics behave for finite samples. Typical values of study sizes, sample sizes and effect sizes found in meta-analytic studies are outlined below for the simulation. By comparing the derived theoretical moments to the empirical (simulated) ones for the distributions of both diffuse and focused test statistics, we can learn how those test statistics behave in finite samples. CHAPTER II LITERATURE REVIEW This section begins with a brief review of three different approaches applied in meta-analysis. Notation is given for the population parameters and the significance value for a t test statistic of the single population parameter, 6. The following sections introduce the significance value, the inverse-nonnal-transformed significance value, and the quantitative summaries based on the inverse-nonnal-transformed significance values. Meta—Analysis in General Cooper and Hedges (1994) described research syntheses or research integration as attempts to discover consistencies and account for variability in similar-appearing studies. Cooper and Hedges (1994) also explained that research synthesis attempts to integrate empirical research for the purpose of creating generalizations. “Meta-analysis” refers to the analysis of analyses, the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings (Glass, 1976). Therefore meta-analysis is the same as the quantitative synthesis of research. Three different approaches have been applied in meta-analysis or research synthesis: vote- counting (including vote-counting estimation procedures), analyses of significance levels, and analyses of effect sizes. Early approaches to the quantification of research domains focused on vote- counting methods. In vote-counting, researchers sort the results of the studies into positive-significant, not-significant, and negative-significant categories. Conclusions are then based on the resulting tallies. This approach is no longer recommended because of poor statistical properties associated with its use. Hedges and Olkin (1980) found the power of this procedure to be low, and to actually decrease as the number of studies reviewed increases. A fast growth in the interest and use of analyses of significance levels has occurred in educational research since Rosenthal (1978) and Rosenthal and Rubin (1979) reviewed methods for combining and comparing sample significance values which test essentially the same directional hypothesis. This approach has an advantage over simple vote-counting, in that it uses information available in the sample significance values. The third approach, analyses of effect size, however, can provide the most information to the researcher about the size and strength of effects. Analyses of significance levels provide less information about the patterns of outcomes than do the simplest analyses of effect sizes (Becker, 1987). Therefore analyses of effect size are preferred among these three approaches in meta-analysis. However, when a researcher cannot obtain effect sizes, but can get sample significance values, combined significance tests can be used. Also, Becker explained that “when studies have examined a common substantive issue, but their results are represented by a variety of different effect- magnitude measures, combined significance tests may be the best summaries available” (Becker, 1994, p. 228). For these reasons, it is important to understand how analyses based on significance values behave. Notation In this section, I describe each of the terms and notation used for study results and particularly for significance values and summaries based on them. Let k be the total number of individual studies included in a meta-analytic review. Suppose each study included in the meta-analysis has the same hypothesis about the population parameter (expressed as 6). In this study, I will consider study results from one-sample and two- sample t-test statistics. Therefore the following population parameters will be examined: 1) the population standardized mean, 6 =u/o; and 2) the population standardized mean difference, 6' =(llrll2)/0 . For consistency in notation, the subscript i with any statistic or parameter (such as a p value, effect size, sample size, etc.) indicates its association with the ith study, for i=1, , k studies. Let the score Yi}. of the jth person in the ith study be normally distributed with a mean u, and the variance 0?. Suppose each study has the null hypothesis H 26:0, 0 l with the alternative hypothesis, H1: 6, > 0, where 6,. = u, /o,. . For an example, a test statistic t, may be used to test the null hypothesis for the ith study. We may have t, = $707,- — u, )/S,. in the case of the one sample t test for example, where Y. is the sample mean and S, is the sample standard I deviation for the ith study. S For this outcome t.- , the p value in study 1' (p,) is the probability of obtaining a t statistic larger than the sample t statistic in the ith study when the null hypothesis is true. And here all p values are independent. Specifically, p, =f:f(t,v,.)dt=1- ,(t,, v,) =P(t > g), (2.1) where t, is the sample statistic value given the statistical null condition, v, = df and f (t) is the probability density function of test statistic I under the null hypothesis. F,( t,) is the null-case cumulative distribution function of t, evaluated at t,. When only large positive values of any statistic are considered extreme, the p value is the upper-tail probability calculated assuming the null hypothesis is true. For the meta-analytic summaries examined in this study, the asymptotic distribution of the transformed sample significance value Z( pl. ) is studied. Here, Z( p, ) = Z .- is defined as the standard normal deviate for the p value. In this study Z( p, ) is associated with the upper tail probability P,- , so it is that -"( pl. ), the negative of the inverse normal transformation of p, ,where (I) represents the standard normal cumulative distribution function. Thus if p, = .025, Z( p, ) = Z(.025) = 1.96, or if p, = .001, Z( pl. ) = Z(.001) = 3.09, etc.. The p value as a Statistic The p value is the probability of obtaining a test statistic as unusual or extreme as that calculated, given that the null hypothesis is true. Generally, the smaller the p value 10 is, the stronger is the evidence against the null hypothesis provided by the sample data. Modern interpretations of the p value are foreshadowed in the early writings of R. A. Fisher in 1922, who referred to the p value as “the probability that a worse fit should be obtained from a random sample of a population of the type considered” (p. 314). (Fisher cited in Lambert, 1978, p. 5). All p values associated with continuous test statistics are uniformly distributed on the interval between zero and one under the null hypothesis. That is, fp(p)=1, Ospsl, where fp (p) is the probability density function of p under the null hypothesis. All p values throughout the zero-to-one range have equal chances to occur under any null hypothesis. Under alternative hypotheses, the sampling distributions, however, are not uniform, making small p values more likely. Suppose there is a study to test a null hypothesis about the population effect size, 6 , such as H.,: 6 = 0 versus the alternative hypothesis, HA: 6 > 0. Suppose also that the true population effect size, 6 , is equal to .5. Then the researcher has a greater chance of obtaining small p values than he or she would have under the null hypothesis. As a result, small p values are taken to mean that the results from a test of statistical significance are not likely to be from a population in which the null hypothesis is true. 11 The cumulative distribution function of p can be written as FPO?) = P0 >t0) =1- FA (1‘3“ (1 - P»: where t is considered as a function of p such as t = F," (1 — p) obtained from (2.1) and F A is the cumulative distribution of t under the alternative hypothesis. Therefore, the density function of the p value under HA can be written as dF.“(1 - 1)) mm = fA(F."(1- 12)) dp where f A (t) is the probability density function oft under the alternative hypothesis and t = E" (1 - p). If the null hypothesis that the population effect size equals zero is true, then the density function of p under the alternative hypothesis will be the same as the one under the null hypothesis, which is the uniform distribution on [0, 1]. However if the null hypothesis is not true, the density functions of p under the alternative hypothesis will be complicated. As a result, under the alternative hypothesis, the distribution of p values is complex, and depends on the specific test statistic used, the size of population parameter(s) being tested, and sample size. The Transformation Z(p) Z(p) is the standard normal deviate associated with the one-sided p value, specifically - 2) " 1p = 2 1,6, = 0 Diffuse a, = a, = a, ..... = a, The statistical hypotheses about the phenomenon of interest in the populations from which the samples are obtained involve the parameters, not the sample statistics. The null models for the summaries in Table 1 are for testing not whether the sample significance levels themselves are equal (or not), but testing whether the effect sizes in the populations are equal or not. However, the test of whether population parameters 19 (Sue of f Cl‘ r5 . lit) 1 $11. (such as population effect sizes) are equal cannot be cleanly made by testing the equality of p values, because p values are affected by sample size. Clearly there has been confusion regarding what can be tested with these summaries. Hsu (1980) noted that Rosenthal and Rubin’s (1979) tests of equality of p values are not correct tests for the hypotheses about equality of effect sizes for studies of unequal size because of the dependence of the significance levels on n,. The null hypothesis for the focused test would be that the effects from a series of studies weighted by sample size did not differ, or that a linear combinations of the effects weighted by sample size, amd weight values (7») equals zero. The null hypothesis for the diffuse test would be that effects from a series of studies, when weighted by sample size, were equal. However, the issue still remains of exactly how to weight by sample size. One plausible model uses the effect sizes weighted by the square root of each sample size because of the form of Rosenthal and Rubin’s expected value of Z(p), as discussed below. Simulation study showed that the null models for the focused test described in Table 4 are reasonable, but for the diffuse test, weighting by the square root of each sample size led to a very conservative (low a) test. 20 l'fi Table 4 Null Models involving Sample Size for the Tests in Table 1 Test Null Model Focused (k = 2) £5 1= ”252 Focused (k > 2) 1P =i)‘1\/n—i6i =0 i-l Diffuse $7151 = $6, =...= nk6,, The Hypotheses for Related Summaries All the summaries in Table 2 are based on the fact that the sample significance levels are uniformly distributed under the null hypothesis of “no effect” in any study. That is, combined significance tests are tests of Hoz61=62 = -~--=6,, =0, for i=1 through k studies and where 6,. is the population parameter of effect size for each study. Therefore if the p value for a combined significance test for a set of studies in meta-analysis is small enough, then a researcher concludes that the likelihood of all population effects equaling zero is small. 21 CHAPTER III DISTRIBUTIONS or Z(p) In this section, the central and the noncentral asymptotic distributions of the inverse-nonnal-transformed significance value are presented, which are used to get the noncentral asymptotic distributions for diffuse and focused tests. The exact distribution of Z(p) can be obtained by exact derivation, and the large-sample approximation to the distribution of Z(p) can be obtained by the delta method (Rao, 1972) and other limit theorems. The exact distribution of Z(p) under the null hypothesis is much simpler than the exact distribution of Z(p) under the alternatives. Therefore the delta method was used by Lambert and Rosenthal and Rubin to get the large-sample approximation to the distribution of Z(p) under alternative hypotheses. Under the Null Hypothesis The density function of the p value, f(p), is one on the interval zero to one when the null model is true. Now we want to know the probability density function of y, g(y), wherey = Z(p) = "(p). 22 The cumulative distribution function G of a random variable Y is, Gy(y) = P(-<1>"(p) s Y) = P0) 5 (y)- Therefore, the probability density function of y is, fly) = 1*ld[-(y)]/dyl = J; we": under H0 and Z"(y) = -(y), because Z(p) equals -"(p) where Z(p) is associated with an upper tail p value. This is the probability density function of the normal distribution with a mean of zero and variance of one, that is, the standard normal distribution, Z , with -112 2 densit of 1 -e y J2rt Because all continuous p values have the uniform distribution under the null hypothesis regardless of the different tests used across individual studies, the distribution of Z(p) under the null hypothesis is always the standard normal distribution with a mean of zero and variance of one. Under Alternative Hypothesis Since functions of the significance values depend on the significance value directly and exact distributions of the significance value under the alternative hypothesis are complex, the exact distributions of functions of p values under alternative hypotheses are also complicated. The asymptotic distributions of the inverse-nonnal-transformed 23 significance values, which are used to get the asymptotic distributions of the quantitative summaries, are discussed next. Distribution Based on Rosenthal and Rubin’s Result Rosenthal and Rubin (1979) treated the asymptotic distribution of the inverse- normal-transformed significance values from one-sample t statistics as normal with a mean of Jr—z6 and a variance of approximately one for any value of 6, without examining the accuracy of that normal approximation. Rosenthal and Rubin showed a more detailed distribution in the technical discussion of their paper, specifically, A (Z(p)-J716(1-n62/ 4f)) ~ N ( 0, (1+(1/2—n62)/ f)), (3.1) where f is the degrees of freedom in each study. So for Z(p) to be normally distributed with a mean of 7176 and variance of 1, f must be large enough to make the terms of this result including f small. Distributions of the tests that Rosenthal and Rubin proposed may be inaccurate when f is not large. Becker (1985, 1991) examined the accuracy of Rosenthal and Rubin’s normal approximation by comparing its cumulative distribution function (cdf) to the exact cdf of the p value from the one-sample t statistic. The simulation study in her research (Becker, 1991) showed that the normal approximation with mean 6 and variance 1 suggested by Rosenthal and Rubin (1979) was quite inaccurate when the effect sizes were moderate to large, and became increasingly worse as the sample size increased. 24 Distribution Based on Lambert’s Results Lambert (1978) and Lambert and Hall (1982) examined the properties of p values as indices for comparing the test statistics with which the p values are associated ( i.e., the t,- above). Lambert (1978) presented an asymptotic normal approximation to the distribution of the inverse normal transformed p value, Z(p). Using her notation, T is a sample statistic and p is the observed probability value associated with test statistic T for a sample size n, under the null hypothesis that the population effect-size is zero. Lambert (1978) considered the case where for any fixed nonnull value 6 , A Jar—b(6» ~ N (0. 02(5)), where b(6 ) is constant 6 as n —> co and 02(6) = 1+62 /2 . For instance, b(6 ) can be III/0 , that is, the effect-size 6 for one-sample t statistics. For the asymptotic distribution of the standard normal deviate, Z(p), associated with the one-sided 1) under alternative hypotheses, Lambert (1978) also showed that (wow-(275» 1 N(0. mo». where c(6) is Bahadur’s (1960) half slope for the test statistic T, 712(5):: [c'(6)o(6)/ 20(6 )12, and 0(6) is shown above. In the case of p values from one-sample t tests, then, the large-sample approximation to the distribution of Z(p) is (Z(p)—J2me)» 3 N (0, 112(6)), 52(1+oZ/2) (1+62)2ln(1+62) where c(6) - ln(1 +6 2)/ 2 , and 112(6) = (See also Becker, 1991). 25 1 Becker (1985, 1991) tested the small sample accuracy of the asymptotic distribution of the inverse-normal transformed p value proposed by Lambert and Lambert and Hall, by comparing the approximate cumulative distribution function (cdf) values to the computed exact cdf of the p value associated with the one-sample t statistic. The results showed that as both the sample size and the population parameter increased, Lambert’s Z approximation grew closer to the exact distribution of p values. She found that the inverse normal approximation, that is, Lambert’s Z approximation was quite accurate for reasonable values of 6 and all sample sizes. Becker’s finding that Lambert’s variance term is never larger than that assumed by Rosenthal and Rubin, suggests that tests based on Rosenthal and Rubin’s approximation may be overly conservative. In addition, Becker (1991) applied the results given by Lambert and Hall to significance values from two-sample t tests. She presented formulas for means and variances of normal approximations in that case. The asymptotic distribution of Z(p) when n=nE +nc,¢” =12” /n and ¢C =21" /n is (Z(p>—Jn 5 622 < > ‘5 Therefore, the variance is 2 i 1 1 TI (61) 0 J5 J5 Ji— 0 TI2(62) __L . «5. 1 2 2 i2 2 =(7—2") 7] (MM-‘5) TI (52) (u’(51)+n2(52)). va—i 30 For p values from two-sample t statistics or other tests we would substitute the appropriate formulas for c and n2 in place of those used here. Theoretical Distribution for the Focused Test when k=3 or more For the focused test, F(Z), when k=3 or more and Z = (Z1 ,Z,,...Zk )' , and A, is a weight for the ith study, IS? The expected value when one-sample t statistics are used is 2 )‘1 Vznic(6i) \IZNZ F(Z) = E (F (2)) = where c(6,) = %log(1+ 6,.2 ). The variance is obtained using 3—Fz- l“ -a—F - k" 9‘ 2.; “WW" 31 Specially, we have In2(o,) 0 ' 2 J21. A. L x. .- _ 0 112(6k)‘ Therefore, the variance for the focused test is 2’32"“61‘) 2’32 ' Table 6 shows the theoretical means of the focused test for different statistics, and Table 7 shows the theoretical variances of the focused test for different statistics. Table 6 Theoretical Means of the Distribution for The Focused Test Focused Test Mean =2 One-sample t J”! log(1 + 6,2 ) _ $721080 + 522) (4.1) Two-sample 1 . . 1 . . , (4.2) t 5 \/n1 10g(1 + ¢lh ¢1C61 2) " 33 \/"2 103(1 + (I); ¢2L62 2) k z 3 One-sample t 2 )‘r J". log(1 + 6.2 ) (4.3) J 2 13 Two-sample . - 2 (4.4) t 2 )‘r \fili log(1 + 4):]: ¢IC5 i ) Ii 32 Table 7 Theoretical Variances of the Distribution for The Focused Test Focused Test Variance k=2 One-samplet 1 512(1+o,2/2) 622(1+622/2) (4.5) 2 (1+ 612)zlog(1+ of) + (1+ 5,2)21og(1+ of) Two-sample 1 ¢f¢f63<1+¢f¢f652/2) (46) t 5 (1+ ¢."¢f'6{’ )2 log(1+ ¢f¢f6{’) + ¢2EII>26622 (1 + ¢2E¢2C 62.? /2) I (1 + ‘I’zb ¢2C522 )2 108(1 + (I): (826522) k 2 3 One-sample t 1 2 52(1), 52 /2) (4.7) 2 ' 2A5 ( ‘2 2 I 2 2A: (1+5i)10g(1+§.-) TWO-sample 1 .5 p6 i2 1 E _C 5 12 2 (4.8) 1 _.2(2( 4) 4) (+4» ¢. /) Z )‘72 (1 + ¢iE¢iC6 ‘iz )2 log(1 + ¢iE¢iC6 .3) Theoretical Distribution for The Diffuse Test The diffuse test (Test C in Table 1) is the total sum of squares among the Z(p) values. For this reason, applying the multivariate delta method may not give the most accurate theoretical moments and asymptotic distribution for the diffuse test. Because the diffuse test is a function of Z(p,) (or Z i) values which is not linear, but quadratic, the distribution of the diffuse test may not be close to normal unless the samples included in the meta-analytic review are quite large. To obtain the distribution of the diffuse test, therefore, I apply results on quadratic forms in normal variables (Johnson & Kotz, 1970, p. 149; Stapleton, 1995, p. 65), starting with the normal theory for the distribution of the Z, values (Lambert, 1978). For 33 convenience, Q(Z) is used to represent the diffuse test in this section. The formula for the diffuse test, Q(Z), is k _ 2 Q(Z)= 21(21' —Z) 2 where k is the number of studies in the meta-analytic review, Z= (Z, , . . ., Z, )'and _ k Z = 22, /k. The quadratic form Q(Z)=Q( Z1 , ..., Z k ) for the diffuse test can be defined in terms of a symmetric matrix A, specifically k k Q(Z)=Q(Zl’ "'2 Z,)=21210,,Z,Z, I- )- =Z'AZ, where A is kx k symmetric matrix with elements 0,, , and Z is defined above. The symmetric matrix A in the diffuse test is 'k—l 1 1 ' T "I ‘75 1 k-l 1 '75 T ‘Z 1 1 k-l ”I ’I T 34 j i i 4 5 J 41.21.:th Mr": M; . r“ ,1. '21- Z 2 'Central case. Let Z = be a vector following a multivariate normal Zk distribution with expected value vector with E=E(Z), given in Table 5 and variance- covariance matrix, V, which is the diagonal matrix V= diag(1r|2 (6, ),Ir]2 (62 ),,, , , 112(6, )) where E, and n 2 (6,) are obtained from Lambert’s results. Further assume that E, =0 for all i ; this is the central case. Now the distribution of Q(Z), which is, Pr[Q(Z)sy] (02, where u) , is the same function of the expected values of the effect sizes as W is of the 2’s. That is, (1) is defined as (n = P’L" 13(2): P'L"E_,, where E(Z,)= Jn, log(‘1 + 6,2) for the ith study, for p values from one-sample t statistics. Therefore, if the a, are all one, then Q(W) has the noncentral x2 distribution with k k degrees of freedom and noncentrality parameter, 2 00,2 . i-l Now, I want to obtain the distribution of Q(W) where a, are not all one. A number of papers about exact and approximate methods for deriving the distribution of quadratic forms in normal variables have been published by Box (1954), Grad and Solomon (1955), Gurland (1955), Imhof (1961), Johnson and Kotz (1968), and others. 37 Some of these papers give tables of exact significance points of the distributions of quadratic forms for selected noncentrality parameters and weights (7), values). Solomon and Stephens (1977) also suggested two new approximations to the distribution of quadratic forms in normal variables. In this study, I use one of two approximations to the distribution of the diffuse test statistic based on results from Solomon and Stephens (1977), specifically, the three-moment chi-square fit. Solomon and Stephens (1977) reported that the three-moment chi-square fit gives excellent results in the upper tail and also appears superior to other approximations in the lower tail. The three-moment chi-square fit. Solomon and Stephens (1977) proposed that the distribution of the quadratic form Q(W)= file-(W. +w.>2, where the W, are independent standard normal variable (i.e., mean 0 and variance 1), and where the a, are non-negative constants. That is, the noncentral distribution of the diffuse test is the same as this except for the sign of to, values. However, since the expected value of each W. is zero and n), values are squared, the difference about the sign I of u), is not problematic. The distribution of the quadratic form can be fitted by Q(W) = A X ' when X has the xi, distribution and the constants A, P, and r can be found by matching moments as described below. Solomon and Stephens (1977) stated that the moments of Q(W) would be matched as 38 II = A2'{F(r +v)}/C, n,’ = A24'{I‘(2r + v)}/C, and [13: = A38'{I’(3r + v)}/C, where P/2 = v, and where C=F(v). Using expectation algebra, I obtained formulas for the three moments of Q(W): p. , u,’,and (23'. First, I: I1- : 20,1504], _wi)2 i-l k = 2ai(1+wi2) 2 i-l since E(W,) is equal to zero and E(W,)2 is equal to one. Next, I consider k He'= EIEMWI "0002]2 i-l I: k =EI<;a.-(W. -w.>-><;a.(m -w,.)2)1, l- 1- under two conditions; when i = j and when i a: j. k Ifi=jthen “'2’ = 2a,2E(((W, -(D,-)2)2) i-I k = Za,.z(3+6u),2 +u),"). i-l k k If i=j then 119' = EzaiajEKW, -00.-)2(W,- -w,)2) i-lj-l k k = 22a,a,(1+w,2)(1+u)f). i-l j-l 39 And finally, these are three conditions in which to evaluate 113' = EIEMWI #00213 ll bvdtr Ma- EaiajalEIO/Vi -wi)2(VVj —(1),)2(I'V, —(1),)2]. jll- (- — uni If i = j = lthen k 113' = 2a,.3(15+45u),2 + 15(1),4 +u),(’). i-I Ifi=j=lthen k k I13, = Z 2 “1201((3 + 60):? + (014 )(1 + (912)) - i-ll-l Ifiaejaelthen k k k I43, = 2 E Eaiajal((1+wi2)(1+wf)(1+ (1)12 )) 1-1 j-lI-l Solomon and Stephens also defined R2 = u,’ / u’= CI‘(2r+v)/{I‘(r+v)}2, (4.11) and R3 = (1.; / n3: C2F(3r+v)/{F(r+v)}3 (4.12) (Solomon & Stephens, 1977, p. 4). I can calculate R2 and R3 based on the parameters used in this simulation study. Therefore, given R2 and R3 , equations (4.11) and (4.12) above can be solved for r and v using computer routines. I set up a new function, D, to find r and v. D = [cr(2r+v)/{r(r+v)}2 - (Lg/(1212 +[C21’(3r+V)/{I“(r+v)}2 - (lg/(1312. (4.13) 40 I find the values of r and V by making the value of the D function close to or equal to zero using a program written in S-PLUS. Finally once r and v are available I can obtain A from the expression 11 = A2'{F(r+v)}/C. (4.14) As a result, I can obtain X, which is the value of v based on the pth percentile value of chi-square distribution with P=2v degrees of freedom obtained from the value of v based on (4.13) and the value of A based on (4.14). That value which is then raised to the rth power (i.e., X’) and multiplied by A can be used as the critical pth percentile value for the diffuse test. I then evaluate the proportions of simulated diffuse test values exceeding that pth percentile in the simulated distribution. Below I examine the accuracy of the asymptotic noncentral distributions through a simulation study showing how they behave in small samples. 41 CHAPTER V. SMALL SAMPLE BEHAVIOR OF THE SUMMARIES In this chapter the accuracy of the noncentral asymptotic sampling distributions of diffuse and focused tests in finite samples is examined through a simulation study. Empirical results from the simulation study provide a way to select a test for different research-synthesis situations, and also can be used to show the differences in power among different tests and analyses of Z( p) values used in meta-analysis. Parameters in Simulation Study I have chosen the values of simulation factors in this simulation study based on Becker’s (1985) and Chang’s (1992) simulation study results, and the evidence in recent research reviews. First of all, the factors in this simulation study are the following: 1) the number of studies, k; 2) the magnitudes of effect sizes, 6,, and 6,'; 3) the total sample size across studies, N; 4) the sampling fractions, 11:,, which represent the distribution of the total sample srze Into the k Ind1v1dual studIes, as It ,. = N , where n,1s study sample mm; and 5) the weight values, A, for i=1 to k. Values for the Parameters used in the Simulation Study To get more information concerning the simulation factors, I searched meta-analytic reviews in two major journals in education and psychology: Psychological Bulletin and 42 Review of Educational Research, from the beginning of 1995 into 1996. A total of 6 volumes of Review of Educational Research and 12 volumes of Psychological Bulletin were examined. The total number of research studies in those 18 volumes of the journals was 110. Thirty meta—analytic reviews were among the 110 research reports. All 30 meta-analytic reviews reported the total number of individual studies included in each review, but not all reported the total sample sizes included in each meta-analytic review. The sample sizes of individual studies included in each synthesis were not reported in any of the studies. Number of studies. In these recent meta-analyses, the numbers of studies included in each review have increased (compared to the counts found by Becker and Chang). Half of the meta-analytic reviews I found included over 50 studies. No meta-analytic review in the recent journals included only two studies in its review. However, because I examine the properties of the tests suggested by Rosenthal and Rubin ( 1979), I still need to consider the case of two studies. The selected values of k are therefore 2, 5, 10, and 50, because summaries for larger k values should behave similarly to those for k: 50 studies, but even larger k values result in computational complexity. Sample sizes. The values of total sample sizes and sampling fractions used in simulation are based on the values from 30 meta-analytic reviews examined and from Becker’s and Chang’s simulation studies, because they selected the values on the basis of 43 empirical study. The total sample sizes are equal to N = kn , and the selected n values are 20, 40, 80, and 160. Sampling fractions. I examine two different sets of sampling fractions 1:, , where i=1 through k for each k value. Table 8 shows the two sets of fractions to be examined within parentheses, for each k values. In the first set (n, = 1/ k), all studies have equal sample sizes (and equal sampling fractions), and in the second set, the studies have different sample sizes computed according to unequal sampling fractions. Table 8 Sampling Fractions for Simulation Study k It, where [=1 k 2 (5.5),(3.7) .5 (2.2.2.2.2)(;15.2.2.2.25) 10 (1.1.1.1.1.1.1.1.1.1) (05.06.07.08.08.08.09.11.16.22) 50 (02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02 .02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02.02 .02.02.02.02.02.02.02) (007.007.009.009.01.01.01.01.01.01.01.01.01.01.01.012.012.012.012 .012.012.014.014.014.014.014.014.016.016.016.016.016.016.02.02.02 .02 .02 .02 .02 .022 .022 .022 .03 .05 .05 .05 .06 .06 .1) For example, when the number of studies in a meta-analytic review is 2, then I have 8 different sets of actual sample sizes. We have four different total sample-size values (N = 40, 80, 160, and 320) and 2 pairs of sampling fractions (.5 .5) and (.3 .7). This leads to eight sample-size pairs (n, ,n2 ), specifically (20, 20), (40, 40), (80, 80), (160, 160), (12, 28), (26, 54), (52, 108), and (106, 214). A detailed list of the sample sizes used is shown in Appendix B. Very small sample sizes (as small as 3, 5, and 6) are indicated in the cases for two-sample 1 statistics. Even though those sample sizes are very small, I include those cases in this study because in some areas (such as physical education and special education) some studies have very small samples. Effect-size parameters. Four sets of effect-size parameters, 6, and 6,’ for i =1 to k and for k in this simulation study are chosen as follows: Set A: 1) All studies have the same effect size value of zero, Set B: 2) All studies have the same effect size, but the effect-size value differs from zero, Set C: 3) Half of the effect sizes equal zero, and the other half of the effect sizes have a second nonzero value, and finally 45 Set D: 4) All studies are divided into four or five groups and within each group the effect— size parameters have the same value. For k = 2 case, two non-zero values were used. Table 9 shows the values of the effect size that I used in this dissertation. Table 9 Different Patterns of Population Effect Sizes k Type of Effect: (6, ) where i=1 k 2 A: (0 0) B: (.2 .2) C: (0 .3) D: (.1 .4) 5 A: (0 0 0 0 0) B: (.2 .2 .2 .2 .2) c: (0 0 0.3.3) D: (0.1 .2.3 .4) 10 A:(0000000000) B:(.2.2.2.2.2.2.2.2.2.2) C: (0000003 .3 .3 .3 .3) D:(00.1.1.2.2.3.3.4.4) 50 A:(0000000000000000000000000 0000000000000000000000000) B: (.2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2) C: (0000000000000000000000000 00000.3 .3 .3 .3 .3 .3 .3.3 .3.3.3 .3 .3 .3.3 .3 .3 .3 .3 .3) 46 On the other hand, I set up other sets of effect-size parameters to check about the null hypothesis as shown in Table 3 and 4. Two sets of effect-size parameters are chosen as follows for only k=5: Set E: The values for effect-size parameters for k=5 were chosen as .2666667 .2 .2 .2 .16. So, all studies have same 11,6, values. Set F: The values for effect-size parameters for k=5 were chosen as .231 .2 .2 .2 .1788. So, all studies have the same ,/n,6, values. Within-study sampling ratio. In the simulation of two-sample r values, the total sample size within each study is divided into two sub-samples, n = n” + n“. Let the ratio of the sample size of the experimental group over the total sample size of study be represented by q), = "‘Z. Three sampling ratios are used, (I) = 05 , (I) = 0.7 , and (I) = 1 (representing the one-sample t test). For the two-sample cases, samples are equal within each study when 4), = 05 , while samples are unequal within each study when (I), = 0.7 . For simplicity, the within-study ratio of sample sizes, 4), , are the same across studies. Therefore, as an example, when k=2 and N=40 with It , =Jt , =.5 , then each study has n, = 20, and have n,” = n,“ = 10 when the sampling ratio is 0.5. The detailed list of sample sizes is shown in Appendix A. 47 Weight values. For the focused test, weight values A," and A," are chosen. A," represents weights for group differences and X," represents linear effects such as change over time. For k = 2, only one set of weights is possible. Table 10 Weight Values for Focused Test k x, 2 A," :(-1 1) A,“ :(-1 1) 5 x," :(-2 -10 1 2) A,“ :(-2 -2 -2 3 3) 10 2.," :(-4.5 -3.5 -25 -1.5 -05 .5 1.5 2.5 3.5 4.5) A,“ :(-1-1-1-1—1 11111) 50 2,!- :(-2.7—2.7 -27 -27 -2.7 -21 -21 2.1-2.1 -2.1 45454545450909090909030303 —0.3 -03 0.3 0.3 0.3 0.3 0.3 0.9 0.9 0.9 0.9 0.9 1.5 1.5 1.5 1.5 152121212121272727272n A." ;(-1-1-1-1-1-1-1-1 -1-1 —1 -1 -1-1-1 -1 -1 -1—1-1 —1-1-1 -1-11111111111111111111111111) I have the 4 different values of k, and the two sets of sampling fractions. Crossing the numbers of k values and sampling fractions ( 4*2), I have 8 sets of meta-analyses. Also I have 4 values of the total sample sizes and 4 sets of population effect-size patterns. Crossing the numbers of sample sizes and effect-size patterns (4*4), I have 16 sets of meta-analyses. Finally, crossing these two numbers of sets (8*16), I generate 128 sets of 48 meta-analyses. This is done for 3 cases ((1) values): one-sample t, and balanced and unbalanced two-sample t statistics. As a result, the total number of configurations of parameter values in the simulation study is 128 * 3 = 384 cases for checking the accuracy of the diffuse test, as shown in Table 11. The different levels for (I) represent that the results are from one- sample, balanced two-sample, and unbalanced two-sample t statistics. For the focused test we also consider the two sets of weight values, 21,, leading to 384 * 2 = 768 sets of meta-analysis. Table 11 Total Number of Configurations k 6or 6' N n o 4 cases 4 4 cases 2 levels 3 levels (2, 5, 10, 50) patterns (20k, 40k, 80k,160k) Equal /Unequal ((.5 .5), (.3 .7), (0 1)) Simulation Procedures For the simulation study in this dissertation, I used SAS/IML (Statistical Analysis System/Interactive Matrix Language) to generate the simulated meta-analyses and compute the test statistics, and I used the S-PLUS and SPSS (Statistical Package for the Social Sciences) statistical packages to check the accuracy of the asymptotic distributions. The following procedures were followed for the simulations for both diffuse and focused tests: 1) Select a set of parameter values k, N, n, 6, or 6* and o. 2) Generate random values of X from the standard normal distribution. 49 3) Generate S2 from a chi-squared distribution with df = n-l (for the one-sample t case) and 52* from a chi-square distribution with (if = nE-nC-Z for the two-sample t case. 4) Compute t variates from the values generated in steps 1), and 2), using I, = (J26 + X) / ,IS%_ , for the one-sample t case, and \/n”n(' /(n"~ +n“‘) *6. + X * = t. I 82. ‘ ‘ \//n" +n‘ —2 5) Obtain the k significance values (p, values) from the central t distribution for the tor , for the two-sample 1 case. t* variates computed in (3). 6) Obtain the standard normal deviates, Z(p,) values, for i=1 through k. 7) Finally, compute the diffuse test and focused test values using the k values of Z(p,). For each combination of parameters , I carried out 2000 replications of steps 1) through 7) above. Each replication represents a simulated synthesis of k studies. As a result, I have 2000 simulated syntheses for each combination of the parameters. To verify that these steps were properly carried out, I printed out each step for a small number of replications and checked each simulation step in the SAS program by hand to examine whether the generated and computed data were obtained correctly. The computer code is available upon request. 50 Accuracy of the Asymptotic Distributions for The Focused Test Statistic Tests for Normalitj Since the distribution formed by 2000 replications of the focused test statistics in each given set of population parameters is expected to be the large sample m distribution, 1 did normality checks for each simulated distribution. In SAS, the NORMAL option in the UNIVARIATE procedure computes a test statistic, called Shapiro and Wilk’s test statistic (SAS Procedures Guide, 1991), by which data are tested against a normal distribution with mean and variance equal to the sample mean and variance. In Appendix B I reported p-values of tests for normality when the p-value for that test of normality was less than .05. These are shown in the column labeled “Fit” (representing fit of the simulated distribution to a normal distribution) in Appendix B. Only 35 cases among 768 configurations of the simulation factors (less than 5%) showed a rejection of the null hypothesis that the simulated focused test values were a random sample from a normal distribution. These 35 included 15 results from the 384 combinations of parameters using weights for linear effects (about 4%), and 20 cases from the 384 combinations using weights representing group differences (about 5%). The simulated distributions of the focused tests in each configuration of parameters in this simulation study appeared to be normal, as expected. Table 12 shows the counts of statistically significant (”non-normal”) distributions. The frequencies were counted separately for the different values of k, of weight values (linear and group effects) and for the different types of statistics that led to the focused test statistics. No distributions showed significant deviations from normality for k=5. The chi-squared test for the association between number of studies and significant results 51 for normality was significant (x2 = 14.46, df = 3, p < .05). The obtained number of significant results of test for normality did not differ according to weight values, sample sizes, sampling fractions, or different types of statistics The chi-squared test for the association between the patterns of effect-size parameters and significant results for normality, however, was significant (x2 = 15.41, df = 3, p < .05). For the different sets of effect-size parameters (set A through set D), respectively, 4, 17, 11 and 3 cases each from 192 tests (=768 / 4) for normality were statistically significant. Set B of effect-size parameters, where all effect sizes were equal .2, showed more significantly non-nonnal cases than the other sets of effect-size parameters. In addition, for a distribution to be normal, values of skewness and kurtosis need to be close to zero. The empirical values of skewness and kurtosis were not different from zero. In most cases, the empirical values of skewness and kurtosis were less than .1, which was not quite different from zero. The maximum values of skewness and kurtosis were .207 for skewness and .371 for kurtosis. Both values were obtained when k=5 from balanced two-sample t statistics were used. However, the test for normality for the distribution that had the maximum values of skewness and kurtosis was not even statistically significant. 52 Table 12 Number of Significant Normal Tests for the Focused Tests Two-sample t k One-sample t Balanced Unbalanced Total Linear Effect 2 3 2 2 7 5 10 1 2 2 5 50 1 1 1 3 Group Effect 2 2 2 4 5 10 3 4 3 10 50 2 2 2 6 Total 10 13 12 35 Note. Each cell has 32 tests for normality. The tests for normality, however, did not provide any more information about the accuracy of the large-sample normal distribution except showing whether the simulated distribution was a normal distribution. Therefore, I compared the empirical and asymptotic distributions of the focused test statistics, comparing the empirical proportions between two distributions and using the Kolmogorov-Smirnov test. Empirical Pr0portions The tests for normality showed that the simulated distribution in each given set of population parameters for the focused test was consistent with the large sample norm—ail distribution (for more than 95% of 768 configurations). However this information did not provide the accuracy of the asymptotic distributions of the focused tests. Therefore I obtained empirical exceedance pr0portions to compare the simulated and the theoretical distributions. Since the distributions of the focused tests appeared normal, they should be 53 symmetrical. Therefore, I used five upper percentage points -- .01, .05, .1, .25, and .5 -- as cut points to compare the simulated and the theoretical distributions. First I obtained the percentile values in the theoretical distribution at five cut points (cc) .01, .05, .1, .25, and .5. Then I computed the proportions of simulated focused test statistic values beyond the critical values for those five cut points. The critical values were obtained as Z value * \[theoretical variance + theoretical mean, where Z value was obtained from the standard normal distribution at each cut point. Figure 5.1.1 shows the distribution of the focused test for k=10 when p values from balanced two-sample t statistics were used to test for group effects. Each study had the same sample size 40 and the pattern of effect-size parameters was B. Since this distribution had most agreement between the empirical proportions and cut points, I showed this distribution as an example of how I got the empirical proportions and compared them with the five cut points. The percentile values used as the critical values in this simulated distribution were 2.31, 1.64, 1.28, .67, and .00 at five cut points .01, .05, .1, .25, and .5. 54 Figure 5.1.1 The Distribution of the Focused Test Statistic for k=10 from Balanced Two-Sample t Statistics 300 “best” fitting distribution 200- Std. Dev=.96 Mean=.03 o - N=2000.00 ' 3.0012501200115011. . . . . . .oo .50 .oo L28 231 Figure 5.1.2 shows the distribution of the focused test for k=50 when p values from balanced two-sample t statistics were used to test group effects. Again, the total sample size was 40*k, but n’s were unequal across studies and the pattern of effect-size parameters was C. This distribution had the least agreement between the empirical proportions and cut points, and the test for normality was statistically significant (p=.006). Figure 5.1.2 displayed the least normal distribution among the 768 distributions based on the test for normality, but it still looked quite normal. The 55 percentile values used as the critical values in this simulated distribution were 6.16, 5.49, 5.12, 4.52, and 3.85 at the five cut points .01, .05, .1, .25, and .5. Figure 5.1.2 The Distribution of the Focused Test Statistic for k=50 from Balanced Two-Sample t Statistics 300 “worst” fitting distribution _ .465 200+ / \ y 3 .2. V 4 74 08 100-( Z Std. Dev: .99 Mean = 3.76 0 , N = 2000.00 ' .0050 1.001.502.00250300‘3502. 02.503005506001550 .00 3.85 4.52 5.12 5.49 6.16 I tested whether significant discrepancies existed between the two distributions for each configuration of parameters, using the normal approximation to the binomial distribution for the proportion. That is, to see whether each empirical proportion of values exceeding each percentile point differs significantly from the appropriate value of .01, .05, .1, .25, or .5, I calculated the confidence intervals for these five cut points using simultaneous tests at the .05 level. 56 I calculated the standard error of the binomial value for each cut point (cc) as SE = Jcc(1— cc) / nrep, . For example, when the comparison was made at the cut point .01, then the standard error was .01 * (.99) 2000 = .002. Then I obtained confidence intervals for each of the five cut points. I considered two ways of obtaining confidence intervals. The first one was to calculate the confidence interval using the confidence comparisons using the usual “nominal” .05 level for each test (the Z value of 1.96 from the standard normal distribution times the standard error). Second, I used a familywise error rate, which set .05 as the probability of making one or more Type I errors in the set of comparison tests. Since five obtained percentages for each distribution were not fully independent, the tests for empirical proportions within each distribution could not be regarded as independent. Therefore I used the standardized value of 2.5758 at the .05 level found in special tables for using Bonferroni tests for comparisons (Hays, 1994, p.1007). There were 3840 comparisons of empirical proportions for all distributions (multiplying 768 distributions by five cut points). A total of 425 cases (11%) from 3840 comparisons were statistically significant when I did comparisons using the usual nominal .05 level for each test. On the other hand, a total of 84 cases (2.2%) from 3840 comparisons were statistically significant using the familywise rate. Table 13 shows the simultaneous confidence intervals for the five cut points, and Table 14 shows the results 57 of the simultaneous tests to compare the empirical proportions. Appendix B shows the detailed test results for each configuration of parameters. Table 13 95% Confidence Intervals for Simultaneous Comparisons of Empirical Proportions for the Focused Test Cut Point Lower Limit Upper Limit .01 .00427 .01573 .05 .03745 .06255 .1 .08272 .117279 .25 .22506 .27494 .5 .47120 .52880 Table 14 Number of Significant Results for Empirical Proportions at Five Cut Points Two-sample t Cut Points One-sample t Balanced Unbalanced Total Linear Effect .01 1 2 3 .05 2 2 3 7 .1 1 3 3 7 .25 1 2 6 9 .5 3 2 5 10 Group Effect .01 1 2 2 5 .05 2 6 7 15 .1 1 7 9 17 .25 1 2 3 6 .5 2 3 5 12 29 43 84 Note. Each cell has 128 sets. 58 The chi-squared test was applied to determine whether the obtained numbers of significant results of tests for empirical proportions differ at the five upper percentage points (cut points). The chi-squared test was not statistically significant (x2 =9.91 df = 4, p > .05). That is, the obtained number of significant results was not different according to what upper percentage point was examined in the focused distribution. As shown in Table 14, 12 cases (about 1%) were statistically significant for the tests of empirical proportions when one-sample t statistics led to the focused test statistics. However when two-sample t statistics were used, a total of 72 cases (about 3%) were statistically significant. Therefore, this result showed that the large-sample approximation to the distribution of the focused test was more accurate when one-sample t statistics led to the focused test statistics. The Kolmogorov-Smimov Tests This test is a goodness-of-fit test to see whether a sample distribution fits the frequencies expected given a normal theoretical distribution. The distributions of the focused tests are expected to be normal, and I checked that the distributions of the focused tests were normal using the tests for normality. The theoretical distributions of the focused tests were symmetrical as were most of the simulated distributions. The empirical proportions at five cut points which focused more on the upper tail area seemed to have mostly agreed with theoretical values as examined by simultaneous tests. As a result, I chose the class intervals of five upper percentage points (.01, .05, .1, .25, and .5) for the Kolmogorov-Smimov test in this study to compare the simulated and the 59 theoretical distributions further, since the behaviors of the Kolmogorov-Smimov test were expected to be the same as ones when the class intervals included the whole range (.01 up to .99) of each distribution of the focused tests. The Kolmogorov-Smirnov test statistic is D, = maximum | Fs(x) - F ,(x) | for — 00 < x < 00 where F 5(x) = cumulative relative frequency in the simulated distribution, and F ,(x) = cumulative relative frequency in the theoretical distribution, where x is the percentile value at each of five cut points (upper percentage points). The statistic D, is the largest absolute difference between the simulated and theoretical cumulative relative frequencies that occur (here, at the five selected cut points). The critical value of D, at the .05 level (Amssey, 1951) is D, = my" , rep Many statistics books provide a table which gives the upper percentage points of the Kolmogorov-Smimov test, and I obtained the value of 1.36 at the .05 level for the Kolmogorov-Smirnov test when It”, was over 35 from Hays (1994). Since I had 2000 replications, the critical value of D“, was .03. The procedure for the Kolmogorov-Smirnov tests was as follows: A. I obtained the percentile values from the standard normal distribution at five out points: .01, .05, .1, .25, and .5. 60 B. The percentile values obtained in A were transformed by multiplying the theoretically derived standard deviation of the focused test and then adding the theoretical mean obtained in each configuration of population parameters (i.e., formulas (7) through (14)): Percentile value * Jtheoretical variance + theoretical mean. C. Then these values were used as the percentile values in the theoretical distribution on the focused test scale. D. I counted the frequencies of simulated focused test statistics exceeding those five percentile values obtained from C, and computed relative frequencies. Then I computed the differences between cumulative relative frequencies (the F S (x) values) in the simulated distribution and the theoretical distributions at the five cut points. Finally I got the maximum value of the five absolute differences between the cumulative frequencies of the two distributions. The calculated D, value from each combination of parameters was compared with the critical value, 0.03, to see whether it was significant beyond the .05 level. A total of 768 Kolmogorov-Smirnov tests were done to compare the simulated and theoretical distributions for the focused test. 34 cases among 768 configurations of population parameters (less than 5%) were statistically significant. More detailed information about the results of the Kolmogorov-Smimov tests was summarized further for the different configurations of parameters. In addition the obtained statistical values of the Kolmogorov-Smimov tests are reported in Appendix C. 61 Different types of statistics. The Pearson chi-square test was applied to determine whether the obtained number of significant results of Kolmogorov-Smirnov tests differed for the two different types of statistics which led to the focused test statistics. The number of significant results of Kolmogorov-Smirnov tests differed significantly for distributions based on one—sample t and two-sample t (x2 = 3.94, df =1, p < .05). The large-sample approximation to the distribution of the focused test statistic was more accurate when the focused test statistics were obtained from one-sample t statistics, with 5 percent of the two-sample t distributions showing lack of fit to the normal distributions versus only 2 percent for one-sample t. Table 15 Results of Kolmogorov-Smimov Tests for Different Statistics Significant Result One-sample t Two-sample t Total Yes 6 28 34 No 250 484 734 Total 256 512 768 Number of studies. Table 16 shows the counts of significant tests according to number of studies (k). Since none of Kolmogorov-Smirnov tests were significant for k = 2 or 5, the large-sample approximation to the distribution of the focused test statistic was very accurate for those k values regardless of the form of the t statistic and type of weight values. The significant results of the Kolmogorov-Smimov test, however, appeared for larger k. The chi-squared test by k was statistically significant (x2 = 47.64, df = 3, p 62 < .05). For larger k the inaccuracies also seemed associated with what type oft statistic led to the focused test statistics. For the cases where balanced or unbalanced two-sample t statistics were used, the frequencies of significant result increased when k increased. Results of Kolmogorov-Smimov Tests by k Table 16 Two-sample t k One-sample t Balanced Unbalanced Total Linear Effect 2 5 10 2 1 3 6 50 2 6 6 14 Group Effect 2 5 10 2 2 4 50 2 4 4 10 6 13 15 34 Note: Each cell has 32 comparisons of distribution fit. Weight values. The counts of significant results for linear and group-effect respectively were 20 and 14. The chi-squared test was not statistically significant (x2 = 1.10, df = 1, p > .05). Therefore, there were no differences in counts of significant results according to the different weight values. Sample sizes. When sample sizes decreased, the number of significant results of Kolmogorov-Smimov tests increased, as expected. Especially for smaller sample sizes, the inaccuracies also depended on what type oft statistic led to the focused test statistics. The chi-squared test was applied to determine whether the obtained number of 63 significant results of Kolmogorov-Smirnov tests differed according to sample size. The chi-squared test was statistically significant (x2 = 14.89, df = 3, p < .05), indicating that accuracy of the large-sample approximation did depend on the sample size as shown in Table 17. Further tests were done for focused tests using each type of weighting. The chi-squared test was not statistically significant (x2 = 7.60, df = 3, p > .05) for the distributions of tests of linear effects, while significant (x2 = 8.60, df = 3, p < .05) for group-effect tests. The large-sample approximation to the distribution of the focused test statistic was most accurate when larger sample sizes were used for tests of linear effects. Table 17 Results of Kolmogorov-Smimov Tests by N Two-sample t N One-sample t Balanced Unbalanced Total Linear Effect 20k 1 4 5 10 40k 1 1 2 80k 1 1 2 4 160k 2 1 1 4 Group Effect 20k 4 8 40k 1 1 1 3 80k 1 1 2 160k 1 1 Total 6 13 15 34 Note. Each cell has 32 comparisons of distribution fit. Effect-size patterns. Table 18 shows that there were no significant results of Kolmogorov-Smirnov tests for the effect-size patterns A and B. That is, the large-sample approximation to the distribution of the focused test was very accurate when the effect- 64 size patterns were A and B. The chi-squared test was applied to determine whether the obtained number of significant results of Kolmogorov-Smimov tests differed for the four sets of effect-size parameters. The chi-squared test was statistically significant (x2 = 51.33, df = 3, p < .05) when tests of linear and group effects were considered together. When the set of effect-size parameters was C (where half of the effect sizes were zero and the other half were .3), the discrepancies between simulated and theoretical distributions of the focused test statistics increased. A total of 25 of the 34 significant results (about 74%) were from the effect-size Pattern C. Table 18 Results of Kolmogorov-Smimov Tests by Effect-Size Pattern Two-sample t 6 One-sample t Balanced Unbalanced Total Linear Effect A 0 B 0 C 3 5 5 13 D 1 2 4 7 Group Effect A 0 B 0 C 2 12 D 1 1 2 Total 6 13 15 34 Note: Each cell has 32 comparisons of distribution fit. Sampling fractions. The chi-squared test also was applied to determine whether the number of significant results of Kolmogorov-Smirnov tests obtained differed for balanced versus unbalanced sampling fractions. The chi-squared test was not statistically 65 significant (x2 =.12, df = 1, p < .05). Therefore the accuracy of the large-sample approximation did not depend on balance in the study sizes. Table 19 Results of Kolmogorov-Smimov Tests by Sampling Fractions Two-sample t Jr One-sample t Balanced Unbalanced Total Linear Effect Equal 2 6 6 14 Unequal 2 1 3 6 Group Effect Equal 2 2 4 Unequal 2 4 4 10 6 13 15 34 Note. Each cell has 64 comparisons of distribution fit. Within-study samplirg ratio. In Table 11 I described three sets of within-study sampling fractions, representing using one-sample t (0 1), and balanced (.5 .5) and unbalanced (.3 .7) two-sample t statistics. The result of the chi-square test for one-sample and two-sample t statistics showed that the large-sample approximation to the distribution of the focused test statistic was more accurate when the focused statistics were obtained from one-sample t than two-sample t statistics. To follow up, I did a chi-square test for distributions of tests from balanced and unbalanced two-sample t statistics. Table 19 shows these counts. The chi-squared test was not statistically significant (x2 =0.99, df = 1, p > .05). The accuracy of the asymptotic distribution was not affected by within-study sampling ratios for tests based on two-sample ts.. 66 Comparisons of Moments I calculated asymptotic means and variances for the focused tests for each combination of parameters (these were derived by applying the multivariate delta method with Lambert’s approximation, and are shown in formulas (4.1) through (4.8)). Appendix B shows all values of the theoretical and empirical means and variances for each combination of parameters, as well as the absolute values of the mean differences. Figure 5.1.3 through Figure 5.1.6 show the magnitudes of the differences between the means of the simulated and theoretical distributions for the different configurations of parameters. Figure 5.1.3 shows that the magnitudes of mean differences between simulated and theoretical distributions for linear-effect tests were bigger when the number of studies (k) was larger and sample sizes were smaller. The case with k=50 and N=20k showed the biggest range of mean difference values. Also (in Figure 5.1.4) when the number of studies was large and sample sizes were smaller for tests of group effects, the magnitude of mean differences were bigger than for smaller numbers of studies and bigger sample sizes. Still, most differences were less than .05, for all parameter combinations. Figure 5 .1.5 showed that the magnitudes of mean differences between simulated and theoretical distributions for linear-effects were bigger for sets of effect-size parameters C and D than for A and B. Also (in Figure 5.1.6) when the numbers of studies were larger for tests of group effects, the ranges of mean difference values were bigger for the sets of effect-size parameters C and D. 67 Figure 5.1.3 Mean Differences for Linear Effect by k and N 0.0 . Ii _ Mean Differences Sample Sizes -.1 ,. DZOK am as“ -.2 .160K Case 24 2474 24 24 24'24 24 24 24'24 24 24 24'24 24 ' k=2 k=5 k=10 k=50 Figure 5 .1.4 Mean Differences for Group Effect by k and N 0.0 Mean Difference ‘- Sample Sizes -.1 DZOK D40K new -.2 .160K Case 24 24'24 24 24 24124 24 24 24'24 24 24 24'24 24 ' k=2 k=5 k=10 k=50 68 Figure 5.1.5 Mean Differences for Linear Effect by k and 6 0.0 ‘ Mean Differences Effect Size -.1 , DAN Zero CIA" Equal “Half Diff :2 'IIAMENH case 24 24'24 24 24 24 '24 24 24 24'24 24 24 24'24 24 k=2 k=5 k=10 k=50 Figure 5.1.6 Mean Differences for Group Effect by k and 6 0.0 . Mean Differen Effect Size -.1 . DA" Zero -.2 Case 24 24 '24 24 24 24'24 24 24 2724 24 24 24'24 24 W k=2 k=5 k=10 k=50 69 Z test for mean differences. Differences between simulated and theoretical mean values were examined using a Z test for each combination of parameters. The Z test was (simulated mean - theoretical mean) / J(theoretical variance) / (number of replications) . A total of 89 cases among 768 combinations ( about 11%) showed that the empirical and the theoretical means differed significantly beyond the 5% level. When k=2, 5, 10 and 50 respectively, totals of 17 (8%), 5 (3%), 24 (12%), and 43 (22%) cases showed statistically significant mean differences. Again the discrepancies are most common for larger values of k. And 39 cases among 89 cases (about 50%) were found for effect-size pattern C, where half of the effect sizes values are zero and the others have the same nonzero value. Appendix B shows the detailed test results for each configuration. Table 20 Results of Z Tests by k Two-sample t k One-sample t Balanced Unbalanced Total Linear Effect 2 2 2 3 7 5 10 4 5 5 14 50 5 9 10 24 Group Effect 2 3 3 4 10 5 1 2 2 5 10 2 4 4 10 50 6 7 6 19 23 32 34 89 Note: Each cell has 32 comparisons of distribution fit. 70 Table 21 Comparisons for the Theoretical Means with Rosenthal and Rubin’s Means for k=5 Mean N d Empirical Theoretical R&R1 R& R2 |Diff1| |Diff2| |Dif13| _ Equal 20* k A -0.005 0.000 0.000 0.000 0.005 0.005 0.005 B -0.032 0.000 0.000 0.000 0.032 0.032 0.032 C 1.229 1.245 1.273 1.243 0.017 0.044 0.014 D 1.333 1.364 1.414 1.357 0.030 0.081 0.024 40*k A 0.022 0.000 0.000 0.000 0.022 0.022 0.022 B 0.002 0.000 0.000 0.000 0.002 0.002 0.002 C 1.752 1.761 1.800 1.758 0.009 0.048 0.006 D 1.945 1.929 2.000 1.921 0.017 0.055 0.024 80*k A -0.025 0.000 0.000 0.000 0.025 0.025 0.025 B -0.001 0.000 0.000 0.000 0.001 0.001 0.001 C 2.489 2.491 2.546 2.488 0.002 0.057 0.001 D 2.723 2.727 2.828 2.718 0.005 0.106 0.005 160*k A 0.019 0.000 0.000 0.000 0.019 0.019 0.019 B 0.000 0.000 0.000 0.000 0.000 0.000 0.000 C 3.495 3.523 3.600 3.518 0.028 0.105 0.024 D 3.856 3.857 4.000 3.845 0.001 0.144 0.011 Unequal 20*k A -0.024 0.000 0.000 0.000 0.024 0.024 0.024 B 0.177 0.141 0.143 0.141 0.036 0.035 0.036 C 1.340 1.343 1.373 1.341 0.003 0.033 0.001 D 1.516 1.492 1.548 1.485 0.023 0.032 0.030 40*k A -0.010 0.000 0.000 0.000 0.010 0.010 0.010 B 0.174 0.200 0.202 0.200 0.026 0.028 0.026 C 1.871 1.900 1.942 1.897 0.029 0.070 0.026 D 2.083 2.111 2.189 2.103 0.027 0.105 0.019 80*k A -0.010 0.000 0.000 0.000 0.010 0.010 0.010 B 0.254 0.282 0.285 0.282 0.028 0.031 0.028 C 2.686 2.687 2.746 2.683 0.001 0.060 0.002 D 2.972 2.985 3.096 2.975 0.013 0.124 0.003 160*k A -0.022 0.000 0.000 0.000 0.022 0.022 0.022 B 0.396 0.399 0.403 0.399 0.003 0.007 0.003 C 3.788 3.800 3.883 3.795 0.012 0.095 0.007 D 4.218 4.221 4.378 4.208 0.003 0.159 0.011 71 Table 22 Comparisons for the Theoretical Variances with Rosenthal and Rubin’s Variances for k=5 Variance N d Empirical Theoretical R&Rl R&R2 |Diff1| |Diff2| IDiff3| Equal 20*]: A 0.981 1.000 1.000 1.026 0.981 0.981 0.955 B 0.974 0.962 1.000 0.984 1.012 0.974 0.989 C 0.919 0.959 1.000 0.979 0.958 0.919 0.939 D 0.928 0.937 1.000 0.948 0.990 0.928 0.978 40*k A 0.990 1.000 1.000 1.013 0.990 0.990 0.977 B 0.982 0.962 1.000 0.972 1.021 0.982 1.011 C 0.957 0.959 1.000 0.967 0.998 0.957 0.990 D 0.943 0.937 1.000 0.937 1.006 0.943 1.006 80*k A 1.018 1.000 1.000 1.006 1.018 1.018 1.011 B 0.944 0.962 1.000 0.966 0.981 0.944 0.977 C 0.948 0.959 1.000 0.961 0.989 0.948 0.987 D 0.914 0.937 1.000 0.931 0.976 0.914 0.981 160*k A 0.934 1.000 1.000 1.003 0.934 0.934 0.931 B 0.981 0.962 1.000 0.963 1.020 0.981 1.019 C 0.941 0.959 1.000 0.958 0.981 0.941 0.982 D 0.949 0.937 1.000 0.929 1.013 0.949 1.022 Unequal 20*k A 1.006 1.000 1.000 1.028 1.006 1.006 0.978 B 0.988 0.962 1.000 0.986 1.027 0.988 1.002 C 1.026 0.959 1.000 0.981 1.070 1.026 1.046 D 0.893 0.937 1.000 0.951 0.953 0.893 0.939 40 *k A 0.979 1.000 1.000 1.014 0.979 0.979 0.965 B 0.995 0.962 1.000 0.972 1.034 0.995 1.023 C 1.013 0.959 1.000 0.968 1.056 1.013 1.047 D 0.981 0.937 1.000 0.938 1.047 0.981 1.046 80*k A 0.976 1.000 1.000 1.007 0.976 0.976 0.970 B 0.967 0.962 1.000 0.966 1.005 0.967 1.001 C 0.933 0.959 1.000 0.961 0.973 0.933 0.971 D 0.937 0.937 1.000 0.932 1.001 0.937 1.006 160*k A 0.954 1.000 1.000 1.003 0.954 0.954 0.950 B 0.964 0.962 1.000 0.963 1.002 0.964 1.001 C 0.956 0.959 1.000 0.958 0.997 0.956 0.998 D 0.928 0.937 1.000 0.929 0.991 0.928 0.999 As an example when k=5, Table 21 and 22 present the mean differences and variance ratios of the moments calculated using Lambert’s theory and two distributions suggested 72 by Rosenthal and Rubin’s theory. In Table 21 and 22 the column labeled “R&Rl” refers to moments based on the distribution of Z(p) with mean 4/176 and variance one, and the column labeled “R&R2” refers to moments based on the distribution with mean and variance described in formula 3.1, proposed by Rosenthal and Rubin for the distribution of the focused test statistics. The columns labeled “IDifflI” and “Ratiol” through “IDiff3I and “Ratiol” are the absolute mean differences and variance ratios between empirical theoretical moments using Lambert’s theory, or Rosenthal and Rubin’s theory respectively. All of the empirical means were farther from the ones calculated based on Rosenthal and Rubin’s first theoretical result than ones based on Lambert’s and Rosenthal and Rubin’s second result. The variances calculated based on Rosenthal and Rubin’s first distribution theory result were mostly bigger than ones calculated by Lambert’s theory or Rosenthal and Rubin’s more complicated theory. As expected, for the larger sample sizes, the mean and variances based on moments of the distribution described in formula 3.1 were closer to the empirical ones. Overall, the obtained numbers of significant results for comparisons of empirical proportions were not different according to what upper percentage point was examined in the focused distribution. The large-sample approximation to the distribution of the focused test was more accurate when one-sample t statistics led to the focused test statistics. The accuracy of the asymptotic distribution was not affected by within-study sampling ratios, sampling fractions, and weight values, as expected. When sample size increased, the accuracy of the asymptotic distribution was increased, but while the number of studies increased, the accuracy of the asymptotic distribution decreased. 73 Generally the distribution of the focused test was more accurate for the group effect-size pattern than for the linear effect—size pattern. Power Analysis for The Focused Test Statistic Power Comparisons Another way of checking the accuracy of the large-sample approximation and learning about test behavior is to compare the simulated and theoretical power values. In this dissertation, I examine the power values at 01 = .05, which is the level frequently used in educational and psychological research, and the more lenient .10. The procedures used to calculate the power values were similar to those used to get the empirical proportions exceeding five cut points. But the difference was that the power values were the proportions exceeding the percentile values at or = .05 obtained from the focused-test distribution under the null hypothesis (the standard normal distribution) not from the theoretical distribution under the alternative hypothesis. The absolute maximum difference between simulated and theoretical power values for 768 configurations of parameters was 0.047 when k=50, N=20k, and the set of effect size pattern D for two-sample t statistics. Appendix D shows the empirical and theoretical power values and the differences when a = .05 and 01 = .10 74 Power Results for The Focused Test Statistic To understand whether the conditions of the simulation study led to differences in the power values, analysis of variance was conducted, treating the 768 power estimates as data points. Analysis of variance (ANOVA) was done for power values at a = 0.05 with factors for the weight values, number of studies, sample sizes, sampling fractions, different sets of effect-size parameters, and within-study sampling ratio for the focused test statistics. Table 23 Analysis of Variance for Power Values of the Focused Test Sum of Squares df Mean Square F Sig Main Combined 74.127 13 5.702 204.953 .000 Effects 1» 1.535E-02 1 1.535E-02 .552 .458 k 12.822 3 4.274 153.618 .000 N 5.853 3 1.951 70.122 .000 It 1.057 1 1.057 38.009 .000 6 49.865 3 16.622 597.131 .000 o 4.515 2 2.257 597.445 .000 Model 74.127 13 5.702 204.953 .000 Residual 20.977 754 2.782E-02 Total 95.104 767 .124 Power values of the focused test statistics were explained mostly by the sets of effect-size parameters (R2 =.52) and number of studies (R2 =.13). Weight values representing linear and group effects did not affect the power of the focused test statistics. Sample size and within-study sampling ratio also explained the power values of the focused test statistics. However since the results from ANOVA shown in Table 23 aggregated information across types of tests leading to the p values, it is not sufficient to 75 fully characterize the power values of the focused test statistics. Therefore, I did separate ANOVAs for the power values for tests based on different types of statistics. Table 24 and Table 25 show the results of ANOVA when one-sample and two sample I statistics respectively led to the focused test statistics, and there was no difference between the results for one-sample and two-sample t statistics. All main effects of k, N , 6, all two- way interactions, and all three-way interactions were statistically significant in both ANOVAs. The plot of distribution of residuals gave the appearance of a sample that might have come from a normal distribution, but the plot of residuals against power values showed that the residuals were not homoscedastic. The results of the ANOVA differed somewhat for tests from different types of statistics. The amount explained by the ANOVA model was about 83% when one-sample t statistics gave rise to the focused statistics, and 75 % for two-sample t statistics. Both ANOVA tables showed that the power values differed according to the three main factors: number of studies, sample size, and the sets of effect-size patterns. The two-way and three-way interactions were all statistically significant. It showed that varying differences of the power values existed between the means of the power values representing different set of effect-size parameters, depending on the particular sample sizes or numbers of studies applied. Tables 26 and 27 show means of power values, showing interactions among these three factors in detail. 76 Table 24 Analysis of Variance for Power Values for One-Sample t Sum of Squares df Mean Square F Sig Main Effects 34.878 9 3.875 266.885 .000 6 29.676 3 9.892 681.243 .000 N 1.898 3 .633 43.568 .000 k 3.304 3 1.101 75.843 .000 Two-way Interactions 3.004 27 .111 7.661 .000 6 xN 1.106 9 .123 8.464 .000 6 x k 1.502 9 .167 11.492 .000 k x N .396 9 4.396E-02 3.027 .002 Three-way Interaction .919 27 3.405E-02 2.345 .000 Model 38.801 63 .616 42.415 .000 Residual 2.788 192 1.452E-02 Total 41.589 255 .163 Table 25 Analysis of Variance for Power Values for Balanced and Unbalanced Two-Sample t Sum of Squares df Mean Square F Sig Main Effects 36.712 9 4.079 785.710 .000 6 23.028 4 7.676 1478.527 .000 N 3.987 3 1.329 255.996 .000 k 9.697 3 3.232 622.607 .000 Two-way Interactions 9.268 27 .343 66.118 .000 6 x N 2.715 9 .302 58.097 .000 6 x k 6.139 9 .682 131.383 .000 k x N .415 9 4.607E—02 8.874 .001 Three-way Interaction .742 27 2.748E-02 5.294 .000 Model 46.722 63 .742 142.849 .000 Residual 2.326 448 5.192E-03 Total 49.048 511 9.598E-02 77 When number of studies and sample sizes increased, generally the power values increased, as expected. This is shown in Tables 26 and 27, and Figure 5.1.7. Comparing the sets of effect-size parameters, the power values were usually a little bigger when the pattern of effect-size parameters represented group effects (D) than when the pattern of effect-size parameters represented linear effects (C), for k=5 or larger and all sample sizes. Table 26 Means of Simulated Power Values for One-Sample t Statistics by k x N x 6 Effect-Size Pattern k bi IX 13 (I I) 2 20k .052 .063 .26 .273 40k .047 .063 .408 .431 80k .053 .078 .64 .669 160k .055 .096 .881 .891 5 20k .049 .052 .396 .369 40k .047 .055 .626 .592 80k .052 .068 .873 .846 160k .049 .072 .989 .979 10 20k .051 .084 .681 .637 40k .051 .112 .9 .877 80k .049 .148 .991 .981 160k .052 .223 1 1 50 20k .046 .248 .998 .961 40k .05 .368 1 .999 80k .049 .48 1 1 160k .05 .521 1 Note. A: all equal 0. 3: all equal .2. C: half 0 and half .3 D: all different values 78 Table 27 Means of Simulated Power Values for Balanced and Unbalanced Two-Sample 1 Statistics bykxNx6 k IV 14 13 CI 1) 2 20k .046 .052 .112 .119 40k .047 .056 .159 .173 80k .048 .061 .238 .269 160k .048 .07 .397 .418 5 20k .042 .052 .15 .15 40k .047 .05 .236 .228 80k .047 .056 .379 .369 160k .048 .061 .602 .584 10 20k .045 .058 .252 .246 40k .048 .068 .414 .405 80k .05 .083 .652 .63 160k .049 .11 .89 .862 50 20k .044 .111 .739 .55 40k .048 .158 .937 .799 80k .05 .242 .996 .958 160k .048 .367 1 .998 Figure 5.1.8 also shows that the power values had more variation for the set D effects than for set C, for k=2 or 50, but this was reversed for k=5 or 10. Also recall that the set of effect-size parameters B with equal sampling fractions represented the null hypothesis (all J; 6, values were equal), so power values for those cases should be closer to .05. Finally the power values when one-sample t statistics led to the focused test statistics had bigger values than those when two-sample t statistics led to the focused test statistics. 79 Figure 5.1.7 Power Values for the Focused Test based on Two-Sample t Statistics by N x k N Dn=20k E3n=4°k -n=80k 22 lll'lGOk N 32 32j32 32 32 32'32 32 32 3232 32 32 32T32 32 k=2 k=5 k=10 k=50 Figure 5.1.8 Power Values for the Focused Test based on Two-Sample t Statistics by 6 x k L2 10‘ delta DA [33 .c .0 32 32'32 32 k=2 32 32'32 32 k=5 80 32 3232 32 k=10 32323232 k=50 Figure 5.1.9 Power Values for the Focused Test based on Two-Sample t Statistics by 6 Pattern xN .6 . POWER .4 , N L—Jn=20k .2 , E l [:ln=40k 0,0 , -—- - -n_80k -.2 .160k 32 3232 32 32 32'32 32 32 32'32 32 32 32'32 32 A B C D Figure 5.1.9 shows that the power values of the focused tests increased within each set of effect-size parameters when the sample sizes increased. Comparing the sets of effect-size parameters for the same sample size, there was more variation of the power values in the set C than the set D within same sample size. This figure supports the presence of interaction effects among numbers of studies, sample sizes, and the sets of effect-size parameters. Under the null hypothesis. The simulated and theoretical power values near the null hypothesis for the focused test should be very close to the or level. To explore the controversy over the nature of the null hypothesis, I examined the power values at a = .05 for k=5 when one-sample t statistics led to the focused test statistics. The results 81 supported the evidence that the null hypothesis involving effect size weighted by squared root of sample size was correct as shown in Table 4. To do this, I used additional sets of effect-size values E and F, described in the simulation procedure and just below. Table 28 Power Values under Possible Null Hypothesis by N and 6 for k=5 from One-Sample t Linear Group n N 6 Empirical Theoretical |Diftj Emp. The. |Diff] Equal 20* k B .046 .047 .001 .041 .047 .006 40* k B .047 .047 .000 .050 .047 .003 80* k B .052 .047 .005 .048 .047 .001 160*k B .050 .047 .003 .045 .047 .002 Unequal 20*k A .049 .050 .001 .047 .050 .004 B .067 .063 .004 .055 .058 .003 E .037 .034 .003 .048 .038 .010 F .043 .047 .004 .038 .047 .009 40*k A .051 .050 .001 .050 .050 .000 B .057 .070 .014 .066 .063 .003 E .029 .030 .001 .032 .035 .003 F .055 .047 .008 .043 .047 .004 80*k A .047 .050 .004 .050 .050 .000 B .089 .082 .006 .084 .070 .014 E .023 .025 .002 .024 .030 .006 F .042 .047 .005 .05 1 .047 .004 160* A .047 .050 .003 .044 .050 .006 B .102 .102 .000 .089 .084 .006 E .020 .019 .001 .029 .025 .003 F .048 .047 .001 .054 .047 .006 Table 28 shows the power values for different sets of effect-size parameters that represented three different cases. First the cases for It equal and effect-size set B described cases where n,62 =...=n,6, or £6, =...= 11,6, . 82 Second, the cases with n unequal and the sets B and E effects represented cases where Third the cases with It unequal and the set of effects labeled F represented cases where fie, =...= n,6,. Set A is always a null case, because all 6 values equal zero. Table 28 shows that for equal sample sizes (equal Jr), the empirical power values for set B effects are quite close to .05, though the value predicted by theory is slightly below .05. These values are consistent with any of three possible null models (all 6, equal, all n,6, equal, or all Jr: 6, equal). The results also showed that the power values for unequal sample sizes and the set B were usually larger than .05, which is consistent with this case (all 6, equal) being an altemative-hypothesis condition when ns are unequal. The values for set E were always less than .05, which does not support a view of all n,6, equal as a null case. Although it also does not seem good if E is an alternative case, either. The power values were much closer to .05 for the cases with n: unequal and the set F of effects, where the effect-size parameters are equal when weighted by the square root of their sample sizes. This leads empirical support to the null models for the focused test in Table 4, and suggests that those shown in Table 3 are only appropriate when sample sizes are equal. Accuracy of the Asymptotic Distributions for The Diffuse Test Statistic Empirical Pr0portions To check the accuracy of the large-sample approximation to the distributions of the diffuse test I again calculated the empirical exceedance proportions. Comparing the 83 simulated proportions and the eleven significance levels (the theoretical exceedance pr0portions) showed the discrepancies (magnitudes of the differences) between the simulated and the theoretical distributions for each configuration of parameters. I used eleven upper percentage points (01) .01, .025, .05, .1, .25, .5, .75, .9, .95, .975, and .99 as cut points to compare the simulated and the theoretical distributions. Then I examined which of these proportions showed statistically significant differences between theoretical and simulated distributions. The procedures to obtain the empirical proportions were as follows: A. I obtained the percentile values C(a ) from the central x2 distribution at eleven points, for a = .01, .025, .05, .1, .25, .5, .75, .9, .95, .975, and .99, where a = P()(2 5 C(01)). The percentile values obtained in (A) were used to obtain percentile points in the theoretically derived distribution, following steps shown in (4.11) through (4.13) in Chapter IV. Specifically, I computed AX', where X = C ( or ) and A and r were obtained for each configuration of population parameters. Then these transformed values were used as cut points or critical values to compare the simulated distribution and asymptotic distribution. The eleven cut points obtained from the theoretical distribution (described in (B)) were used to make comparisons of frequencies between the two distributions in 12 categories. I calculated the empirical proportions, dividing the frequencies (observed) of simulated diffuse test statistics in each of the 12 categories by 2000. 84 Figure 5.2.1 shows the distribution of the diffuse test statistic for k=5 when p values from one-sample t statistics were used to test for the set of effect-size parameters C. Each study had the same sample size 20. Since this distribution had most agreement between the empirical proportions and cut points, I showed this distribution as an example of how I got the empirical proportions and compared with the cut points. The percentile values used as the critical values in this simulated distribution were 12.8, 10.8, 9.2, 7.5, 5.2, 3.2, 1.9, 1.0, 0.7, 0.5, and 0.3 at eleven cut points .01, .025, .05, .1, .25, .5, .75, .9, .95, .975, and.99. Figure 5.2.1 The Distribution of the Diffuse Test Statistic for k=5 from One-Sample t Statistics 400 a “best” fitting -— distribution 300. .50 2004 H 100. i , i i i l g , E I “05 Std. Dev=2.91 .- E , i i i '01 Mean=4.2 E l o g , : ; . I : N=2ooo.oo 03 "0.0 ‘20 2.0 “0.00.0 100120140160130200220 ' 07 L9 3.2 7.5 10.8 12.8 0.5 1.0 5.2 9.2 Figure 5.2.2 shows the distribution of the focused test for k=5 when p values from balanced two-sample t statistics were used to test for the set of effect-size parameters C. Again, the total sample size was 20*k, but ns were unequal across studies. This 85 distribution had the least agreement between the empirical proportions and cut points, but it still looked quite similar to be chi-squared distribution. The percentile values used as the critical values in this simulated distribution were 14.8, 12.5, 10.6, 8.7, 6.0, 3.8, 2.2, 1.2, 0.8, 0.6, and 0.3 at the eleven cut points .01, .025, .05, .1, .25, .5, .75, .9, .95, .975, and.99. Figure 5.2.2 The Distribution of the Diffuse Test Statistic for k=5 from Balanced Two-Sample t Statistics 400 g .472 (should be .50) “WOYSI” fitting 300 1 __— distribution 2001 100, .069 (should be .10) . I i .037 (should be .05) Std, Dev = 2.37 7‘ i I 1 Mean = 4.2 4 ‘ l I f ‘00 2.0 '4.0 15.0 '80 '10.o‘12.o14033013020022.0240 0.3 0.6 1.2 3.8 3.7 10.6 12.5 14.8 0.8 2.2 6.1 I also calculated the standard error to obtain the confidence interval for each of eleven probability points using the standard error (SE) of the binomial value, SE = flu - a) / rim . For example, when the comparison was made at the probability point .01, then the standard error was 86 01* (.99) = .0022 2000 ’ and the 95 percent confidence interval for probability point .01 with a = .05, was .01 x 1.96 * .0022, or from .0057 to .0143. Finally I obtained the confidence interval for each of 11 cut points. I considered two ways of obtaining the confidence interval as I did for the focused test. The first one was to calculate each confidence interval using the confidence limit with the usual nominal .05 level for each test. The second way of test for comparisons was that I used a .05 familywise error rate, which was the probability of making one or more Type I errors in the set of comparison tests. Since the eleven percentages for each distribution were not fully independent, the tests for empirical proportions within each distribution could not be regarded as independent. There were 4224 comparisons of empirical proportions for all distributions (multiplying 384 distributions by 11 cut points). A total of 945 cases (22%) from 4224 comparisons were statistically significant when I did comparisons using the usual .05 level for each test. On the other hand, a total of 540 cases (13%) from 4224 comparisons were statistically significant using the familywise rate, which seemed more appropriate for this study making more than one test on the same data. Table 29 shows the simultaneous confidence intervals for the 1 1 cut points, and Table 30 shows the results of the simultaneous tests to compare the empirical proportions. Appendix B shows the 87 expected and simulated proportions for other combinations of various values of N, k, and the effect-size patterns. Table 29 95% Confidence Intervals for Comparisons of Empirical Proportions for the Focused Test Cut Points Lower Limit Upper Limit .01 .00370 .01630 .025 .0151 1 .03489 .05 .03619 .06381 .10 .08099 .11901 .25 .22256 .27744 .5 .46832 .53168 .75 .72256 .77744 .90 .88099 .91901 .95 .93619 .96381 .975 .96511 .98489 .99 .98370 .99630 Table 30 Number of Significant Results for Empirical Preportions at Eleven Cut Points Two-sample t Cut Points One-sampler Balanced Unbalanced Total .01 2 7 8 17 .025 2 16 12 30 .05 2 23 24 49 .1 2 27 26 55 .25 4 31 34 69 .5 2 32 36 70 .75 2 31 27 60 .90 4 26 24 54 .95 8 22 19 49 .975 8 19 19 46 .99 13 14 14 41 49 248 243 540 Note. Each cell has 128 sets. 88 The chi-squared test was applied to determine whether the obtained numbers of significant results of tests for empirical proportions differ at the eleven percentage points (cut points). The chi-squared test was statistically significant (x2 =57.94 df = 10, p < .05). That is, the obtained number of significant results was different according to what upper percentage point was examined in the distribution of the diffuse test. Table 30 shows that there seems to be more discrepancies in the middle among eleven cut points. Table 31 shows the expected proportions and simulated proportions at eleven probability points for a subset of 32 distributions for k=2 when one-sample t statistics led to the diffuse test. It shows where the discrepancies between simulated and theoretical distributions exist when k=2 for the case of tests from one-sample t statistics, and gives more detail on accuracy of 321arge-sample approximations to the distribution of the diffuse test. A total of 28 out of 352 simulated proportions (about 8%) were significantly different from those of the theoretical distribution. This table shows that all of the discrepancies existed in the lower tail, so the large-sample approximation to the distributions in the upper tail (which was more interesting to examine in this study) was quite accurate. On the other hand, when larger numbers of studies and small sample sizes were used to obtain diffuse statistics, the asymptotic distributions tended to have thinner upper tails and much fatter lower tails than the distribution predicted by the theory. 89 Table 31 Accuracy of the Asymptotic Distribution of the Diffuse Test for k=2 N 6 Empirical Proportions from Upper—Tail .999 .975 .95 .9 .75 .5 .25 .10 .05 .025 .01 Equal Sampling Fractions 20k A 0.989 0.974 0.957 0.909 0.767 0.511 0.258 0.095 0.043 0.025 0.008 B 0.992 0.976 0.955 0.903 0.749 0.484 0.241 0.106 0.049 0.026 0.01 l C 0.991" 0.974 0.954 0.904 0.757 0.507 0.251 0.095 0.047 0.023 0.006 D 0.993 0.975 0.955 0.903 0.752 0.518 0.253 0.100 0.044 0.021 0.008 40k A 0.992 0.975 0.955 0.905 0.760 0.514 0.257 0.103 0.051 0.022 0.009 B 0991* 0.971“ 0.946“ 0.891 0.739 0.503 0.268 0.] 14 0.057 0.026 0.012 C 0.982“ 0.963" 0.929" 0.881 0.736 0.503 0.247 0.083 0.041 0.022 0.007 D 0.981 0.960 0.930 0.882 0.742 0.503 0.250 0.090 0.042 0.022 0.007 80k A 0.992 0.975 0.950 0.898 0.738 0.513 0.257 0.103 0.051 0.023 0.01 l B 0.988 0.970 0.943 0.889 0.751 0.482 0.252 0.099 0.050 0.026 0.012 C 0.972“ 0.948“ 0.919“ 0.867“ 0.745 0.494 0.238 0.089 0.046 0.023 0.008 I) 0.975‘ 0.955‘ 0.931 “ 0.884 0.748 0.502 0.262 0.105 0.054 0.033 0.010 160k A 0.991 0.981 0.956 0.911 0.766 0.523 0.249 0.103 0.051 0.025 0.013 B 0.99 '1 0.976 0.951 0.899 0.745 0.495 0.248 0.101 0.051 0.027 0.012 C 0.980“ 0.969 0.942 0.885 0.743 0.500 0.256 0.107 0.053 0.030 0.01 l D 0977" 0.962" 0.940 0.892 0.751 0.520 0.247 0.099 0.050 0.026 0.01 l Unequal Sampling Fractions 20k A 0.993 0.978 0.957 0.909 0.766 0.507 0.248 0.105 0.054 0.027 0.01 l B 0.991 0.976 0.954 0.902 0.757 0.497 0.244 0.092 0.042 0.020 0.009 C 0.988 0.971 0.946 0.900 0.755 0.510 0.239 0.089 0.044 0.024 0.012 D 0.983“ 0.968 0.935" 0.884 0.728 0.479 0.239 0.092 0.041 0.026 0.010 40k A 0.991 0.980 0.950 0.891 0.752 0.51 1 0.255 0.102 0.054 0.027 0.009 B 0.992 0.979 0.953 0.895 0.752 0.497 0.248 0.100 0.049 0.028 0.012 C 0.969“ 0.951“ 0.923“ 0.873“ 0.737 0.500 0.243 0.098 0.050 0.029 0.010 I) 0.981“ 0.960“ 0.929“ 0.889 0.750 0.507 0.255 0.1 10 0.059 0.034 0.014 80k A 0.987 0.975 0.944 0.895 0.740 0.503 0.259 0.104 0.058 0.027 0.015 B 0.993 0.977 0.954 0.899 0.758 0.496 0.244 0.096 0.043 0.022 0.010 C 0.974" 0.960“ 0.940 0.896 0.771 0.522 0.248 0.099 0.053 0.023 0.008 D 0978" 0.966 0.947 0.901 0.764 0.510 0.251 0.094 0.043 0.024 0.010 160k A 0.990 0.969 0.939 0.889 0.729 0.480 0.234 0.091 0.042 0.020 0.007 B 0.995 0.982 0.954 0.912 0.753 0.498 0.234 0.086 0.046 0.021 0.010 C 0.992 0.977 0.955 0.909 0.762 0.500 0.247 0.101 0.049 0.028 0.013 I) 0.988 0.978 0.954 0.908 0.760 0.512 0.250 0.094 0.048 0.025 0.008 *Statistically Significant at a: .05. 90 I reported the empirical proportions from upper-tail values to three decimal places in Table 31 and Appendix E. Because of rounding, some identically printed values were marked as statistically significant, and not, in Appendix E. For example, the confidence interval for the upper percentage point .01 was from .00370 to .01630. There were two values rounded to .004, reported in two different places for the cut point .01. One of the .004 values was statistically significant and other was not. The actual value of the .004 that was marked was .0035 which was not included within the confidence interval for .01. The same explanation would be applied to marked and unmarked equal printed values in Appendixes B and E. Goodness-of—Fit Tests The simulated distribution formed by 2000 replications of diffuse test statistics in each given set of population parameters is expected to be a modified chi-squared distribution, as predicted by the theory for the diffuse test. Therefore, I used a x 2 goodness-of—fit test to compare each simulated distribution (for a given set of parameters) with the distribution predicted by theory. I calculated the Pearson chi-square statistic, x 2 , 2 =12(fi—mi)2 j'] m}- where m , = the expected frequency of the theoretical distribution in category j, and 91 f, = the obtained frequency from the simulated distribution in category j for j=1 through 12 categories. The procedures for the x2 goodness-of-fit test were as follows: A. I obtained the percentile values C (or ) from the central x2 distribution at eleven points, for a = .01, .025, .05, .1, .25, .5, .75, .9, .95, .975, and .99, where or = P()(2 5 C(01)). B. The percentile values obtained in (A) were used to obtain percentile points in the theoretically derived distribution, following steps shown in (4.11) through (4.13) in Chapter IV. Specifically, I computed AX', where X = C(a ) and A and r were obtained for each configuration of population parameters. Then these transformed values were used as cut points or critical values to compare the simulated distribution and asymptotic distribution. C. The eleven cut points obtained from the theoretical distribution (described in (B)) were used to make comparisons of frequencies for the two distributions in 12 categories. I counted the frequencies (observed) of simulated diffuse test statistics in each of total 12 categories obtained from the theoretical distribution. D. I had the expected frequencies of the theoretical distribution in 12 categories, specifically 20, 30, 50, 100, 300, 500, 500, 300, 100, 50, 30, and 20, based on 2000 replications. Finally I calculated the Pearson chi-square statistic using the obtained theoretical and empirical frequencies. The number of degrees of freedom for the goodness-of—fit test was 11 in this study. The critical value of the x2 goodness-of-fit test with or =.01 and 11 degrees of freedom 92 was 24.72. Therefore, the x2 value obtained from the simulated distribution in each configuration of parameters was compared with the critical value 24.72 to see whether it was significant beyond the .01 level. I found 75 cases among 384 configurations (about 19%) statistically significant, using 12 categories. However, in this study I examined the empirical proportions and I found most of the discrepancies existed in lower tail, which was less interesting to examine than the upper tail. Solomon and Stephens (1977) made comments about the accuracy of various approximations, and noted that all approximations presented in their paper generally gave excellent accuracy in the upper tail, but all approximations became relatively less good in the lower tail. Even though Solomon and Stephens claimed that the three-moment chi- square fit was better than the other approximations, still the three-moment approximation (used here) would be worse in the lower tail. For the asymptotic distribution of the diffuse tests, I found that most of the discrepancies came from the lower tail. Therefore, I did another set of x 2 goodness-of-fit tests, using the upper 90% of the distribution to check the accuracy of the large-sample approximations to the diffuse test in this study. I found 48 cases among 384 configurations (about 13%) statistically significant. The statistical values of the x2 goodness-of—fit tests are shown in Appendix F. More detailed information about the results of the x2 goodness—of-fit tests is summarized below for the different parameter configurations. Different Types of Statistics. The number of significant results for the x2 goodness-of—fit tests differed when one-sample t and two-sample 1‘ statistics led to the 93 diffuse test statistics. For one-sample 1 statistics, none of the tests was statistically significant. For two-sample t statistics (balanced or unbalanced), about 19 % of the x2 tests were statistically significant. That is, the accuracy of the large-sample approximation to the distribution of the diffuse test clearly depended on the different types of statistics. The chi-squared test comparing the distributions based on one and two-sample is was statistically significant (x2 = 27.43, df =1, p < .05). The large-sample approximation to the distribution of the diffuse test statistic was more accurate when the diffuse test statistics were obtained from one-sample t statistics. Table 32 Results of x2 Goodness-of-Fit Test for Different Statistics Significant Result One—sample t Two-sample 1 Total Cases Yes 0 48 48 No 128 208 336 Total Cases 128 256 384 Number of Studies. The chi-squared test was applied to determine whether the number of significant results obtained differed according to number of studies across the 3 test types. The chi-squared test was also statistically significant (x2 =53.14, df = 3, p < .05). When one-sample t statistics led to the diffuse test, the large-sample distribution of the diffuse test was very well approximated, without regard to the number of studies. However, when two-sample t statistics (balanced or unbalanced) led to the diffuse test statistics, the frequencies of significant results of the x2 goodness-of-fit tests increased when the numbers of studies increased. When the number of studies was 5, 10, and 50, 3%, 15%, and 30% of tests, respectively, were statistically significant. Therefore, the theory for the distributions of the diffuse tests was less accurate when the number of studies increased. Table 33 Number of Significant x2 Goodness-of—Fit Tests by Number of Studies (k) Two - sample t k One-sample t Balanced Unbalanced Total 2 0 5 1 2 3 10 8 7 15 50 15 15 30 Total 24 24 48 Note: Each cell has 32 comparisons of distribution fit. Sample Sizes. As expected, the large-sample approximations to the distribution of diffuse test were more accurate as the sample sizes increased. The chi-squared test of these data in Table 33 was statistically significant (x2 = 67.62, df = 3, p < .05). Table 34 Number of Significant x2 Goodness-of—Fit Tests by N Two — sample t Sample Size One-sample Balanced Unbalanced Total ,7 20k 16 17 33 40k 8 6 14 80k 160k 1 1 Total 24 24 48 Note: Each cell has 32 comparisons of distribution fit. When one-sample t statistics led to the diffuse test, the large-sample distribution of the diffuse test was very well approximated, without regard to the sample sizes. However, when two-sample t statistics (balanced or unbalanced) led to the diffuse test 95 statistics, the frequencies of significant results of the x2 goodness-of—fit tests decreased when the sample sizes increased. When the total sample sizes were 20k, 40k, 80k, and 160k, 34%, 14%, 0%, and 1% of tests were statistically significant. Therefore, the accuracy for the distributions of the diffuse tests was less accurate when sample sizes decreased. Sampling Fractions. The pattern of sampling fractions (balanced and unbalanced) did not seem to relate to distribution fit. The chi-squared test was not statistically significant (x2 = 0.10, df = 1, p > .05). The number of significant results of Kolmogorov-Smimov tests did not differ by the pattern of sampling fractions. Table 35 Number of Significant x 2 Goodness-of—F it Tests by Sampling Fractions Sampling Two - sample t Fractions One-sample t Balanced Unbalanced Total Equal 12 11 23 Unequal 12 13 25 Total 24 24 48 Note: Each cell has 64 comparisons of distribution fit. Effect-Size Patterns. Table 35 showed the frequencies of significant x2 goodness-of-fit tests according to the effect-size patterns. The chi-squared test was not statistically significant (x2 = .019, df = 3, p > .05). About 14% of the tests were statistically significant for each set of effect-size parameters. Therefore, the accuracy for the distributions of the diffuse tests did not depend on the set of effect-size parameters. 96 Table 36 Number of Significant x2 Tests by Effect-Size Pattern for the Diffuse-Test Distributions Effect-Size Two — sample t Pattern One-sample t Balanced Unbalanced Total A 6 7 13 B 6 6 12 C 6 6 12 D 6 5 1 1 Total 24 24 48 Note: Each cell has 32 comparisons of distribution fit. Within-Study Sampling Ratio. In Table 11 I described three sets of sampling fractions, representing using one-sample t (0 1), balanced (.5 .5) and unbalanced (.3 .7) two sample t statistics. The result of the chi-square test for one-sample and two-sample t statistics showed that the large-sample approximation to the distribution of the diffuse test statistic was more accurate for one-sample t than for two-sample t statistics. To follow up, I did a chi-square test for distribution of tests from balanced and unbalanced two-sample t statistics. The chi-squared test was not statistically significant (x 2 =0, df = 1, p > .05). The accuracy of the asymptotic distribution was not affected by within-study sampling ratios. Combination of Three Parameters (7i, 0, ID. When two-sample t statistics were the basis of the diffuse test, the total sample sizes were set for each study according to the sampling fractions (11:), and then divided again according to the value of it), to get t statistics within each study. Even with equal sampling fractions (.5 .5) when total sample 97 sizes were 20k or 40k, the large-sample approximations to the distribution of the diffuse test were not accurate for larger k, as shown in Table 33 through Table 35. In addition, the within-study sample size used to obtain two-sample t statistics in the unbalanced case could be as small as 2 or 3 for larger k. This may explain why the large-sample approximation to the distribution of the diffuse test was less accurate when the sample sizes were 20k or 40k, especially for larger k. However, when the total sample sizes were 80k and 160k, the large-sample approximation to the distribution of diffuse test was fairly accurate, and did not depend on the number of studies. As a result, there were fewer discrepancies between the simulated and the theoretical distribution for sample sizes more than 80k when balanced and unbalanced two-sample t statistics were the basis of the diffuse test. Appendix F shows the detailed results of x2 goodness-of—fit tests for all configurations of parameters. Overall, the obtained numbers of significant results were different according to what upper percentage point was examined in the diffuse distribution. The large-sample approximation to the distribution of the diffuse test statistic was less accurate in the lower tails of the distributions than in the upper tails. In the upper tail the distribution of the diffuse test statistic was well approximated by the modified chi-squared distribution predicted by the theory, with few exceptions. The accuracy of the large-sample approximation to the distribution of the diffuse test depended on the type of t-statistic that led to the p values. More significant misfit (empirical versus theoretical) appeared for larger k. When sample size decreased, the number of significant results of fit tests increased, regardless of the effect-size pattern. The accuracy of the asymptotic 98 distribution was not affected by within-study sampling ratios and sampling fractions. Finally the empirical distributions tended to have thinner upper tails and much fatter lower tails than the asymptotic distributions predicted by the theory when larger numbers of studies and small sample sizes were used to obtain diffuse statistics. Power Analysis for The Diffuse Test Statistic Power Comparisons Another way of checking the accuracy of the large-sample approximation is to compare the simulated and theoretical power values. Again, I examined power values at 01 = .05 and 01 = .10. The procedures to calculate the power values were similar to those used to get the empirical proportions exceeding eleven cut points. But the difference was that the power values were the proportions exceeding the null-hypothesis percentile values at a = .05, not percentiles from the theoretical distribution under the alternative hypothesis. That is, the percentile values used as cut points for the diffuse test scale were the values from the central chi-square distribution. The absolute maximum difference between simulated and theoretical power values for 384 configurations of parameters was 0.064 when k=50, N=20k, and the set of effect sizes showed pattern C for two-sample t statistics. Appendix G shows the empirical and theoretical power values and the differences for the diffuse test statistic when a = .05 and 01 = .10. 99 Results for The Diffuse Test Statistic To understand whether the conditions of the simulation study for the diffuse test led to differences in the power values, analysis of variance also was conducted, treating 384 power estimates as data points. Analysis of variance (ANOVA) was done for power values at the 01 = 0.05 level, by number of studies (k), sample sizes (N), sampling fractions (7r), different sets of effect-size parameters (6), and within-study sampling ratios (4) ). Power values of the diffuse test statistics were explained mostly by the sets of effect-size parameters (R2 =.33) and sample sizes (R2 =.13). The number of studies (R2 =.06) and within-study sampling ratio (R2 =.15) also explained the power values of the diffuse test statistics. However since the results of the ANOVA shown in Table 37 aggregated across the three types of statistics, two additional ANOVAs are given. Table 37 Analysis of Variance for Power Values for the Diffuse Test Sum of Squares df Mean Square F Sig Main Combined 19.571 12 1.631 57.086 .000 Effects k 1.816 3 .605 21.190 .000 N 4.038 3 1 .346 47.109 .000 it .356 1 .356 12.451 .000 6 9.915 3 3.305 111.686 .000 (I) 3.446 2 1.723 60.310 .000 Model 19.571 12 1.63 1 57.086 .000 Residual 10.599 371 2.857E-02 Total 30.171 383 7.877E-04 100 Table 38 Analysis of Variance for Power Values for One—Sample t Statistics Sum of Squares df Mean Square F Sig Main Effects 13.200 9 1.467 106.512 .000 6 9.819 3 3.273 237.693 .000 N 2.237 3 .746 54.159 .000 k 1.144 3 .381 27.682 .000 Two-way Interactions 2.327 27 8.620E-02 6.260 .000 6 x N 1.608 9 179 12.976 .000 6 x k .632 9 7.017e-02 5.096 .000 k x N 8.791E-02 9 9.768E-03 .709 .698 Three-way Interaction .413 27 1.529E-02 1.111 .357 Model 15.940 63 .253 18.375 .000 Residual .881 64 1.377E-02 Total 16.821 127 .132 Table 39 Analysis of Variance for Power Values for Balanced and Unbalanced Two-Sample t Statistics Sum of Squares df Mean Square F Sig Main Effects 5.522 9 .614 160.322 .000 6 2.701 3 .900 235.230 .000 N 2.012 3 .671 175.279 .000 k .809 3 .270 70.455 .000 Two-way Interactions 3.170 27 .117 30.678 .000 6 x N 1.656 9 .184 48.079 .000 6 x k .893 9 9.920E-02 25.923 .000 k x N .621 9 6.900E-02 18.031 .000 Three-way Interaction .496 27 1.836e-02 4.797 .000 Model 9.187 63 .146 38.107 .000 Residual .735 192 3.827E-03 Total 9.922 255 3.891E-02 101 The amount of variation in power explained by ANOVA differed for the two different types of statistics. The amount explained by the ANOVA model in Table 37 was about 78% when one-sample t statistics gave rise to the diffuse statistics, but was only 55 % for two-sample t statistic in Table 38. Two-way and three-way interactions were not statistically significant for one-sample t statistics, while statistically significant for two—sample t statistics. As a result, the variation in power worked differently in the cases of one-sample t statistics and two-sample t statistics. Tables 40 and 41 show means of power values for each configuration of number of studies, sample sizes, and the sets of effect-size parameters in detail for one-sample and two-sample ts, respectively. 102 Means of Power Values of the Diffuse Test Statistic for One—Sample t Statistics Table 40 by kxNx6 Effect-Size Pattern (6) N A B C D 2 20k 0.052 0.05 0.164 0.172 40k 0.050 0.051 0.298 0.312 80k 0.053 0.055 0.532 0.545 160k 0.050 0.069 0.816 0.822 5 20k 0.052 0.043 0.18 0.156 40k 0.056 0.045 0.342 0.31 80k 0.047 0.049 0.635 0.603 160k 0.049 0.049 0.939 0.914 10 20k 0.052 0.042 0.26 0.268 40k 0.043 0.05 0.545 0.566 80k 0.053 0.071 0.888 0.874 160k 0.049 0.101 0.997 0.993 50 20k 0.053 0.056 0.687 0.46 40k 0.053 0.109 0.968 0.793 80k 0.049 0.233 1 0.983 160k 0.052 0.454 1 1 When one-sample t statistics led to the diffuse test statistics, the power values generally were higher than ones for two-sample 1‘ statistics, as shown in Table 40 and 41. For the sets of effect-size parameters B, C, and D, when the sample sizes increased, the power values within each number of studies increased regardless of whether one-sample or two-sample t statistics led to the diffuse test statistic, as shown also in Figure 5.2.3. 103 Table 41 Means of Power Values of the Diffuse Test Statistic for Two-Sample t Statistics by kxNx6 Effect-Size Pattern (6) N A B C D 2 20k 0.046 0.043 0.067 0.065 40k 0.046 0.046 0.111 0.113 80k 0.051 0.05 0.162 0.179 160k 0.049 0.05 2 0.281 0.307 5 20k 0.042 0.039 0.071 0.053 40k 0.049 0.041 0.099 0.096 80k 0.046 0.052 0.163 0.164 160k 0.047 0.049 0.337 0.308 10 20k 0.037 0.031 0.062 0.068 40k 0.037 0.041 0.122 0.132 80k 0.051 0.051 0.25 0.273 160k 0.046 0.058 0.532 0.553 50 20k 0.025 0.02 0.081 0.056 40k 0.035 0.045 0.281 0.18 80k 0.04 0.068 0.677 0.443 160k 0.048 0.117 0.96 0.788 Comparing the sets of effect-size parameters, the set C had slightly lower power values than the power values for the set D for both one-sample and two-sample t statistics when k=2. However, when the number of studies increased the power values for set C of effect-size parameters grew higher than the power values for set D, as shown in Figure 5.2.4. 104 Figure 5.2.3 Power Values for the Diffuse Test based on Two-Sample t Statistics by N x k N [:3ZM I:I“°k new n2 lll'lGOk a 8'3 3 8 3'3 3 a 3'3 0 a 8'8 8 k=2 k=5 k=10 k=50 Figure 5.2.4 Power Values for the Diffuse Test based on Two-Sample t Statistics by kx 6 I2 LO . .8. INDVWER .6. a 0's 3 a 0'3 8 a 8's 8 a 3'3 3 k=2 k=5 k=10 k=50 105 Figure 5.2.5 Power Values for the Diffuse Test based on Two-Sample t Statistics by N x 6 .6 u .4 , N POWER 2% ,2 ‘ 1:] [34011 0.0 4 neck -.2 fi .160k Within each set of effect-size parameters, when sample sizes increased the power values increased, as shown in Figure 5.2.5 for two-sample t statistics. However, for the set A and set B effects, with equal sampling fractions (representing the null hypothesis case), power values did not change consistently when sample sizes or number of studies increased when one-sample t statistics were the basis of the diffuse test statistic. This is reasonable, as the values should all be close to or (here, close to .05). Finally the accuracy of the large-sample distributions of the diffuse tests was less good for the cases of larger numbers of studies and small sample sizes. As a result, the power values obtained for larger numbers of studies and small sample sizes were less informative than ones obtained for other configurations of parameters. 106 It Under the null hypothesis. The simulated and theoretical power values near the null hypothesis for the diffuse test should be very close to the a level. Table 42 shows the power values at the a = .05 level for k=5 when one-sample t statistics led to the diffuse test statistics, and Table 43 shows the results for two-sample t for k=5. Tables 42 and 43 take closer looks at the power values for different sets of effect-size parameters representing the null hypotheses. However, again there is some controversy over what the null model should be. The effect-size values are shown in the note to Table 42. Table 42 Empirical Power Values under Several Null-Hypothesis Scenarios by N and 6 for k=5 from One-Sample t Set of Effect-Size Parameters (6) II N A B E F G H Equal 20* k .053 .044 .046a .049a .040a .041“l 40* k .046 .043 .050 a .055 a .053 ’ .041a 80* k .047 .044 .055 a .044 a .078 a .041 ‘ 160*k .050 .046 .086 a .065 a . 161 a .070 Unequal 20*k .043 .043a .041 .046 .033 .032 40*k .047 .0472’ .043 .044 .034 .032 80*k .054 .054“ .049 .045 .038 .033 160* .052 .052" .049 .042 .049 .032 Note. A: (0 0 0 0 0) B: (.2 .2 .2 .2 .2) E: (.26667 .2 .2 .2 .16) F: (.231 .2 .2 .2 .1788) G: (.46667 .35 .35 .35 .28) H: (.4041 .35 .35 .35 .313) a Power values under the alternative hypothesis. First the cases for 11: equal and for the effects in sets A and B described the cases where either 71,62 =...=n,6, or £6, =...= n,6, . 107 In Second, the cases with It unequal and with effect sets E and G satisfied the condition that The difference between the sets E and G was that the range of effect-size parameters for the set G was bigger than that for E. Third the cases for it unequal and the sets F and H represented the case where Jib, =...= n,6,. Again the range of effect-size values was wider in set H than F. Other pairings of sampling fractions and effect-size sets represent alternative hypotheses cases (e.g., unequal n with set B, and equal it with sets E through H). These cases are marked with a subscript a in Tables 42 and 43. Power values did not vary not in a consistent way when the sample sizes increased and different sets of effect-size parameters were the basis of the diffuse test statistic. The set A of effect sizes is always under the null hypothesis, for any cases of the null hypotheses, so the power values look acceptable for those cases, and actually better as sample size increased. The set B of effect sizes is under the null hypothesis only for equal sample sizes. Therefore, that case is consistent with all possible null hypotheses. When the sample size increased, the power values for the set B under the null model increased. In Table 42 the power values were higher for the set B under the alternative hypothesis (unequal sampling fractions) than under the null hypothesis, but still were quite low to detect the effect size differences. For the sets E and G with unequal 11: under the second possiblenull model, the power values were mostly lower than .05 and much less for the set G except when the sample size was very high. On the other hand, for the 108 SCI 00 1111 th sets E and G with equal I: under the alternative model, the power values were higher than ones under the null model and increased when the sample sizes increased (the same as for the set B under the alternative model). If the diffuse test statistic tests that n,6, are equal, then the test has a lower type I error rate than .05. For the sets F and H with unequal 1r under the third possible null model, the power values were quite low and even did not seem to approach .05 when the sample sizes increased. On the other hand, for the sets F and H with equal it under the alternative model, the power values were lower than ones for the sets E and G. Overall the results for the sets E, F, G, and H suggest that the diffuse test was not detecting the deviations from all effect-size values equal. Further examination was done for two-sample t statistics shown in Table 43. For two-sample t statistics, there were inconsistent results about the power values under the null hypotheses to similar those for one-sample t statistics. For the set A, the power values were close to .05 under any null hypotheses, but did not seem to relate to the sample size, unlike the results obtained from one-sample t statistics. For the set B with equal or unequal sampling fractions, the results were same as those obtained from one-sample t statistics. For the sets E and G under the null model, the power values seemed to approach .05 when the sample size increased. Under the alternative model for the sets E and G, the power values generally quite low, and even not always greater than the power values under the null model. 109 Table 43 Empirical Power Values under Several Null-Hypothesis Scenarios by N and 6 for k=5 from Balanced Two-Sample t Set of Effect-Size Parameters (6) it N A B E F G H Equal 20* k .043 .040 0.037a 0.043a 0.039a 0.0393 40* k .041 .041 0.049a 0.0443 0.051“ 0.0448’ 80* k .046 .049 0.048al 0.049’1 0.0528 0.056" 160*k .048 .050 0.0581’ 0.049" 0.0753 0.0522' Unequal 20*k .041 .043al 0.046 0.033 0.034 0.044 40*k .057 .0473 0.038 0.041 0.037 0.030 80*k .046 .0543 0.059 0.047 0.047 0.052 160* .046 .05 23 0.049 0.047 0.046 0.048 For the sets F and H, the power values were higher (better) than those from one- sample 1 statistics, and approached .05 as the sample sizes increased for the null model. However the power values under the alternative model for the sets F and H were quite low, even lower than .05. Overall the results for the sets E, F, G, and H suggest that the diffuse test was not detecting the deviations from the case where all effect-size values were equal. As a result, comparing the power values for the sets E and G with ones for the sets F and H for both cases (where one-sample tor two-sample t statistics led to the diffuse test), power values under the null hypothesis of all effect sizes weighted by their own sample sizes equal seemed to approach .05 more closely than ones under the null hypothesis for all effect sizes weighted by square root of their sample size. 110 CHAPTER V] CONCLUSIONS Summary In this dissertation, 1 derived the asymptotic distribution for the focused test by the multivariate delta method. I also derived the approximation to the asymptotic distribution for the diffuse test by applying rules for quadratic forms and using Solomon and Stephens’s (1977) three-moment chi-square fit. I studied the accuracy of the large- sample approximations to the distributions of both the focused and diffuse tests through a simulation study. For the focused test, less than five percent of simulated distributions showed discrepancies between the simulated and the theoretical distributions. That is, the large-sample approximation to the distribution showed quite accurate small-sample behavior, so the distribution of the focused-test statistic was well approximated by the theory. Particularly when one-sample t statistics gave rise to the focused test statistics, the large-sample approximation to the distribution was quite accurate regardless of the different configurations of simulation parameters. On the other hand, for the diffuse test, the large-sample approximation to the distribution was somewhat less accurate in the lower tails of the distributions than in the upper tails. However, in the upper tail the distribution of the diffuse test statistic was well approximated by the modified chi-squared distribution predicted by the theory, with few exceptions. When large numbers of studies occurred with small sample sizes within each study, the simulated distributions tended to have thinner upper tails and much fatter lower tails than the asymptotic distribution predicted by the theory. 111 stud the for 0b 165 of Overall, the asymptotic distributions for the focused and diffuse test statistics I studied in this dissertation in most cases proved good approximations. The differences in the power values between the simulated and theoretical distributions were mostly small for the focused test and the diffuse tests with few exceptions. Therefore the power values obtained from these asymptotic distributions should be useful for the focused and diffuse test statistics. The power study for the focused test statistic showed that when the number of studies and sample sizes increased, the power for the focused tests increased under the alternative hypotheses regardless of other configurations of p0pulation parameters. Comparing the sets of effect-size parameters, the power values when the pattern of effect- size parameters represented group effects (set D) were usually a little bigger than those when the pattern of effect-size parameters represented linear effects (set C), for k=5 or larger and all sample sizes. However, this is purely a result of the set of 6 values that were chosen. Generally the power values when one-sample t statistics led to the focused test statistics had bigger values than those when two-sample t statistics led to the focused test statistics. In addition, the power study for the focused test supported that the null hypotheses for the effect sizes should use effects weighted by the square root of their own sample sizes. The test was clearly not detecting many deviations from the cases where all effect-size values are equal or where the effect-size values were unweighted. The power study for the diffuse test also showed that when number of studies and sample sizes increased the power for the diffuse tests increased under the alternative hypotheses regardless of other configurations of population parameters. Comparing the sets of effect-size parameters, the effect-size pattern representing the linear difference 112 '1) (set C) again had slightly lower power values than those for the effect-size pattern representing the group differences (set D) for both one-sample and two-sample t statistics but only when k=2. However, when the number of studies increased the power values for effect-size pattern C of linear effects grew higher than the power values for the group-differences effect-size pattern D. On the other hand the power study for the diffuse test did not fully support using the null hypothesis of equality of effect sizes weighted by the square roots of their own sample sizes. Clearly, more studies are needed to learn more about the null hypothesis for the diffuse test, and its behavior under alternative hypotheses. However, again the test was not detecting many deviations from the cases where all effect-size values are equal. Practical Implications Some researchers have applied diffuse or focused tests with effect-size analyses in meta-analysis. However, applying both approaches in meta-analysis does not provide more information, but increases the chance that the reviewer would make a Type I error because the analyses are not independent. So diffuse and focused tests using the inverse- normally transformed p values are recommended as useful tests only when the information for effect-size analyses is not available. Since functions of the significance values depend on the significance value directly and exact distributions of the significance value under the alternative hypothesis are complex, the exact distributions of functions of p values under alternative hypotheses are also complicated. As a result, to compute the exact power values for diffuse and focused test statistics would be very complex. However, asymptotic distributions for the tests are 113 useful because they are simpler to calculate than exact distributions. Researchers who do not have strong a mathematical background can apply asymptotic distributions more easily than calculating exact distributions, and the asymptotic distribution for the focused test is relatively easy to use. Still it is of use to understand the distributions of both the focused and diffuse tests theoretically. This study showed that the theoretical moments of the focused test suggested by Rosenthal and Rubin were very inaccurate compared to the ones based on work by Lambert. This probably means that both the focused and diffuse test which Rosenthal and Rubin proposed are overly conservative, especially because Rosenthal and Rubin’s variance term was always higher than the ones computed from Lambert’s distribution. Findings of empirical null-case rejection rates and power values that were less than the nominal 01:.05 are consistent with that conclusion. This study supported the evidence that the null hypotheses for the focused tests were not just contrasting the effect sizes using specified weight values ( the A, weights), but were contrasting the effect sizes weighted by the square roots of their own sample sizes with the 1., weight values. However, this simulation study showed that there was confusion regarding the null hypothesis of the diffuse test. For several different collections of effect-size parameters and sample sizes, the power values of the diffuse test varied non systematically, sometimes dropping well below .05. At this point the use of the diffuse test seems risky, at best. For the case of confounding sample size and effect size, and that when sample sizes vary but effect sizes do not, the tests seem likely 114 to become large, which would mean rejecting some null models even when all effect- sizes are equal. Suggestions for Further Research In this study, I examined the diffuse and focused tests suggested by Rosenthal and Rubin (1979) which only involved inverse-normally transformed p values. Since Rosenthal and Rubin did not limit the application of diffuse and focused tests to p values from specific types of statistics, I studied two cases, where one-sample t statistics and two-sample 1 statistics led to the diffuse and focused test statistics. The results of this simulation study showed that there were differences in accuracy of the large sample distribution depending on the origin of the p values used in the tests. This means that if the tests involve inverse-normally transformed p values obtained from other different types of statistics, the asymptotic distributions and their accuracy may also differ. This provides one area of possible future research. For further research the asymptotic distribution of Z(p) suggested by Lambert (1978) can also be used to get the asymptotic distribution for other statistics which use Z(p) such as the “file drawer” number, and other functions of p values. If the tests involve differently transformed p values rather than inverse-normally transformed p values, we also may need to have large-sample distributions for the statistics that are the functions of those p values. Finally this study showed that the power values for the diffuse test did not change consistently according to the different configurations of population parameters under null 115 and alternative hypotheses. However, the present study was not meant as an extensive study of the power of these two tests, but rather as an explanation of distribution accuracy. Further work to examine a variety of different effect-size configurations would provide a better understanding of the two tests and their ability to detect scenarios of interest to research reviewers. Even for the focused test, the sets of effects studied here show performance for only a few of the outcomes that could interest reviewers. The results of the limited power study in this research seemed to eliminate the model of equal effects as a reasonable null model, but did not differentiate between the alternative two possible null hypotheses. A more extensive power study is needed to know about what sets of effects the diffuse test can detect, if it is to be used for hypothesis testing in meta- analysis at all 116 APPENDIX A SAMPLE SIZES USED IN SIMULATION STUDY 117 Table 44 Sample Sizes for Simulation Study n o N=20k N=40k N=80k N=160k (.5 .5) (.5 .5) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) (.3 .7) (7 13) (13 27) (27 53) (53 107) (7 13) (13 27) (27 53) (53 107) (.3 .7) (.5 .5) (6 6) (13 13) (26 26) (53 53) (14 14) (27 27) (54 54) (107 107) (.3 .7) (4 8) (9 17) (16 36) (34 72) (10 18) (17 37) (36 72) (72 142) (.2 .2 .2 .2 .2) (.5 .5) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) (.3 .7) (7 13) (13 27) (27 53) (53 107) (7 13) (13 27) (27 53) (53 107) (7 13) (13 27) (27 53) (53 107) (7 13) (13 27) (27 53) (53 107) (7 I3) (13 27) (27 53) (53 107) (.15 .2 .2 .2 .25) (.5 .5) (8 7) (15 15) (30 30) (60 60) (10 10) (20 20) (40 40) (80 80) (10 10) (20 20) (40 40) (8O 80) (10 10) (20 20) (40 40) (80 80) (13 12) (25 25) (50 50) (100 100) (.3 .7) (5 10) (10 20) (20 40) (40 80) (7 13) (13 27) (26 54) (53 107) (7 13) (13 27) (26 54) (53 107) (7 13) (13 27) (26 54) (53 107) (8 17) (16 34) (32 68) (67 133) 118 Table 44 (Cont’d) k It (I) N =20k N=40k N=80k N=160k 10 (.1.1.l.1.1.1.1.1.1.1) (5.5) n,=(1010) (20 20) (4040) (8080) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n.=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n7=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,,,=(10 10) (20 20) (40 40) (80 80) (.3 .7) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (7 13) (13.27) (27 53) (53 107) (7 13) (13,27) (27 53) (53 107) (.05 .06 .07 .08 .08 (.5 .5) (5 5) (10 10) (20 20) (40 40) .08 .09 .11 .16 .22) (6 6) (12 12) (24 24) (48 48) (7 7) (14 14) (28 28) (56 56) (8 8) (16 16) (32 32) (64 64) (8 8) (16 16) (32 32) (64 64) (8 8) (16 16) (32 32) (64 64) (9 9) (18 18) (36 36) (72 72) (11 11) (22 22) (44 44) (88 88) (16 16) (32 32) (64 64) (128 128) (22 22) (44 44) (88 88) (176 176) (.3 .7) (3 7) (7 13) (13 27) (27 53) (4 8) (8 16) (16 32) (32 64) (5 9) (9 19) (19 37) (37 75) (6 10) (11 21) (21 43) (43 85) (6 10) (11 21) (21 43) (43 85) (6 10) (11 21) (21 43) (43 85) (6 12) (12 24) (24 48) (48 96) (7 15) (15 29) (29 59) (59 117) (11 21) (2143) (43 85) (85 171) (15 29) (29 58) (59 117) (117 235) 119 Table 44 (Cont’d) k 7: (I) N=20k N=40k N=80k N=160k 50 (.02 .02 .02 .02 .02 .02 .02 (.5 .5) n,=(10 10) (20 20) (40 40) (80 80) .02 .02 .02 .02 .02 .02 .02 n,=(10 10) (20 20) (40 40) (80 80) .02 .02 .02 .02 .02 .02 .02 n,=(10 10) (20 20) (40 40) (80 80) .02 .02 .02 .02 .02 .02 .02 n,=(10 10) (20 20) (40 40) (80 80) .02 .02 .02 .02 .02 .02 .02 n,=(10 10) (20 20) (40 40) (80 80) .02 .02 .02 .02 .02 .02 .02 n,=(10 10) (20 20) (40 40) (80 80) .02 .02 .02 .02 .02 .02 .02 n7=(10 10) (20 20) (40 40) (80 80) .02) n,=( 10 10) (20 20) (40 40) (80 80) n,=(10 10) (20 20) (40 40) (80 80) n,0=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,_,=(10 10) (20 20) (40 40) (8O 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,,=(10 10) (20 20) (40 40) (80 80) n,,=( 10 10) (20 20) (40 40) (8O 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n20=(10 10) (20 20) (40 40) (8O 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n24=(10 10) (20 20) (40 40) (80 80) n,.=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n30=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n3,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n40=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n44=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) n,,=(10 10) (20 20) (40 40) (80 80) _ n50=(10 10) (20 20) (40 40) (80 80) 120 Table 44 (Cont’d) k 7! 0 N=20k N=40k N=80k N=160k 50 (.02 .02 .02 .02 .02 .02 .02 (.3 .7) n,= (7 13) (13 27) (27 53) (53 107) .02 .02 .02 .02 .02 .02 .02 .1,: (7 13) (13 27) (27 53) (53 107) .02 .02 .02 .02 .02 .02 .02 .2,: (7 13) (13 27) (27 53) (53 107) .02 .02 .02 .02 .02 .0202 11,: (7 13) (13 27) (27 53) (53 107) .02 .02 .02 .02 .02 .02 .02 n,= (7 13) (13 27) (27 53) (53 107) .02 .02 .02 .02 .02 .02 .02 "6: (7 13) (13 27) (27 53) (53 107) .02 .02 .02 .02 .02 .02 .02 .1,: (7 13) (13 27) (27 53) (53 107) .02) 11,: (7 13) (13 27) (27 53) (53 107) .1,: (7 13) (13 27) (27 53) (53 107) um: (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,_,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) 11,0: (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n22: (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,,= (7 13) (13 27) (27 53) (53 107) n27: (7 13) (13 27) (27 53) (53 107) my: (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,_,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) nw= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) 7142: (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) in“: (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n47: (7 13) (13 27) (27 53) (53 107) n“: (7 13) (13 27) (27 53) (53 107) n49: (7 13) (13 27) (27 53) (53 107) rim: (7 13) (13 27) (27 53) (53 107) 121 Table 44 (cont’d) k 71: 0 N =20k N =40k N=80k N=160k 50 (.007 .007 .009 .009 .01 (.5 .5) n,= (3 4) (7 7) (14 14) (28 28) .01 .01 .01 .01 .01 .01 . .1,: (3 4) (7 7) (14 14) (28 28) 01 .01 .01 .01 .012 .012 n3: (4 5) (9 9) (1818) (3636) 012.012.012.012 .01 71,: (4 5) (9 9) (1818) (3636) 4.014 014.014.0140 .1,: (5 5) (1010) (20 20) (40 40) 14 .016.016 016.016. 11,: (5 5) (1010) (20 20) (40 40) 016 .016 .02 .02 .02 .02 n7: (5 5) (10 10) (20 20) (40 40) .02 .02 .02 .022 .022 .0 .1,: (5 5) (10 10) (20 20) (40 40) 22 .03 .05 .05 .05 .06 .0 n9: (5 5) (10 10) (20 20) (40 40) 6.1) mo: (5 5) (1010) (20 20) (40 40) n,,= (5 5) (10 10) (20 20) (40 40) n,,= (5 5) (10 10) (20 20) (40 40) n13: (5 5) (10 10) (20 20) (40 40) n,,= (5 5) (10 10) (20 20) (40 40) n,,= (5 5) (10 10) (20 20) (40 40) um: (6 6) (12 12) (24 24) (48 48) n”: (6 6) (12 12) (24 24) (48 48) n,,,= (6 6) (12 12) (24 24) (48 48) n,,= (6 6) (12 12) (24 24) (48 48) rim: (6 6) (12 12) (24 24) (48 48) n,,= (6 6) (12 12) (24 24) (48 48) n,,= (7 7) (14 14) (28 28) (56 56) n,,= (7 7) (14 14) (28 28) (56 56) n,,= (7 7) (14 14) (28 28) (56 56) n,.= (7 7) (14 14) (28 28) (56 56) n..= (7 7) (14 14) (28 28) (56 56) n27: (7 7) (14 14) (28 28) (56 56) 1:2,: (8 8) (16 16) (32 32) (64 64) n,,= (8 8) (16 16) (32 32) (64 64) 11,0: (8 8) (16 16) (32 32) (64 64) n,,= (8 8) (16 16) (32 32) (64 64) n,,= (8 8) (16 16) (32 32) (64 64) n,,= (8 8) (16 16) (32 32) (64 64) n,,= (10 10) (20 20) (40 40) (80 80) n,_.= (10 10) (20 20) (40 40) (80 80) n,,= (10 10) (20 20) (40 40) (80 80) n,,= (10 10) (20 20) (40 40) (80 80) r133: (10 10) (20 20) (40 40) (80 80) n,,= (10 10) (20 20) (40 40) (80 80) 71,0: (10 10) (20 20) (40 40) (80 80) n,,= (11 11) (22 22) (44 44) (88 88) ’14,: (11 11) (22 22) (44 44) (88 88) n,_,= (11 11) (22 22) (44 44) (88 88) n“: (15 15) (30 30) (60 60) (120 120) n,,= (25 25) (50 50) (100 100) (200 200) n“: (25 25) (50 50) (100 100) (200 200) n47: (25 25) (50 50) (100 100) (200 200) n..= (30 30) (60 60) (120 120) (240 240) n,,= (30 30) (60 60) (120 120) (240 240) \ um: (50 50) (100 100) (200 200) (400 400) 122 Table 44 (Cont’d) k 7! o N =20k N =40k N =80k N =160k 50 (.007 .007 .009 .009 .01 (.3 .7) n,= (2 5) (5 9) (9 19) (19 37) .01 .01 .01 .01 .01 .01 . .1,: (2 5) (5 9) (919) (19 37) 01 .01 .01 01.012012 .2,: (3 6) (711) (12 24) (24 48) 01201201201201 .1,: (3 6) (711) (1224) (2448) 4 014.014 .014 .014 .0 n,= (3 7) (713) (13 27) (27 53) 14016016016016. .2,: (3 7) (713) (1327) (2753) 016 .016 .02 .02 .02 .02 .1,: (3 7) (7 13) (13 27) (27 53) .02 .02 .02 .022 .022 .0 71,: (3 7) (7 13) (13 27) (27 53) 22.03 .05 .05 .05 .06 .0 n9: (3 7) (7 13) (13 27) (27 53) 6.1) um: (3 7) (7 13) (13 27) (27 53) n,,= (3 7) (713) (13 27) (27 53) n,,= (3 7) (7 13) (13 27) (27 53) n,,= (3 7) (7 13) (13 27) (27 53) n,,= (3 7) (7 13) (13 27) (27 53) n,,= (3 7) (7 13) (13 27) (27 53) n,,= (4 8) (8 16) (16 32) (32 64) n,,= (4 8) (8 16) (16 32) (32 64) n,,,= (4 8) (8 16) (16 32) (32 64) n,.,= (4 8) (8 16) (16 32) (32 64) rim: (4 8) (8 16) (16 32) (32 64) n,,= (4 8) (8 16) (16 32) (32 64) n,,= (5 9) (9 19) (19 37) (39 79) 1:1,: (5 9) (9 19) (19 37) (39 79) "2,: (5 9) (9 19) (19 37) (39 79) n,,= (5 9) (9 19) (19 37) (39 79) 1.2,: (5 9) (9 19) (19 37) (39 79) n27: (5 9) (9 19) (19 37) (39 79) 1:2,: (5 11) (11 21) (21 43) (43 85) n,,= (5 11) (11 21) (21 43) (43 85) n30: (5 11) (11 21) (21 43) (43 85) n,,= (5 11) (11 21) (2143) (43 85) n,,= (5 11) (11 21) (21 43) (43 85) n,,= (5 11) (11 21) (21 43) (43 85) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) n,,= (7 13) (13 27) (27 53) (53 107) 11,0: (7 13) (13 27) (27 53) (53 107) n“: (7 I5) (15 29) (29 59) (65 131) n,,= (7 15) (15 29) (29 59) (65 131) n,_,= (7 15) (15 29) (29 59) (65 131) n“: (10 20) (20 40) (40 80) (80 160) n,,= (17 33) (33 67) (67 133) (133 267) It“: (17 33) (33 67) (67 133) (133 267) r147: (17 33) (33 67) (67 133) (133 267) It“: (20 40) (40 80) (80 160) 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Hoo.o HHH.o Hoo.o NHo.o ooo.o woo woo 5o Noo.o H.H.oo ooo.o NHo.o NHo.o Hoo.o Hoo.o oNo.o ooN.o ooN.o 0 HHo.o 22 «NS ooo.o HHH.o ooo.o Noo.o oNN.o om.o Noo.o HHH.o ooo.o Noo.o ooo.o ooo.o Hoo.o mom.o ooo.o u ooo.o Noo.o ooo.o oNo.o ooo.o oNo.o NHo.o Hoo.o ooo.o oNo.o mH.o.o oNo.o Hoo.o ooo.o ooo.o ooo.o Hoo.o oNo.o m ooo.o ooHo Noo.o H.Hoo omoo Hoo.o ooo.o ooHo Noo.o HHo.o ooo.o Hoo.o Hoo.o 83 83 Hoo.o ooo.o Hoo.o < «.8 H25 ER: .2: oo.: ER: .2: as: ER: .2: .oEmJEe .2: .95 ER: .2: .95 EE. .2: as: o 2 HH So u .0 modu H0 Loo?" .0 mod" 5 oH.oH .0 mod" .0 30:33:: Boom—om H «38325 H 2952-03— omuHH H8 Hmo.H. omo.H:nH 0.: H8 82o> H030: ooH Boob 185 References Amssey, F. J. (1951). The Kolmogorov-Smimov test for goodness of fit. Journal of the American Statistical Association, @, 70. Bahadur, R. R. (1960). On the asymptotic efficiency of tests and estimates. Sankhya, 2, 229-252. Becker, B. J. ( 1985). Applying tests of combined significance: Hypotheses andjower considerations. 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