A COMPREHENSIVE EXAMINATION OF PROCEDURES FOR TESTING THE SIGNIFICANCE OF A CORRELATION MATRIX AND ITS ELEMENTS by RACHEL TANYA FOULADI B.A., The University of British Columbia, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER'S OF ARTS in THE FACULTY OF GRADUATE STUDIES Department of Psychology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1991 © Rachel Tanya Fouladi, 1991 In presenting this degree at the thesis in partial fulfilment of University of British Columbia, I agree freely available for reference copying of department this or publication of and study. this his or her representatives. Department of Psychology The University of British Columbia Vancouver, Canada Date DE-6 (2/88) September, 1991 that the may be It thesis for financial gain shall not permission. requirements I further agree thesis for scholarly purposes by the that advanced Library shall make it by the understood be an permission for extensive granted is for allowed that without head of my copying or my written ii ABSTRACT Correlational techniques are important tools in multivariate behavioural and social science exploratory research. A wide array of procedures have been proposed for testing (a) whether any of the variables are related, and (b) which variables are related. In the current study, the performance of the procedures currently available for testing these distinct questions is assessed on the primary Neyman-Pearson criterion for an optimal test. According to this criterion, an optimal procedure is the most powerful procedure that controls experimentwise Type I error rate at or below the nominal level. The findings of the first part of this study addressing how to test complete multivariate independence suggest that the statistic traditionally used (QBA) is not the optimal test, and that one of several recently derived statistics (QSE> QSA> QF) should be used. Computational efficiency of the procedures is also considered with the resulting recommendation of the use of QSA- The second part of this study addresses how to test which variables are correlated; the findings suggest the use of a multi-stage order statistics approach with z-tests (CF). The conditions necessary to ensure maximal power when addressing these questions are also considered. iii T A B L E OF CONTENTS ABSTRACT ii T A B L E OF CONTENTS iii .' LIST OF TABLES vii LIST OF FIGURES xvii ACKNOWLEDGEMENTS xviii GENERAL INTRODUCTION l Correlational Techniques PART I TESTING WHETHER A L L THE CORRELATIONS ARE ZERO 1. Introduction 1.1 Likelihoodratiocriterion based tests Modifications in the multiplier Box 1949 Box 1949, Bartlett 1950 Modifications in the root Lawley 1940 Modifications in the multiplier and root Bartlett 1950 1.2 Quadratic form tests derived from asymptotic normal theory 2 6 6 7 9 9 10 11 11 11 11 12 iv Tests based on the asymptotic normal distribution of correlations Browne 1977 13 13 Tests based on the normalizing property of the Fisher transform Steiger 1980, Burt 1954 Steiger and Fouladi 1991a, 1991b 14 14 16 1.3 Summary of LR and NT based tests 1.4 Other tests of multivariate independence An exact test: Mathai and Katiyar 1979 20 21 21 A normal approximation: Mudholkar, Trivedi, and Lin 1982 21 Test on rmax A sum of independent chi-squares: Steiger and Fouladi 1991c 1.5 Overview of simulation studies of tests of multivariate independence 23 1.6 Current state of affairs 26 : 22 22 2. Methods Procedures for generation of sample correlation matrices Simulation experiments Measures of experimentwise Type I error rate control 26 26 28 30 3. Results 30 When P is equal to I When P is not equal to 1 4. Conclusions PART II IDENTIFYING WHICH CORRELATIONS ARE NON-ZERO 1. Introduction 1.1 Simultaneous test procedures t-tests at nominal alpha Reduced alpha t-test procedures: Bonferroni (RB), DunnSidak(RD) 1.2 Sequential test procedures Procedures using t-tests 30 32 33 34 34 35 35 37 39 42 V Multi-stage reduced alpha: Bonferroni, Dunn-Sidak Procedures using order statistics theory Silver 1988 Fouladi . Steiger 1.3 Overview of papers on the topic of tests on correlations 1.4 Current state of affairs 2. Methods Procedure for generation of sample correlation matrices 57 59 59 Simulation experiments 60 Measures of performance 61 3. Results 63 P equal to 1 65 No preliminary test of the null hypothesis With a preliminary test of omnibus hypothesis P not equal to 1 With no preliminary test of the omnibus hypothesis With a preliminary test of the null hypothesis Detection theory measures of performance Examining experimentwise Type I error control and Tp 65 66 66 66 67 67 68 3. Conclusions G E N E R A L 42 45 47 49 55 56 D I S C U S S I O N 68 7 0 1. A discussion of the results of this study in the context of prior research 1.1 Experimentwise Type I error control Considering the tests of the omnibus null hypothesis Considering tests of the elements of a correlation matrix 1.2 Power issues 1.3 Important design issues 70 70 71 72 73 74 2. Extensions and suggestions for future study 75 2.1 Tests of alternative null hypotheses 75 vi Generalizations of the omnibus tests Generalization of the tests on individual correlations 2.2 Extension to non-normal data 3. Final note 75 77 77 78 BIBLIOGRAPHY 81 APPENDICES 88 Appendix A Extensions of Hotelling's approximations obtained by Steiger and Fouladi (1991a) 89 Appendix B The configuration of population matrices 95 Appendix C Setting the story straight 97 TABLES 98 FIGURES 182 LIST OF TABLES Table 1 The relationship between the number of variables, p, and the number of pairwise correlations, v , in a matrix 99 Table 2 Asymptotic chi-square tests of complete independence with v degrees of freedom 99 Table 3 Summary table of reviews of tests of LR and NT tests of multivariate independence 99 Table 4 Table of sample sizes, N, for sample correlation matrices generated under P of order p and specified N:p ratios 100 Table 5.1 Table of empirical Type I error rates for tests of complete multivariate independence at a = .05 101 Table 5.2 Table of empirical Type I error rates for tests of complete multivariate independence at a = .01 101 Table 6.1 Chi-square goodness of fit values for tests of complete multivariate independence at 102 p p a = .05 under a true null hypothesis, df = 1, X i,.05 3-84, X i,.01 = 6-64, X i,.001 2 = 2 2 = 10-83; X i5,.ooi = 37.70 2 Table 6.2 Chi-square goodness of fit values for tests of complete multivariate independence at a = .01 under a true null hypothesis 102 viii Table 7.1.1.1 Empirical power of tests of complete multivariate independence at a = .05, p = 5, mNz = .1 103 Table 7.1.1.2 Empirical power of tests of complete multivariate independence at oc = .05, p = 5, mNz = .3 104 Table 7.1.1.3 Empirical power of tests of complete multivariate independence at a = .05, p = 5, mNz = .5 105 Table 7.1.2.1 Empirical power of tests of complete multivariate independence at a = .05, p = 10, mNz = .1 106 Table 7.1.2.2 Empirical power of tests of complete multivariate independence at a = .05, p = 10, mNz = .3 107 Table 7.1.2.3 Empirical power of tests of complete multivariate independence at a = .05, p = 10, mNz = .5 108 Table 7.2.1.1 Empirical power of tests of complete multivariate independence at a = .01, p = 5 mNz = .1 109 Table 7.2.1.2 Empirical power of tests of complete multivariate independence at a = .01, p = 5 mNz = .3 110 Table 7.2.1.3 Empirical power of tests of complete multivariate independence at a = .01, p = 5 mNz = .5 Ill Table 7.2.2.1 Empirical power of tests of complete multivariate independence at a = .01, p = 10, mNz =. 1 112 ix Table 7.2.2.2 113 Empirical power of tests of complete multivariate independence at a = .01, p = 10, mNz = .3 Table 7.2.2.3 114 Empirical power of tests of complete multivariate independence at a = .01, p = 10, mNz = .5 Table 8 115 Experimentwise Type I error rate for testing the correlations between p uncorrelated variables at the nominal level a = .05 under the assumption of independent tests: c a =l-(l-<x )Vp EX c Table 9 116 Table of critical values that an observed value must exceed for each procedure to test the significance of the pairwise correlations from a matrix of order p = 5 Table 10.1.1.1 117 Table of empirical results for tests of correlations with no preliminary omnibus test when P = I, p = 5 Table 10.1.1.2 118 Table of empirical results for tests of correlations with no preliminary omnibus test when P = I, p = 10 Table 10.1.2.1 119 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P = I, p = 5 Table 10.1.2.2 120 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P = I, p = 10 Table 10.2.1.1.1.1 121 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .1, pNz = .2 Table 10.2.1.1.1.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .1, pNz = .4 122 X Table 10.2.1.1.1.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .1, pNz = .6 123 Table 10.2.1.1.1.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .1, pNz = .8 124 Table 10.2.1.1.2.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .3, pNz = .2 125 Table 10.2.1.1.2.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P I, p = 5, mNz = .3, pNz = .4 126 Table 10.2.1.1.2.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .3, pNz = .6 127 Table 10.2.1.1.2.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .3, pNz = .8 128 Table 10.2.1.1.3.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .5, pNz = .2 129 Table 10.2.1.1.3.2.. Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .5, pNz = .4 130 Table 10.2.1.1.3.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .5, pNz = .6 131 Table 10.2.1.1.3.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P 561, p = 5, mNz = .5, pNz = .8 132 xi Table 10.2.1.2.1.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .1, pNz = .2 133 Table 10.2.1.2.1.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .1, pNz = .4 134 Table 10.2.1.2.1.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .1, pNz = .6 135 Table 10.2.1.2.1.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .1, pNz = .8 136 Table 10.2.1.2.2.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .3, pNz = .2 137 Table 10.2.1.2.2.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .3, pNz = .4 138 Table 10.2.1.2.2.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .3, pNz = .6 139 Table 10.2.1.2.2.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P 561, p = 10, mNz = .3, pNz = .8 140 Table 10.2.1.2.3.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .5, pNz = .2 141 Table 10.2.1.2.3.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .5, pNz = .4 142 Table 10.2.1.2.3.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .5, pNz = .6 143 Table 10.2.1.2.3.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 10, mNz = .5, pNz = .8 144 Table 10.2.2.1.1.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz = .2 145 Table 10.2.2.1.1.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz = .4 146 Table 10.2.2.1.1.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz = .6 147 Table 10.2.2.1.1.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz = .8 148 Table 10.2.2.1.2.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .2 149 Table 10.2.2.1.2.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .4 150 Table 10.2.2.1.2.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .6 151 Table 10.2.2.1.2.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .8 152 xiii Table 10.2.2.1.3.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .2 153 Table 10.2.2.1.3.2 154 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .4 Table 10.2.2.1.3.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .6 155 Table 10.2.2.1.3.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .8 156 Table 10.2.2.2.1.1 157 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz = .2 Table 10.2.2.2.1.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz = .4 158 Table 10.2.2.2.1.3 159 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz = .6 Table 10.2.2.2.1.4 160 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz = .8 Table 10.2.2.2.2.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .3, pNz = .2 161 Table 10.2.2.2.2.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .3, pNz = .4 162 Table 10.2.2.2.2.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P I, p = 10, mNz = .3, pNz = .6 163 Table 10.2.2.2.2.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .3, pNz = .8 164 Table 10.2.2.2.3.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .5, pNz = .2 165 Table 10.2.2.2.3.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .5, pNz = .4 166 Table 10.2.2.2.3.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .5, pNz = .6 167 Table 10.2.2.2.3.4... Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .5, pNz = .8 168 Table 11.1 dEX for tests on correlations when P = I, across N:p x O x p (8crit .05 = .00427, 5crit .01 = .00562, 5crit .001 = .00717) 169 Table 11.2 SEX for tests on correlations with no preliminary omnibus test when P * I, p = 5 across N:p x mNz x pNz (5crit .05 = .00427, Scrit .01 = .00562, Scrit .001 = .00717) 170 Table 11.3 SEX for tests on correlations with a preliminary omnibus test when P * I, p = 5 across N:p x mNz x pNz (Scrit .05 = .00427, Scrit .01 = .00562, Scrit .001 = .00717) 171 XV Table 11.4 SEX for tests on correlations with no preliminary omnibus test when P * I, p = 10 across N:p x mNz x pNz (Scrit .05 = .00427, Scrit .01 = .00562, 5crit .001 = .00717) 172 Table 11.5 SEX for tests on correlations with a preliminary omnibus test when P * I, p = 10 across N:p x mNz x pNz (Scrit .05 = .00427, Scrit .01 = .00562, Scrit .001 = .00717) 173 Table 12.1 Table of parametric measure of sensitivity obtained for tests of correlations with no preliminary omnibus test when P * L p=5, N:p x mNz x pNz 174 Table 12.2 Table of parametric measure of sensitivity obtained for tests of correlations with a preliminary omnibus test when P * I, p = 5,N:px mNz x pNz 175 Table 12.3 Table of parametric measure of sensitivity obtained for tests of correlations with no preliminary omnibus test when P * I, p = 10, N:p x mNz x pNz 176 Table 12.4 Table of parametric measure of sensitivity obtained for tests of correlations with a preliminary omnibus test when P * I, p = 10, N:p x mNz x pNz 177 Table 13.1 Table of parametric measure of criterion obtained for tests of correlations with no preliminary omnibus test when P* I, p = 5, N:p x mNz x pNz 178 Table 13.2 Table of parametric measures of criterion obtained for tests of correlations with a prelimary omnibus test when P* I, p = 5, N:p x mNz x pNz 179 Table 13.3 Table of parametric measures of criterion obtained for tests of correlations with no preliminary omnibus test when P* I, p = 10, N:p x mNz x pNz 180 Table 13.4 Table of parametric measures of criterion obtained for tests of correlations with a preliminary omnibus test when I, p = 10, N:p x mNz x pNz xvii LIST OF FIGURES Figure 1 Empirical experimentwise Type I error rate for each procedure controlling ct^x 183 a t o r below the nominal level, O = y, p = 10, mNz = .3 for N:p = 2 and 20 Figure 2 184 Empirical experimentwise Type I error rate for CF when the omnibus null hypothesis is false, O = y, p = 10, mNz = .3 Figure 3 Empirical hit rate for CF when the omnibus null hypothesis is false, O = y, p = 10, mNz = .3 185 xviii ACKNOWLEDGEMENTS A few years of effort and consideration have gone into the completion of this project. Without the inspiration, guidance, and encouragement of Professor James H. Steiger, this study would never have been completed. I would like to thank Dr. Steiger for being available for continual interface. If I had not been able to get in touch with my supervisor and discuss my work, many problems would have taken exponentially longer to resolve. I would also like to thank Dr. Steiger's family for fielding many phone-calls and relaying accurate messages. I sincerely thank the members of my committee. Professors Steiger, Hakstian, and Wiggins have helped me bring this study to a finish. Their time, input, consideration, and support are deeply and lastingly appreciated. I thank my family for the love and support they have given me throughout my as yet short life. They have always encouraged me to continue and succeed in my studies. Without their love and support, this endeavor could not have been a reality. I wish to thank my father and mother for helping me construct an environment in which I could work with optimal efficiency. I would like to thank my partner for being so caring and supportive, but, especially for being such a good cook. I thank everyone who has contributed to this thesis. Without everybody's support, this thesis would not have been finished on time. The love and consideration that I enjoy has made this all worthwhile. 1 G E N E R A L I N T R O D U C T I O N Correlational methods are becoming increasingly popular in the behavioural and social sciences. A primary reason for this trend is the growing recognition of the legitimacy of correlational and quasi-experimental research. Numerous conditions can limit a researcher's ability to conduct experimental studies. Experimental research is often not feasible because the variables of interest can not be manipulated; this can be because of (a) the intrinsic nature of the variables or (b) ethical considerations. Alternative methods are required under these circumstances; correlational studies can be appropriate when constraints such as these apply. Correlational techniques are also useful in exploratory studies. Many researchers gather observational data for one or more occasions on several quantitative variables and then attempt to uncover important relations among the variables with exploratory data analysis. A variety of questions may be of concern to the researcher. But, the first question to be addressed is often "Given the sample data, are any of the variables related in the population?" If this is answered affirmatively, the next question becomes "Which variables are related?" Procedures that enable the researcher to answer these questions accurately are extremely important. The availability of appropriate statistical techniques is essential when the interest of the researcher is to explore the relationships between variables. 2 Correlational Techniques A variety of procedures have been proposed and used in the analysis of correlational data. Since the early 1900's, many authors have studied the sampling distribution of the correlation coefficient r based on a sample of size N drawn from a bivariate normal distribution with population correlation coefficient p. The exact sampling distribution of the sample correlation coefficient was first derived by Fisher (1915). He later proposed (Fisher, 1921) the normalizing and variance stabilizing transformation which is approximated by a normal distribution with mean z(p) and variance 1/(N - 3), for use in a significance test of the null hypothesis for a single correlation, H :p = 0 0 vs. where p is the bivariate correlation in the population. Other kinds of approximations to the distribution of r have been suggested (Sankaran, 1958; Ruben, 1966; Kraemer, 1973; Samiuddin, 1970; Konishi, 1978). Recent studies on transformations of the correlation coefficient in the presence of outliers and mixed distributions have indicated that in general, of currently available statistics, the Fisher z and the Samiuddin t , s {r-pylN-2 are the transformations of choice for the test of a single correlation (Srivastava & Lee, 1983, 1984). 3 When p is equal to 0, t is distributed exactly as Student's / with N - 2 degrees of freedom; s when p is not equal to 0, t$ is approximately Student's t with N - 2 degrees of freedom (Samiuddin, 1970). Kraemer (1973) considered a more general formula; t$ is a particular case of her equation. Thus, either z, 2 or F i 2 (since fj equals F) transformations can be used for a 2 significance test of a single bivariate correlation. However, when a researcher collects data on more than two quantitative variables for a single sample and the correlations between these variables are of interest to the researcher, the performance of individual significance tests on each of the correlations is not advisable. As Larzelere and Mulaik (1977) indicated, the performance of a large number of significance tests on correlations is analogous to the fallacy of multiple /-tests on means that was pointed out by Fisher (1925). In the same way that the experimentwise Type I error rate increases with the number of tests performed on a group of means, the probability of making at least one false rejection increases with multiple tests on a group of correlations. It is now common practice to take precautions when conducting tests on multiple means. Similar protection is just as important when tests on several correlations are performed. When data are collected on p variables (p > 2), the number of non-repeating pairwise correlations is pip - l)/2. Thus, the number of correlations that may be of interest, v , p increases rapidly as p increases. In a significant proportion of correlational studies, especially ones that are exploratory, the researcher is often concerned with examining all non-repeating pairwise correlations; hence, in many studies, v equals pip - l)/2. p Table 1 shows the relationship between p and the number of pairwise correlations for several common values of p. Given the sheer number of pairwise correlations under increasing p and keeping in mind the problem of inflated experimentwise Type I error rate 4 with multiple tests, it is obvious that the question of how the researcher should proceed in testing whether the p variables are significantly correlated is very important. Indeed, how to test whether the v , where v = p(p - l)/2, correlations between p variables p p on a single sample of size N are significantly different from zero and to simultaneously control experimentwise error rates has been a question of concern for the better part of this century and continues to be so. For some of the same reasons why the analysis of variance is commonly used as an omnibus test on means, overall tests on the correlation matrix have been examined as one way of obtaining experimentwise Type I error rate protection when testing the simultaneous hypothesis that all the correlations are zero. These overall tests address the first exploratory question of whether any of the correlations are non-zero in the population. These tests, however, do not address the second exploratory question of which correlations are non-zero in the population. Another category of tests is necessary to answer the latter question. In the present study, the procedures available to answer these two distinct questions concerning significance tests on correlation matrices and their elements are reviewed. In Part I, the question of how to test the null hypothesis of complete multivariate independence is examined; section 1 consists of a comprehensive review of the statistics available for this test; section 2 details the method of the Monte Carlo simulation experiment conducted to examine the performance of the statistics currently available; sections 3 and 4 describe some of results of this examination. In Part II, the question of how to test which correlations are significantly different from zero is examined; section 1 consists of a review of some of the procedures available to address this question; section 2 details the method of the Monte Carlo simulation experiment conducted to examine the performance of these procedures; sections 3 and 4 describe some of results of this examination of how to test which correlations are significant. Parts I and II are followed by a general discussion of the results 5 of the current study in the context of previous research; in a final note, tentative recommendations are made on how to proceed in testing the significance of a correlation matrix and its elements. 6 PARTI TESTING WHETHER ALL THE CORRELATIONS ARE ZERO 1. Introduction Let R = {r,y} be a pxp sample correlation matrix based on N vectors of observations from a multivariate normal distribution with population correlation matrix P = {p,y}. There are several procedures to test the hypothesis that all of the correlations between p variables are zero, i.e., = 0, V H: 0 i *j vs Hy. pij * 0 , for at least one p,y, i*j , or expressed in equivalent matrix terms, H :V = 1 0 vs. // P*I, i : where I is the identity matrix. The procedures discussed in this section all test the null hypothesis of complete multivariate independence. The procedures vary substantially in accuracy and ease of computation. Most are seldom if ever described in textbooks. These procedures may be conceptualized in several different ways. Some procedures are exact, or very close to it, but are very difficult to compute. Others are based on asymptotic approximations and are somewhat less accurate. 7 One class of tests is based on the likelihood ratio criterion. Another class of tests is based on asymptotic normal theory for correlations and their differentiable functions; these tests are in general much easier to compute but have not yet become popular. 1.1 Likelihood ratio criterion based tests Historically, the procedures that have been used to test the null hypothesis of multivariate independence have been of a general class of likelihood ratio tests. Under the assumption thatNobservation vectors xt,...,x# have been drawn independently from the p-dimensional multinomial population Np(u,Z), where £ is nonsingular, then the maximized value in the parameter space of the distribution is 1 (N exp --j-tr X( I" x X X X I ~ ) ^ X 1 Under the null hypothesis, the maximized likelihood of the sample is The Wilks (1935) generalized likelihood criterion is the ratio A — — L(Q) , which by substitution equals ,N/2 (OyGp) Wilks's lambda has the simplified form A = |R| W2 , where |R| is the determinant of the positive semi-definite sample correlation matrix of p variables. Neyman and Pearson (1928, 1931) showed that under the null hypothesis, the following function of the generalized likelihood ratio criterion, Q' = -2 In A , is a large-sample chi-square distribution with v degrees of freedom. p The Neyman-Pearson likelihood ratio method in conjunction with Wilks's lambda results the following relationship, Q' = A/(-ln|R|) , where Q' may be used to obtain a decision rule in a test of multivariate independence. Unfortunately, the distribution of this likelihood ratio statistic departs rather significantly from a chi-square distribution at small samples. Many authors have indicated that this should be considered a large sample test and is too inaccurate for small samples (e.g., Morrison, 1976). There have been numerous attempts to deal with this problem. 9 The modifications to the likelihood ratio test that have been proposed have varied along two dimensions. (1) Changes in the multiplier, N, have been suggested in order to improve the performance of the statistic. (2) Changes in the root, - ln|R|, have also been put forth. In general, the changes in the root that have been suggested have the desirable property of increased ease of computation since calculating the determinant of a correlation matrix can be quite tedious. Modifications in the multiplier Box 1949 The use of the likelihood ratio statistic was modified in an approach discussed by Box (1949). In the modified approach, U is a function of X, U = X' 2 N , and W is a function of U, w = r/N-D/2 In a similar development as the derivation of Q, Q is obtained, g=2(-lnW) , which in terms of U is Q=(N-l)(-lnU) . Substitution of U yields G=(tf-l)(-ln|R|) , where Q is asymptotically distributed as a chi-square with v degrees of freedom. p 10 Box 1949, Bartlett 1950 Box (1949) extended an approach adopted by Bartlett (1937, 1938) in which, from the moments of a more sufficient form of the likelihood test of multivariate independence, he developed a scale factor which is related to sample size. He suggested the use of the 2p+5 multiplying factor, m = l - , with Q and indicated that the product of m and Q is distributed asymptotically in TV as a chi-square with v degrees of freedom. Thus, the use of p mQ, where mQ = m(N-l)(-\n\R\) = (^-l-^)(-ln|R|) , is suggested as an asymptotic test for multivariate independence. In order to improve the performance of the likelihood ratio test at small samples, Bartlett (1950, 1954) suggested the use of this modified test statistic. Bartlett indicated that improved approximation to the limiting distribution may be achieved by adopting the above modification in the multiplier (Box, 1949; Bartlett, 1950,1954). Although Box suggested this modification to the likelihood ratio test, Bartlett is often credited with this statistic. This statistic is commonly referred to as "Bartlett's test". Thus, in keeping with convention, the notation, Qg^, is adopted to refer to this statistic. 11 Modifications in the root Lawley 1940 Lawley (1940) showed through a multiple MacLaurin expansion that -ln|R| is approximately equal to r ij . Hence, the test of multivariate independence that develops from Lawley's result is the following quadratic form statistic, where Qi is asymptotically distributed as a chi-square with v degrees of freedom. p Modifications in the multiplier and root Bartlett 1950 Bartlett (1950) suggested that a more appropriate version of Lawley's (1940) test is a statistic that includes the MacLaurin approximation and the multiplying coefficient proposed by Box and Bartlett; the resulting quadratic form statistic is where, as before, QJJ^ is asymptotically distributed as a chi-square with v degrees of p freedom. This test capitalizes on the modification proposed by Lawley and produces a computationally simpler variant than Bartlett's test. Morrison has indicated that this approximation is best for small correlations, and that for a large sample size, the decision should be the same as that obtainedfromthe determinental formula (Morrison, 1976). Even 12 though it was Bartlett who suggested the modifications, the resulting statistic is commonly attributed to Lawley. 1.2 Quadratic form tests derived from asymptotic normal theory Steiger and Browne (1984) presented a general class of tests based on asymptotic theory for the distribution of elements of a correlation matrix and their functions. Asymptotic theory is available for the joint distribution of correlation coefficients under an assumption of multivariate normality (Hsu, 1949; Olkin & Siotani, 1976; Pearson & Filon, 1898), and without this assumption (Hsu, 1949; Steiger & Hakstian, 1982; Steiger & Browne, 1984). Available asymptotic theory for simple correlation coefficients and functions of a correlation matrix and some useful asymptotic theory of testing linear hypotheses can be applied to obtain simplified tests of the hypothesis of multivariate independence. Let x be apxl random vector variate with expected value \x, covariance matrix X, and population correlation matrix P. Let p be a v^xl vector of unique off-diagonal elements of P. Let r be the corresponding random vector of elements of the sample correlation matrix R, based on a sample size of N on x. Define r* = (N- \)W(r - p). r* has an asymptotic multivariate normal distribution with mean vector 0, and variance-covariance matrix *¥*, where the elements of F* are given by V ¥ *jkjm = I [(p jl ~ P jkPkl ){Pkm ~ PklPlm ) + (pj -P jlPlm ){Pkl ~ PkjP jl) m + {P jl ~ PimPml )(pkm ~ PkjPjm ) + {pj -P jkPkm ){Pkl ~ PkmPml)] m (Pearson & Filon, 1898). The large sample distribution of r is hence, approximately multivariate normal with mean vector p and variance-covariance matrix ¥ = F*//V. V 13 Tests based on the asymptotic normal distribution of correlations Browne 1977 Browne (1977) described a very general procedure for testing pattern hypotheses on correlations. Steiger and Browne (1984) showed that pattern hypotheses can be expressed as linear hypotheses on p of the form H: 0 Mp = h , where M is a specified gxv matrix of rank g, and h is a specified gxl vector. If ¥ * is any p consistent estimate of ¥*, then the asymptotic distribution under HQ of the statistic, x Q* = (N-l)(Mr-h)'(M P*M)~ (Mr-h) , v 1 is chi-square with g degrees of freedom. While in general, M and h can have arbitrarily chosen elements, in the case of a test of multivariate independence: g equals v , M is the identity matrix, and h is the null vector. p The resulting test statistic, 2 * = (/V-l)r F*~ r , /x 1 is asymptotically distributed as a chi-square with v degrees of freedom. p Under the assumption that *F* is the identity matrix, ¥ * is taken to be I. Where in general r is asymptotically N^p,*?), under the null hypothesis, r is approximately Np(0,D) where D is [1/0V- 1)]I. Thus, (N- l)!/ r is asymptotically N (0,I) and (N- l^r'iN2 p asymptotically distributed as a chi-square. Therefore, the test statistic, V) l r is l 2 14 Q* = (N-l)r'r , is distributed asymptotically as a chi-square with v degrees of freedom. Let this special p case be referred to as Q R. B The derivation of this statistic as presented by Browne (1977) is by the method of generalized least squares which is asymptotically equivalent to methods of maximum likelihood but computationally more efficient. For the special case of concern, testing the null hypothesis that all the correlations are zero, the method of generalized least squares results in the same test suggested by Lawley's MacLaurin series expansion of - lnjRj, Q . L Thus, for a test of complete multivariate independence, note that Q BR = Q^. Tests based on the normalizing property of the Fisher transform Steiger 1980, Burt 1954 Steiger (1980a) showed how to modify Browne's general procedure by use of asymptotic theory for monotonic, differentiable functions of a correlation coefficient. Let f(r) be a qxl vector of functions of the elements of r, differentiable at r = p. Then (Nl)!/ (f(r) - f(p)) has an asymptotic distribution which is multivariate normal with mean 2 vector 0, and variance-covariance matrix P*, where Y* = A ¥*A ,X and A' = ^/^\ _^r Let z be a VpXl vector of Fisher transforms of the elements of r, where z = {z,y}. r r Straightforward application of the multivariate delta theorem (Olkin & Siotani, 1964) yields the result that z , with A' = ^/^.\ _^, has an asymptotic distribution which is multivariate r r normal with mean Zp and variance-covariance matrix T*/(N - 3), where T* has typical elements given by 15 YjkM (i-p^Xi-A) (Dunn & Clark, 1969). Capitalizing on the normalizing and variance stabilizing effect of the Fisher transform, Steiger (1980a) presented the derivation of an alternative statistic. In the case of the Fisher transform, he obtained the quadratic form Q* = (N-3)z; f * z _ 1 r . Under the null hypothesis and the assumption that P* is I, (N- 3) z has an asymptotic V 1/2 r distribution which is multivariate normal with mean vector 0 and variance-covariance matrix I. As a special case of the general test proposed by Steiger, one obtains for a test of the hypothesis of multivariate independence. Q^j is asymptotically distributed as a chi-square with v degrees of freedom. p In the back pages of a factor analysis paper recently cited in Reddon (1987), Cyril Burt (1952) proposed a statistic without proof to be used for tests of residual correlations after extraction of factors. Taking this statistic and considering it under the condition of extracting no factors, one obtains a test, QB\J, that the correlation matrix has only unique factors, i.e., represents p uncorrected variates. The special case of this asymptotic chisquare test statistic is equivalent to the test of multivariate independence, Q^j, described by Steiger (1980a, 1980b). Q JJ B equals Q^j with v degrees of freedom. p 16 The theoretical justification of the statistic as discussed by Steiger is that, since the Fisher transforms of the correlations are more nearly normally distributed than the correlations, the square of the normalized Fisher z/y will be more nearly chi-square distributed and so will their sum. For this statistic, Steiger uses the approximation to the inverse of the variance of the Fisher transforms, N - 3, suggested by Hotelling (1953). Steiger and Fouladi 1991a, 1991b Generations of psychometricians have believed the commonly used approximate value of the variance of the Fisher transforms, (N- 3) , to be accurate enough for practical use. _1 However, analysis and simulation studies have indicated that this is not quite the case (Konishi, 1978; Steiger & Fouladi, 1991a). The results of the examination of the quality of the approximation suggest that a test using known theory for the moments of the Fisher transform and the squared Fisher transform will likely result in improved performance. Konishi (1978) citing Johnson and Kotz (1970) indicated that the variance and the mean of the Fisher transform may be approximated by z(p)+p/2n+p(p+p2)/8/z and {l+(4— 2 p )/2«+(22-6p -3p )/6« }/«, respectively where n equals A - 1. When p equals zero, the 2 2 2 2 7 mean is zero and the variance is {l+4/2«+22/6« }/n. Unfortunately, this approximation is 2 not accurate enough for use in practical application (Steiger & Fouladi, 1991a). Steiger and Fouladi (1991a) derived exact expressions for the cumulants and moments of the Fisher transform and the squared Fisher transform when p = 0. They suggested that use of these results may produce significant improvements in the performance of tests of pattern hypotheses on correlation matrices. A test of multivariate independence, Q$£, results from some of these exact expressions. Let K"2 and /Qj be the second and fourth cumulants of the Fisher transform, 17 & A — 2 2^ 12 r N even , odd , 2 K K- ^4 \r 3 — 120 4 7T 12^ = 4 * N even , odd , where W-4 1 2 ;=1 ,-1 < N even y N-3 1 2 T — iv* odd r A (2/-iy From the second and fourth cumulants of the Fisher transform, the first and second cumulants of the squared Fisher transform were obtained, where K (Z ) = K (Z) 2 1 K (Z ) 2 2 2 = K (Z) + 4 , 2K (Z) 2 The mean and variance of the squared Fisher transform are thus, Li\(z ) = K (z) , 2 2 and H' (Z ) 2 2 = K4(Z) + 2K (Z) 2 + K (Z) 2 2 Ideally, as a chi-square with one degree of freedom, (N - 3)z should have a mean of one 2 and a variance of two. However, for small samples, the mean and variance of (N - 3)z 2 18 deviate non-trivially from these values. The effect of this deviation becomes more severe when more correlations are tested. If the actual mean and variance of a squared Fisher transform are known, then a single z 2 may be corrected, by a linear transformation, so that its mean and variance are precisely correct. If one assumes the v Zy values are essentially independent, then both the mean 2 p and variance of Q^mny also be corrected. The current mean of Qgj is (N - 3)v K ; the p 2 desired mean is v . The current variance of Q$ (assuming independence of z 's) is (N p T y 3) v (K + 2K"2>; the desired variance is 2v . 2 p p 4 QSE corrects the mean and variance of Q^j with the following linear transformation, QSE = SEQsT m + SE ' a where the multiplicative and additive correction constants based on the exact results for the first two moments of the squared Fisher transforms are and a SE = p l-ms (Nv E 3)K (Z ) 2 [ In the limit, under increasing sample size, m$£ approaches one and a$ approaches zero. E Hence, Q$£ is asymptotically distributed as a chi-square with v degrees of freedom. p Steiger and Fouladi suggested that use of the correction constants will likely result in improved small sample performance. However, for greater computational ease, the use of series approximations is preferable to the exact expressions given above. 19 In a classic paper in the annals of correlational statistics, Hotelling (1953) derived series expansion formulae for moments of the correlation coefficient, and moments of the Fisher transformation about its mean. Unfortunately, these formulae are not accurate enough to be used in place of the exact formulae (Steiger & Fouladi, 1991b). Steiger and Fouladi (1991b) extended Hotelling's series expansion formulae. Analytic expansion of the approximations to the moments yields several more terms. The expanded approximations for the first-four moments of the Fisher transform and the lambda's are included in Appendix A. Monte Carlo simulation experiments have shown that these results are a significant improvement over approximations that are in current use and are accurate to a high level of precision (Steiger & Fouladi, 1991b). The corrections to the series approximations for the first two moments of the squared Fisher transforms when p = 0 are given below. Defining v,- = E[(z(r))'], we have 2 v 1 (N-\) 1+ (N-l) + 3(N-]) 2 6 127 + — + (N-l) 1 (N-l) 2 10 + 15(N-1) 3 , 14 3+ —-; (N-l) + (N-l) 124 4247 ^ 5 (N-l) J 4 46 2 (N-IY ^ 15(N-IT Hence, the series approximations for the mean and variance of the squared Fisher transforms are and V4 - Vj , respectively. Q 2 SA corrects the mean and variance of Qg with a linear T transformation using the above series approximations. Specifically, QsA where the multiplicative constant is = SAQsT m + SA > a is given by 20 2 ™5A = and the additive constant is a SA = v ,[l-m 4(^-3)v ] • / In the limit, under increasing sample size, S 2 approaches one and a$ approaches zero. A Hence, Q$ is asymptotically distributed as a chi-square with v degrees of freedom. A p 1.3 Summary of LR and NT based tests All of the statistics based on the generalized likelihood criterion and asymptotic normal theory for testing complete independence among p variables are asymptotic chi-square statistics with v degrees of freedom. For each of these statistics, the test of the null p hypothesis, P = I, rejects when the observed value exceeds X v ,l-a • 2 P The LR and NT tests of multivariate independence reviewed in this paper vary along two dimensions: (a) the multiplier, (b) the root. These tests are tabled by multiplier and root in Table 2. The standard likelihood ratio criterion, based on the natural logarithm of the observed correlation matrix, gives an overall test procedure. However, the statistics from the class of tests derived from the standard likelihood ratio criterion necessitate certain considerations. (1) The observed correlation matrix is singular if the number of variables,/?, is greater than or equal to N, the sample size. The determinant of a correlation matrix that is singular is not defined. Thus, the tests utilizing the determinant can only be used when N is greater than p. (2) The likelihood ratio statistic and modifications using - lnJR| as a root have been derived as asymptotic tests. Further caution with respect to the use of the test comes from 21 the results of Monte Carlo simulations indicating that the statistic is not a good approximation to a chi-square unless N is large. Studies have indicated that the various modifications have different sample size requirements. Asymptotic normal theory tests are easier to compute than likelihood ratio based tests. However, results obtained in Monte Carlo simulation studies on normal theory tests currently available show that these statistics do not perform as well as expected under certain conditions. Steiger and Fouladi have suggested that the recently derived Q^ and E Q$A result in improved performance. 1.4 Other tests of multivariate independence In recent years, other classes of tests have been proposed. For various reasons, these tests have remained more or less unknown. An exact test: Mathai and Katiyar 1979 In an important but often overlooked paper, Mathai and Katiyar (1979) derived an exact test of the hypothesis of multivariate independence. The mathematical and computational aspects of the work of Mathai and Katiyar are quite complicated. The test is extremely difficult to describe, and just as difficult to calculate. Computer software is essential for the performance of the test. This software is not currently being distributed in a form that is accessible to the general research community. A normal approximation: Mudholkar, Trivedi, and Lin 1982 Mudholkar, Trivedi, and Lin (1982) proposed a normal approximation to test the null hypothesis. The computations of this procedure, though rather complicated, are less so than those of Mathai and Katiyar (1979). This procedure and the approximation are reportedly 22 quite accurate. However, test results with large sample sizes or matrices have not been published. The approximation has not been used widely in practice, although the authors have stated that they can provide a FORTRAN program to perform the necessary computations. Test on r m a x Conceptually equivalent to the test of the hypothesis that all the correlations are zero is the test that the largest correlation is zero. One technique involves ordering the observed correlations, then comparing r m a x = maxl ) to a distribution of order statistics from the half-normal distribution. Moran (1980) considered an another method and examined the asymptotic approximation of a test on the largest correlation to the likelihood ratio statistic. In general, these approaches have not received much attention for use as alternative omnibus test procedures. A sum of independent chi-squares: Steiger and Fouladi 1991c Various studies have indicated that correlations between p independent variates are pairwise uncorrected (Cameron & Eagleson, 1985; Geisser & Mantel, 1962) and are asymptotically totally independent and jointly normally distributed (Cameron & Eagleson, 1985). Monte Carlo studies have confirmed that under the null hypothesis, the squared Fisher transforms are essentially uncorrelated (Steiger & Fouladi, 1991c). These results suggest the possibility of a test using the distribution of the sum of several independent functions. 23 Several well-known statistics are available for a test of the hypothesis p = 0. As discussed earlier, a z-test, f-test, or F-test can be used to test the significance of a bivariate correlation. The F-test of this null hypothesis for a single correlation is given by the square of the Student's t statistic, i.e., F(r) = (/V-2) < r ^ 2 1-r 2 where, when p = 0, F(r) is distributed as an F with one and N- 2 degrees of freedom. Let O(F) be the cumulative probability distribution function for F. The following function, X(r) = -21n(l-<D(F(/-)) , is exactly distributed as a chi-square with two degrees of freedom (see, e.g., Hedges & Olkin). For each correlation, asymptotically independent chi-square statistics may be computed from the probability level of each corresponding F statistic. The sum of these v p approximately independent chi-squares gives the statistic where Qp is asymptotically distributed as a chi-square with 2v = pip - 1) degrees of p freedom. It follows that Qp can be used as a test of multivariate independence. 1.5 Overview of simulation studies of tests of multivariate independence The statistic most frequently used for tests of multivariate independence is QBA- However, as has been shown, a number of alternative procedures are available. To date, one of the best and certainly one of the easiest to use is £>sr- Steiger (1980a) demonstrated that this statistic 24 controls Type I error better than Q BA at small-to-moderate sample sizes. This result was confirmed by Wilson and Martin (1983), Reddon (1987), Silver and Dunlap (Silver, 1988; Silver & Dunlap, 1989). The importance of examining whether tests on several correlations control experimentwise error rates is gaining increasing recognition. There have been several papers in recent years that have purported to have examined this issue. The quality of these papers has been less than optimal (Wilson & Martin, 1983; Reddon, 1987; Silver & Dunlap, 1989), however. The studies comparing the performance of tests of multivariate independence have been compromised for a variety of reasons: (1) inadequate number of Monte Carlo replications, (2) unrepresentative sampling of the parameter space and the class of hypotheses of interest, (3) limited scope, and (4) theoretical inaccuracies. Table 3 is a summary table of reviews of LR and NT tests of multivariate independence. In the most recently published examination of the question of how to test the significance of correlation matrices, Silver and Dunlap (1989) considered Brien's (1984) application of an analysis of variance approach to tests of equal correlations in a correlation matrix for a possible test of multivariate independence. Brien proposed a procedure for examining linear hypotheses on correlations. One special case of his test is a "grand mean" test (in the ANOVA sense) that the mean of all the correlations is zero. This statistic is ^2 (N-3) QAN where is asymptotically distributed as a chi-square with one degree of freedom. In the original paper, Brien (1984) suggested that a test of significance of the grand mean component is equivalent to a test of the hypothesis of independence under the assumption 25 of equal correlations. Silver and Dunlap failed to note this essential assumption in their statement that a test for the grand mean tests the null hypothesis that P = I versus the alternative P * I. The failure to acknowledge the requisite assumption is a major oversight. But, most importantly, this is not an assumption that is generally made or justifiable when testing complete multivariate independence. In general, Brien's statistic is a test that the grand mean of the correlations is zero, not that all the correlations are zero. A test of the hypothesis that the mean of a set of numbers is zero is not equivalent to a test that all the numbers in a set are zero. Thus, contrary to what Silver and Dunlap stated, (?AW * sn o t a validtestof multivariate independence. Silver and Dunlap performed a Monte Carlo comparison of QBA> QST> QAN> QBR procedures and concluded that QA/V d the best overall performance. However, their Monte Carlo fta populations involved only non-negative correlations! In general, researchers designing Monte Carlo studies have not employed negative correlations. Analysis of the Brien statistic proposed by Silver and Dunlap as a general test of multivariate independence indicates that the chi-square value is sensitive to signs of the correlations. Given this fact, the exclusion of negative values from the range of population correlation coefficients is a serious oversight. This problem is a classic example of what happens when the Monte Carlo examination of the parameter space is non-representative and this non-representativeness is unjustified. In general, when designing Monte Carlo simulation experiments, it is advisable to make some attempt to evaluate the performance of the statistics under conditions that are representative of the parameter space and the class of hypotheses of interest. This is a very complex and important issue. When the conditions are not representative, this nonrepresentativeness must be justified. 1.6 Current state of affairs As has been shown, when observations are independent and identically distributed multivariate normal random variables and the measure of association is the Pearson product moment correlation, there are a wide variety of procedures available for testing the degree of relationship. Since the Mathai and Katiyar exact procedure and the Mudholkar, Trivedi, and Lin normal approximation technique are effectively unavailable, in general they have not been included in Monte Carlo examinations of tests of multivariate independence. Similarly, they are not the subject of the present study. To date, Monte Carlo simulations of available tests show that Q^j has the best overall performance; however, the relative performance of the recently derived statistics, Q$£, Q$A, and Qp, is as yet unconfirmed. The present study is a comprehensive comparison of the performance of the currently available and the recently derived statistics. 2, Methods In the present paper, a Monte Carlo simulation experiment was conducted in order to compare the experimentwise error rate control of tests of complete multivariate independence. A FORTRAN program was written for this set of experiments. The program uses several subroutines from a library of statistical modules implemented by James H. Steiger and Rachel T. Fouladi. Procedures for generation of samnle correlation matrices Several techniques for the generation of sample correlation matrices are available. One way to obtain sample correlation matrices is to generate samples of random variables from a multivariate normal distribution with a specified population correlation matrix and then 27 calculate the sample correlation matrices in the usual manner. In general, a considerable amount of computation is involved in this procedure. This is particularly salient when samples are large since this procedure requires the generation of Np normal variates. Given the computational intensity of this technique to generate sample correlation matrices, the use of a more economical procedure is advised. Browne (1968) described an alternative procedure to generate sample correlation matrices that is not as computationally intensive as the standard procedure. Let T be a lower triangular matrix of order p, whose non-zero elements are independently distributed variates of the following form: tjj is distributed as chi with (N- /) degrees of freedom, tjj is distributed as N(0,1) (i > J), tjj equals 0 (i < j). Random matrix H, where H = T T , is distributed as Wishart distribution with covariance 1 matrix I and N- 1 degrees of freedom (Odell & Feiveson, 1966). p The approach described by Browne capitalizes on the fact that random matrices H can be used to construct random matrices, A, distributed as Wishart distributions with covariance matrix X and N- \ degrees of freedom. Since £ is positive definite or semi-definite, there is a matrix O such that Z equals 00*. For computational simplicity, O is chosen to be lower triangular with positive diagonal elements computed using the square root method of triangulating a matrix (Dwyer, 1945). Then A = OHO' = (OT)(OT) . Let S = A/N and R = (Diag A)" /2A(Diag A ) / . The resulting S and f 1 -1 2 R are distributed as the maximum likelihood estimates of X and P, based on samples of N observations from a p variate normal distribution with population covariance matrix, E, and correlation matrix, P. 28 Using these results, the Browne procedure for obtaining a random correlation matrix directly, without generating a random sample from the multivariate normal distribution, reduces the number of random variables to be generated from Np random normal deviates to v random normal deviates and p variates from chi distributions. In contrast with the p standard method, there is also a considerable reduction in the computation involved in calculating the correlation matrix using the Browne method. In preliminary runs, it was verified that the fast method (Browne, 1968) produced similar results to the slower standard technique based on generating independent normal observations. Independent normal observations for the standard procedure were generated using the Kinderman-Ramage (1976) procedure using mixture distributions (Kennedy & Gentie, 1980). The Kinderman-Ramage technique uses a decomposition of the half-normal into a triangular density, three nearly linear densities, and a tail region. Simulation experiments For the main body of the study, the matrices were generated using the fast method. For the examination of Type I error control, sample correlation matrices were generated from population correlation matrix P where P was equal to I; that is, the sample matrices were generated under the condition of a true null hypothesis that all of the bivariate correlations are zero in the population. P was varied along one dimension: order pip = 5, 10, 15). For the examination of Type II error control, sample correlation matrices were generated from population correlation matrix P where P was not equal to I; that is, the sample matrices were generated under the condition of a false null hypothesis. P was varied along several dimensions: (a) order p ip = 5, 10), (b) proportion of non-null correlations ipNz = .2, .4, .6, .8), (c) magnitude of non-null correlations (mNz = .1, .3, .5), and (d) configuration, 29 M, of the pattern of non-zero correlations (U = upper submatrix, R = random location). The magnitude of the non-null correlations was selected on the basis of maintaining a positive definite P; non-null correlations in increments of .1 greater than .5 for the proportion of non-null correlations selected yield non-positive definite matrices. The locations for the non-zero correlations were determined according to the configuration of P. For matrices with configuration U, the upper submatrix was filled with Vj, where Vi = pNz(v ), non-zero correlations of a specific magnitude. For matrices with configuration p R, V/ non-zero correlations of a specific magnitude were located randomly according to locations generated using a uniform random number generator. The configurations of the random pattern matrices for varying proportions of non-zero correlations generated through this procedure are shown in Appendix B. Sample correlation matrices were generated at various sample sizes for specific ratios of sample size to number of variables (N:p = 2, 4, 10, 20, 40). Table 4 shows the sample sizes at which correlation matrices of order p were sampled. For each sample correlation matrix, the test statistic for each test of multivariate independence (Q , BA Q R> L=B QlA^ QST> QsE' QSA> QF) w a s calculated; the percentile point of the observed value was computed and the decisions for the hypothesis tests at the nominal levels .05 and .01 were recorded. Under each condition, experiments for tests under a true null hypothesis were replicated 100,000 times; experiments for tests under a false null hypothesis were replicated 10,000 times. The number of rejections was tabulated and transformed into proportion rejected. The experimentwise Type I error rates for each statistic, that is, the proportion of false rejections yielded for each omnibus test statistic in 100,000 replications under each factorial condition where P equals I, were assessed. Similarly, type II error rates, that is, one minus the proportion of true 30 rejections yielded for each omnibus test statistic in 10,000 replications under each factorial condition where P equals I, were also assessed. Measures of experimentwise Type I error rate control Chi-square goodness of fit values based on a normal approximation to the binomial were computed for each statistic in each condition. The chi-square statistic used was (a;-a) 9 2 where a, is the empirical estimate of Type I error and R is the number of replications. 3. Results The empirical results for the performance of the tests of complete multivariate independence QBA> QL=B& QLA' QST' QSE' QSA> a n d QF presented in this section. When, P js equal to I Empirical Type I error rate performance of each test of complete multivariate independence was assessed with 100,000 replications under p (5,10, 15) x N:p (2,4,10, 20, 40) x a (.05, .01) factorial conditions. Table 5 gives the empirical experimentwise Type I error performance of the statistics under these factorial conditions. Chi-square goodness of fit values for experimentwise Type I error performance were computed for each p x N:p x a factorial cell for each omnibus statistic. These values are given in Table 6. 31 Tables 5 and 6 also show the overall pattern for the actual experimentwise Type I error rates of each statistic. The results indicate that the experimentwise Type I error rates for Q R and Qm are B consistently at or below the nominal level. Q R is conservative for small and moderate-toB large ratios of sample size to number of variables; Q R performs at the nominal level for B extremely large N:p ratios. In general, Q^A is conservative for all levels of N:p ratio tested. The tabled results show that the Type I error rate performance of Q BA across most conditions. Q BA is at the nominal level does not perform at the nominal level for sample matrices generated at small N:p ratios. Under these conditions, Q BA does not control the experimentwise Type I error rate at or below the nominal level; Q BA is consistently liberal for small N:p. The results show that actual experimentwise Type I error rate for Qgj exceeds the nominal level for small ratios of sample size to number of variables. Q$j performs at the nominal level for moderate and large N:p. Q$E performs at the nominal level across most conditions. For a nominal alpha equal to .05, the experimentwise Type I error rate of the statistic is at the nominal level except when N:p is at the smallest level. For a nominal alpha equal to .01, the experimentwise Type I error rate of the statistic is at the nominal level except when N:p is at the two smallest levels. The pattern of Type I error performance of <2s£ is the same for and Qp. Overall, the analyses show a similar pattern across all levels of a and all levels of p. This consistent pattern is the following where (a) Q R and QJJ^ are extremely conservative, (b) B Q-BA-> Q-ST' QSE> QSA> QF control experimentwise Type I error loosely at the nominal level when N:p exceeds two, and (c) Q BA is more liberal that Q$p, Q$ , and Qp at small N:p. A 32 Given the overall results, it can be said that, in general, the tests Qgp, QSA> a n d QF satisfy the Neyman-Pearson criterion of an optimal test with a caveat that these statistics are liberal for N:p less than or equal to two. When p is not equaltoI Empirical performance of Q , QL R, BA =B QIA, QST> QSE> QSA> a n d QF w a ssessed with s a 10,000 replications under p (5,10) x N:p (2,4,10,20, 40) x a (.05, .01) x pNz (.2, .4, .6, .8) x mNz (.1, .3, .5) x M (U, R) factorial conditions. The empirical power of the statistics as tests of complete multivariate independence under these factorial conditions is given in Table 7. These tables show that the overall pattern of results is generally the same for matrices P with matrix configurations U and R. There is, however, some differential performance between the performance of Q BA at these levels. The tables show that power is very weak for tests of matrices with small proportions and low magnitudes of non-zero correlations. Within each table, the results show that power increases as the ratio of N to p increases; as the proportion of non-zero correlations, pNz, increases, the power of each statistic increases. As the magnitude of the non-zero correlations, mNz, increases, the power of each statistic increases. Comparison between results of tests performed on matrices of different order shows that power increases as the order of the matrix increases. The results show that Qr^ and Q independence. £>S£ a n a " QSA h BR are the least powerful statistics for tests of multivariate a v e t n e s a m e power and tend to be more powerful than Qp as mNz departs increasingly from zero. Qp tends to be more powerful than Q . Exceptions to BA this overall pattern are evidenced when the N:p ratio is low. The tables show that when N:p 33 equals two, Qg is sometimes more powerful than Qp, Qg , or Q$p. The results show that A A Qgj- is more powerful than all of the procedures. 4, Conclusions The results of Part I of the current study show that of currendy available omnibus statistics for testing complete multivariate independence, Qp, Q$ , and Q$p have the best overall A performance in terms of Type I error control. Of these recently derived statistics, Q$ and A Q$ are the most powerful. Thus, Q$ and Q$p are the optimal procedures in terms of the E A Neyman-Pearson definition. The empirical results further suggest that if an experimenter suspects that the non-zero correlations in the population are few and small, then as large an N:p ratio as possible should be used since this will increase the power for tests under weak power conditions. 34 PART II IDENTIFYING WHICH CORRELATIONS ARE NON-ZERO 1, Introduction Testing the null hypothesis of complete multivariate independence is a primary test in social science correlational research. Rejecting this hypothesis suggests that in the population, some of the p variables are likely correlated. Depending on how the researcher has selected the variables, the question of interest varies. But, beyond examining the structure of a correlation matrix, a main question a researcher may ask is: which of the correlations are significantly differentfromzero; that is, which of the p variables are significantly correlated. The omnibus test does not answer this question. Like the omnibus ANOVA test procedure, the test of multivariate independence is an overall test. The overall test on the correlation matrix indicates whether there is any dependency among the variables. Just as follow-up test procedures have been suggested and are routinely conducted in the context of tests on means, similar follow-up procedures can be conducted when testing correlations. In the following, let v be the number of zero correlations in the set of v bivariate 0 p population correlation coefficients. Let v be the number of non-zero population { coefficients, thus, v, = v„ - v . 0 35 A variety of procedures have been proposed for use in testing which correlations are nonzero. The test statistics traditionally used for significance tests of correlations have been ttests or z-tests using the Fisher transforms of the correlations. Recently, however, the use of order statistics has also been proposed for this purpose. In the following, we examine the use of these statistics applied in simultaneous and sequential test procedures. 1.1 Simultaneous test procedures Tests of single bivariate correlations have been examined in the presence of outliers and mixed distributions. Research on the performance of these statistics has shown that overall the Student's f-test is the preferred procedure (Konishi, 1978; Srivastava & Lee, 1983, 1984). In order to test which of several correlations are non-zero, individual tests of the form H: 0i =0 vs. P / are performed on each sample correlation r,-, i = l...v . One dimension on which r-tests vary p is the level at which the correlations are tested. /-tests at nominal alpha One approach to test which v elements in a correlation matrix are non-zero is to test each p correlation r,-, / = l...v at some typical nominal level, such as .05. The /-test of each p correlation rejects the corresponding null hypothesis when 1 1 l-^-,N-2 2 where a is the nominal level at which each correlation is tested. c 36 The main problem with testing each correlation at the nominal level is elevated experimentwise Type I error rate, which in the context of testing "which correlations are significantly differentfromzero" is the probability of falsely rejecting at least one correlation in the set of correlations under examination. The probability of making at least one Type I error on the set of v correlations is p EX ~ ^(at * a e a s t o n e TyP I e e r r o r m v p tests ) = 1 - Pr(no Type I errors in v tests) p = 1 - Pr(no Type I errors in v and Vj tests) 0 = 1 - Pr(no Type I errors in v tests) . 0 When all the Mests are independent, the experimentwise Type I error rate is o v c*EX = l-YlQ-<Xci)> i"=l and if each comparison is tested at a common level, i.e., a =a , V /, then the cl c experimentwise Type I error rate is a =l-(l-a ) ° v EX c . Table 8 shows the experimentwise Type I error rate for testing the correlations between p uncorrected variables (i.e., v = v ) at the nominal level, a = .05, under the assumption of p 0 c independent tests. The elevated experimentwise Type I error rate that resultsfromtesting each correlation at a nominal level typically employed in individual tests is a serious problem. This procedure of testing each correlation at the nominal level violates the convention of conducting test procedures that satisfy the Neyman-Pearson criterion. According to this decision theory criterion, an optimal test is defined as the least conservative procedure that controls the 37 overall experimentwise Type I error rate at or below the nominal level. In spite of the inflated experimentwise Type I error rate, multiple tests at the nominal level are in fact conducted quite frequently. As specified by the Neyman-Pearson decision theory definition of an optimal test, an important criterion in assessing the quality of a test is whether a procedure maintains the experimentwise Type I error rate at the nominal level. Since conducting a set of f-tests at the nominal level does not satisfy this criterion, alternative procedures to control the actual experimentwise Type I error rate at or below the nominal level have been considered. Reduced alpha /-test procedures: Bonferroni (RB), Dunn-Sidak (RD) One familiar test procedure to identify which correlations are significant is to test each correlation at a single reduced level. Typically, the reduced level is at the Bonferroni level or the Dunn-Sidak level. This is in order to ensure that the experimentwise Type I error rate is controlled at or below the nominal level. It is known that the probability of a Type I error for the set of v /-tests is never greater than p the sum of the per comparison Type I error rates a j, since the probability of making at least c one Type I error is a E X U Type I error on comparison,- = Pr v V P = 2 Pr(Type I error on comparison/) /=i - Pr(Type I error on two or more comparisons) V P = ^oc ci - Pr(Type I error on two or more comparisons) 38 Thus, the sum of the probability of making a Type I error on each comparison is the upper limit of the probability of making at least one Type I error in the family of correlations; i.e., i If each comparison is conducted at the same level a j where a = cc , Vz, then the c ci c probability of at least one Type I error for the set is never greater than v a ; i.e., p a <v a EX p c . c In order to control the experimentwise Type I error rate at or below the nominal level, a, the Bonferroni level, a , may be used for each comparison, where cB cB=— a a V ' P A test using a reduced Bonferroni level rejects when 2 The Dunn-Sidak level specified by the Dunn-Sidak inequality may also be taken for each comparison in order to control the experimentwise Type I error rate at or below the nominal level. The Dunn-Sidak per comparison level is a j), where c a =l-(l-a)l/vp . cD A test using a reduced Dunn-Sidak level rejects when 2 Traditionally, the Bonferroni level is used due to greater computational simplicity even though tests at the reduced Dunn-Sidak level yield an experimentwise error rate close to the 39 desired level. However, since the Dunn-Sidak inequality is based on the assumption of independence, non-zero covariance between the correlations may result in experimentwise Type I error rates that exceed the nominal level (Larzelere & Mulaik, 1977); however, Hochberg and Tamhane have suggested that tests at the Bonferroni and the Dunn-Sidak levels control experimentwise alpha at the nominal level strongly (Hochberg & Tamhane, 1987). 1.2 Sequential test procedures Sequential test procedures have been proposed for use in testing which correlations are zero. Most of these procedures use a step-down testing approach. A step-down procedure begins by testing the overall intersection hypothesis, HQ= f] //o/,and then steps down through the iel hierarchy of implied hypotheses. If any hypothesis is not rejected, then all of its implied hypotheses are retained without further tests; thus, a hypothesis is tested if and only if all of its implying hypotheses are rejected. Typically a step-down procedure uses a non-increasing sequence of critical constants for successive steps of testing (Hochberg & Tamhane, 1987, p. 53). Step-down procedures are generally more powerful than the corresponding single-step procedures. Marcus, Peritz, and Gabriel (1976) proposed a general method for constructing step-down test procedures. This method is referred to as the closure method. Hochberg and Tamhane have referred to procedures resulting from the closure method as closed testing procedures. Let {///(l ^ / ^ v )J be a finite family of hypotheses on scalar or vector parametric p functions. The closure of this family is formed by taking all nonempty intersections Hp= fl Hj for P c jl,2,...,v }. If an a-level test of each hypothesis Hp is available, then p 40 the closed testing procedure rejects any hypothesis H if and only if HQ is rejected by its P associated or-level test for all Q • P. Hochberg and Tamhane described a proof by Marcus, Peritz, and Gabriel showing that a closed testing procedure strongly controls the Type I experimentwise error rate. The proof follows below. Proof. (Marcus, Peritz & Gabriel, 1976; Hochberg & Tamhane, 1987). Let {//,-,/ e P} be any collection of true null hypotheses and let Hp= f] Hj where P is some unknown subset of {1,2,...,v}. If P is empty, then there can be no Type I error, so let P be nonempty. Let A p be the event that at least one true /// is rejected and B be the event that Hp is rejected. The closed testing procedure rejects a true /// if and only if all hypotheses implying //,-, in particular Hp, are tested at level or and are rejected, and the test of /// is also significant at level or. So A = A n B and thus under Hp, a E X = PrM} = PrlAnfl} = PT{B}PT{A\B} < a. This inequality follows since Pr{B} = or when H is true. Because the above expression P holds under any Hp, the experimentwise Type I error is strongly controlled at level or. A major drawback of the closed testing procedure is the feasibility of testing all of the Hi hypotheses. The number of tests in a closed testing procedure increases exponentially with the number of elements being tested. Therefore, a shortcut version of the closed testing procedure is advocated. According to Hochberg and Tamhane, a shortcut version of the closed testing procedure was formally proposed by Holm (1979) who labeled it a sequentially rejective procedure; however, it was also independently proposed earlier in special contexts by Hartley (1955), Williams (1971), and Naik (1975). Consider a closed testing procedure that uses union- 41 intersection statistics for testing all intersection hypotheses {//,-,/ e P] (that is, the test statistic for every intersection hypothesis Hp is derived from those for the Hjs by the union intersection method). Such a closed testing procedure can be applied in a shortcut manner. Union-intersection tests have the property that whenever any intersection hypothesis Hp is rejected, at least one of the Hjs implied by Hp is rejected. Therefore, in order to make a rejection decision on any H[, it is not necessary to test all Hp containing that ///; one simply needs to test only ///. However, it is essential that the shortcut procedure be conducted in a step-down manner by ordering and then testing the Hjs. This procedure ensures that a hypothesis is automatically retained if any intersection hypothesis implying that hypothesis is retained; this is referred to as the coherence condition. If the union-intersection related test statistics for Hp are of the form Z = max /> Z, (which P ig is the case if the rejection regions for the individual Hjs are of the form Z/ > Q, then the coherence requirement can be ensured if the Hjs are tested in the order of the magnitudes of the corresponding test statistics Z/'s from the largest to the smallest. Thus, the hypothesis corresponding to the largest Z/ is testedfirst.(Note that testing this hypothesis is equivalent to testing the overall intersection hypothesis.) If it is rejected, then any intersection hypothesis containing that hypothesis will clearly be rejected and therefore that hypothesis can be set aside as being rejected without further tests. Next, the hypothesis corresponding to the second largest Z/ is tested. This procedure is continued until some Z/ is found to be not significant. At that point, all the hypotheses whose test statistic values are less than or equal to the current Z; are automatically retained. Suppose that at some step in the procedure, the hypotheses ///, / e P, remain to be tested. Then an a-level test is obtained by comparing the test statistic Zp = max,-/> Z,- against the 6 upper point of its distribution under Hp = f] Hj. ieP 42 According to Hochberg and Tamhane, the short-cut procedure controls the experimentwise Type I error rate at the nominal level or if the individual tests at different steps are of level or. They suggested that this relationship between experimentwise Type I error and the nominal level at which the individual tests are conducted holds rigorously when the tests are independent; however, when the tests are not independent, they speculated that this relationship may not hold. Procedures using t-tests Multi-stage reduced alpha: Bonferroni, Dunn-Sidak A multi-stage reduced alpha level test procedure for tests on correlations was proposed by Larzelere and Mulaik (1977). The sequence of tests recommended by Larzelere and Mulaik constitutes a sequentially rejective closed testing procedure. The procedure consists of a series of tests on the correlations. According to Larzelere and Mulaik, at each stage of this procedure, all of the correlations which have not been declared significant are tested at the reduced alpha level determined by the number of correlations to be tested at that stage; i.e., /// (1 < / < v ) is tested at the con-esponding reduced level. p The multi-stage procedure begins by testing all of the correlations at the strict reduced level and continues in a step-down manner until there are no rejections at a particular stage. Larzelere and Mulaik suggested using sequentially determined Bonferroni reduced alpha levels for each of the tests. The recommended procedure is described by the following sequence of tests. In this algorithm, v is the number of correlations being tested; k is the p number of correlations rejected. At the start of the procedure, none of the correlations have been tested and therefore the number of rejections, k, is zero. At each stage /, for / = l...v , p test each correlation at the two-tailed a cBi level, where a cBi = ccl(v - k). p Let be the 43 number of rejections made during each stage. If k( is greater than zero, increment k by k( and proceed to the next stage testing the remaining correlations at the updated level; if k[ equals zero then do not proceed to the next stage. At the ith stage, this algorithm rejects a test when 2 An equivalent procedure and more simple to describe is a sequential stepped-down test on sorted correlations; the idea behind the procedure is the same as for Holm's (1979) sequentially rejective Bonferroni procedure. The absolute values of the observed correlations are sorted. The largest of the sorted values is tested at the two-tailed a/v level, p the second largest is tested at the two-tailed cc/(v - 1) level, the third largest is tested at the p two-tailed a/(v - 2), the ith largest is tested at the two-tailed a #; level, where CC BJ = oc/(v c p p C - i + 1), and so on until one of the correlations fails to reject. In general, this algorithm rejects the /th test when 2 The following shows the equivalence of the two algorithms described above. If at stage 1, k[ correlations are rejected at the ajv level, then those k\ correlations are the largest of the v p p correlations and would have been tested at the a/v , cd(y - 1 ) , a l ( y - k\+Y) levels. If p p p these correlations are rejected at the ajv level, then it follows directly that they are also p rejected at the ajv , aj(y - 1 ) , a l ( v - £]+l) levels. Following this rationale, the p p p equivalence of the two algorithms extends to stage 2 and so on. It is clear that the multi-stage reduced alpha procedure constitutes a closed testing procedure. The multi-stage reduced alpha procedure can also be implemented using tests at 44 sequentially determined Dunn-Sidak alpha levels. Using the second algorithm, the DunnSidak alpha level for the /th test is a c D i = \-(\-a) p The ith test rejects when \ i\ \ t > CCcDi 2 • Hochberg and Tamhane suggested that the experimentwise Type I error rate is strongly controlled for this procedure using either the sequentially determined Bonferroni or DunnSidak alpha levels. Larzelere and Mulaik suggested, however, that the dependency structure of the correlations may drive the actual experimentwise alpha level above the nominal level when the sequentially determined Dunn-Sidak levels are used. Until recently in the social sciences, these multi-stage procedures had not received much attention. It is interesting to consider some of the reasons why the multi-stage procedure is gaining renewed examination (Crosbie, 1985; Silver, 1988). One factor that plays a major role in determining whether a procedure is used or not is feasibility. In general, it is simply not feasible to conduct a multi-stage test procedure by hand since / reference tables with detailed increments in probability levels are not currently available. However, with computers and accurate ^-routines, critical values for any cc can be obtained. c Crosbie (1985) implemented a Pascal program to execute the multi-stage procedure. The program is not widely available, nor has its accuracy yet been evaluated externally. Crosbie used a four-term approximation due to Zelen and Severo (1964) instead of Peiser's (1943) tapproximation which was used by Larzelere and Mulaik. More accurate t routines are currently available. Hence, Crosbie's program is probably not as accurate as it could be. 45 Silver (1988) attempted to examine the error control of the multi-stage procedure. Silver conducted a Monte Carlo simulation; however, his assessment of the performance of the procedure was compromised by his failure to examine the techniques under a representative choice of parameters. Procedures using order statistics theory In this section, the application of order statistics theory to tests on correlations is considered. This results in an alternative class of procedures for identifying which correlations are significant. Suppose thatXi, X^X , v are v independent observations, each with common cumulative distribution function F(x). If the observations are rearranged in ascending order of magnitude and written as such thatX(i) corresponds to the smallest observation, and so on, then is called the ith order statistic. Alternative notation for order statistics is sometimes employed where X\ < X2 ^ . . :v In this notation, X ; :v :v X. v v . are the order statistics in random samples of size v drawn from a continuous population having F(x) for its cumulative distribution function. Because the notation X^ is less informative than X since the symbol , includes information on the Y i:v : v total number of ordered observations, the latter notation will be employed. The distributions of order statistics are generally very hard to describe. Unlike statistics which are based on symmetric functions of the observations, order statistics do not lend themselves to simple study. They have the distinctive property that if an ordinary random 46 sample of values is rearranged as order statistics, the values of the order statistics are no longer independent nor identically distributed even though the original observations were independently identically distributed (Kendall & Stewart, 1969). The distribution of each order statistic depends on: (1) the rank of the observation, (2) the number of observations, and (3) the parent distribution. In what follows, we will use F Xx) /:v (/ = 1... v) to denote the cumulative distribution function of the ith order statistic X . i:v Effective application of order statistics techniques usually requires adequate computer software or extensive reference tables since each ranked observation has its own distribution. Few approximations have been implemented and reference tables are not widely available. Undeterred by their generally complex nature, several authors have considered the application of some of the concepts of order statistics to tests on correlations (e.g., Stavig & Acock, 1976; Moran, 1980). These examinations have had virtually no impact on practice in the field. Stavig and Acock (1976) introduced a technique to test a set of correlations called the rank adjusted method; this testing procedure is based on a normal approximation to the distribution of order statistics. The validity of their approach is questionable, however (Larzelere & Mulaik, 1977; Silver, 1988). Moran (1980) examined the distribution of the largest element of a set of correlation coefficients. He made several suggestions for the use of a test of the largest correlation. However, he did not extend any suggestions for use in a test of several correlations. More recently, several authors have reexamined the use of order statistics for testing the individual elements in a correlation matrix. 47 Silver 1988 Silver (1988) proposed a sequential procedure for testing which correlations are non-zero using an order statistics approach. In this procedure, the absolute values of the standardized Fisher transforms of the observed correlations,V/V - 3jz,| , i = l...v , are ordered. The p largest observation, X Vp VpJ is compared to the critical value corresponding to the largest of v ordered half-normal observations, Z p where Z . Vp:Vp Vp Vp = £ such that F . (Q = 1 - a. Vp Vp The second largest observation,X _i , is compared to the critical value, Z _\. , Vp :Vp Vp corresponding to the second largest of v ordered observations, where Z p that F _i {£) Vp :Vp value, Z _ Vp 2:Vp _i Vp :Vp Vp = £ such = 1 - a. The third largest observation, X _2 , is compared to the critical Vp where Z _ = Vp 2:Vp £ such that F _2 (Q Vp :Vp = 1 - or, and so on. :Vp In Silver's procedure, the null hypothesis that a correlation is zero in the population is rejected at the a level if the order statistic corresponding to the absolute value of the standardized Fisher transform of the observed correlation, X _j- , exceeds the critical Vp value Z _ , Vp i:Vp Vp i.e., X v -i:v p > Z p v -i:v p p • Silver proposed two ways to apply the sequentially determined tests. The first procedure involves comparing all of the correlations with their respective critical values; let this procedure be referred to as OR. The second procedure involves conducting the procedure in a step-down manner, i.e., when an observation fails to reject, stop and declare all subsequent observations as having failed to reject; let this procedure be referred to as OS. Silver conducted a Monte Carlo experiment to examine the performance of these procedures. On the basis of his result, he stated that his order statistics procedure is too conservative when assessing correlations via the step-down method, OS, and suggested the use of the OR procedure for tests on correlations, 48 Silver stated that the procedures were implemented using the half-normal distribution as an approximation of the order statistic distributions to obtain the critical values of each order statistic. The adequacy of the approximation and the theoretical basis of the use of order statistics proposed by Silver are questionable, however. The problems with Silver's proposal include: 1) A problem of coherence: OR vs OS. Analysis of the procedures shows that the OR procedure sometimes leads to rejections on observations when larger ones have failed to be rejected; thus, OR is not a coherent procedure. Intuitively, it does not make sense to state that a correlation is significant if larger correlations have not been rejected. Unlike OR, the step-down approach, OS, is consistent on this dimension and is therefore a coherent procedure. 2) A problem with reference distributions. The reference distributions for the ith of v observations with cumulative distribution functions F , :Vp p are the correct ones to use under complete multivariate independence. These distributions are derived under the assumption that the v observations have been independently sampledfromthe same distribution. Thus, p the use of the reference distributions is asymptotically appropriate under a true null hypothesis. However, in general, one does not know the configuration of the population matrix. The critical values for Silver's procedure to test individual correlations are based on the assumption that all the null hypotheses are true. This assumption is retained even after some of the null hypotheses have been rejected. If the omnibus null hypothesis is false, the observations do not all have the same distribution, nor are they (even asymptotically) independent. Hence, under a false null hypothesis, the use of the reference distributions suggested by Silver for testing the v ordered observations is not justified. p 49 Let the population correlation matrix be P where P is not equal to I; thus, v does not equal 0 v . Let r ,-, for i = l...v , denote the v observed correlations corresponding to the 0 p 0 0 correlations that are zero in the population, and r ,-, for i = l...v denote the v, observed 1 lt correlations corresponding to the correlations that are non-zero in the population. Let X°J :VQ denote the tth ordered observation corresponding to the v zero population correlations, and 0 X j. 1 v denote the ith ordered observation corresponding to the v non-zero population 2 correlations. Under the procedure proposed by Silver, the ordered observations for r ,-, / = l...v with v 0 0 not equal to v , are not compared to the appropriate critical values. For instance, if X p 0 0 is . VQ VQ the;'th of v order statistic for the complete set of observations, i.e., Xj. , the order statistic p v is not compared to the critical value corresponding to the largest of v , Z . , but is 0 V q V o compared to the critical value corresponding to the jth of v , Zj; . The appropriate critical p ;v values are never less than and, indeed, are usually greater than the critical values specified by Silver's procedure. Hence, it is likely that use of this procedure results in an excessive number of false rejections. Fouladi A coherent step-down order statistics closed testing procedure for testing the significance of v ordered values of observations using appropriate ordered statistics reference distributions p entails the following steps. First, test the largest value, X . , against the critical value Vp corresponding to the largest of v observations, Z p Vp:Vp Vp where Z . Vp Vp - t\ such that Fy y (0 p: p = 1 - or. If the test fails to yield a rejection the test procedure stops; otherwise, test the next observation. To test the second largest of the v observations, compare the observed value p against the critical value corresponding to the largest of (v^ - 1) observations where Z _ Vp i vp_\ = £ such that Fvp-l:Vp-l(Q = 1 - Of- If the second test fails to yield a rejection, the : test procedure stops; otherwise, test the next observation, and so on. 50 In general, the test of the ith largest of the v observations is a comparison of the observed p value against the critical value corresponding to the largest of (v - /+1), where p l = £ such that F _ i i+ Vp i+ : vp_,+i(0 = - - Th 1 a e t e s t Z _i _ Vp i+ :Vp procedure continues in a step-down manner until one of the tests fails to reject. The logic of this procedure comes directly from the theory justifying the closed testing procedure and order statistics theory. A primary assumption of order statistics theory is that the original un-ordered values are independent observations drawn from the same population. Consider v non-negative observations. At the start of the procedure, one has these v nonp p negative observations. In order to test the intersection hypothesis that all v observations are p zero andfromthe same population, it is standard to test the equivalent hypothesis that the largest observation is zero. To test the largest observation, the observed value is compared against the critical value from the reference distribution for the largest of v . If one rejects p the hypothesis that the largest one is zero in the population, then one is simultaneously rejecting the hypothesis that all of the observations are zero. After rejecting the hypothesis that the largest of v observations, X - , is zero, then V p Vp Vp observations, where v' = iy - 1), remain to be tested. If one can no longer state that all of p the observations arefromthe hypothesized distribution given arejectionof the largest of v p observations, the following steps must be taken in order to ensure a valid test. To test whether the remaining v' correlations are significantly different from zero, the largest of the v' observations (which is the same observation as the second largest of the original v p observations) is tested against the critical value from the appropriate reference distribution; the reference distribution for this test is the distribution for the largest of V observations, Fy' '(;c). The procedure continues in a step-down sequential manner testing each :v observation against the corresponding reference distribution until one of the tests fails to reject. 51 In general, for a test of the /th observation of the original v observations, the reference p distribution is the distribution for the largest of v', F '-y(x) where v'=v -i v p + \. Unlike the procedure proposed by Silver, this procedure relies only on the use of the reference distributions corresponding to the extremes of an ordered set. Where F (x) (/ i:v 1...V) denotes the cumulative distribution function of the /th order statistic X cumulative distribution function of the largest order statistic X v v i:VJ the is given by F (x) = Pi{X <x) (v \ = Pr f](Xi<x) \i=l v:v v:v Since a primary assumption of order statistics is that the original un-ordered observations are independent, the cumulative distribution function can be expressed as i=l = f[F(x) 1=1 = [F(x)] . v To test the largest of an ordered set at the nominal level, F (X ) v v:v = [(F(x)] = l-cc . v Thus, F(x) = (l-a) l/v . Under the assumption of independent observations, this step-down order statistics procedure satisfies the conditions of a closed method testing procedure described in Hochberg and Tamhane. The tests meet the condition of coherence and since in the sequence of tests the 52 largest observation of the remaining set is tested at the nominal level (i.e., each intersection hypothesis is tested at the nominal level), according to the Marcus, Peritz, and Gabriel proof, the experimentwise Type I error rate should be controlled at or below the nominal level. This step-down order statistics procedure may be applied in the context of tests on correlations. It is well known that correlations and their Fisher transforms are asymptotically normally distributed. Under the null hypothesis, the standardized Fisher transforms are very nearly distributed as a standard normal and the squared standardized Fisher transforms are very nearly distributed as chi-square with one degree of freedom. If all v correlations are not zero, then the observations are not all from the same distribution p since the distribution of a transform of a correlation coefficient is a function of p. Thus, a procedure for testing correlations requires sequentially determined reference distributions for extremes. To test the significance of a set of correlations, the above order statistics procedure can be applied to the ordered squared standardized Fisher transforms of the correlations. Since the parent population of the ordered observations is approximately chi-square with one degree of freedom, the critical values for the largest of v observations may be generated by an accurate inverse chi-square distribution function. Similarly, the above order statistics procedure can be applied to test the significance of a set of correlations using the ordered f transformations of the correlations. Since the parent 2 population of the ordered observations is approximately F with one and N - 2 degrees of freedom, the critical values for the largest of v observations may be generated by an accurate inverse F distribution function. Since extensive tables of chi-square and F distributions are not widely available, the order statistics procedure using tests on the squared standardized Fisher transforms or on the 53 squared t's cannot be conducted by hand. The order statistics procedure on chi-square observations can be conducted with an accurate inverse chi-square distribution program. Similarly, the order statistics procedure on observations distributed as F's can also be conducted with an accurate inverse F distribution program. An equivalent procedure that does not require computer software and for which tables are widely available involves tests on the ordered absolute values of the standardized Fisher transforms of the sample correlations. A one-tail test of a chi-square observation at the nominal level is the same as a two-tail test of a standard normal observation. If the observations are not restricted to being nonnegative, a test of the intersection hypothesis is expressed in terms of the extremes of the observations. The test of the intersection hypothesis that all the observations are zero, is, in order statistics terms, a test of the intersection hypothesis that the extremes of the order statistics are zero. A test of the hypothesis that the extremes of the order statistics are zero rejects when the largest ordered observation is greater than Z v v and the smallest ordered observation is less than Z\.\. Thus, the experimentwise Type I error rate of this test is a EX = Pr(X <Z nX 1:v l:v = 1 -Pr(X = l-Pr(Z 1:v 1:v > Z ) v:v >Z 1:v v:v nX ^X <Z / v:v v:v < Z ) v:v Vi) . For normally distributed observations, the critical value Zj.^is non-positive and the critical value Zyyis non-negative; the critical values also have equal absolute values, i.e., Zj.y = —Z . v:v Therefore, 54 a =l-Pr<\X \<Z . EX i Vi) . v v Since the unordered observations are assumed to be independent, the experimentwise Type I error rate can be expressed as v a£X=l-Il (W^ v:v) Z P r i=l = l-[Pr(|X |<Z )] / V v:v The above yields Pr(\X \<Z ) = {\-a )l/v i v:v EX and since Pr(lX |<Z ) = l - 2 ( l - F ( Z ) ) , / v:v v:v the cumulative probability for the critical value Zy-yis l/v i - ( i - q C T ) 2 Reference to tables for the standard normal gives the critical value Z vv for each test for a specified experimentwise Type I error rate. Let Qr^ be the inverse distribution function of the standard normal. The critical values Zy-y, where Z v : v = 0>-1 1- l - ( l - a )l/v 55 can be computed for each v, for vfrom v to 1, as necessary. Each intersection hypothesis p rejects at the nominal level a when v-v X i-(l-g)" • >z v 2 Thus, a direct relationship between this sequential step-down testing of the extremes of ordered standardized Fisher transforms of a set of correlations and a sequentially rejective closed Dunn-Sidak testing procedure using z-tests is made. Indeed, the two procedures are exactly equivalent. This is a relationship that is not clarified in the popular literature. This test procedure is easily implemented with use of an accurate inverse normal distribution function. As stated above, however, a primary assumption of order statistics is that the observations are independent. Once one rejects a hypothesis ///, one can no longer state that the correlations are asymptotically independent. Thus, this procedure may not control experimentwise Type I error rate at or below the nominal level. Steiger An alternative to sequentially testing the intersection hypothesis that v, v=v ...l, p observations are all zero by use of order statistics is testing the hypotheses using the asymptotic chi-square statistic proposed by Steiger (1980a, 1980b). The improved series general approximation formula for the mean and variance of the squared Fisher transforms (Steiger & Fouladi, 1991b) may be applied to this approach yielding the general formula: QsA= SAQsT+aSA m where a =v[l-m (N-3)v ], SA SA 2 . 56 of which the test for complete multivariate independence is a special case since v is restricted to v . p Testing each hypothesis at the nominal level in a sequential step-down will yield a powerful test procedure, but since this chi-square statistic may reject a hypothesis without rejecting all of its implying hypotheses the error rate experimentwise will not be controlled at or below the nominal level. 1.3 Overview of papers on the topic of tests on correlations Few papers have been published on the topic of tests on correlations. The papers that have appeared in journals have generally suggested rather loosely that a problem of excessive family-wise Type I error rates exists with testing each correlation at the nominal level (e.g., Collis & Rosenblood, 1985). It has also been stated that conducting tests on each correlation at the reduced alpha level will constrain family-wise Type I error rate below the nominal level. Larzelere and Mulaik (1977) described the multi-stage reduced alpha procedure and on the basis of results in Larzelere's unpublished masters' thesis suggested that it would be more powerful than the reduced-alpha procedure and would constrain family-wise Type I error rates at or below the nominal level. Several published papers have been written suggesting the use of the multi-stage reduced alpha procedure, but none of these has included Monte Carlo simulation results (Collis & Rosenblood, 1985; Crosbie, 1986). Silver's unpublished dissertation (1988) included a simulation study comparing the error rates of certain tests on correlations. The comparison of a procedure using approximations to order statistics and the multi-stage Bonferroni procedure yielded results showing that the approximate order statistics approach suggested by Silver was more powerful than Larzelere's multi-stage Bonferroni procedure and maintained Type I error rates below the 57 nominal level. However, the procedures were examined under conditions that are not representative of the population parameter space. The population matrices were either identity or non-identity such that all of the off-diagonal elements were equal to a specified non-zero value. Consideration of the order statistics approach, as discussed above, suggests that the family-wise Type I error rates may be sensitive to the proportion of non-zero correlations in the population matrix and use of the procedure proposed by Silver may be inappropriate. Silver (1988) also proposed to examine the utility of conducting a preliminary test of complete multivariate independence prior to follow-up procedures testing individual correlations. He concluded that "testing the global hypothesis P = I is a moot point in examining individual correlations in a matrix" (p. 39). Since he used an incorrect omnibus test, and failed to examine a representative set of parameters, further examination of this issue is necessary. 1.4 Current state of affairs The issue of maintaining control of experimentwise Type I error is of primary importance. The necessity of making this clear is quite salient. As students of the social sciences proceed through their graduate training, many are exposed to a cookbook method of performing statistical techniques without understanding the theoretical relevance of what they are learning. The performance of an ANOVA prior to follow-up Mests on means has been ritualized, but an appreciation of the theoretical issues is missing. Without this understanding, students are unable to extend what they have learned to novel situations. Thus, many do not appreciate why it is necessary to ensure experimentwise error rate protection in the context of correlations. This is reflected in the surplus of papers that report significance at the nominal level for individual correlations. However, it is not appropriate to conduct tests at reduced alpha level in certain situations either; researchers not 58 understanding the issues will conduct experiments with a large number of correlations, thereby, running excessively conservative tests. Most of the alternative procedures for testing which correlations are significant have not appeared in the popular literature. One reason may be that most of these procedures cannot be conducted by hand because complete reference tables are not available. As more textbooks incorporate statistical software tutorial examples, these procedures may be implemented increasingly. Current statistical software packages do not include the follow-up procedures for tests on correlations that have been discussed in this paper. Some of the statistical packages include an option for printing the p-levels of each correlation. But, there has been no instruction as to how to proceed with the p-levels. At best an almost arbitrary decision is made to consider all correlations with p-levels less than a specified value to be significant. Under such circumstances, a researcher has no idea what the experimentwise Type I error rate might be. To date, few Monte Carlo simulation studies have adequately examined the relative performance of the procedures for identifying which correlations are significantly different from zero. The present study is a preliminary comprehensive comparison of the performance of the procedures currently under discussion. In what follows we will signify < the various procedures with the following notation, NT: procedure with f-tests at the nominal level, MB: sequential step-down testing of //,• (i = l...Vp) using the multi-stage Bonferroni procedure, RB: procedure with f-tests at the reduced Bonferroni level, MD: sequential step-down testing of //,• (/ = 1... v ) using the multi-stage Dunn-Sidak p procedure, 59 RD: procedure with /-tests at the reduced Dunn-Sidak level, OR: Silver order statistics approach (not step-down), OS: Silver order statistics approach (step-down), OF: sequential step-down testing of (i = l...v ) using the Fouladi order statistics approach p with critical values generated by Silver and Dunlap's routine, CF: sequential step-down testing of H (i = l...v ) using the Fouladi order statistics approach { p with critical values generated by an inverse normal distribution function, CS: sequential testing of //,• (i = 1...v ) p using Q$ , A and examine their performance as tests of individual correlations both with and without a preliminary test of the overall null hypothesis. 2. Methods In the present paper, a Monte Carlo simulation experiment was conducted in order to compare the error rate control, sensitivity, and bias of procedures (NT, MB, RB, MD, RD, OR, OS, OF, CF, and CS) for testing which elements in a correlation matrix are different from zero in the population. A FORTRAN program was written utilizing a variety of Applied Statistics algorithms and subroutinesfroma library of statistical subroutines implemented by James H. Steiger and Rachel T. Fouladi. Procedure for generation of sample correlation matrices Sample correlation matrices from multivariate normal distribution with specified population correlation matrices were generated using the Browne procedure (1968). 60 Simulation experiments. For the examination of Type I error control, sample correlation matrices were generated from a population correlation matrix P where P was equal to I; that is, the sample matrices were generated under the condition of a true null hypothesis that all of the bivariate correlations are zero in the population. For the examination of Type I and Type II error control when all of the correlations are not zero in the population, sample correlation matrices were generated from a positive definite population correlation matrix P, where P was not equal to I; that is, the sample matrices were generated under the condition of a false null hypothesis. P was varied along several dimensions: (a) order p (p = 5,10), (b) proportion of non-null correlations (pNz = .2, .4, .6, .8), and (c) magnitude of non-null correlations (mNz =.1, .3, .5). The magnitude of the nonnull correlations were selected on the basis of maintaining a positive definite P; non-null correlations in increments of .1 greater than .5 for the proportion of non-null correlations selected yield non-positive definite matrices. Procedures for testing which correlations are significant were conducted under two conditions: (a) after rejection of the hypothesis of complete multivariate independence, and (b) with no prior test of the hypothesis of complete multivariate independence. The statistic QSA s used to test the hypothesis P = I; Q$A was selected to test the omnibus hypothesis wa because of its control of experimentwise Type I error at or below the nominal level across all conditions and its power. For each sample correlation matrix, the procedures for testing which correlations are significant (NT, MB, RB, MD, RD, OR, OS, OF, CF, and CS) were conducted as necessary; the number of rejections, the number of false positives, the number of true positives, the number of false negatives, the number of true negatives, and whether a Type I error was committed were recorded for each procedure. 61 Each experiment was replicated 10,000 times at specific ratios of sample size to number of variables (N:p = 2,4, 10,20,40). Under each condition, the Type I experimentwise error rates, that is the proportion of 10,000 replications in which at least one r was falsely declared significant, were computed. 0 The mean number of rejections, false positives, true positives (and therefore false negatives, true negatives) were calculated and transformed into proportions; the standard deviations for each of these proportions were also computed. Measures of performance Under each condition, the observed difference between empirical and nominal experimentwise Type I error, S x = EX ~ -> A E A w a s compared with the critical difference, £ i , derived from a non-null consistent chi-square statistic; i.e., cr t where a is the nominal experimentwise Type I error rate and R is the number of replications. From this information, whether a procedure satisfied the primary condition of the NeymanPearson criterion of controlling experimentwise Type I error rate at or below the nominal level was assessed. Additional descriptive measures of performance based on signal detection theory were also computed on each procedure. Detection theory measures of performance were included as possible additional useful information on which to base a recommendation of a procedure for "detecting" which correlations are non-zero in the population. Detection theory is commonly used to evaluate the performance of a detector of signals in noise. The popular 62 detection theory measures of performance are based on the assumption of Gaussian signals embedded in Gaussian noise; the signals and noise are further assumed to have equal variance. If one considers non-zero population correlations to be signals and zero population correlations to be noise, then procedures to test which correlations are non-zero can be conceptualized as detectors. Since a vector of correlation coefficients is asymptotically multinomial though the assumption of homoscedasticity does not necessarily hold, detection theory measures can be considered suggestive of the relative performance of the test procedures as detectors of nonzero population correlations (signals) from zero population correlations (noise). From mean hit and mean false alarm proportions, Tp and Fp, measures of sensitivity and criterion (or bias) were computed. Let <J> be the inverse distribution function of the _ 1 standard normal. Parametric measures of sensitivity and bias are d' and p\ where , cr=Qr\\-Fp)-QT\\-Tp) and f(<tT\\-Tp);0,\) 0= -4—; Since the standard assumptions of detection theory do not necessarily hold, nonparametric measures of sensitivity A', 1 (Tp-Fp)(\ + Tp-Fp) 2 4Tp(\-Fp) | and criterion B", Tp(\-Tp)-Fp{\-Fp) Tp(\-Tp) + Fp(\-Fp) 1 63 were also computed (Coren, Porac, & Ward, 1984, p. 26). The ranges for the measures of sensitivity (from zero sensitivity to perfect performance) are (a) non-negative for d', and (b) .5 to 1 for A'. The ranges for the measures of criterion are (a) non-negative for p\ and (b) -1 to 1 for B". Lax criterion (bias toward rejecting) is indicated by values of /3 less than 1 or values of B" less than zero. Stringent criterion (bias toward not rejecting) is indicated by values of j3 greater than 1 or values of B" greater than zero. The relative performance of each procedure was considered. Keeping with conventional criteria, the procedure with the greatest experimentwise Type I error rate at or below the nominal level and the highest hit rate was declared to be optimal. 3. Results Table 9 gives the critical r's that an observed value must exceed for each of the procedures that examine the individual correlations directly; the critical chi-square values that each Q $ must exceed for CS is also included in this list. This table shows the following relationship between the rejection values for each procedure for which a critical /• can be computed directly RB<RD<MB<MD<CF,OF<OS<OR<NT. This relationship is reflected in the measures of performance for the procedures with "critical r's". The relative performance of CS, however, cannot be deduced from this table. Empirical measures of performance were obtained for procedures NT, MB, RB, MD, RD, OR, OS, OF, CF, and CS. Empirical performance was assessed with 10,000 replications under p (5, 10) x N:p (2,4,10,20,40) x pNz (.2, .4, .6, .8) x mNz (.1, .3, .5) x O (n = with A 64 no omnibus test, y = with a preliminary omnibus test at a = .05) factorial conditions. The complete set of empirical measures EX A = empirical experimentwise Type I error, i.e., the proportion of 10,000 simulation experiments on which at least one Type I error was made, Rej = the mean proportion of v correlations rejected in 10,000 replications, p Fp = the mean proportion of v r° correlations rejected in 10,000 replications, 0 Fn = the mean proportion of v, r correlations that failed to be rejected in 10,000 1 replications, Tp = the mean proportion of v, r correlations rejected in 10,000 replications, 1 Tn = the mean proportion of v r° correlations that failed to be rejected, 0 sRej - the standard deviation of Rej sFp/Tn = the standard deviation of Fp and Tn, sFn/Tp = the standard deviation of Fn and Tp, d' = a parametric detection theory measure of sensitivity, = a parametric detection theory measure of criterion, A' = a nonparametric measure of sensitivity, B" = a nonparametric measure of criterion, are included in Table 10. Some of the primary measures of performance d', p) are presented in this section. The tabled results show the equivalence of two methods to generate critical values for Fouladi's step-down order statistics approach. The performance of OF and CF is essentially identical. Thus, Silver's routine to generate critical values of order statistics distributions is valid for testing extremes of observations with a half-normal parent distribution.- The overall validity of Silver's approximation is, however, not known. In the following, reference will 65 only be made to CF, Fouladi's step-down order statistics approach using the inverse normal distribution function to generate critical values. CS was included in the simulation study to demonstrate the importance of satisfying the conditions of a closed testing procedure in order to control experimentwise Type I error at or below the nominal level. The empirical results show that, indeed, the shortcut version of a procedure using the normal theory statistic Q$ as a test of the intersection hypotheses A does not control actual experimentwise Type I error. Because CS was included to demonstrate the importance of satisfying the requisite conditions of a closed testing procedure, the results will not be discussed further. In the following, the results obtained in the examination of procedures that have been proposed to be useful in identifying which correlations are significant are reviewed. P equal to I Table 11.1 gives the difference between the empirical experimentwise Type I error performance of the test procedures and the nominal Type I error rate of the test procedures when all the correlations are zero in the population, i.e., P is the identity matrix. Using a non-null consistent chi-square statistic based on 10,000 replications, critical values <5j for cr t the difference o^, where 8 x = &EX ~ E .00562, and 5 crit A > w e r e obtained. 8 CTlt 95 = .00427, 8 CTlt 01 = .ooi = .00717. No preliminary test of the null hypothesis The results show that when no preliminary omnibus test is conducted, the experimentwise Type I error rate is controlled at or below the nominal level by MB and RB. Experimentwise Type I error is not always controlled at or below the nominal level across all conditions when P is the identity matrix by RD, MD, OS, and CF; these procedures fail 66 to control actual a when N:p is low. Experimentwise Type I error is never controlled at or below the nominal level for OR and NT. With a preliminary test of omnibus hypothesis The results show that when a preliminary omnibus test is conducted, the experimentwise Type I error rate is controlled at or below the nominal level for all procedures except NT and OR when N:p is small or moderate. P not equal to I Tables 11.2-5 give the difference between the empirical experimentwise Type I error performance and the nominal Type I error rate of the test procedures when not all the population correlation coefficients are zero, i.e., P is not an identity matrix. Using a nonnull consistent chi-square statistic based on 10,000 replications, critical values <5 i for the cr t difference 8 x, where 8 E EX = cc x - &, were obtained. 8 E CIlt 0 5 = .00427, 8 CTlt 01 = .00562, Scrit . 0 0 1 =-00717. With no preliminary test of the omnibus hypothesis The results show that across all levels of N:p, pNz, mNz, and p, when no preliminary omnibus test is conducted and P does not equal I, the experimentwise Type I error rate is controlled at or below the nominal level for MB and RB, MD and RD, and CF; in contrast, procedures OS, OR, and NT do not control experimentwise Type I error rate at or below the nominal level. However, when N:p equals 2 and pNz equals .2, CF does not control actual experimentwise Type I error rate at or below the nominal level. 67 The overall results show that of the procedures that have overall control of experimentwise Type I error, RB and RD are the most conservative, MB and MD are the second most conservative procedures, and CF is the least conservative procedure. With a preliminary test of the null hypothesis The results show that across all levels of N:p, pNz, mNz, and p when a preliminary omnibus test is conducted, the experimentwise Type I error rate is controlled at or below the nominal level for MB and RB, MD and RD, and CF; procedures OS, OR, and NT do not control experimentwise Type I error rate at or below the nominal level. The overall results indicate that of the procedures that control experimentwise Type I error at or below the nominal level (RB, RD, MB, MD, and CF), the strictly reduced alpha level procedures RB and RD are the most conservative, the sequentially determined reduced alpha level procedures MB and MD are the second most conservative procedures, and CF is the least conservative procedure. Figure 1 illustrates this consistent pattern among the procedures that control of experimentwise Type I error. The expeiimentwise Type I error of CF is shown to be consistently greater than that of MD, MB, RD, and RB; though, as N:p increases, the difference between the level of performance of MB, MD, and CF decreases. Detection theory measures of performance. Parametric measures of sensitivity and criterion are included in Tables 12 and 13. Comparison of the sensitivity of the procedures indicates that MB, RB, MD, RD, and CF are more sensitive than NT, OR, and OS. Comparison of measures of criterion indicates that RB, RD, MB, and MD have stricter criteria than CF which in turn has much more stringent criterion than NT, OR, and OS; NT, OR, and OS are, in fact, often quite lax. 68 Comparison between the measures of performance of these procedures when (a) an omnibus test is not conducted and (b) the overall test is conducted, shows that, in general, the procedures are more sensitive when the omnibus test is not conducted. The procedures have comparable bias under both conditions. A problem with using these detection theory measures to indicate which test is optimal is suggested in the tabled results. The results show that measures of sensitivity gain when experimentwise Type I error rate is too low. Examining experimentwise Type I error control and To In general, a procedure is considered optimal if the procedure has the highest hit rate (Tp) of the procedures controlling experimentwise Type I error. Of the procedures that control empirical alpha at or below the nominal level (RB, RD, MB, MD, and CF), CF consistently has the highest hit rate. According to this objective of maximizing hit rate under the constraint that experimentwise Type I error be controlled, CF is defined as the optimal procedure. 3. Conclusions The empirical results provide evidence that multiple tests (each at the nominal level) on a set of correlations does not control experimentwise Type I error. Further evidence showed that none of the procedures recommended by Silver control experimentwise alpha at or below the nominal level. Thus, none of these procedures should be used in the context of identifying which correlations are significant. 69 Reduced-level tests have generally been considered to be the only available testing procedure that controls empirical alpha at or below the nominal level. This procedure, however, tends to be conservative, particularly for moderate-to-large con-elation matrices. The alternative multi-stage reduced alpha procedures perform much better than the corresponding single level reduced alpha tests especially for moderate-to-large N:p (see Figure 1 for example). The results of the current study show, however, that the sequential step-down order statistics procedure (CF) outperforms the other multi-stage procedures. The results show that of the test procedures examined, the sequential step-down order statistics approach CF strictly satisfies the Neyman-Pearson criterion of an optimal test when conducted in conjunction with a preliminary omnibus test on a correlation matrix of order p, p > 2. When the researcher has a priori knowledge that P does not equal I, the results of the current study suggest that CF can be conducted without a preliminary omnibus test. Detection theory measures showed that the sensitivity of CF is very weak for tests of matrices with low magnitudes of non-zero correlations. Figure 3 illustrates the hit rate of the procedure for N:p x pNz when mNz = .3. This figure shows the importance of having adequate sample size to obtain acceptable power. To ensure optimal detection of non-zero correlations, a researcher should have as large an N:p as possible. 70 G E N E R A L DISCUSSION Correlational studies constitute a significant portion of behavioural and social science research. The primary role of inferential statistics in social science methodology is reflected in the fact that most papers include at least one significance test. As has been shown in the preceding sections of this paper, a wide array of correlational procedures can be used for testing the exploratory questions (a) whether any of the variables under examination are related in the population and (b) which correlations are non-zero in the population. The present study examined the relative performance of the procedures for these two questions. In this final section, the general results of this study in the context of prior research are reviewed. The importance of design issues in Monte Carlo research is considered. Possible application of the recendy derived procedures to tests other than the ones addressed in this study is also suggested. 1. A discussion of the results of this studv in the context of prior research U Experimentwise Type \ error control A primary criterion for the evaluation of a statistic is experimentwise Type I error control. Many papers in statistics and in the behavioural and social sciences are devoted to discussing the general importance of experimentwise Type I error control. The primary purpose of this study was to evaluate the performance of significance tests on a correlation matrix and its elements on this dimension. 71 Considering the tests of the omnibus null hypothesis Traditionally, the statistic used for a test of complete multivariate independence is from the class of likelihood ratio tests. This statistic is Bartlett's modified likelihood ratio test, Q Q B A B A . has been used because, of the likelihood ratio tests, it has the best control for experimentwise Type I error. However, the results of several Monte Carlo simulation studies have shown that Q B A does not control experimentwise Type I error rate at or below the nominal level when the ratio of sample size to the order of the matrix being tested, N:p, \ is small. The current study examined the experimentwise Type I error rate of several statistics from the class of likelihood ratio tests: Q R, QU^, and Q B the likelihood ratio statistics examined, Q B A B A . This study confirms that, indeed, of (1) has the best experimentwise Type I error rate control, (2) does not control alpha at the nominal level for small N:p, and (3) controls alpha tightly at the nominal level for adequate N:p. Alternative statistics are available for the test of the omnibus null hypothesis. Steiger's (1980) normal theory statistic, Q$j, has repeatedly been shown to have better control of experimentwise Type I error. In spite of these assessments, this statistic has not been widely used. Recent analytic results suggest, however, that the popular approximation of the variance of the Fisher transforms used in Q$j is not optimally accurate. The results of this study show that normal theory statistics proposed by Steiger and Fouladi (1991a, 1991b), incorporating the exact variance of the squared Fisher transforms (Q$E) of the variance (QSA)> D o m o r a more accurate approximation result in even better control of experimentwise Type I error. Steiger and Fouladi's Qp also controls the experimentwise Type I error rate at the same level as Q SE and Q . SA 72 The results of the Monte Carlo simulation studies suggest that for overall optimal control of experimentwise Type I error, Q$ , Q$ , or Q should be used. However, Q E A F BA can be used if N:p is moderate-to-large. Considering tests of the elements of a correlation matrix Several procedures have been suggested for testing the significance of the elements of a correlation matrix. However, few valid and decisive recommendations have been made on how to proceed in the context of testing several correlations. This study showed that Mests at the nominal level, the procedures proposed by Silver, OS and OR, and sequential application of Q$ (CS) do not control experimentwise alpha at the A nominal level. These procedures should not be used. The results of the present study showed that the remaining procedures including tests at stricdy reduced alpha levels (RB and RD), tests at sequentially determined alpha levels (MB and MD), and step-down testing of the extremes (CF), all control experimentwise Type I error rates at or below the nominal level. Of these procedures, CF is the optimal test. Overall, the use of CF is recommended for testing which con-elations are significant. The results show that this procedure can be used both with and without a preliminary test of the omnibus null hypothesis when N:p is moderate-to-large. When N:p is small, however, a preliminary omnibus test is recommended. If the omnibus test rejects, proceed to conduct follow-up CF testing; otherwise, declare all individual correlations not significantly different from zero. A preliminary omnibus test is required because the results obtained showed that when P equals I, experimentwise Type I error rate is conuolled for CF if a preceding omnibus test is conducted, but not if no preliminary omnibus test is conducted. 1.2 Power issues Overall, the results of the current study show that experimentwise power is low when the magnitude and proportion of non-zero correlations in the population are small. Power is also a function of sample size. In order to ensure maximal power under relatively low power conditions, it is recommended that a researcher have as large a sample size as possible when examining either exploratory question. Previous studies comparing the power of Q^j and Q B A have consistently shown Q$j to be more powerful; the current study confirms this result. However, since Q$j does not control Type I error as well as its derivative statistics Q^E and Q $ , it cannot be recommended as a A general test of complete multivariate independence. Of the omnibus tests that control experimentwise Type I error best, Q$E and Q $ are shown to be more powerful than the A others. Since the experimentwise Type I error rate for CF is always controlled at or below the nominal level for moderate-to-large N:p, both with and without a preliminary omnibus test, and since the procedure has greater power when conducted without a preliminary omnibus test, if the researcher is not primarily interested in the overall structure of the correlation matrix but is primarily interested in identifying which pairs of variables are significantly correlated, the following strategy is recommended. If N:p is small, conduct a preliminary overall test and proceed to follow-up CF testing should the omnibus hypothesis reject; if N:p is moderate or large, conduct tests of the individual correlations using CF without a preliminary omnibus test. 74 1.3 Important design issues. In this study, the procedures were examined under a variety of conditions. The conditions were selected in order to try to have as representative a sampling of the parameter space as possible. Even though Browne's procedure was used to generate sample con-elation matrices, the computational time factor of simulating many replications meant that matrices of smaller order than sometimes seen in behavioral correlational research were investigated. The major trends manifested in these data were consistent, and the results obtained probably generalize to matrices of larger order. In general, Monte Carlo simulation studies have only examined the performance of tests under conditions when (a) P is equal to I or (b) P is equal to A, where A is the correlation matrix with all off-diagonal correlations non-zero and equal in magnitude. In the present study, the performance of the test procedures was examined under additional conditions. These additional conditions were included because analysis showed (a) alternative configurations would yield different results for one omnibus test procedure, QBA-> A N D (°) some of the follow-up procedures are differentially sensitive to differing proportions of nonzero correlations. The empirical results showed differences on both (a) and (b). The results of this study confirm the importance of examining the parameter space prior to designing and running the Monte Carlo simulation experiments. Silver and Dunlap (Silver, 1988; Silver & Dunlap, 1989) failed to run their experiments under conditions to which some of their procedures were sensitive. Hence, the results of their studies do not accurately represent the general relative performance between the statistics they examined. The serious consequence of failing to conduct a study under representative conditions is exemplified by the fact that, contrary to their conclusions, the cuirent study shows that the procedures 75 recommended by them do not control experimentwise Type I error rate at or below the nominal level. The present study did not include an examination of the performance of the statistics under the condition when the non-identity population correlation matrix has a range of non-zero correlations. However, the major trends in the results of this study were very consistent across all magnitudes of non-zero correlations tested; these results probably generalize quite well. 2. Extensions and suggestions for future study In this study, the performance of several statistics was examined in the context of testing the simplest form of a null hypothesis on data collected from a multivariate normal population. Tests of significance of all of the correlations in a matrix were examined. Several of the procedures examined in this study can be extended to tests of alternative null hypotheses. 2.1 Tests of alternative null hypotheses Generalizations of the omnibus tests Steiger and Browne (1984) provide a general procedure for comparing correlation coefficients. The procedure allows a wide variety of computationally efficient significance tests. Their results indicate that asymptotic tests for any linear hypothesis can be expressed with differentiable monotonic functions of p, where f(r) is gxl vector of functions of the elements of r, differentiable at r = p, such that (A - l^CfCr) - f(p)) has an asymptotic 7 ryo, T*). For tests of linear hypotheses of the form 76 HQ\ Mz = h , r where the vector of transforms is v xl vector z , and M is a specified gxv matrix of rank g, p p r and h is a specified gxl vector, if t * is any consistent estimate of P*, then according to their development, the asymptotic distribution under HQ of the statistic, g* = ( N - 3 ) ( M ^ - h ) (Mf*M)~ (Mz.-h) , is chi-square with g degrees of freedom. The results in Steiger (1980a) and Steiger and Browne (1984) suggest that this normal theory statistic can be extended to test any linear combination hypothesis. Similarly, Q$ E and Q SA can be used to test linear hypotheses by applying their respective linear transforms toQ*. Hence, the normal theory statistics Q$ and Q$ can be extended to tests of pattern matrices E in the same way that Q ST A is extended to tests of pattern matrices in Steiger (1980a). Q$ and E QSA can be applied to tests of any set or subset of elements from an intercorrelation matrix or an intracorrelation matrix. Thus, with the recenUy derived normal theory procedures, Qg and Q$ , (a) the significance E A of any linear combination of a set of correlations can be tested, (b) pattern hypotheses for a set of correlations can be tested, and (c) tests between sets of correlations can be tested. The statistic Q can be applied to tests of any set or subset of intercorrelations or F intracorrelations. Steiger (1980a) showed that the relative performance of the general statistic is consistent across tests of several alternative pattern hypotheses. It is likely that Q$ and Q$ as E transforms of this general statistic will perform similarly. A 77 However, the current study did not examine the performance of the recently derived statistics for any of the tests described above. Any recommendation for the use of these tests under these conditions should be accompanied by Monte Carlo simulation results. Generalization of the tests on individual correlations In the current study, the performance of the sequential step-down implementation of an order statistics approach to testing extremes (CF) was examined where the observations of interest were all pairwise correlations in matrix of order p. However, it is important not to forget that this closed testing procedure applies to any set of independent identically distributed observations. Thus, in the context of tests on correlations, the significance of the elements from any set or subset of an intercorrelation or intracorrelation matrix can be tested. This sequential step-down procedure (CF) can also be extended to tests of contrasts between the elements of two or more sets or subsets of correlation matrices. It is likely that the sequential step-down procedure can be applied freely in this context as long as the observations (contrasts or transformed contrasts) are independently and identically distributed with a known parent distribution. In this case, the observed statistics for a test of the contrast are ordered and the extremes are tested. 2.2 Extension to non-normal data Steiger and Hakstian (1982) demonstrated that although a vector of correlation coefficients remains asymptotically multinomial under non-normal parents, the asymptotic variancecovariance structure of the correlation coefficients changes, with non-normality (from the Pearson-Filon values), as a function of the moments of the parent distribution. They suggested that quadratic form test statistics that rely on the Pearson-Filon formula for 78 estimation of the variance-covariance structure of the correlations may be distributed quite differently from corresponding statistics in the multinomial case. Their results showed that normal theory tests retain their correct asymptotic distributions under non-normal parent distributions when the correlations are all equal to zero. However, when sample size is small-to-moderate, Steiger and Hakstian emphasized that a normal theory test should be approached with caution. Given the results in Steiger and Hakstian, Q S E and Q$A are probably somewhat robust as tests under non-normal conditions. However, before a recommendation can be made for use of a statistic as a test when the assumption of normality is violated, a Monte Carlo simulation experiment under a variety of non-normal conditions must be conducted. 3. Final note Several statistics were examined in the current study. Deciding which procedure is most suitable for a significance test involves considering a myriad of dimensions. Convention dictates, however, that first and foremost is the issue of experimentwise Type I error control. The primary purpose of this study was to determine the relative performance of several procedures on this dimension. Of the statistics currently available for testing the hypothesis of multivariate independence with multinomial data, normal theory statistics Q$£ and Q$ , and the statistic Qp had the A best overall performance. However, the results also showed that even though these statistics perform better than Qg for small sample size, Q^ A QSA> an( * QF a r e suu " "asymptotic in N". Furthermore, the results show that for adequate N:p, Qg has tighter control over A experimentwise Type I error than Q$ , Q$ , and Qp. This is because, unlike Q$ , Q$ , and E Qp, Q BA A E is not based on the assumption of asymptotic independence (p increasing) of A 79 correlations under a true null hypothesis. Thus, Q$g, Q$ , and Qp are also "asymptotic in A p". In general, the use of Q$£ or Q$ is suggested for optimal Type I error control and power; A however, if testing requires "exact" control of experimentwise Type I error and N:p is adequate, then Q BA should be used. If testing requires precise control of experimentwise Type I error and N:p is not moderate-to-large, then an exact test of complete multivariate independence should be conducted (e.g., Mathai & Katiyar, 1979). Even though independence of correlations was assumed for the order statistics closed testing procedure CF, dependency among correlations did not appear to detract from the performance of this procedure recommended for testing which correlations are significant. The current study has shown that the order statistics closed testing procedure CF is the optimal test procedure for testing individual correlations. However, further analysis of the issue of non-independence is recommended. Unfortunately, optimality is not the only factor that affects whether a test procedure will be widely employed. Other factors also determine whether a researcher uses a procedure. (1) The researcher must be aware of a procedure before he or she can use it. (2) The experimenter must be able to perform the required computations. Indeed, knowing which procedure is best is not enough. To be useful, a procedure must be computable by hand, trivially programmable, or implemented in pre-packaged software. For most social scientists, the primary sources of such awareness are (a) the general statistical training received in graduate school, (b) procedures given in common "advanced" texts, and (c) procedures implemented in best-selling general purpose computer packages. However, many useful procedures of value are not contained in or mentioned by any of these sources; in addition, many classic books and computer programs implement procedures which are either erroneous or suboptimal. 80 At present, the recently derived and here recommended statistics for a test of overall independence and for the significance test of the individual correlations are not available in popular statistical packages, nor are they in any statistical reference book. But perhaps, in spite of this, the results of this comprehensive examination of procedures for testing the significance of correlation matrices and their elements will become known and the recommended statistics will be applied widely and effectively. 81 BIBLIOGRAPHY Bartlett, M.S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society, Ser. A, 30, 327. Bartlett, M.S. (1938). Further aspects of the theory of multiple regression. 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Educational and Psychological Measurement, 43,11-14. 88 APPENDICES 89 Appendix A Extensions of Hotelling's approximations obtained bv Steiger and Fouladi (1991a) la. V!. p 5p + p> llp + 2p +3p~ N-l 1^2 ' 8(JV-1) ' 16(N-1) j Vi = | | 2 83p + 13p -27p + 75p 128(A/-1) 3 5 7 3 143p+20p + 138p -780p +735p 256(A/-1) 3 + 5 7 9 4 625p +113p - 990p + 14250p - 33075p +19845p* ^ 1024(/V-1) 3 | 5 7 9 1 5 lb. v . 2 v = 2 8-p" 88-9p^-9p 4(/V-l) 24(iV-l) 4 [ N-l 2 384-19p +2p -75p 64(/V-l) 2 4 6 3 16256 -225p - 375p + 8025p -11025p 1920(N-1) 2 | 4 6 8 4 5120 + 113p + 20p - 7194p + 26460p -19845p ^ 512(/V-1) 2 [ 4 6 8 5 10 90 lc. V3. V3 3p N-l 39p + 6 p 2(N-1) S(N-lf + 362p + ( 45p +63p 3 32(N-1) 2809p + ( 3 5 3 296p -327p -H170p 3 5 12S(N-1) 4 47461p + 5 7 2 0 p + 1 1 1 9 0 p 3 ( 7 - 7 3 3 5 0 p + 7 7 1 7 5 p9^ 5 7 1280(7V-1) 5 Id. v . 4 v = 4 3 N-l 28-3p^ ( IJA'-I) 2(tf-l) 2 736-84p -51p 2 16(N-1) 4 3 31744-3016p 2 -864p 256(/V-l) + 4 -3480p 4 543616 - 4 1 3 1 0 p - 1 7 5 9 5 p + 8 7 3 0 0 p 2 [ 6 4 1920(A -1) / 5 6 -165375p 8 91 p 2 p - 9 p p + 42p -75p 8(W-1) 16(/V-1) 3 3 5 ( 2 -5p-657p + 3825p -3675p 3 | 5 ^(iV-l) 7 3 -23p + 2580p + 3621 Op + 91140p - 59535p 3 5 7 9 256(A -1) + / 4 53p - 20817p + 623250p - 3079650p + 4862025p - 2401245P 3 t 5 7 9 1024(W-1) 5 23p 4(N-l) -97p + 309p S(N-l) 2 + 2 + 2 1541p -15210p +20901p 64(W-1) 2 ( 4 4 6 3 -6121p +142851p -509067p + 431985p 2 4 6 8 11 92 2c. % 15p 123p^ 798p 2 8(/V-l) ( 1 3 + (N-iy L -978p + 19449p -40311p 32(7V-1) 3 [ 5 2 7809p - 364308p +1953909p - 221661 Op 128(N-1) 3 | 5 7 3 -31305p + 3125790p - 32698Q80p + 89257770p - 67864095p 3 5 7 256(N-1) + 4 2d. 1 XA — -12 + 237p 2(JV-1) z [ ^ nT (N-l) 2 + 192-11484p +39789p 16(N-1) 2 | 4 2 -768 + 107391p - 961884p + 1529325p 32(N-1) 2 | 4 6 3 6144 - 1842030p + 32147595p - 122970780p + 120297375p ^ 128(A/-1) 2 [ 4 6 8 4 2e. X5. 1 l35p X =(N-l)' V 2 f 5 3255p-22605p 8(/V-l) 3 ( -60810p-H090470p -2603445p 32(/V-l) 3 5 2 1042725p-36421425p + 209106600p -272870100p ^ 128(N-1) • 3 | 5 4 7 93 A*=- 15 + (N-l)' -360 + 7515p^ 4(N-l) 6720 - 3 6 1 8 9 0 p + 1440945p 2 16(N-1) 4 2 1800 + 1 2 0 7 8 2 2 5 p - 1 1 5 6 2 9 7 5 0 p + 2 1 1 3 1 4 0 6 0 p 2 | 4 64(/V-l) 3 2g. A ^ A 7 1 = (N-l) 'l365p I 4 2 8(N-1) -2188830p + 41926185p -114961140p ^ 3 | 32(/V-l) 5 2 2h. Ag. 1 105+• -1260 + 3 0 1 3 5 p J (N-l) 84000 - 4 8 2 8 7 4 0 p + 2 2 0 7 2 3 6 5 p ^ 2 4 8(7V - 1 ) 2 2 i . A9. 16065p Ao — • 2 (N-lf | 1286145p-11760525p S(N-l) 2 j . A 10- A 1 0 =• (N-l) 5 - 7 5 6 0 0 + 2 0 7 4 2 7 5 p2 945 + 4(/V-l) A 3A 6 ^ 2(iV-l) 6 95 Appendix B The configuration of population matrices A. When p = 5 A/ = U pNz 1.0 1.0 .2 .4 .6 .x 1.0 .x .0 1.0 .0 .0 .0 1.0 .0 .0 .0 .0 1.0 .0 .0 .x .0 1.0 .0 1.0 .0 .0 1.0 .x .0 .0 1.0 1.0 .x .x .x .0 1.0 .x 1.0 .0 .0 1.0 .0 .0 .0 1.0 1.0 .x .0 .x .0 1.0 .0 1.0 .0 .0 1.0 .x .x .0 1.0 1.0 .x .x .x .0 1.0 .x 1.0 .x .x 1.0 .0 .0 .0 1.0 1.0 .x .x .0 .0 1.0 .x 1.0 .x .x 1.0 .0 .x .0 1.0 1.0 .x 1.0 .x .x 1.0 .0 .0 .0 1.0 1.0 .x .x .x .0 1.0 .x 1.0 .x .x 1.0 .x .0 .x 1.0 1.0 .8 M =R .x .x .x .0 96 2. When p = 10 M=R M=U pNz 1.0 .X .2 1.0 .X .X .X .X .X .X .X .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .X .4 1.0 .X 1.0 1.0 1.0 .0 1.0 .0 .0 1.0 .0 .0 .0 1.0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 .0 1.0 .X .X .X .X .X .X .X .X .X .X .X .X .X 1.0 1.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .X • 1.0 .6 .X .X .X 1.0 .X .X 1.0 .X .X .X .X .X .X .X .X .X .X .X 1.0 .X .0 .0 .0 .0 1.0 .X 1.0 .0 1.0 .0 .0 1.0 .0 .0 .0 1.0 .0 .0 .0 .0 1.0 1.0 1.0 .0 1.0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .X .X .X .X .X .X .X .X .X .X .X •X 1.0 .X .8 • 1.0 .0 1.0 .X .0 1.0 .0 . X .0 1.0 .X .0 .0 .0 1.0 .X .0 . X .0 . X 1.0 .0 .X . X . X .0 . X 1.0 .X .0 .0 .0 .0 . X . X 1.0 .X .0 .0 . X .0 .0 .0 .0 1.0 .0 .0 .X . X .0 . X .0 .0 .0 1.0 1.0 .0 1.0 .0 . X 1.0 .0 . X .0 1.0 .X .X .0 .0 1.0 .X .0 . X .0 . X 1.0 .0 .0 . X .0 . X . X 1.0 .X .X .X .X .X .X .X .X .0 .0 .0 .0 .X .X .X .X 1.0 .0 .0 .0 1.0 .X .X .X .X .X 1.0 1.0 1.0 .X 1.0 .X .X .X .X' .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X .X 1.0 1.0 1.0 1.0 .X 1.0 1.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .X 1.0 .0 1.0 .0 .0 1.0 .X .0 .0 1.0 .X .0 .0 . X 1.0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 . X . X 1.0 .0 .X .0 .0 .0 .0 .0 1.0 .0 .0 .0 . X .0 .0 . X .0 1.0 .X .0 .0 .0 .0 .0 .0 .0 .0 1.0 .X .X .X .X .X .X .X 1.0 .X .X 1.0 .X .X .X .0 1.0 .X .X .X .X .X .X .X .X .0 .X .X .X .X .X .X .0 .X .0 .X .X .X .X .X .X .X .X .0 .X .X .0 .0 .X .X .X 1.0 1.0 .0 1.0 1.0 .0 1.0 .X .X 1.0 Appendix C Setting the storv straight Several statistics were examined in this study. In this paper, I have tried to delineate the relationship between these statistics as accurately as possible. The results that I have referenced here have for the most part come directly from my reading of the original sources. I have avoided the primary referencing of secondary sources as much as possible. A comparison of the results in primary sources with the results in secondary sources produced some esoteric findings. I would like to set the story straight in this section. 1) Lawley's statistic is commonly described as equalling Qy^. However, the original Lawley test is Q . QIA is the statistic resulting from substitution of Bartlett's multiplier for Q E BA to Ql suggested by Bartlett (1950). The results of the current study show that (2LA performs worse than Q . Thus, a disservice is being rendered to Lawley by crediting him with a E statistic that performs worse than his own statistic. 2) Wilks's result in the original paper has the multiplier N - 1, some sources have the result described as having the multiplier N. This error is likely attributable to the inconsistent notation for sample size and sample size minus 1. In the first part of Wilks's paper, sample size is defined as n; half way through the paper, Wilks's changes notation and lets n equal sample size minus 1. Wilks's result appears after he changes notation. The confusion is further enhanced by the subsequent example where for computational considerations, n equals sample size. 98 TABLES 99 TABLES Table 1 The relationship between the number of variables,/?, and the number of pairwise correlations, V p , in a matrix 2 1 p 5 10 45 10 20 190 15 105 100 4950 Table 2 Asymptotic chi-square tests of complete independence with v degrees of freedom p N -lnlRI Root ZSrii Q 2 N- 1 Multiplier N- 1 N-3 (2p+5)/6 Q QBA QL=QBR QLA SA SE QST=QBU M M QSE QSA Table 3 Summary table of reviews of tests of LR and NT tests of multivariate independence # replications in a typical condition Steiger (1980a) conditions representative of parameter space 10000 theoretical inaccuracies scope Q vs. Q B A vs. QSTVS- QBR- Wilson & Martin (1983) Reddon (1987) Silver & Dunlap (1990) 20 2500 10000 NO QBA v s QBA v s QBA™. QBR v s - QST - QST (2sr - QAN vs YES 100 Table of 4 sample sizes, N, for sample correlation matrices generated under P of order p and specified N:p ratios P 5 io 15 2 4 N:p 10 20 40 10 20 30 20 40 60 50 100 150 100 200 300 200 400 600 101 Table 5.1 Table of empirical Type I error rates for tests of complete multivariate independence at a = .05 p N:p a 5 2 4 10 20 40 2 4 10 20 40 2 4 10 20 40 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 5 5 5 5 10 10 10 10 10 15 15 15 15 15 QBA QST QSE QLA QBR QF QSA .05847 .05084 .04983 .04897 .05152 .06668 .05233 .05000 .04999 .05041 .07403 .05312 .04991 .05022 .05087 .05718 .05369 .05021 .05061 .05195 .05574 .05315 .05171 .05059 .05059 .05537 .05189 .05046 .05138 .05145 .05109 .05065 .04862 .04992 .05168 .05232 .05142 .05085 .05020 .05041 .05290 .05054 .05001 .05114 .05126 .00210 .01491 .03179 .04052 .04726 .00102 .01067 .02933 .03833 .04443 .00024 .00590 .02452 .03600 .04360 .02998 .03924 .04393 .04756 .05053 .04103 .04559 .04803 .04903 .04976 .04455 .04629 .04820 .05026 .05093 .05206 .05037 .04864 .04950 .05190 .05255 .05160 .05062 .05041 .05084 .05266 .05090 .04989 .05098 .05131 .05128 .05065 .04862 .04992 .05168 .05232 .05142 .05085 .05020 .05041 .05290 .05054 .05001 .05114 .05126 Table 5.2 Table of empirical Type I error rates for tests of complete multivariate independence at a = .01. P 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 N:p a 2 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 4 10 20 40 2 4 10 20 40 2 4 10 20 40 QBA QST QSE QLA QBR QF QSA .01304 .01021 .00954 .01010 .01036 .01569 .01055 .01031 .01000 .00986 .01712 .01018 .00990 .01072 .01036 .01712 .01375 .01067 .01068 .01069 .01504 .01218 .01127 .01027 .01017 .01327 .01077 .01070 .01090 .01068 .01450 .01230 .01017 .01054 .01058 .01345 .01152 .01106 .01016 .01012 .01214 .01028 .01047 .01083 .01063 .00010 .00178 .00506 .00745 .00889 .00007 .00142 .00529 .00714 .00814 .00001 .00058 .00382 .00714 .00839 .00501 .00666 .00782 .00929 .00978 .00777 .00865 .00996 .00962 .00988 .00847 .00855 .00978 .01040 .01045 .01356 .01184 .01010 .01037 .01079 .01324 .01154 .01078 .01015 .01007 .01213 .01028 .01037 .01093 .01037 .01459 .01230 .01017 .01054 .01058 .01345 .01152 .01106 .01016 .01012 .01214 .01028 .01047 .01083 .01063 Table 6.1 Chi-square goodness offitvalues for tests of complete multivariate independence at a .05 under a true null hypothesis, df = 1, % i 05 = 3.84, X*1,M - 6-64, X 1,.001 = 10.83 2 2 v X 15,.001 = 37.70 2 p N:p a QBA QST QSE 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 2 4 10 20 40 2 4 10 20 40 2 4 10 20 40 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 130.32 1.46 0.06 2.28 4.73 447.06 10.95 0.00 0.00 0.35 842.37 19.35 0.02 0.10 1.57 1460.61 95.63 26.80 0.09 0.77 7.72 62.60 19.72 5.96 0.72 0.72 55.13 7.26 0.44 3.91 4.31 291.79 2.45 0.88 4.12 0.01 5.76 10.86 4.13 1.50 0.08 0.35 16.79 0.61 0.00 2.68 3.26 53.48 QLA 109487.54 8383.26 1077.36 231.16 16.67 235440.19 14653.53 1500.71 369.47 73.08 1031938.33 33158.52 2714.31 564.78 98.23 1439797.14 QBR QF QSA 1378.21 307.10 87.73 13.14 0.59 204.49 44.70 8.49 2.02 0.12 69.78 31.18 7.06 0.14 1.79 2156.54 8.60 0.29 4.00 0.53 7.34 13.06 5.23 0.80 0.35 1.46 14.18 1.68 0.03 1.99 3.53 63.05 3.37 0.88 4.12 0.01 5.76 10.86 4.13 1.50 0.08 0.35 16.79 0.61 0.00 2.68 3.26 54.39 Table 6.2 Chi-square goodness of fit values for tests of complete multivariate independence at a .01 under a true null hypothesis p N:p a QBA QST QSE 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 2 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 71.81 0.44 2.24 0.10 1.26 209.64 2.90 0.94 0.00 0.20 301.27 0.32 0.10 4.89 1.26 597.37 301.27 103.70 4.25 4.38 4.50 171.47 39.50 14.47 0.72 0.29 81.66 5.57 4.63 7.51 4.38 748.30 141.71 43.54 0.29 2.80 3.21 89.70 20.29 10.27 0.25 0.14 38.19 0.77 2.13 6.43 3.77 363.51 4 10 20 40 2 4 10 20 40 2 4 10 20 40 QLA 98019.80 3802.75 484.74 87.94 13.98 140874.00 5191.63 421.59 115.38 42.85 998010.98 15308.26 1003.63 115.38 31.16 1263524.07 QBR 499.51 168.62 61.25 5.48 0.50 64.50 21.25 0.02 1.52 0.15 27.87 24.80 0.50 1.55 1.96 879.49 QF 94.75 28.94 0.10 1.33 5.85 80.35 20.79 5.71 0.22 0.05 37.86 0.77 1.33 8.00 1.33 287.39 QSA 146.54 43.54 0.29 2.80 3.21 89.70 20.29 10.27 0.25 0.14 38.19 0.77 2.13 6.43 3.77 368.33 103 Table 7.1.1.1 Empirical power of tests of complete multivariate independence at a = .05, p = 5, mNz = .1 P N.p M pNz QSE QLA QBR QF QSA .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .0616 .0638 .0609 .0622 .0760 .0757 .1201 .1160 .2181 .2116 .0629 .0627 .0658 .0648 .0793 .0799 .1222 .1159 .2200 .2144 .0578 .0565 .0609 .0617 .0778 .0788 .1211 ' .1149 .2189 .2129 .0024 .0026 .0172 .0182 .0542 .0541 .1026 .0986 .2052 .2003 .0345 .0354 .0467 .0514 .0709 .0726 .1172 .1110 .2143 .2096 .0558 .0585 .0614 .0627 .0782 .0776 .1180 .1140 .2059 .2056 .0580 .0567 .0609 .0617 .0778 .0788 .1211 .1149 .2189 .2129 R .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .0664 .0665 .0710 .0708 .1090 .1144 .2033 .2009 .4155 .4429 .0680 .0679 .0793 .0756 .1174 .1155 .2151 .2022 .4342 .4404 .0623 .0607 .0740 .0719 .1159 .1124 .2134 .2006 .4327 .4395 .0028 .0023 .0234 .0241 .0853 .0818 .1892 .1758 .4161 .4241 .0376 .0387 .0583 .0585 .1081 .1051 .2077 .1938 .4282 .4356 .0612 .0609 .0772 .0716 .1149 .1124 .2122 .1968 .4259 .4356 .0624 .0610 .0740 .0719 .1159 .1124 .2134 .2006 .4327 .4395 U .6 .6 .6 .6 .6 .6 .6 .6 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .0680 .0683 .0810 .0801 .1359 .1469 .2788 .2942 .5718 .6149 .0707 .0714 .0949 .0897 .1603 .1569 .3170 .3119 .6154 .6308 .0661 .0659 .0903 .0848 .1575 .1544 .3144 .3098 .6145 .6299 .0403 .0409 .0730 .0666 .1481 .1434 .3086 .3038 .6108 .6262 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .0743 .0722 .0935 .0947 .1768 .1758 .3730 .3762 .7317 .7323 .0760 .0754 .1052 .1083 .2033 .2034 .4122 .4150 .7617 .7631 .0679 .0686 .1011 .1039 .1998 .2006 .4101 .4125 .7612 .7619 .0030 .0030 .0311 .0295 .1190 .1158 .2894 .2838 .6005 .6140 .0039 .0035 .0394 .0405 .1596 .1586 .3806 .3842 .7494 .7506 .0655 .0653 .0919 .0857 .1574 .1534 .3171 .3126 .6178 .6286 .0696 .0666 .1030 .1045 .2010 .2027 .4180 .4174 .7688 .7722 .0661 .0659 .0903 .0848 .1575 .1544 .3144 .3098 .6145 .6299 .0681 .0688 .1011 .1039 .1998 .2006 .4101 .4125 .7612 .7619 U 5 5 5 5 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 QST .2 U 5 5 QBA .2 .2 .2 .2 .2 .2 .2 .2 .2 2 2 5 5 5 5 a R R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R .6 .6 .0433 .0443 .0835 .0884 .1904 .1912 .4043 .4077 .7588 .7606 104 Table 7.1.1.2 Empirical power of tests of complete multivariate independence at a = .05, p = 5, mNz - .3 p N:p 5 2 5 5 5 5 5 5 5 5 5 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 M pNz a .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 QBA QST QSE QLA QBR QF QSA .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .0929 .0962 .1788 .1676 .4986 .4787 .8868 .8619 .9985 .9969 .1012 .1040 .1875 .1830 .4830 .4931 .8723 .8644 .9979 .9975 .0929 .0951 .1804 .1760 .4791 .4889 .8705 .8633 .9979 .9975 .0057 .0059 .0727 .0733 .3857 .4020 .8475 .8352 .9973 .9965 .0583 .0622 .1475 .1421 .4463 .4572 .8618 .8536 .9976 .9971 .0933 .0957 .1761 .1707 .4395 .4526 .8398 .8305 .9963 .9947 .0930 .0954 .1804 .1760 .4791 .4889 .8705 .8633 .9979 .9975 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .1337 .1479 .3215 .3714 .8102 .8890 .9924 .9990 1.0000 1.0000 .1638 .1592 .3609 .3599 .8241 .8635 .9936 .9983 1.0000 1.0000 .1500 .1480 .3505 .3472 .8213 .8610 .9934 .9983 1.0000 1.0000 .0136 .0112 .1942 .1754 .7716 .8126 .9911 .9978 1.0000 1.0000 .1056' .1524 .1004 .1508 .3119 .3483 .2989 .3386 .8119 .8083 .8510 .8529 .9930 .9927 .9983 .9981 1.0000 1.0000 1.0000 1.0000 .1506 .1482 .3505 .3472 .8213 .8610 .9934 .9983 1.0000 1.0000 U .6 .6 .6 .6 .6 R .6 U .6 .6 .1588 .1907 .3903 .4899 .8663 .9576 .9968 .9999 1.0000 1.0000 .2437 .2311 .4932 .5140 .9098 .9563 .9984 .9998 1.0000 1.0000 .2276 .2147 .4866 .5027 .9088 .9557 .9984 .9998 1.0000 1.0000 .0381 .0280 .3433 .3312 .8872 .9370 .9981 .9998 1.0000 1.0000 .1802 .1605 .4634 .4701 .9046 .9517 .9983 .9998 1.0000 1.0000 .2407 .2244 .4958 .5092 .9112 .9566 .9984 1.0000 1.0000 1.0000 .2276 .2151 .4866 .5027 .9088 .9557 .9984 .9998 1.0000 1.0000 .2198 .2192 .5549 .5550 .9721 .9758 1.0000 1.0000 1.0000 1.0000 .3020 .3063 .6203 .6214 .9771 .9804 1.0000 1.0000 1.0000 1.0000 .2861 .2889 .6115 .6113 .9764 .9802 1.0000 1.0000 1.0000 1.0000 .0576 .0595 .4653 .4635 .9686 .9743 1.0000 1.0000 1.0000 1.0000 .2357 .2369 .5892 .5917 .9757 .9792 1.0000 1.0000 1.0000 1.0000 .3022 .3057 .6248 .6275 .9791 .9824 1.0000 1.0000 1.0000 1.0000 .2870 .2894 .6115 .6113 .9764 .9802 1.0000 1.0000 1.0000 1.0000 U R U R U R U R U R U R U R U R U R U R U R .6 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 U .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 R U R R U R U R U R U R U R .6 105 Table 7.1.1.3 Empirical power of tests of complete multivariate independence at a = .05, p = 5, mNz = .5 p N:p 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 M pNz a QBA QST QSE QLA QBR .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .2330 .2011 .6522 .5305 .9957 .9743 1.0000 1.0000 1.0000 1.0000 .2348 .2393 .5683 .5679 .9896 .9776 1.0000 1.0000 1.0000 1.0000 .2169 .2239 .5563 .5578 .9893 .9767 1.0000 1.0000 1.0000 1.0000 .0179 .0189 .2862 .2900 .9768 .9610 1.0000 1.0000 1.0000 1.0000 .1374 .1437 .4562 .4604 .9852 .9711 1.0000 1.0000 1.0000 1.0000 .2092 .2132 .5079 .5075 .9804 .9655 1.0000 1.0000 1.0000 1.0000 .2173 .2247 .5563 .5578 .9893 .9767 1.0000 1.0000 1.0000 1.0000 R .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .4249 .6407 .9007 .9930 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4602 .4741 .8885 .9329 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4414 .4519 .8834 .9276 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .0751 .0680 .7384 .7765 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .3484 .3302 .8551 .9019 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4427 .4433 .8761 .9179 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4416 .4528 .8834 .9276 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 U .6 R .6 .6 .6 .6 .6 .6 .6 .6 .6 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .4337 .7678 .8533 .9988 .9993 1.0000 1.0000 1.0000 1.0000 1.0000 .5866 .6260 .9176 .9861 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 .2539 .2016 .8566 .9431 .9997 1.0000 1.0000 1.0000 1.0000 1.0000 .5878 .6264 .9176 .9861 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .7445 .7431 .9942 .9961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7278 .7275 .9858 .9863 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .3697 .3749 .9645 .9666 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5460 .5530 .9106 .9838 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 .6934 .6905 .9849 .9860 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .6069 .6384 .9210 .9883 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .6045 .6454 .9202 .9873 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 .7419 .7427 .9863 .9873 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7518 .7502 .9881 .9887 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7280 .7282 .9858 .9863 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 U R U R U R U R U R U R U R U R U R U U R U R U R U R U R U R U R U R U R QF QSA 106 Table 7.1.2.1 Empirical power of tests of complete multivariate independence at a = .05, p - 10, mNz =. 1 p N:p 10 10 10 10 10 10 10 10 10 10 2 4 4 10 10 20 20 40 40 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 2 M U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R pNz a .2 .2 QF QSA .0633 .0632 .0967 .0907 .2262 .2242 .4912 .5010 .8757 .9022 .0814 .0795 .1070 .1027 .2298 .2250 .4789 .4925 .8604 .8878 .0794 .0799 .1066 .1014 .2332 .2323 .4971 .5067 .8768 .9035 .0057 .0047 .0722 .0639 .4049 .4009 .8261 .8671 .9967 .9991 .0911 .0879 .1770 .1669 .4741 .4853 .8482 .8843 .9970 .9992 .1108 .1057 .1913 .1802 .4818 .4895 .8467 .8841 .9973 .9993 .1094 .1059 .1900 .1829 .4810 .4943 .8502 .8867 .9971 .9992 .1472 .1413 .2780 .2759 .6782 .6984 .9602 .9800 .9999 1.0000 .0103 .0099 .1319 .1229 .6095 .6279 .9508 .9735 .9999 1.0000 .1257 .1194 .2646 .2600 .6739 .6925 .9594 .9796 .9999 1.0000 .1499 .1435 .2808 .2756 .6830 .7004 .9614 .9806 .9999 1.0000 .1473 .1413 .2780 .2759 .6782 .6984 .9602 .9800 .9999 1.0000 .1872 .1793 .3645 .3630 .8050 .8308 .9892 .9960 1.0000 1.0000 .0212 .0175 .2005 .1952 .7560 .7794 .9867 .9943 1.0000 1.0000 .1646 .1553 .3518 .3479 .8018 .8284 .9891 .9958 1.0000 1.0000 .1903 .1808 .3709 .3694 .8112 .8378 .9904 .9962 1.0000 1.0000 .1872 .1793 .3645 .3630 .8050 .8308 .9892 .9960 1.0000 1.0000 QBA QST QSE QLA QBR .2 .2 .2 .2 .2 .2 .2 .2 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .0866 .0860 .0930 .0954 .1981 .2222 .4349 .4965 .8368 .9034 .0841 .0848 .1097 .1036 .2345 .2340 .4991 .5081 .8769 .9039 .0794 .0799 .1066 .1014 .2332 .2323 .4971 .5067 .8768 .9035 .0031 .0026 .0295 .0272 .1686 .1654 .4527 .4623 .8672 .8930 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .1052 .1084 .1459 .1630 .3945 .4680 .7822 .8761 .9935 .9992 .1162 .1120 .1938 .1860 .4834 .4966 .8506 .8872 .9971 .9992 .1094 .1059 .1900 .1829 .4810 .4943 .8502 .8867 .9971 .9992 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .05 .05 .05 .05 .05 .05 05 .05 .05 .05 .1217 .1278 .1933 .2182 .5529 .6383 .9199 .9697 .9998 1.0000 .1530 .1470 .2830 .2811 .6800 .7002 .9604 .9802 .9999 1.0000 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .1366 .1476 .2414 .2815 .6774 .7630 .9718 .9915 1.0000 1.0000 .1939 .1852 .3703 .3695 .8062 .8320 .9892 .9960 1.0000 1.0000 107 Table 7.1.2.2 Empirical power of tests of complete multivariate independence at a= ,05 p = 10, y mNz - .3 p N:p 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 M pNz a QUA QST QSE QlA QBR .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .2941 .4175 .6595 .8808 .9953 1.0000 1.0000 1.0000 1.0000 1.0000 .5205 .8161 .9266 .9994 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .6307 .8975 .9661 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .6898 .9602 .9795 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4108 .4174 .7690 .8456 .9978 .9999 1.0000 1.0000 1.0000 1.0000 .7003 .7712 .9674 .9980 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8252 .8985 .9910 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8861 .9475 .9961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4013 .4066 .7656 .8422 .9977 .9999 1.0000 1.0000 1.0000 1.0000 .6921 .7612 .9661 .9979 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8210 .8931 .9908 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8819 .9450 .9959 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .0764 .0529 .5901 .6229 .9956 .9999 1.0000 1.0000 1.0000 1.0000 .3069 .2738 .9122 .9840 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5309 .5626 .9754 .9967 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .6789 .7233 .9915 .9995 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .3616 .3545 .7484 .8184 .9976 .9999 1.0000 1.0000 1.0000 1.0000 .6655 .7272 .9639 .9976 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8059 .8812 .9899 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8758 .9421 .9960 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R QF .3963 .3943 .7523 .8179 .9962 .9999 1.0000 1.0000 1.0000 1.0000 .6944 .7584 .9662 .9980 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8261 .8987 .9907 .9995 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8875 .9520 .9965 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QSA\ .40131 .4066 .7656 .8422 .9977 .9999 1.0000 1.0000 1.0000 1.0000 .6921 .7612 .9661 .9979 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8210 .8931 .9908 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8819 .9450 .9959 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 108 Table 7.1.2.3 Empirical power of tests of complete multivariate independence at a = .05, p = 10, mNz = .5 p N:p M pNz a 10 10 10 10 10 10 10 10 10 10 2 2 U .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 4 4 10 10 20 20 40 40 U 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 10 10 10 10 10 10 10 10 10 10 2 2 4 4 10 10 20 20 40 40 R R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R QF QSA QBA QST QSE QLA QBR .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .8536 .9961 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9014 .8526 .9993 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8970 .8446 .9993 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5438 .2862 .9958 .9931 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8684 .7583 .9992 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8835 .7974 .9989 .9993 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8970 .8446 .9993 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .9989 .9599 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9906 .9156 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9896 .9108 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9227 .4757 1.0000 .9993 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9885 .8844 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9910 .9034 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9896 .9108 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .05 .05 .05 .05 .05 .05 05 .05 .05 .05 .9974 .9983 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9974 .9981 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9731 .9472 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9969 .9977 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9974 .9974 .9983 .9981 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.00000 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .9939 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9804 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9981 .9985 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9981 .9984 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9871 .9657 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9976 .9981 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9981 .9983 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9981 .9984 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 109 Table 7.2.1.1 Empirical power of tests of complete multivariate independence at a = .01, p = 5, mNz = .1 p N:p 5 5 5 5 5 5 5 5 5 5 2 2 U 4 4 10 10 20 20 40 40 U 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 U 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 U 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 U M R R U R U R U R R U R U R U R U R R pNz a .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 QBA QST QSE QLA QBR .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0137 .0131 .0120 .0136 .0175 .0171 .0336 .0325 .0735 .0761 .0173 .0192 .0156 .0178 .0210 .0207 .0382 .0354 .0753 .0785 .0148 .0161 .0136 .0152 .0202 .0201 .0379 .0347 .0747 .0776 .0001 .0000 .0025 .0032 .0113 .0113 .0290 .0264 .0668 .0689 .0062 .0055 .0080 .0088 .0159 .0163 .0345 .0313 .0712 .0736 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0143 .0144 .0159 .0162 .0308 .0293 .0728 .0704 .2011 .2159 .0202 .0205 .0210 .0215 .0373 .0351 .0831 .0703 .2201 .2154 .6 .6 .6 .6 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0159 .0158 .0199 .0199 .0440 .0440 .1162 .1203 .3393 .3747 .0242 .0222 .0290 .0267 .0606 .0558 .1502 .1405 .3978 .4011 .0174 .0179 .0189 .0192 .0355 .0343 .0821 .0690 .2188 .2144 .0204 .0187 .0266 .0242 .0592 .0543 .1489 .1388 .3962 .3996 .0002 .0002 .0048 .0043 .0220 .0199 .0668 .0568 .2034 .1986 .0002 .0003 .0063 .0054 .0399 .0337 .1283 .1186 .3781 .3826 .0067 .0061 .0112 .0112 .0303 .0286 .0770 .0640 .2124 .2095 .0074 .0074 .0174 .0160 .0520 .0462 .1420 .1327 .3892 .3944 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0165 .0170 .0240 .0242 .0613 .0634 .1813 .1825 .5071 .5132 .0268 .0272 .0363 .0356 .0839 .0850 .2234 .2199 .5580 .5623 .0219 .0224 .0342 .0331 .0819 .0829 .2211 .2177 .5567 .5609 .0004 .0005 .0092 .0090 .0533 .0552 .1934 .1933 .5394 .5444 .0089 .0077 .0226 .0223 .0716 .0732 .2133 .2101 .5519 .5572 U .6 .6 U .6 .6 U .6 .6 R R R U R R U R U R U R U R QF QSA .0149 .0163 .0140 .0149 .0184 .0203 .0366 .0325 .0706 .0727 .0182 .0168 .0186 .0191 .0360 .0337 .0816 .0686 .2118 .2094 .0205 .0188 .0270 .0242 .0614 .0547 .1515 .1422 .4025 .4073 .0223 .0230 .0334 .0334 .0835 .0846 .2272 .2251 .5692 .5748 .0149 .0163 .0136 .0152 .0202 .0201 .0379 .0347 .0747 .0776 .0175 .0180 .0189 .0192 .0355 .0343 .0821 .0690 .2188 .2144 .0206 .0187 .0266 .0242 .0592 .0543 .1489 .1388 .3962 .3996 .0222 .0226 .0342 .0331 .0819 .0829 .2211 .2177 .5567 .5609 110 Table 72.12 Empirical power of tests of complete multivariate independence at a = .01, p - 5, mNz = .3 p N:p M pNz a 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 u .2 R .2 U 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 U 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 QF QSA .0115 .0122 .0394 .0425 .2097 .2113 .6644 .6528 .9855 .9832 .0305 .0307 .0613 .0637 .2195 .2208 .6341 .6219 .9760 .9749 .0297 .0308 .0654 .0666 .2484 .2561 .6968 .6795 .9865 .9848 .0018 .0015 .0631 .0494 .5369 .5658 .9637 .9847 1.0000 1.0000 .0263 .0230 .1300 .1097 .5965 .6323 .9690 .9873 1.0000 1.0000 .0591 .0522 .1725 .1520 .6159 .6490 .9672 .9866 . .9999 1.0000 .0604 .0527 .1764 .1580 .6303 .6684 .9717 .9892 1.0000 1.0000 .1051 .0960 .3126 .2991 .8068 .8636 .9934 .9990 1.0000 1.0000 .0068 .0045 .1716 .1387 .7609 .8145 .9918 .9987 1.0000 1.0000 .0606 .0468 .2680 .2405 .7935 .8525 .9932 .9990 1.0000 1.0000 .1111 .0971 .3203 .3039 .8142 .8687 .9939 .9991 1.0000 1.0000 .1052 .0960 .3126 .2992 .8068 .8636 .9934 .9990 1.0000 1.0000 .1416 .1455 .4231 .4188 .9347 .9351 1.0000 .9999 1.0000 1.0000 .0139 .0135 .2604 .2613 .9110 .9103 1.0000 .9999 1.0000 1.0000 .0898 .0919 .3779 .3773 .9278 .9300 1.0000 .9999 1.0000 1.0000 .1515 .1569 .4394 .4377 .9401 .9438 1.0000 1.0000 1.0000 1.0000 .1418 .1458 .4231 .4189 .9347 .9351 1.0000 .9999 1.0000 1.0000 QBA QST QSE QLA QBR .2 .2 .2 .2 .2 .2 .2 .2 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0232 .0240 .0550 .0544 .2631 .2400 .7255 .6727 .9893 .9845 .0361 .0350 .0707 .0717 .2538 .2608 .6999 .6823 .9865 .9848 .0296 .0307 .0653 .0666 .2484 .2561 .6968 .6795 .9865 .9848 .0005 .0004 .0121 .0165 .1673 .1684 .6333 .6248 .9840 .9814 R .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0372 .0420 .1342 .1520 .5936 .7167 .9691 .9931 1.0000 1.0000 .0677 .0612 .1848 .1665 .6358 .6743 .9721 .9893 1.0000 1.0000 .0602 .0525 .1764 .1580 .6303 .6684 .9717 .9892 1.0000 1.0000 U .6 .6 .6 .6 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0487 .0564 .1919 .2521 .7069 .8674 .9871 .9991 1.0000 1.0000 .1167 .1074 .3213 .3104 .8109 .8669 .9935 .9990 1.0000 1.0000 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0765 .0750 .3193 .3240 .9136 .9160 1.0000 .9999 1.0000 1.0000 .1588 .1587 .4353 .4311 .9365 .9372 1.0000 .9999 1.0000 1.0000 R U R U R U R R U R U R U R U R U R .6 .6 .6 U R .6 .6 U .6 R U R U R U R U R U R U R Ill Table 7.2.1.3 Empirical power of tests of complete multivariate independence at or = .01, p = 5, mNz = .5 p N:p 5 5 5 5 5 5 5 5 5 5 2 2 U 4 4 10 10 20 20 40 40 U 5 .5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 U 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 5 5 5 5 5 5 5 5 5 5 2 2 4 4 10 10 20 20 40 40 M R R U R U R U R R U R U R U R U R U R U R U R U R U R U R U R U R U R U R pNz or .2 .2 QF QBA QST QSE QLA QBR QSA .2 .2 .2 .2 .2 .2 .2 .2 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .0718 .0645 .3751 .2795 .9769 .9141 1.0000 .9997 1.0000 1.0000 .1003 .1034 .3263 .3300 .9452 .9235 1.0000 .9998 1.0000 1.0000 .0882 .0878 .3118 .3166 .9431 .9219 1.0000 .9998 1.0000 1.0000 .0016 .0026 .0913 .0921 .8736 .8472 1.0000 .9996 1.0000 1.0000 .0340 .0359 .1920 .1936 .9103 .8859 1.0000 .9998 1.0000 1.0000 .0817 .0825 .2661 .2717 .9031 .8766 1.0000 .9994 1.0000 1.0000 .0886 .0882 .3118 .3166 .9431 .9219 1.0000 .9998 1.0000 1.0000 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .1821 .3359 .7418 .9606 .9997 1.0000 1.0000 1.0000 1.0000 1.0000 .2711 .2543 .7351 .7813 .9995 1.0000 1.0000 1.0000 1.0000 1.0000 .2498 .2331 .7220 .7647 .9994 1.0000 1.0000 1.0000 1.0000 1.0000 .0142 .0112 .4565 .4424 .9987 1.0000 1.0000 1.0000 1.0000 1.0000 .1283 .1175 .6332 .6468 .9992 1.0000 1.0000 1.0000 1.0000 1.0000 .2417 .2218 .7035 .7336 .9994 1.0000 1.0000 1.0000 1.0000 1.0000 .2500 .2334 .7220 .7647 .9994 1.0000 1.0000 1.0000 1.0000 1.0000 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .2046 .4958 .6969 .9905 .9969 1.0000 1.0000 1.0000 1.0000 1.0000 .4455 .4394 .8417 .9307 .9993 1.0000 1.0000 1.0000 1.0000 1.0000 .4255 .4154 .8371 .9244 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 .0822 .0550 .7205 .7829 .9986 1.0000 1.0000 1.0000 1.0000 1.0000 .3334 .2861 .8120 .8976 .9990 1.0000 1.0000 1.0000 1.0000 1.0000 .4435 .4195 .8438 .9307 .9993 1.0000 1.0000 1.0000 1.0000 1.0000 .4260 .4159 .8371 .9244 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .4916 .4940 .9785 .9791 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5799 .5794 .9549 .9574 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5534 .5565 .9522 .9538 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .1588 .1619 .8920 .8941 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4595 .4665 .9409 .9442 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5810 .5812 .9580 .9613 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5542 .5572 .9522 .9538 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 112 Table 7.2.2.1 Empirical power of tests of complete multivariate independence at a= .01, p = 10, mNz = .1 p 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 N:p 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 M U R U pNz a .2 .2 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 R .2 .2 U .2 R .2 .2 .2 .2 .2 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R QBA .0220 .0232 .0229 .0220 .0733 .0792 .2203 .2659 .6500 .7518 .0273 .0292 .0430 .0481 .1867 .2316 .5808 .7051 .9714 .9942 .0343 .0357 .0657 .0775 .3222 .3948 .8027 .8962 .9983 .9997 .0426 .0449 .0928 .1116 .4525 .5492 .9111 .9669 .9999 .9999 QST QSE .0280 .0260 .0331 .0301 .1001 .0908 .2867 .2784 .7280 .7560 .0412 .0373 .0778 .0697 .2858 .2645 .6992 .7325 .9859 .9943 .0608 .0565 .1392 .1304 .4873 .4976 .8969 .9304 .9996 .9998 .0906 .0820 .2066 .2016 .6579 .6776 .9673 .9829 1.0000 1.0000 .0255 .0242 .0317 .0289 .0982 .0894 .2859 .2774 .7272 .7555 .0385 .0352 .0743 .0666 .2832 .2615 .6979 .7313 .9858 .9943 .0558 .0531 .1340 .1264 .4846 .4928 .8965 .9300 .9996 .9998 .0849 .0767 .2021 .1967 .6546 .6749 .9667 .9828 1.0000 1.0000 QlA .0002 .0002 .0054 .0043 .0629 .0528 .2479 .2379 .7067 .7332 .0007 .0003 .0196 .0144 .2103 .1923 .6583 .6921 .9842 .9935 .0024 .0019 .0513 .0412 .4069 .4060 .8795 .9144 .9996 .9998 .0058 .0047 .0909 .0802 .5825 .5957 .9598 .9778 1.0000 1.0000 QRR QF .0148 .0248 .0152 .0239 .0255 .0311 .0224 .0277 .0922 .0946 .0812 .0834 .2800 .2718 .2698 .2601 .7233 .6975 .7510 .7211 .0260 .0392 .0235 .0340 .0752 .0645 .0544 .0661 .2729 .2814 .2518 .2625 .6925 .6925 .7267 .7238 .9854 .9846 .9942 .9941 .0413 .0593 .0371 .0540 .1186 .1378 .1110 .1276 .4751 .4900 .4828 . .4983 .8949 .8988 .9292 .9320 .9996 .9996 .9998 .9998 .0656 .0858 .0600 .0801 .1862 .2077 .1802 .2019 .6472 .6642 .6667 .6832 .9701 .9661 .9826 .9838 1.0000 1.0000 1.0000 1.0000 QSA\ .0255 .0242 .0317 .02«9 .0982 .0894 .2859 .2774 .7272 .7555 .0385 .0352 .0743 .0666 .2832 .2615 .6979 .7313 .9858 .9943 .0558 .0531 .1340 .1264 .4846 .4928 .8965 .9300 .9996 .9998 .0849 .0767 .2021 .1967 .6546 .6749 .9667 .9828 1.0000 1.0000 113 Table 7.2.2.2 Empirical power of tests of complete multivariate independence at a = .01, p = 10, mNz = .3. p N:p 10 2 10 2 4 10 10 4 10 10 10 10 10 20 10 20 10 40 10 40 10 2 10 2 10 4 10 4 10 10 10 10 10 20 10 20 10 40 10 40 10 2 10 2 10 4 10 4 10 10 10 10 10 20 10 20 10 40 10 40 10 2 10 2 4 10 10 4 10 10 10 10 10 20 10 20 10 40 10 40 M U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R pNz a QBA .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .6 .6 .6 .6 .6 .6 \6 .6 .6 .6 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .1158 .1912 .4253 .7015 .9803 .9998 1.0000 1.0000 1.0000 1.0000 .2878 .5936 .8106 .9970 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4054 .7363 .9035 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .4819 .8652 .9341 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QST QSE .2424 .2316 .2235 .2128 .6142 .6077 .6584 .6499 .9902 .9906 .9996 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5414 .5278 .5769 .5600 .9144 .9166 .9852 .9849 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7070 .7153 .7865 .7745 .9752 .9763 .9967 .9967 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8101 .8150 .8776 .8706 .9914 .9915 .9994 .9994 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QlA .0246 .0131 .3794 .3593 .9832 .9991 1.0000 1.0000 1.0000 1.0000 .1647 .1146 .8175 .9185 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .3779 .3633 .9422 .9868 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .5464 .5606 .9785 .9967 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QBR .1842 .1540 .5683 .5965 .9884 .9996 1.0000 1.0000 1.0000 1.0000 .4793 .4924 .9049 .9804 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .6801 .7417 .9731 .9963 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7887 .8531 .9904 .9993 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QF .2210 .1989 .5829 .6113 .9862 .9995 1.0000 1.0000 1.0000 1.0000 .5331 .5562 .9122 .9841 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7178 .7852 .9769 .9973 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8188 .8839 .9920 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QSA\ .2316] .2129 .6077 .6499 .9902 .9996 1.0000 1.0000 1.0000 1.0000 .52781 .5602 .9144 .9849 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7070 .7746 .9752 .9967 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8101 .8706 .9914 .9994 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000J 114 Table 7.2.2.3 Empirical power of tests of complete multivariate independence at a = .01,/? = 10, mNz = .5 p 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 N:p 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 2 2 4 4 10 10 20 20 40 40 M pNz a QRA QST QSE QlA QBR .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .6 .6 .6 .6 .6 .6 .6 .6 .6 .6 .8 .8 .8 .8 .8 .8 .8 .8 .8 .8 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .6711 .9682 .9979 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9911 .8480 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9757 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9426 1.0000 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7993 .6744 .9964 .9977 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9748 .7845 1.0000 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9909 .9920 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9955 .9927 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7917 .6607 .9961 .9975 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9735 .7738 1.0000 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9906 .9913 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9954 .9917 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .3260 .1096 .9790 .9482 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .8352 .2577 1.0000 .9928 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9461 .8616 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9748 .9005 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .7314 .5168 .9944 .9907 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9665 .7035 1.0000 .9989 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9884 .9859 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9950 .9905 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R QF .7657 .5987 .9938 .9918 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9741 .7586 1.0000 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9915 .9896 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9961 .9934 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 QSA .7917 .6607 .9961 .9975 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9735 .7739 1.0000 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9906 .9913 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9954 .9917 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 115 Table 8 Experimentwise Type I error rate for testing the correlations between p uncorrected variables at the nominal level a = .05 under the assumption of independent tests: a c i-(i-ai.)v p P V EX a 2 5 10 15 20 100 1 10 45 105 190 4950 .0500 .4013 .9006 .9954 .9999 1.000 EX - Table 9 Table of critical values that an observed value must exceed for each procedure to test the significance of the pairwise correlations from a matrix of order p-5 i N.p 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 rNT .6319 .6319 .6319 .6319 .6319 .6319 .6319 .6319 .6319 .6319 .4438 .4438 .4438 .4438 .4438 .4438 .4438 .4438 .4438 .4438 .2787 .2787 .2787 .2787 2787 .2787 .2787 .2787 .2787 .2787 .1966 .1966 .1966 .1966 .1966 .1966 .1966 .1966 .1966 .1966 .1388 .1388 .1388 .1388 .1388 .1388 .1388 .1388 .1388 .1388 pMB .0250 .0125 .0083 .0063 .0050 .0042 .0036 .0031 .0028 .0025 .0250 .0125 .0083 .0063 .0050 .0042 .0036 .0031 .0028 .0025 .0250 .0125 .0083 .0063 .0050 .0042 .0036 .0031 .0028 .0025 .0250 .0125 .0083 .0063 .0050 .0042 .0036 .0031 .0028 .0025 .0250 .0125 .0083 .0063 .0050 .0042 .0036 .0031 .0028 .0025 rMB .6319 .6973 .7294 .7499 .7646 .7759 .7850 .7926 .7990 .8046 .4438 .4993 5282 5473 .5614 5725 .5816 5893 5959 .6018 .2787 .3168 .3371 .3508 .3610 .3692 .3759 .3816 .3866 .3909 .1966 .2241 .2389 .2490 .2565 .2625 .2675 .2717 .2754 .2786 .1388 .1585 .1691 .1763 .1818 .1861 .1897 .1927 .1954 .1978 pMD .0250 .0127 .0085 .0064 .0051 .0043 .0037 .0032 .0028 .0026 .0250 .0127 .0085 .0064 .0051 .0043 .0037 .0032 .0028 .0026 .0250 .0127 .0085 .0064 .0051 .0043 .0037 .0032 .0028 .0026 .0250 .0127 .0085 .0064 .0051 .0043 .0037 .0032 .0028 .0026 .0250 .0127 .0085 .0064 .0051 .0043 .0037 .0032 .0028 .0026 rMD .6319 .6962 .7281 .7486 .7633 .7746 .7837 .7913 .7978 .8034 .4438 .4984 .5270 .5461 .5602 .5713 .5804 .5880 .5947 .6005 .2787 .3161 .3363 .3499 .3601 .3682 .3749 .3806 .3856 .3900 .1966 .2236 .2383 .2483 .2558 .2618 .2668 .2710 .2747 .2779 .1388 .1581 .1687 .1759 .1813 .1856 .1892 .1922 .1949 .1973 rOR/OS .1242 .1926 .2534 .3119 .3704 .4310 .4961 .5692 .6580 .7850 .0800 .1245 .1647 .2041 .2445 .2876 .3356 .3925 .4672 .5909 .0481 .0751 .0996 .1239 .1490 .1761 , .2069 .2444 .2955 .3871 .0335 .0523 .0695 .0864 .1041 .1232 .1451 .1719 .2089 .2768 .0235 .0367 .0488 .0607 .0732 .0867 .1022 .1212 .1477 .1969 rOF .6296 .6886 .7175 .7359 .7491 .7592 .7674 .7742 .7800 .7850 .4425 .4948 .5220 5400 .5532 .5636 .5721 .5793 .5855 5909 .2783 .3151 .3349 .3482 .3581 .3660 .3725 .3780 .3828 .3871 .1964 .2233 .2378 .2477 .2551 .2610 .2659 .2700 .2736 .2768 .1387 .1580 .1685 .1756 .1810 .1853 .1888 .1919 .1945 .1969 tCF .6296 .6886 .7175 .7359 .7491 .7592 .7674 .7742 .7800 .7850 .4425 .4948 .5220 .5400 5532 5636 .5721 .5793 .5855 .5909 .2783 .3151 .3349 .3482 .3581 .3660 .3725 .3780 .3828 .3871 .1964 .2233 .2378 .2477 .2551 .2610 .2659 .2700 .2736 .2768 .1387 .1580 .1685 .1756 .1810 .1853 .1888 .1919 .1945 .1969 xCS 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 14.0671 15.5073 16.9190 18.3070 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 14.0671 15.5073 16.9190 18.3070 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 14.0671 15.5073 16.9190 18.3070 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 14.0671 15.5073 16.9190 18.3070 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 14.0671 15.5073 16.9190 18.3070 Table 10.1.1.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P = I, p = 5 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .4112 .05058 .0507 .00519 .0507 .00516 .0525 .00537 .0525 .00536 .2290 .04967 .0737 .01132 .0737 .00756 .0737 .00756 .0542 .00652 .4078 .05052 .0480 .00490 .0480 .00488 .0492 .00503 .0492 .00500 .2329 .05040 .0597 .00924 .0597 .00613 .0597 .00613 .0532 .00603 .3993 .05011 .0460 .00474 .0460 .00471 .0469 .00483 .0469 .00480 .2318 .04996 .0496 .00822 .0496 .00513 .0496 .00513 .0491 .00557 .3949 .04923 .0475 .00489 .0475 .00486 .0483 .00498 .0483 .00494 .2337 .04910 .0502 .00840 .0502 .00519 .0502 .00519 .0507 .00568 .4077 .05056 .0488 .00502 .0488 .00499 .0501 .00516 .0501 .00512 .2355 .05050 .0509 .00814 .0509 .00524 .0509 .00524 .0486 .00537 Fp Fn .05058 1.00000 .005191.00000 .005161.00000 .00537 1.00000 .005361.00000 .04967 1.00000 .011321.00000 .00756 1.00000 .00756 1.00000 .00652 1.00000 .05052 1.00000 .00490 1.00000 .00488 1.00000 .00503 1.00000 .005001.00000 .05040 1.00000 .00924 1.00000 .00613 1.00000 .00613 1.00000 .006031.00000 .050111.00000 .00474 1.00000 .00471 1.00000 .00483 1.00000 .00480 1.00000 .04996 1.00000 .008221.00000 .00513 1.00000 .00513 1.00000 .00557 1.00000 .04923 1.00000 .00489 1.00000 .00486 1.00000 .00498 1.00000 .00494 1.00000 .04910 1.00000 .00840 1.00000 .005191.00000 .00519 1.00000 .00568 1.00000 .05056 1.00000 .00502 1.00000 .00499 1.00000 .005161.00000 .005121.00000 .05050 1.00000 .008141.00000 .00524 1.00000 .00524 1.00000 .00537 1.00000 Tp Tn .00000 .94942 .00000 .99481 .00000 .99484 .00000 .99463 .00000 .99464 .00000 .95033 .00000 .98868 .00000 .99244 .00000 .99244 .00000 .99348 .00000 .94948 ,00000 .99510 .00000 .99512 .00000 .99497 .00000 .99500 .00000 .94960 .00000 .99076 .00000 .99387 .00000 .99387 .00000 .99397 .00000 .94989 .00000 .99526 .00000 .99529 .00000 .99517 .00000 .99520 .00000 .95004 .00000 .99178 .00000 .99487 .00000 .99487 .00000 .99443 .00000 .95077 .00000 .99511 .00000 .99514 .00000 .99502 .00000 .99506 .00000 .95090 .00000 .99160 .00000 .99481 .00000 .99481 .00000 .99432 .00000 .94944 .00000 .99498 .00000 .99501 .00000 .99484 .00000 .99488 .00000 .94950 .00000 .99186 .00000 .99476 .00000 .99476 .00000 .99463 s:Rej s:FpTn s.FnTp .06888 .02272 .02253 .02307 .02301 .12347 .05636 .02722 .02722 .02986 .06907 .02205 .02191 .02236 .02216 .12010 .04851 .02465 .02465 .02714 .06959 .02194 .02170 .02213 .02189 .11798 .04710 .02286 .02286 .02585 .06883 .02221 .02201 .02243 .02217 .11688 .04823 .02294 .02294 .02588 .06889 .02247 .02227 .02279 .02254 .11897 .04517 .02295 .02295 .02486 .06888 .02272 .02253 .02307 .02301 .12347 .05636 .02722 .02722 .02986 .06907 .02205 .02191 .02236 .02216 .12010 .04851 .02465 .02465 .02714 .06959 .02194 .02170 .02213 .02189 .11798 .04710 .02286 .02286 .02585 .06883 .02221 .02201 .02243 .02217 .11688 .04823 .02294 .02294 .02588 .06889 .02247 .02227 .02279 .02254 .11897 .04517 .02295 .02295 .02486 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 Table 10.1.1.2 Table of empirical results for tests of correlations with no preliminary omnibus when P = 1,/? = 10 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OEX Rej .9044 .05038 .0496 .00113 .0496 .00113 .0507 .00115 .0507 .00115 .3841 .04816 .0711 .00300 .0711 .00163 .0711 .00163 .0550 .00170 .9042 .04972 .0485 .00110 .0485 .00110 .0498 .00112 .0498 .00112 .3854 .04987 .0591 .00222 .0591 .00134 .0591 .00134 .0478 .00137 .9035 .05025 .0474 .00108 .0474 .00108 .0487 .00111 .0487 .00111 .3884 .05073 .0516 .00205 .0516 .00117 .0516 .00117 .0534 .00160 .8984 .05003 .0517 .00118 .0517 .00117 .0532 .00121 .0532 .00121 .3912 .05092 .0551 .00232 .0551 .00126 .0551 .00126 .0518 .00151 .9047 .04987 .0492 .00112 .0492 .00112 .0500 .00114 .0500 .00114 .3875 .04939 .0507 .00184 .0507 .00115 .0507 .00115 .0492 .00140 Fp Fn .05038 1.00000 .001131.00000 .001131.00000 .001151.00000 .001151.00000 .04816 1.00000 .003001.00000 .001631.00000 .00163 1.00000 .00170 1.00000 .04972 1.00000 .001101.00000 .001101.00000 .001121.00000 .001121.00000 .04987 1.00000 .00222 1.00000 .00134 1.00000 .00134 1.00000 .00137 1.00000 .05025 1.00000 .00108 1.00000 .00108 1.00000 .00111 1.00000 .00111 1.00000 .05073 1.00000 .00205 1.00000 .001171.00000 .001171.00000 .00160 1.00000 .05003 1.00000 .00118 1.00000 .00117 1.00000 .00121 1.00000 .00121 1.00000 .05092 1.00000 .00232 1.00000 .00126 1.00000 .001261.00000 .00151 1.00000 .04987 1.00000 .001121.00000 .001121.00000 .001141.00000 .001141.00000 .04939 1.00000 .00184 1.00000 .001151.00000 .00115 1.00000 .001401.00000 Tp Tn .00000 .94962 .00000 .99887 .00000 .99887 .00000 .99885 .00000 .99885 .00000 .95184 .00000 .99700 .00000 .99837 .00000 .99837 .00000 .99830 .00000 .95028 .00000 .99890 .00000 .99890 .00000 .99888 .00000 .99888 .00000 .95013 .00000 .99778 .00000 .99866 .00000 .99866 .00000 .99863 .00000 .94975 .00000 .99892 .00000 .99892 .00000 .99889 .00000 .99889 .00000 .94927 .00000 .99795 .00000 .99883 .00000 .99883 .00000 .99840 .00000 .94997 .00000 .99882 .00000 .99883 .00000 .99879 .00000 .99879 .00000 .94908 .00000 .99768 .00000 .99874 .00000 .99874 .00000 .99849 .00000 .95013 .00000 .99888 .00000 .99888 .00000 .99886 .00000 .99886 .00000 .95061 .00000 .99816 .00000 .99885 .00000 .99885 .00000 .99860 s:Rej s.FpTn s:FnTp .03295 .00500 .00500 .00506 .00504 .10950 .01901 .00600 .00600 .00819 .03232 .00489 .00489 .00495 .00495 .10880 .01424 .00541 .00541 .00691 .03287 .00489 .00487 .00496 .00495 .11112 .01651 .00509 .00509 .00787 .03274 .00509 .00506 .00517 .00516 .10921 .01824 .00528 .00528 .00740 .03215 .00497 .00497 .00500 .00500 .10702 .01082 .00503 .00503 .00696 .03295 .00500 .00500 .00506 .00504 .10950 .01901 .00600 .00600 .00819 .03232 .00489 .00489 .00495 .00495 .10880 .01424 .00541 .00541 .00691 .03287 .00489 .00487 .00496 .00495 .11112 .01651 .00509 .00509 .00787 .03274 .00509 .00506 .00517 .00516 .10921 .01824 .00528 .00528 .00740 .03215 .00497 .00497 .00500 .00500 .10702 .01082 .00503 .00503 .00696 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 Table 10.1.2.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P = l,p = 5 N:p OLEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 M D 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 M D 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 M D 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn Tp Tn .0541 .01044 .010441.00000 .00000 .98956 .0239 .00251 .00251 1.00000 .00000 .99749 .0239 .00248 .00248 1.00000 .00000 .99752 .0243 .00255 .00255 1.00000 .00000 .99745 .0243 .00254 .00254 1.00000 .00000 .99746 .0542 .02175 .02175 1.00000 .00000 .97825 .0295 .00670 .00670 1.00000 .00000 .99330 .0295 .00314 .00314 1.00000 .00000 .99686 .0295 .00314 .00314 1.00000 .00000 .99686 .0542 .00652 .00652 1.00000 .00000 .99348 .0530 .01021 .01021 1.00000 .00000 .98979 .0251 .00261 .002611.00000 .00000 .99739 .0251 .00259 .00259 1.00000 .00000 .99741 .0255 .00266 .00266 1.00000 .00000 .99734 .0255 .00263 .00263 1.00000 .00000 .99737 .0532 .02020 .02020 1.00000 .00000 .97980 .0280 .00593 .005931.00000 .00000 .99407 .0280 .00296 .002961.00000 .00000 .99704 .0280 .00296 .00296 1.00000 .00000 .99704 .0532 .00603 .006031.00000 .00000 .99397 .0488 .00976 .00976 1.00000 .00000 .99024 .0225 .00239 .00239 1.00000 .00000 .99761 .0225 .00236 .00236 1.00000 .00000 .99764 .0227 .00241 .00241 1.00000 .00000 .99759 .0227 .00238 .00238 1.00000 .00000 .99762 .0490 .01837 .01837 1.00000 .00000 .98163 .0233 .00542 .00542 1.00000 .00000 .99458 .0233 .00250 .00250 1.00000 .00000 .99750 .0233 .00250 .00250 1.00000 .00000 .99750 .0491 .00557 .00557 1.00000 .00000 .99443 .0503 .00989 .00989 1.00000 .00000 .99011 .0239 .00253 .00253 1.00000 .00000 .99747 .0239 .00250 .00250 1.00000 .00000 .99750 .0241 .00256 .002561.00000 .00000 .99744 .0241 .00252 .002521.00000 .00000 .99748 .0507 .01902 .01902 1.00000 .00000 .98098 .0247 .00570 .00570 1.00000 .00000 .99430 .0247 .00264 .00264 1.00000 .00000 .99736 .0247 .00264 .00264 1.00000 .00000 .99736 .0507 .00568 .00568 1.00000 .00000 .99432 .0483 .00933 .00933 1.00000 .00000 .99067 .0228 .00242 .00242 1.00000 .00000 .99758 .0228 .00239 .00239 1.00000 .00000 .99761 .0234 .00249 .00249 1.00000 .00000 .99751 .0234 .00245 .00245 1.00000 .00000 .99755 .0484 .01835 .01835 1.00000 .00000 .98165 .0237 .00525 .00525 1.00000 .00000 .99475 .0237 .00252 .00252 1.00000 .00000 .99748 .0237 .00252 .002521.00000 .00000 .99748 .0486 .00537 .005371.00000 .00000 .99463 s.Rej s.FpTn s.FnTp .04768 .04768 .00000 .01639 .01639 .00000 .01612 .01612 .00000 .01651 .01651 .00000 .01642 .01642 .00000 .10474 .10474 .00000 .05249 .05249 .00000 .01861 .01861 .00000 .01861 .01861 .00000 .02986 .02986 .00000 .04679 .04679 .00000 .01656 .01656 .00000 .01638 .01638 .00000 .01676 .01676 .00000 .01650 .01650 .00000 .09704 .09704 .00000 .04519 .04519 .00000 .01787 .01787 .00000 .01787 .01787 .00000 .02714 .02714 .00000 .04612 .04612 .00000 .01623 .01623 .00000 .01589 .01589 .00000 .01629 .01629 .00000 .01595 .01595 .00000 .09201 .09201 .00000 .04403 .04403 .00000 .01673 .01673 .00000 .01673 .01673 .00000 .02585 .02585 .00000 .04617 .04617 .00000 .01657 .01657 .00000 .01630 .01630 .00000 .01672 .01672 .00000 .01636 .01636 .00000 .09382 .09382 .00000 .04543 .04543 .00000 .01706 .01706 .00000 .01706 .01706 .00000 .02588 .02588 .00000 .04470 .04470 .00000 .01625 .01625 .00000 .01598 .01598 .00000 .01652 .01652 .00000 .01616 .01616 .00000 .09256 .09256 .00000 .04190 .04190 .00000 .01660 .01660 .00000 .01660 .01660 .00000 .02486 .02486 .00000 Table 10.1.2.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P = I, p = 10 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .0550 .00602 .0154 .00037 .0154 .00037 .0155 .00037 .0155 .00037 .0550 .01847 .0193 .00154 .0193 .00047 .0193 .00047 .0550 .00170 .0478 .00517 .0119 .00027 .0119 .00027 .0120 .00028 .0120 .00028 .0478 .01573 .0133 .00090 .0133 .00031 .0133 .00031 .0478 .00137 .0534 .00586 .0143 .00034 .0143 .00034 .0145 .00034 .0145 .00034 .0534 .01810 .0151 .00104 .0151 .00035 .0151 .00035 .0534 .00160 .0518 .00566 .0149 .00035 .0149 .00035 .0152 .00036 .0152 .00036 .0518 .01646 .0155 .00121 .0155 .00037 .0155 .00037 .0518 .00151 .0492 .00532 .0137 .00032 .0137 .00032 .0138 .00032 .0138 .00032 .0492 .01526 .0140 .00079 .0140 .00032 .0140 .00032 .0492 .00140 Fp Fn .00602 1.00000 .000371.00000 .00037 1.00000 .00037 1.00000 .00037 1.00000 .018471.00000 .00154 1.00000 .00047 1.00000 .00047 1.00000 .00170 1.00000 .00517 1.00000 .00027 1.00000 .00027 1.00000 .00028 1.00000 .00028 1.00000 .01573 1.00000 .00090 1.00000 .00031 1.00000 .000311.00000 .00137 1.00000 .00586 1.00000 .00034 1.00000 .00034 1.00000 .00034 1.00000 .00034 1.00000 .01810 1.00000 .00104 1.00000 .00035 1.00000 .00035 1.00000 .001601.00000 .00566 1.00000 .00035 1.00000 .00035 1.00000 .000361.00000 .00036 1.00000 .016461.00000 .00121 1.00000 .000371.00000 .00037 1.00000 .00151 1.00000 .005321.00000 .000321.00000 .00032 1.00000 .00032 1.00000 .00032 1.00000 .015261.00000' .00079 1.00000 .00032 1.00000 .000321.00000 .001401.00000 Tp .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 Tn .99398 .99963 .99963 .99963 .99963 .98153 .99846 .99953 .99953 .99830 .99483 .99973 .99973 .99972 .99972 .98427 .99910 .99969 .99969 .99863 .99414 .99966 .99966 .99966 .99966 .98190 .99896 .99965 .99965 .99840 .99434 .99965 .99965 .99964 .99964 .98354 .99879 .99963 .99963 .99849 .99468 .99968 .99968 .99968 .99968 .98474 .99921 .99968 .99968 .99860 s.Rej s.FpTn s.FnTp .02612 .00303 .00303 .00306 .00304 .08915 .01774 .00353 .00353 .00819 .02407 .00253 .00253 .00254 .00254 .08192 .01265 .00271 .00271 .00691 .02567 .00285 .00285 .00287 .00287 .08920 .01553 .00292 .00292 .00787 .02511 .00295 .00290 .00300 .00298 .08282 .01724 .00307 .00307 .00740 .02420 .00272 .00272 .00273 .00273 .07925 .00906 .00275 .00275 .00696 .02612 .00303 .00303 .00306 .00304 .08915 .01774 .00353 .00353 .00819 .02407 .00253 .00253 .00254 .00254 .08192 .01265 .00271 .00271 .00691 .02567 .00285 .00285 .00287 .00287 .08920 .01553 .00292 .00292 .00787 .02511 .00295 .00290 .00300 .00298 .08282 .01724 .00307 .00307 .00740 .02420 .00272 .00272 .00273 .00273 .07925 .00906 .00275 .00275 .00696 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 Table 10.2.1.1.1.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P = 5, mNz = .1, pNz = .2 N:p OLEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn .3401 .05187 .05036 .94210 .0416 .00535 .00529 .99440 .0415 .00532 .00525 .99440 .0432 .00555 .00550 .99425 .0431 .00551 .00545 .99425 .2097 .05208 .05200 .94760 .0622 .01170 .01150 .98750 .0595 .00762 .00759 .99225 .0595 .00762 .00759 .99225 .0481 .00700 .00690 .99260 .3386 .05468 .05049 .92855 .0399 .00578 .00508 .99140 .0399 .00577 .00506 .99140 .0406 .00588 .00516 .99125 .0405 .00586 .00514 .99125 .2226 .05694 .05583 .93860 .0521 .01128 .01028 .98470 .0478 .00698 .00609 .98945 .0478 .00698 .00609 .98945 .0466 .00705 .00661 .99120 .3285 .06084 .04931 .89305 .0374 .00713 .00478 .98345 .0372 .00707 .00473 .98355 .0378 .00727 .00483 .98295 .0376 .00721 .00479 .98310 .2475 .06652 .06370 .92220 .0472 .01314 .01031 .97555 .0400 .00774 .00513 .98180 .0400 .00774 .00513 .98180 .0495 .00895 .00683 .98255 .3304 .07360 .04939 .82955 .0381 .01079 .00486 .96550 .0378 .01069 .00481 .96580 .0391 .01096 .00499 .96515 .0388 .01089 .00494 .96530 .2970 .08846 .08150 .88370 .0591 .02053 .01359 .95170 .0406 .01135 .00518 .96395 .0406 .01135 .00518 .96395 .0636 .01426 .00883 .96400 .3455 .09941 .05121 .70780 .0400 .02023 .00510 .91925 .0391 .02005 .00495 .91955 .0404 .02043 .00515 .91845 .0395 .02026 .00501 .91875 .3737 .12910 .11073 .79740 .0874 .03993 .02230 .88955 .0409 .02074 .00521 .91715 .0409 .02074 .00521 .91715 .0794 .02598 .01100 .91410 Tp .05790 .00560 .00560 .00575 .00575 .05240 .01250 .00775 .00775 .00740 .07145 .00860 .00860 .00875 .00875 .06140 .01530 .01055 .01055 .00880 .10695 .01655 .01645 .01705 .01690 .07780 .02445 .01820 .01820 .01745 .17045 .03450 .03420 .03485 .03470 .11630 .04830 .03605 .03605 .03600 .29220 .08075 .08045 .08155 .08125 .20260 .11045 .08285 .08285 .08590 Tn .94964 .99471 .99475 .99450 .99455 .94800 .98850 .99241 .99241 .99310 .94951 .99492 .99494 .99484 .99486 .94418 .98973 .99391 .99391 .99339 .95069 .99523 .99528 .99518 .99521 .93630 .98969 .99488 .99488 .99318 .95061 .99514 .99519 .99501 .99506 .91850 .98641 .99483 .99483 .99118 .94879 .99490 .99505 .99485 .99499 .88928 .97770 .99479 .99479 .98900 s.Rej s.FpTn s.FnTp d' .06976 .02303 .02284 .02346 .02321 .12875 .05858 .02739 .02739 .03113 .07150 .02397 .02391 .02420 .02408 .12936 .05861 .02629 .02629 .02969 .07609 .02717 .02682 .02746 .02711 .13938 .06389 .02836 .02836 .03303 .08231 .03281 .03235 .03305 .03275 .16041 .07932 .03359 .03359 .04148 .08984 .04363 .04310 .04384 .04333 .19046 .11315 .04414 .04414 .05375 .07754 .02559 .02538 .02612 .02583 .13179 .05923 .03047 .03047 .03283 .07808 .02511 .02502 .02531 .02519 .13267 .05829 .02742 .02742 .03165 .07747 .02454 .02423 .02466 .02444 .14108 .06017 .02547 .02547 .03123 .07728 .02468 .02450 .02497 .02479 .16197 .06930 .02540 .02540 .03540 .07761 .02523 .02470 .02534 .02490 .19066 .09545 .02548 .02548 .03946 .16417 .05309 .05309 .05378 .05378 .17750 .08772 .06217 .06217 .06241 .18076 .06539 .06539 .06595 .06595 .18744 .09496 .07255 .07255 .06688 .21815 .09139 .09113 .09266 .09228 .20965 .12074 .09550 .09550 .09445 .26754 .12889 .12799 .12947 .12903 .25205 .17167 .13163 .13163 .13400 .32137 .19237 .19159 .19321 .19242 .31963 .25977 .19459 .19459 .20031 .0687 .0200 .0225 .0156 .0188 .0037 .0320 .0077 .0077 .0252 .1751 .1883 .1891 .1887 .1904 .0477 .1539 .2008 .2008 .1037 .4086 .4608 .4620 .4692 .4684 .1044 .3453 .4748 .4748 .3568 .6984 .7670 .7667 .7628 .7644 .2014 .5474 .7654 .7654 .5738 1.0862 1.1689 1.1773 1.1709 1.1783 .3903 .7843 1.1753 1.1753 .9239 P A' B" 1.1167 i;0524 1.0590 1.0403 1.0487 1.0061 1.0750 1.0188 1.0188 1.0637 .5345 .5140 .5157 .5109 .5131 .5020 .5203 .5053 .5053 .5170 .0657 .0285 .0321 .0221 .0266 .0036 .0412 .0105 .0105 .0347 1.3123 1.5940 1.5975 1.5939 1.6008 1.0776 1.4115 1.6215 1.6215 1.2861 1.8065 2.9689 2.9814 3.0173 3.0155 1.1662 2.0953 3.0232 3.0232 2.2623 2.4820 5.4139 5.4249 5.3369 5.3660 1.2978 2.8847 5.3097 5.3097 3.3097 3.2677 10.1732 10.4179 10.1605 10.3794 1.4934 3.5523 10.1725 10.1725 5.4157 .5789 .6034 .6037 .6034 .6041 .5242 .5834 .6069 .6069 .5627 .6499 .6808 .6811 .6823 .6822 .5491 .6481 .6829 .6829 .6549 .7094 .7222 .7222 .7217 .7219 .5843 .6885 .7219 .7219 .6956 .7697 .7532 .7536 .7534 .7537 .6392 .7221 .7538 .7538 .7369 .1611 .2561 .2573 .2562 .2584 .0446 .1940 .2661 .2661 .1409 .3415 .5480 .5496 .5546 .5543 .0921 .4007 .5560 .5560 .4333 .5015 .7463 .7467 .7429 .7442 .1572 .5485 .7419 .7419 .5974 .6195 .8720 .8752 .8719 .8747 .2426 .6368 .8722 .8722 .7566 Table 10.2.1.1.1.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p - 5, mNz - .1, pNz - .4 N:p O.EX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 ' OS 40 OF 40 CF 40 CS 40 .2703 .05338 .0316 .00552 .0315 .00550 .0326 .00571 .0326 .00569 .1831 .05469 .0522 .01267 .0471 .00801 .0471 .00801 .0400 .00759 .2644 .05871 .0311 .00666 .0310 .00660 .0319 .00682 .0319 .00676 .2075 .06480 .0432 .01333 .0359 .00803 .0359 .00803 .0400 .00880 .2586 .07220 .0282 .00959 .0276 .00946 .0286 .00977 .0280 .00965 .2549 .08852 .0470 .01993 .0299 .01028 .0299 .01028 .0468 .01412 .2587 .09603 .0293 .01667 .0288 .01643 .0296 .01695 .0291 .01670 .3525 .14112 .0700 .03882 .0312 .01755 .0312 .01755 .0657 .02716 .2656 .14730 .0302 .03645 .0289 .03579 .0310 .03689 .0297 .03626 5034 .24692 .1357 .09561 .0313 .03728 .0313 .03728 .0897 .06054 Fp .05095 .00532 .00530 .00548 .00548 .05363 .01250 .00793 .00793 .00742 .05000 .00525 .00520 .00538 .00535 .06193 .01097 .00612 .00612 .00740 .04932 .00473 .00462 .00480 .00468 .08185 .01355 .00503 .00503 .00852 .04905 .00498 .00490 .00503 .00495 .12350 .02300 .00532 .00532 .01200 .05038 .00513 .00488 .00527 .00503 .19838 .05125 .00532 .00532 .01645 Fn Tp Tn .94298 .99418 .99420 .99395 .99400 .94373 .98708 .99188 .99188 .99215 .92823 .99123 .99130 .99103 .99113 .93090 .98313 .98910 .98910 .98910 .89348 .98313 .98328 .98278 .98290 .90148 .97050 .98185 .98185 .97748 .83350 .96580 .96628 .96518 .96568 .83245 .93745 .96410 .96410 .95010 .70733 .91658 .91785 .91568 .91690 .68028 .83785 .91478 .91478 .87333 .05703 .00583 .00580 .00605 .00600 .05628 .01293 .00813 .00813 .00785 .07178 .00878 .00870 .00898 .00887 .06910 .01688 .01090 .01090 .01090 .10653 .01688 .01673 .01723 .01710 .09853 .02950 .01815 .01815 .02253 .16650 .03420 .03373 .03483 .03433 .16755 .06255 .03590 .03590 .04990 .29268 .08343 .08215 .08433 .08310 .31973 .16215 .08523 .08523 .12668 .94905 .99468 .99470 .99452 .99452 .94637 .98750 .99207 .99207 .99258 .95000 .99475 .99480 .99462 .99465 .93807 .98903 .99388 .99388 .99260 .95068 .99527 .99538 .99520 .99532 .91815 .98645 .99497 .99497 .99148 .95095 .99502 .99510 .99497 .99505 .87650 .97700 .99468 .99468 .98800 .94962 .99487 .99512 .99473 .99497 .80162 .94875 .99468 .99468 .98355 s.Rej s.FpTn s.FnTp .07035 .02336 .02323 .02376 .02364 .13272 .06175 .02802 .02802 .03255 .07377 .02603 .02570 .02634 .02601 .14065 .06527 .02854 .02854 .03380 .08353 .03145 .03083 .03174 .03118 .16537 .08538 .03250 .03250 .04302 .09504 .04117 .04034 .04150 .04061 .20793 .12382 .04223 .04223 .05845 .10858 .06072 .05915 .06100 .05948 .25497 .19592 .06132 .06132 .08267 .08970 .02957 .02953 .03001 .03001 .14015 .06583 .03588 .03588 .03812 .08951 .02949 .02917 .02984 .02957 .14849 .06461 .03204 .03204 .03809 .08938 .02789 .02745 .02807 .02765 .17240 .07658 .02881 .02881 .03996 .08884 .02897 .02874 .02910 .02888 .21256 .10493 .02995 .02995 .04727 .09008 .02937 .02850 .02972 .02901 .26058 .16083 .02985 .02985 .05482 .11557 .03805 .03797 .03874 .03859 .15558 .07460 .04489 .04489 .04745 .12964 .04735 .04690 .04784 .04733 .16764 .08535 .05250 .05250 .05574 .15941 .06584 .06530 .06650 .06602 .20044 .12013 .06798 .06798 .08031 .19466 .09236 .09108 .09318 .09174 .25763 .18008 .09454 .09454 .11816 .23517 .14291 .13997 .14356 .14074 .32758 .29224 .14438 .14438 .17854 d' .0555 .0320 .0315 .0345 .0316 .0238 .0129 .0087 .0087 .0206 .1822 .1839 .1841 .1835 .1816 .0562 .1685 .2115 . .2115 .1434 .4063 .4717 .4767 .4751 .4807 .1027 .3219 .4800 .4800 .3818 .6860 .7546 .7541 .7594 .7586 .1938 .4617 .7541 .7541 .6113 1.0956 1.1843 1.1932 1.1813 1.1890 .3789 .6472 1.1838 1.1838 .9911 P 1.0933 1.0845 1.0834 1.0912 1.0832 1.0387 1.0294 1.0212 1.0212 1.0513 1.3271 1.5742 1.5758 1.5702 1.5635 1.0886 1.4504 1.6612 1.6612 1.4039 1.8012 3.0424 3.0872 3.0578 3.1063 1.1477 1.9340 3.0650 3.0650 2.3119 2.4583 5.2589 5.2774 5.2905 5.3063 1.2282 2.2584 5.1660 5.1660 3.2968 3.3130 10.3657 10.7122 10.2139 10.5180 1.2832 2.3334 10.2093 10.2093 5.0689 A' .5282 .5219 .5217 .5236 .5217 .5124 .5083 .5059 .5059 .5139 .5816 .6013 .6015 .6009 .6002 .5278 .5890 .6109 .6109 .5812 .6493 .6829 .6840 .6835 .6846 .5469 .6392 .6840 .6840 .6590 .7072 .7209 .7209 .7214 .7213 .5783 .6682 .7207 .7207 .6995 .7707 .7543 .7546 .7543 .7545 .6327 .7002 .7545 .7545 .7456 B" .0531 .0454 .0448 .0488 .0447 .0226 .0165 .0118 .0118 .0282 .1676 .2497 .2501 .2484 .2461 .0509 .2093 .2789 .2789 .1896 .3399 .5577 .5632 .5598 .5658 .0834 .3634 .5613 .5613 .4456 .4969 .7389 .7397 .7407 .7413 .1261 .4459 .7349 .7349 .5999 .6245 .8748 .8789 .8729 .8767 .1553 .4729 .8730 .8730 .7448 123 Table 10.2.1.1.1.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I,p = 5, mNz = S,pNz = .6 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .1851 .05453 .0203 .00587 .0203 .00586 .0208 .00604 .0208 .00602 .1474 .05825 .0378 .01415 .0307 .00846 .0307 .00846 .0273 .00808 .1845 .06247 .0194 .00711 .0194 .00707 .0199 .00726 .0199 .00720 .1818 .07467 .0339 .01631 .0234 .00882 .0234 .00882 .0297 .01117 .1823 .08343 .0190 .01191 .0189 .01178 .0191 .01211 .0190 .01201 .2485 .11644 .0438 .02937 .0200 .01281 .0200 .01281 .0378 .02103 .1810 .11943 .0194 .02217 .0189 .02166 .0196 .02251 .0192 .02196 .3761 .20763 .0799 .06470 .0206 .02322 .0206 .02322 .0599 .04611 .1857 .19607 .0204 .05268 .0195 .05106 .0205 .05345 .0197 .05178 .5847 .38982 .2013 .18549 .0208 .05400 .0208 .05400 .0859 .10953 Fp .05040 .00510 .00510 .00523 .00523 .05623 .01338 .00775 .00775 .00738 .04990 .00493 .00490 .00505 .00503 .07085 .01260 .00600 .00600 .00810 .04940 .00478 .00473 .00480 .00475 .10388 .01860 .00503 .00503 .01003 .04903 .00498 .00485 .00503 .00493 .17450 .03783 .00528 .00528 .01598 .05083 .00523 .00498 .00528 .00505 .30690 .11000 .00535 .00535 .02330 Fn .94272 .99362 .99363 .99342 .99345 .94040 .98533 .99107 .99107 .99145 .92915 .99143 .99148 .99127 .99135 .92278 .98122 .98930 .98930 .98678 .89388 .98333 .98352 .98302 .98315 .87518 .96345 .98200 .98200 .97163 .83363 .96637 .96713 .96583 .96668 .77028 .91738 .96482 .96482 .93380 .70710 .91568 .91822 .91443 .91707 .55490 .76418 .91357 .91357 .83298 Tp .05728 .00638 .00637 .00658 .00655 .05960 .01467 .00893 .00893 .00855 .07085 .00857 .00852 .00873 .00865 .07722 .01878 .01070 .01070 .01322 .10612 .01667 .01648 .01698 .01685 .12482 .03655 .01800 .01800 .02837 .16637 .03363 .03287 .03417 .03332 .22972 .08262 .03518 .03518 .06620 .29290 .08432 .08178 .08557 .08293 .44510 .23582 .08643 .08643 .16702 Tn .94960 .99490 .99490 .99478 .99478 .94378 .98663 .99225 .99225 .99263 .95010 .99508 .99510 .99495 .99498 .92915 .98740 .99400 .99400 .99190 .95060 .99523 .99528 .99520 .99525 .89613 .98140 .99498 .99498 .98998 .95098 .99503 .99515 .99498 .99508 .82550 .96218 .99473 .99473 .98403 .94918 .99478 .99503 .99473 .99495 .69310 .89000 .99465 .99465 .97670 s.FnTp s:Rej s:FpTn .07194 .02405 .02399 .02440 .02429 .13783 .06796 .02899 .02899 .03404 .07781 .02688 .02667 .02719 .02687 .15538 .07923 .03014 .03014 .03945 .09146 .03554 .03498 .03584 .03540 .19545 .11477 .03684 .03684 .05529 .11017 .04893 .04728 .04948 .04762 .25752 .17792 .05025 .05025 .08130 .12988 .07623 .07269 .07680 .07311 .30592 .29798 .07711 .07711 .11620 .11045 .03552 .03552 .03594 .03594 .15676 .07860 .04376 .04376 .04585 .10948 .03528 .03502 .03570 .03544 .17452 .07908 .03923 .03923 .04832 .10896 .03440 .03404 .03449 .03413 .21118 .10139 .03527 .03527 .05190 .10870 .03580 .03538 .03597 .03563 .26987 .14951 .03679 .03679 .06539 .11146 .03663 .03562 .03696 .03605 .32861 .25433 .03720 .03720 .07903 .09554 .03268 .03264 .03323 .03307 .14961 .07448 .03856 .03856 .04271 .10835 .03836 .03804 .03882 .03830 .16804 .09034 .04310 .04310 .05287 .13429 .05452 .05373 .05501 .05449 .21728 .13761 .05656 .05656 .08029 .16797 .07733 .07497 .07828 .07549 .28724 .21434 .07930 .07930 .12128 .20338 .12387 .11848 .12482 .11912 .34361 .35451 .12533 .12533 .17915 d' .0630 .0788 .0778 .0813 .0795 .0291 .0362 .0521 .0521 .0539 .1763 .1972 .1968 .1957 .1938 .0454 .1588 .2113 .2113 .1846 .4033 .4637 .4628 .4694 .4699 .1085 .2914 .4772 .4772 .4201 .6858 .7477 .7461 .7513 .7469 .1968 .3888 .7477 .7477 .6403 1.0920 1.1840 1.1844 1.1887 1.1868 .3666 .5067 1.1893 1.1893 1.0239 P 1.1067 1.2205 1.2176 1.2275 1.2220 1.0469 1.0827 1.1329 1.1329 1.1388 1.3161 1.6316 1.6307 1.6228 1.6163 1.0680 1.4089 1.6628 1.6628 1.5323 1.7938 2.9868 2.9865 3.0211 3.0290 1.1398 1.7588 3.0479 3.0479 2.4317 2.4581 5.1949 5.2140 5.2158 5.2011 1.1793 1.8499 5.1170 5.1170 3.2172 3.2912 10.2855 10.5007 10.3119 10.4712 1.1250 1.6374 10.2603 10.2603 4.5416 A' .5319 .5506 .5501 .5519 .5509 .5151 .5223 .5334 .5334 .5347 .5794 .6072 .6071 .6064 .6057 .5223 .5839 .6110 .6110 .5981 .6485 .6814 .6813 .6824 .6826 .5478 .6273 .6835 .6835 .6663 .7072 .7202 .7201 .7206 .7202 .5768 .6472 .7200 .7200 .7024 .7704 .7544 .7541 .7548 .7543 .6275 .6687 .7549 .7549 .7519 B" .0603 .1111 .1098 .1144 .1119 .0273 .0454 .0703 .0703 .0732 .1627 .2682 .2679 .2655 .2634 .0396 .1940 .2793 .2793 .2376 .3377 .5504 .5503 .5550 .5560 .0798 .3172 .5590 .5590 .4705 .4968 .7357 .7364 .7368 .7359 .1025 .3512 .7322 .7322 .5945 .6222 .8739 .8763 .8743 .8761 .0746 .2960 .8737 .8737 .7188 124 Table 10.2.1.1.1.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P = 5, mNz = A,pNz = .8 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 N T 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 N T 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OLEX Rej Fp .0977 .05596 .0089 .00617 .0089 .00615 .0093 .00637 .0093 .00634 .0973 .06160 .0216 .01455 .0139 .00877 .0139 .00877 .0140 .00859 .1010 .06659 .0094 .00770 .0093 .00764 .0096 .00787 .0095 .00779 .1266 .08413 .0236 .01883 .0121 .00954 .0121 .00954 .0176 .01318 .0949 .09429 .0095 .01453 .0093 .01434 .0098 .01483 .0097 .01467 .2011 .14403 .0363 .03922 .0106 .01571 .0106 .01571 .0226 .02747 .0967 .14325 .0106 .02849 .0104 .02777 .0108 .02899 .0107 .02819 .3427 .27641 .0814 .09421 .0110 .02993 .0110 .02993 .0367 .06639 .0986 .24494 .0096 .06923 .0090 .06623 .0101 .07022 .0092 .06717 .5664 .52793 .2370 .28114 .0104 .07097 .0104 .07097 .0587 .16678 .05070 .00445 .00445 .00465 .00465 .05845 .01270 .00695 .00695 .00735 .05190 .00470 .00465 .00480 .00475 .07790 .01450 .00605 .00605 .00890 .04865 .00475 .00465 .00490 .00485 .12675 .02355 .00530 .00530 .01145 .04960 .00540 .00530 .00550 .00545 .22780 .05645 .00560 .00560 .01875 .05075 .00485 .00455 .00510 .00465 .40995 .17815 .00525 .00525 .03025 Fn .94273 .99340 .99343 .99320 .99324 .93761 .98499 .99078 .99078 .99110 .92974 .99155 .99161 .99136 .99145 .91431 .98009 .98959 .98959 .98575 .89430 .98303 .98324 .98269 .98287 .85165 .95686 .98169 .98169 .96853 .83334 .96574 .96661 .96514 .96613 .71144 .89635 .96399 .96399 .92170 .70651 .91468 .91835 .91350 .91720 .44258 .69311 .91260 .91260 .79909 Tp Tn .05728 .94930 .00660 .99555 .00658 .99555 .00680 .99535 .00676 .99535 .06239 .94155 .01501 .98730 .00923 .99305 .00923 .99305 .00890 .99265 .07026 .94810 .00845 .99530 .00839 .99535 .00864 .99520 .00855 .99525 .08569 .92210 .01991 .98550 .01041 .99395 .01041 .99395 .01425 .99110 .10570 .95135 .01698 .99525 .01676 .99535 .01731 .99510 .01713 .99515 .14835 .87325 .04314 .97645 .01831 .99470 .01831 .99470 .03148 .98855 .16666 .95040 .03426 .99460 .03339 .99470 .03486 .99450 .03388 .99455 .28856 .77220 .10365 .94355 .03601 .99440 .03601 .99440 .07830 .98125 .29349 .94925 .08533 .99515 .08165 .99545 .08650 .99490 .08280 .99535 .55742 .59005 .30689 .82185 .08740 .99475 .08740 .99475 .20091 .96975 s.FnTp d' A' B" s.Rej s.FpTn P .0600 1.1014 .5304 .0574 .07286 .15695 .08279 .02464 .04696 .02867 .1375 1.4195 .5820 .1935 .02448 .04696 .02851 .1362 1.4148 .5813 .1917 .02507 .04800 .02911 .1332 1.4014 .5796 .1867 .02490 .04800 .02899 .1312 1.3946 .5786 .1841 .14251 .18871 .14700 .0329 1.0523 .5168 .0305 .06925 .08994 .07217 .0655 1.1552 .5391 .0823 .02946 .05854 .03390 .1034 1.2827 .5622 .1395 .02946 .05854 .03390 .1034 1.2827 .5622 .1395 .03577 .06302 .03932 .0699 1.1831 .5439 .0946 .08158 .15703 .09339 .1529 1.2674 .5702 .1407 .02819 .04825 .03303 .2083 1.6807 .6119 .2834 .02785 .04800 .03269 .2092 1.6858 .6123 .2849 .02852 .04876 .03343 .2091 1.6816 .6120 .2838 .02808 .04850 .03296 .2090 1.6822 .6121 .2840 .16798 .21791 .17295 .0515 1.0745 .5248 .0434 .08760 .09870 .09160 .1279 1.3115 .5693 .1546 .03135 .05467 .03678 .1981 1.6119 .6058 .2630 .03135 .05467 .03678 .1981 1.6119 .6058 .2630 .04486 .06687 .05136 .1794 1.5054 .5952 .2285 .09699 .15219 .11506 .4084 1.8107 .6499 .3426 .03935 .04850 .04750 .4729 3.0482 .6831 .5585 .03861 .04800 .04673 .4751 3.0733 .6837 .5615 .03971 .04926 .04795 .4701 3.0152 .6824 .5545 .03909 .04901 .04728 .4692 3.0147 .6823 .5543 .21859 .27114 .22782 .0984 1.1135 .5426 .0661 .13640 .12893 .14689 .2700 1.6481 .6185 .2844 .04089 .05121 .04932 .4657 2.9497 .6809 .5465 .04089 .05121 .04932 .4657 2.9497 .6809 .5465 .06421 .07579 .07515 .4156 2.3610 .6641 .4585 .11968 .15360 .14418 .6813 2.4381 .7064 .4932 .05573 .05264 .06805 .7276 4.9036 .7179 .7207 .05361 .05217 .06557 .7224 4.8808 .7174 .7192 .05640 .05310 .06889 .7290 4.8935 .7179 .7203 .05404 .05287 .06605 .7192 4.8183 .7169 .7158 .29018 .34419 .30291 .1885 1.1308 .5723 .0771 .22361 .20197 .23951 .3243 1.5864 .6264 .2712 .05734 .05356 .07004 .7374 4.9457 .7188 .7235 .05734 .05356 .07004 .7374 4.9457 .7188 .7235 .09915 .09708 .11808 .6637 3.1912 .7053 .5937 .14387 .15573 .17520 1.0944 3.2981 .7707 .6229 .08815 .04951 .10918 1.2162 11.0893 .7560 .8835 .08228 .04801 .10212 1.2142 11.3572 .7554 .8861 .08883 .05074 .10996 1.2063 10.7128 .7557 .8793 .08280 .04851 .10277 1.2143 11.2569 .7556 .8851 .31484 .40458 .32721 .3721 1.0156 .6286 .0098 .36687 .34205 .38986 .4177 1.3473 .6440 .1846 .08916 .05146 .11037 1.2020 10.5210 .7556 .8771 .08916 .05146 .11037 1.2020 10.5210 .7556 .8771 .14275 .12293 .17253 1.0388 4.0975 .7564 .6910 125 Table 10.2.1.1.2.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P = 5, mNz = .3, pNz = .2 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OLEX Rej .3341 .06647 .0404 .00811 .0400 .00805 .0421 .00843 .0417 .00837 .2613 .07737 .0692 .02003 .0590 .01153 .0590 .01153 .0579 .01175 .3355 .09161 .0402 .01630 .0400 .01619 .0410 .01662 .0406 .01645 .3449 .11934 .0802 .03566 .0473 .01859 .0473 .01859 .0786 .02234 .3246 .15315 .0391 .05302 .0373 .05224 .0398 .05347 .0379 .05270 .4933 .21752 .1815 .11247 .0424 .05531 .0424 .05531 .0921 .06054 .3261 .21336 .0435 .12583 .0381 .12361 .0447 .12649 .0385 .12415 .6012 .33430 .3774 .24973 .0469 .12755 .0469 .12755 .0869 .12243 .3366 .23888 .0486 .19358 .0391 .19171 .0498 .19387 .0404 .19210 .6511 .38196 .5084 .32576 .0503 .19400 .0503 .19400 .0599 .18362 Fp Fn .05011 .86810 .00513 .97995 .00506 .98000 .00534 .97920 .00529 .97930 .07288 .90465 .01550 .96185 .00756 .97260 .00756 .97260 .00860 .97565 .05055 .74415 .00516 .93915 .00511 .93950 .00528 .93800 .00519 .93850 .10489 .82285 .02169 .90845 .00610 .93145 .00610 .93145 .01135 .93370 .04953 .43235 .00508 .75520 .00480 .75800 .00516 .75330 .00488 .75600 .16113 .55690 .05609 .66200 .00555 .74565 .00555 .74565 .01324 .75025 .04990 .13280 .00569 .39360 .00496 .40180 .00585 .39095 .00501 .39930 .21274 .17945 .12005 .23155 .00615 .38685 .00615 .38685 .01215 .43645 .05063 .00810 .00641 .05775 .00505 .06165 .00656 .05690 .00524 .06045 .22979 .00935 .16021 .01205 .00663 .05650 .00663 .05650 .00856 .11615 Tp Tn .13190 .94989 .02005 .99488 .02000 .99494 .02080 .99466 .02070 .99471 .09535 .92713 .03815 .98450 .02740 .99244 .02740 .99244 .02435 .99140 .25585 .94945 .06085 .99484 .06050 .99489 .06200 .99473 .06150 .99481 .17715 .89511 .09155 .97831 .06855 .99390 .06855 .99390 .06630 .98865 .56765 .95048 .24480 .99492 .24200 .99520 .24670 .99484 .24400 .99513 .44310 .83888 .33800 .94391 .25435 .99445 .25435 .99445 .24975 .98676 .86720 .95010 .60640 .99431 .59820 .99504 .60905 .99415 .60070 .99499 .82055 .78726 .76845 .87995 .61315 .99385 .61315 .99385 .56355 .98785 .99190 .94938 .94225 .99359 .93835 .99495 .94310 .99344 .93955 .99476 .99065 .77021 .98795 .83979 .94350 .99338 .94350 .99338 .88385 .99144 s:FnTp d' s.Rej s.FpTn P .07912 .07848 .23876 .5263 2.0681 .02831 .02516 .09937 .5146 3.2826 .02800 .02496 .09925 .5178 3.3116 .02881 .02564 .10109 .5157 3.2662 .02851 .02553 .10086 .5169 3.2802 .15867 .16110 .23289 .1462 1.2238 .08324 .07839 .15173 .3845 2.1286 .03356 .03058 .11511 .5089 3.0248 .03356 .03058 .11511 .5089 3.0248 .04158 .03785 .11416 .4112 2.4476 .08811 .07887 .30623 .9834 3.0918 .03986 .02556 .16963 1.0171 8.0956 .03946 .02532 .16850 1.0175 8.1293 .04032 .02587 .17105 1.0191 8.0588 .03973 .02549 .16977 1.0208 8.1281 .18959 .18960 .30332 .3279 1.4297 .11050 .09670 .23558 .6888 3.1716 .04235 .02779 .17925 1.0196 7.6575 .04235 .02779 .17925 1.0196 7.6575 .05304 .04168 .18113 .7745 4.3265 .09397 .07938 .34512 1.8199 3.8415 .06548 .02573 .30657 1.8797 21.4449 .06407 .02479 .30240 1.8900 22.3966 .06574 .02592 .30765 1.8798 21.2089 .06435 .02496 .30344 1.8911 22.1877 .22278 .22677 .39985 .8467 1.6155 .18009 .15600 .40764 1.1706 3.2360 .06674 .02705 .31031 1.8787 20.2114 .06674 .02705 .31031 1.8787 20.2114 .07374 .04446 .30440 1.5439 9.3405 .07934 .07968 .23784 2.7591 2.0849 .07293 .02722 .34357 2.8009 23.7231 .07129 .02541 .34032 2.8271 26.9297 .07306 .02762 .34329 2.7979 23.0938 .07126 .02552 .34010 2.8301 26.6469 .22170 .25384 .30238 1.7144 .9019 .20617 .21272 .36541 1.9085 1.5232 .07307 .02834 .34225 2.7910 22.0258 .07307 .02834 .34225 2.7910 22.0258 .07256 .04156 .31176 2.4123 12.4753 .06418 .07884 .06313 4.0432 .2127 .04041 .02907 .16536 4.0625 6.4102 .03940 .02543 .16979 4.1135 8.3406 .04042 .02935 .16421 4.0617 6.2076 .03934 .02597 .16830 4.1107 7.9512 .20613 .25636 .07027 3.0910 .0828 .18736 .23125 .08649 3.2491 .1287 .04040 .02947 .16374 4.0618 6.1217 .04040 .02947 .16374 4.0618 6.1217 .05259 .03615 .21387 3.5785 8.4014 A' B" .6765 .4127 .6898 .5879 .6905 .5911 .6897 .5865 .6900 .5880 .5650 .1215 .6542 .4126 .6860 .5605 .6860 .5605 .6657 .4718 .7547 .5973 .7428 .8351 .7428 .8357 .7430 .8345 .7431 .8359 .6222 .2165 .7086 .5935 .7435 .8266 .7435 .8266 .7211 .6931 .8645 .6781 .8051 .9468 .8046 .9492 .8055 .9462 .8051 .9487 .7431 .2922 .7832 .6173 .8071 .9434 .8071 .9434 .7967 .8697 .9507 .4168 .8987 .9537 .8970 .9597 .8993 .9523 .8976 .9593 .8782 -.0643 .8952 .2549 .9002 .9498 .9002 .9498 .8842 .9069 .9851 -.7136 .9838 .7904 .9832 .8402 .9839 .7833 .9834 .8319 .9390 -.9005 .9559 -.8374 .9840 .7802 .9840 .7802 .9683 .8472 Table 10.2.1.1.2.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P *!,/> = 5, mNz = .3, pNz = A N:p OEX Rej N 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 N 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 N 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 N 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 N 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .2595 .08155 .0334 .01112 .0332 .01101 .0349 .01145 .0347 .01132 .2870 .11327 .0688 .03372 .0458 .01534 .0458 .01534 .0629 .02057 .2614 .13266 .0318 .02842 .0310 .02768 .0323 .02884 .0316 .02814 .4361 .21673 .1163 .08418 .0377 .03250 .0377 .03250 .0904 .05165 .2471 .25692 .0306 .10354 .0261 .09911 .0307 .10450 .0264 .10000 .7010 .46404 .3699 .30613 .0325 .10760 .0325 .10760 .1135 .15556 .2511 .37594 .0383 .25408 .0286 .24247 .0391 .25538 .0291 .24376 .8065 .62774 .6920 57143 .0402 .25710 .0402 .25710 .1029 .28203 .2580 .42667 .0453 .38503 .0295 .37830 .0463 .38559 .0300 .37878 .8267 .65286 .7590 .61963 .0468 .38577 .0468 .38577 .0660 .37660 Fp .05010 .00567 .00563 .00593 .00588 .10112 .02338 .00788 .00788 .01243 .05095 .00550 .00528 .00558 .00538 .17462 .04837 .00658 .00658 .01747 .04878 .00530 .00445 .00532 .00450 .32370 .16872 .00563 .00563 .02212 .04907 .00670 .00497 .00688 .00505 .41073 .33643 .00710 .00710 .01968 .05028 .00817 .00522 .00835 .00532 .42207 .36688 .00843 .00843 .01275 Fn .87128 .98070 .98093 .98028 .98053 .86850 .95078 .97348 .97348 .96723 .74478 .93720 .93873 .93628 .93773 .72010 .86210 .92863 .92863 .89708 .43088 .74910 .75890 .74673 .75675 .32545 .48775 .73945 .73945 .64427 .13375 .37485 .40128 .37188 .39817 .04675 .07608 .36790 .36790 .32445 .00875 .04968 .06207 .04855 .06103 .00095 .00125 .04822 .04822 .07763 Tp Tn .12873 .94990 .01930 .99433 .01908 .99437 .01973 .99407 .01948 .99412 .13150 .89888 .04923 .97662 .02653 .99212 .02653 .99212 .03278 .98757 .25523 .94905 .06280 .99450 .06128 .99472 .06373 .99442 .06228 .99462 .27990 .82538 .13790 .95163 .07137 .99342 .07137 .99342 .10293 .98253 .56913 .95122 .25090 .99470 .24110 .99555 .25328 .99468 .24325 .99550 .67455 .67630 51225 .83128 .26055 .99437 .26055 .99437 .35573 .97788 .86625 .95093 .62515 .99330 59873 .99503 .62813 .99312 .60183 .99495 .95325 .58927 .92393 .66357 .63210 .99290 .63210 .99290 .67555 .98032 .99125 .94972 .95033 .99183 .93793 .99478 .95145 .99165 .93898 .99468 .99905 .57793 .99875 .63312 .95178 .99157 .95178 .99157 .92237 .98725 s.Rej s.FpTn s.FnTp .09023 .03395 .03346 .03444 .03388 .19815 .12252 .03973 .03973 .05741 .11105 .05593 .05374 .05636 .05439 .25517 .19423 .05960 .05960 .08518 .12223 .10147 .09563 .10190 .09598 .26902 .30551 .10297 .10297 .11230 .09076 .11363 .10947 .11340 .10907 .20817 .23765 .11305 .11305 .09502 .05840 .05255 .05388 .05203 .05359 .18914 .19262 .05183 .05183 .05954 .09145 .03075 .03067 .03151 .03129 .20122 .11061 .03654 .03654 .05197 .09311 .03096 .02986 .03117 .03012 .25667 .16451 .03405 .03405 .05954 .09265 .03046 .02758 .03050 .02773 .30700 .27889 .03139 .03139 .06661 .09180 .03439 .02949 .03504 .02971 .31854 .32571 .03566 .03566 .06155 .09269 .03895 .03070 .03937 .03104 .31456 .31995 .03953 .03953 .05144 .17787 .07082 .06986 .07163 .07055 .24060 .16530 .08241 .08241 .10251 .23724 .13022 .12571 .13128 .12730 .32356 .27994 .13821 .13821 .17401 .27221 .24651 .23434 .24762 .23507 .33581 .43296 .24989 .24989 .25228 .18314 .27771 .27056 .27697 .26948 .13487 .21705 .27592 .27592 .21558 .04781 .11679 .12735 .11525 .12643 .01578 .02058 .11465 .11465 .12473 d' .5114 .4638 .4611 .4566 .4544 .1559 .3360 .4798 .4798 .4020 .9776 1.0110 1.0126 1.0132 1.0142 .3529 5711 1.0136 1.0136 .8441 1.8309 1.8840 1.9131 1.8903 1.9162 .9099 .9900 1.8927 1.8927 1.6420 2.7628 2.7920 2.8282 2.7902 2.8305 1.9029 1.8542 2.7896 2.7896 2.5156 4.0181 4.0494 4.0987 4.0523 4.1007 3.3020 3.3635 4.0519 4.0519 3.6550 P 2.0339 2.9065 2.8928 2.8424 2.8330 1.2052 1.8435 2.8382 2.8382 2.2728 3.0685 7.8431 7.9739 7.8276 7.9418 1.3075 2.1934 7.3836 7.3836 4.1538 3.8856 20.9057 23.9137 20.9513 23.7905 1.0022 15835 20.1968 20.1968 7.0667 2.1235 20.2240 26.9005 19.6973 26.4508 .2513 .3921 19.0988 19.0988 7.5296 .2289 4.5967 8.1466 4.4262 7.9041 .0082 .0110 4.3643 4.3643 4.4143 A' B" .6734 .4042 .6800 .5412 .6795 .5392 .6783 .5325 .6779 .5311 .5662 .1137 .6379 .3441 .6804 5350 .6804 .5350 .6603 .4416 .7539 5944 .7425 .8299 .7425 .8326 .7427 .8297 .7427 .8320 .6259 .1661 .6858 .4418 .7432 .8204 .7432 .8204 .7293 .6865 .8653 .6818 .8064 .9454 .8048 .9527 .8071 .9456 .8053 .9525 .7597 .0014 .7710 .2809 .8087 .9435 .8087 .9435 .8197 .8275 .9507 .4258 .9030 .9448 .8971 .9597 .9036 .9431 .8979 .9589 .8724 -.6890 .8803 -.5211 .9046 .9412 .9046 .9412 .9100 .8382 .9850 -.6926 .9853 .7071 .9830 .8363 .9856 .6960 .9832 .8310 .8940 -.9922 .9077 -.9893 .9856 .6918 .9856 .6918 .9769 .7010 Table 10.2.1.1.2.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * L / ? = 5, mNz = .3, pNz = .6 N:p OLEX Fp Fn Rej NT 2 .1746 .09861 .04925 .86848 MB 2 .0221 .01426 .00570 .98003 RB 2 .0220 .01401 .00568 .98043 MD 2 .0228 .01462 .00588 .97955 RD 2 .0227 .01434 .00585 .98000 OR 2 .3013 .16897 .14363 .81413 OS 2 .0697 .05581 .03665 .93142 OF 2 .0304 .01969 .00795 .97248 CF 2 .0304 .01969 .00795 .97248 CS 2 .0544 .03575 .01605 .95112 NT 4 .1772 .17217 .05033 .74660 MB 4 .0198 .04021 .00513 .93640 RB 4 .0191 .03846 .00493 .93918 MD 4 .0202 .04088 .00523 .93535 RD 4 .0196 .03905 .00505 .93828 OR 4 .4932 .33960 .26645 .61163 OS 4 .1643 .15993 .09678 .79797 OF 4 .0235 .04607 .00610 .92728 CF 4 .0235 .04607 .00610 .92728 CS 4 .0823 .09617 .02440 .85598 NT 10 .1709 .35992 .04878 .43265 MB 10 .0210 .15865 .00555 .73928 RB 10 .0169 .14641 .00435 .75888 MD 10 .0214 .16034 .00565 .73653 RD 10 .0170 .14782 .00438 .75655 OR 10 .7779 .66984 .47820 .20240 OS 10 5132 .51031 .32008 .36287 OF 10 .0231 .16486 .00615 .72933 CF 10 .0231 .16486 .00615 .72933 CS 10 .1165 .27527 .03405 56392 NT 20 .1719 .53840 .04925 .13550 MB 20 .0323 .39077 .00885 .35462 RB 20 .0193 .36077 .00508 .40210 MD 20 .0328 .39258 .00898 .35168 RD 20 .0194 .36249 .00510 .39925 OR 20 .8715 .81593 56752 .01847 OS 20 .8101 .78564 .52093 .03788 OF 20 .0336 .39501 .00918 .34777 CF 20 .0336 .39501 .00918 .34777 CS 20 .0990 .46087 .02948 .25153 NT 40 .1760 .61530 .05060 .00823 MB 40 .0442 58080 .01228 .04018 RB 40 .0213 .56533 .00563 .06153 MD 40 .0448 .58120 .01245 .03963 RD 40 .0217 56604 .00575 .06043 OR 40 .8846 .83130 57855 .00020 OS 40 .8462 .81803 .54555 .00032 OF 40 .0453 58145 .01258 .03930 CF 40 .0453 .58145 .01258 .03930 CS 40 .0700 .57913 .02115 .04888 Tp Tn .13152 .95075 .01997 .99430 .01957 .99433 .02045 .99413 .02000 .99415 .18587 .85638 .06858 .96335 .02752 .99205 .02752 .99205 .04888 .98395 .25340 .94968 .06360 .99488 .06082 .99508 .06465 .99478 .06172 .99495 .38837 .73355 .20203 .90322 .07272 .99390 .07272 .99390 .14402 .97560 .56735 .95123 .26072 .99445 .24112 .99565 .26347 .99435 .24345 .99563 .79760 .52180 .63713 .67993 .27067 .99385 .27067 .99385 .43608 .96595 .86450 .95075 .64538 .99115 59790 .99492 .64832 .99103 .60075 .99490 .98153 .43248 .96212 .47908 .65223 .99083 .65223 .99083 .74847 .97053 .99177 .94940 .95982 .98773 .93847 .99438 .96037 .98755 .93957 .99425 .99980 .42145 .99968 .45445 .96070 .98743 .96070 .98743 .95112 .97885 A' B" s.Rej s.FpTn s.FnTp d' P .10585 .11369 .16134 .5329 2.0927 .6780 .4185 .04018 .03847 .06224 .4758 2.9760 .6822 .5508 .03892 .03840 .06019 .4690 2.9368 .6810 5454 .04063 .03901 .06295 .4750 2.9566 .6818 .5485 .03927 .03893 .06069 .4673 2.9123 .6804 .5423 .25351 .26146 .28343 .1709 1.1821 .5691 .1033 .17615 .15687 .20788 .3045 1.6471 .6247 .2881 .04726 .04582 .07281 .4926 2.9049 .6827 .5447 .04726 .04582 .07281 .4926 2.9049 .6827 .5447 .08102 .07189 .11663 .4874 2.5238 .6763 .4929 .14321 .11543 .22706 .9779 3.0872 .7538 .5967 .07388 .03664 .12051 1.0420 8.4341 .7445 .8423 .06892 .03581 .11250 1.0331 8.4387 .7438 .8420 .07480 .03697 .12205 1.0437 8.3965 .7447 .8417 .06956 .03622 .11356 1.0319 8.3477 .7438 .8403 .32338 .33070 .36746 .3400 1.1667 .6200 .0972 .29532 .24854 .35451 .4658 1.6439 .6594 .2969 .07956 .04001 .12983 1.0505 8.0127 .7458 .8350 .07956 .04001 .12983 1.0505 8.0127 .7458 .8350 .13017 .08842 .19737 .9079 3.9619 .7383 .6763 .16567 .11496 .26572 1.8265 3.8891 .8648 .6821 .14990 .03930 .24705 1.8984 20.4726 .8088 .9443 .13466 .03382 .22319 1.9209 24.4052 .8049 .9538 .15056 .03960 .24805 1.9006 20.2592 .8095 .9437 .13500 .03391 .22375 1.9264 24.4076 .8055 .9538 .26548 .35447 .28497 .8878 .7079 .7531 -.2144 .37763 .37451 .43608 .8183 1.0489 .7410 .0302 .15191 .04168 .25022 1.8926 19.0490 .8109 .9399 .15191 .04168 .25022 1.8926 19.0490 .8109 .9399 .16352 .10095 .25744 1.6634 5.2131 .8345 .7641 .11486 .11549 .17437 2.7529 2.1361 .9501 .4288 .16779 .05059 .27541 2.7447 15.5378 .9071 .9262 .15858 .03699 .26311 2.8186 26.4025 .8968 .9589 .16746 .05088 .27485 2.7474 15.3025 .9078 .9249 .15805 .03707 .26225 2.8243 26.2385 .8976 .9586 .15025 .33915 .07644 1.9164 .1151 .8448 -.8625 .19690 .36136 .16212 1.7233 .2069 .8449 -.7451 .16668 .05133 .27357 2.7498 14.9488 .9087 .9229 .16668 .05133 .27357 2.7498 14.9488 .9087 .9229 .12303 .09586 .19082 2.5582 4.7546 .9254 .7362 .05299 .11754 .03992 4.0375 .2159 .9851 -.7094 .06383 .06026 .09684 3.9970 2.7154 .9866 .5217 .07168 .03905 .11637 4.0769 7.5668 .9830 .8234 .06336 .06069 .09593 3.9979 2.6524 .9867 5117 .07110 .03959 .11532 4.0783 7.3167 .9833 .8171 .13308 .33229 .00577 3.3419 .0019 .8552 -.9984 .14051 .35016 .01166 3.3025 .0029 .8634 -.9974 .06322 .06092 .09557 3.9980 2.6116 .9868 .5050 .06322 .06092 .09557 3.9980 2.6116 .9868 .5050 .06378 .08376 .08655 3.6863 1.9953 .9820 .3838 128 Table 10.2.1.1.2.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P*I,p = 5, mNz = .3,pNz = .8 N:p OEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .0928 .11405 .0118 .01748 .0115 .01712 .0121 .01793 .0117 .01748 .2678 .21484 .0643 .07329 .0151 .02379 .0151 .02379 .0357 .04917 .0985 .21273 .0096 .05180 .0090 .04911 .0103 .05270 .0094 .04984 .4944 .45181 .1931 .22921 .0118 .05928 .0118 .05928 .0541 .14081 .0905 .46394 .0137 .21808 .0094 .19478 .0139 .22072 .0098 .19667 .7823 .83714 .6129 .71032 .0150 .22719 .0150 .22719 .0782 .40898 .0957 .70150 .0263 .54470 .0103 .47914 .0268 .54719 .0104 .48144 .8558 .93381 .8332 .92775 .0272 .55010 .0272 .55010 .0800 .65106 .0966 .80350 .0429 .78283 .0098 .75151 .0436 .78334 .0101 .75242 .8656 .94006 .8472 .93820 .0436 .78354 .0436 .78354 .0562 .78479 Fp .04855 .00600 .00585 .00625 .00595 .18130 .04835 .00785 .00785 .01940 .05140 .00480 .00450 .00515 .00470 .35815 .15095 .00600 .00600 .02875 .04750 .00720 .00475 .00730 .00495 .61135 .48660 .00790 .00790 .04150 .05010 .01385 .00525 .01410 .00530 .69240 .67900 .01435 .01435 .04300 .05060 .02295 .00495 .02330 .00510 .70055 .69135 .02330 .02330 .03050 Fn .86958 .97965 .98006 .97915 .97964 .77678 .92048 .97223 .97223 .94339 .74694 .93645 .93974 .93541 .93888 .52478 .75123 .92740 .92740 .83118 .43195 .72920 .75771 .72592 .75540 .10641 .23375 .71799 .71799 .49915 .13565 .32259 .40239 .31954 .39953 .00584 .01006 .31596 .31596 .19693 .00828 .02720 .06185 .02665 .06075 .00006 .00009 .02640 .02640 .02664 Tp Tn .13043 .95145 .02035 .99400 .01994 .99415 .02085 .99375 .02036 .99405 .22323 .81870 .07953 .95165 .02778 .99215 .02778 .99215 .05661 .98060 .25306 .94860 .06355 .99520 .06026 .99550 .06459 .99485 .06113 .99530 .47523 .64185 .24878 .84905 .07260 .99400 .07260 .99400 .16883 .97125 36805 .95250 .27080 .99280 .24229 .99525 .27408 .99270 .24460 .99505 .89359 .38865 .76625 .51340 .28201 .99210 .28201 .99210 .50085 .95850 .86435 .94990 .67741 .98615 .59761 .99475 .68046 .98590 .60048 .99470 .99416 .30760 .98994 .32100 .68404 .98565 .68404 .98565 .80308 .95700 .99173 .94940 .97280 .97705 .93815 .99505 .97335 .97670 .93925 .99490 .99994 .29945 .99991 .30865 .97360 .97670 .97360 .97670 .97336 .96950 s.Rej s.FpTn s.FnTp d' .11483 .04471 .04314 .04538 .04350 .28345 .20763 .05295 .05295 .10119 .15928 .08530 .07806 .08628 .07879 .34340 .35809 .09219 .09219 .16284 .18280 .17984 .15307 .18085 .15343 .19123 .36218 .18268 .18268 .18615 .12142 .19752 .17945 .19680 .17907 .07890 .10196 .19569 .19569 .12120 .04089 .06111 .08008 .06057 .07934 .07143 .07372 .06029 .06029 .05140 .15515 .05536 .05469 .05733 .05514 .32433 .19504 .06453 .06453 .10428 .15878 .04876 .04722 .05048 .04825 .40221 .32726 .05536 .05536 .12349 .15410 .06244 .04902 .06283 .05001 .39005 .43208 .06548 .06548 .14639 .15746 .08622 .05194 .08690 .05217 .36235 .37533 .08786 .08786 .15051 .15825 .11158 .05001 .11229 .05074 .35675 .36807 .11229 .11229 .12931 .13707 .5347 .05393 .4656 .05209 .4660 .05460 .4612 .05256 .4688 .29426 .1491 .21956 .2528 .06355 .5013 .06355 .5013 .11700 .4824 .19388 .9665 .10521 1.0643 .09637 1.0595 .10633 1.0482 .09727 1.0518 .35953 .3013 .37993 .3540 .11316 1.0554 .11316 1.0554 .19439 .9407 .22567 1.8410 .22261 1.8367 .19067 1.8946 .22385 1.8416 .19110 1.8877 .18914 .9630 .37858 .7601 .22600 1.8366 .22600 1.8366 .22730 1.7357 .14835 2.7440 .24436 2.6620 .22408 2.8061 .24346 2.6635 .22359 2.8102 .02993 2.0192 .07155 1.8591 .24206 2.6666 .24206 2.6666 .14686 2.5695 .03447 4.0356 .07039 3.9199 .09975 4.1187 .06957 3.9224 .09881 4.1175 .00279 3.3101 .00331 3.2529 .06922 3.9265 .06922 3.9265 .05586 3.8062 P 2.1047 2.8898 2.9046 2.8451 2.9129 1.1327 1.4740 2.9604 2.9604 2.4121 3.0336 8.9356 9.0809 8.4989 8.8328 1.0662 1.3537 8.1210 8.1210 3.8358 3.9712 16.5755 22.6212 16.4732 21.9169 .4790 .7685 15.5819 15.5819 4.4932 2.1088 10.1483 25.6225 9.9535 25.3602 .0472 .0748 9.7574 9.7574 3.0354 .2169 1.1531 8.5121 1.1192 8.1733 .0007 .0010 1.1105 1.1105 .8935 A' B" .6785 .4212 .6799 .5395 .6802 .5413 .6787 .5335 .6806 .5426 .5598 .0776 .6062 .2281 .6844 .5523 .6844 .5523 .6738 .4747 .7524 .5899 .7459 .8514 .7453 .8534 .7450 .8437 .7449 .8493 .6072 .0407 .6271 .1864 .7461 .8373 .7461 .8373 .7435 .6681 .8657 .6886 .8097 .9301 .8048 .9498 .8105 .9297 .8052 .9481 .7605 -.4284 .7274 -.1649 .8121 .9255 .8121 .9255 .8491 .7255 .9498 .4226 .9131 .8823 .8967 .9575 .9138 .8798 .8974 .9570 .8211 -.9469 .8207 -.9126 .9146 .8772 .9146 .8772 .9352 .5870 .9851 -.7082 .9871 .0826 .9831 .8435 .9872 .0654 .9834 .8367 .8248 -.9994 .8271 -.9992 .9873 .0608 .9873 .0608 .9853 -.0656 Table 10.2.1.1.3.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P = 5, mNz- .5,pNz- .2 N:p CtEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn Tp Tn .3249 .10477 .04985 .67555 .32445 .95015 .0412 .01981 .00531 .92220 .07780 .99469 .0406 .01960 .00524 .92295 .07705 .99476 .0425 .02026 .00550 .92070 .07930 .99450 .0420 .02002 .00541 .92155 .07845 .99459 .3677 .13804 .11776 .78085 .21915 .88224 .1065 .05218 .03158 .86540 .13460 .96843 .0577 .02607 .00750 .89965 .10035 .99250 .0577 .02607 .00750 .89965 .10035 .99250 .0869 .02822 .01313 .91140 .08860 .98687 .3253 .16930 .05081 .35675 .64325 .94919 .0415 .06358 .00549 .70405 .29595 .99451 .0384 .06254 .00501 .70735 .29265 .99499 .0420 .06425 .00556 .70100 .29900 .99444 .0389 .06313 .00509 .70470 .29530 .99491 5198 .25273 .18345 .47015 52985 .81655 .2271 .14898 .07841 56875 .43125 .92159 .0502 .06965 .00668 .67845 .32155 .99333 .0502 .06965 .00668 .67845 .32155 .99333 .1045 .07297 .01536 .69660 .30340 .98464 .3152 .23359 .05020 .03285 .96715 .94980 .0444 .17323 .00601 .15790 .84210 .99399 .0375 .17076 .00494 .16595 .83405 .99506 .0455 .17386 .00615 .15530 .84470 .99385 .0380 .17114 .00501 .16435 .83565 .99499 .6144 .37227 .22483 .03795 .96205 .77517 .4578 .31245 .15279 .04890 .95110 .84721 .0492 .17538 .00669 .14985 .85015 .99331 .0492 .17538 .00669 .14985 .85015 .99331 .0833 .16212 .01205 .23760 .76240 .98795 .3171 .24013 .05023 .00025 .99975 .94978 .0481 .20446 .00655 .00390 .99610 .99345 .0388 .20324 .00515 .00440 .99560 .99485 .0492 .20464 .00675 .00380 .99620 .99325 .0395 .20334 .00525 .00430 .99570 .99475 .6246 .38657 .23325 .00015 .99985 .76675 .4914 .33260 .16581 .00025 .99975 .83419 .0513 .20489 .00705 .00375 .99625 .99295 .0513 .20489 .00705 .00375 .99625 .99295 .0668 .20509 .00981 .01380 .98620 .99019 .3221 .24044 .05055 .00000 1.00000 .94945 .0483 .20516 .00645 .00000 1.00000 .99355 .0393 .20414 .00518 .00000 1.00000 .99483 .0495 .20529 .00661 .00000 1.00000 .99339 .0403 .20424 .00530 .00000 1.00000 .99470 .6266 .38802 .23503 .00000 1.00000 .76498 .4933 .33335 .16669 .00000 1.00000 .83331 .0502 .20537 .00671 .00000 1.00000 .99329 .0502 .20537 .00671 .00000 1.00000 .99329 .0661 .20782 .00979 .00005 .99995 .99021 s:FnTp s.Rej s.FpTn d' P A' B" .09115 .08020 .32034 1.1910 3.4957 .7838 .6446 .04380 .02601 .18855 1.1348 95377 .7511 .8628 .04308 .02584 .18666 1.1345 9.5883 .7511 .8635 .04429 .02654 .19043 1.1329 9.3830 .7512 .8606 .04347 .02623 .18808 1.1327 9.4403 .7511 .8614 .20768 .20819 .32796 .4112 1.4966 .6444 .2445 .13905 .12356 .28052 .7532 3.0524 .7180 .5841 .05016 .03087 .21215 1.1528 8.4960 .7547 .8477 .05016 .03087 .21215 1.1528 8.4960 .7547 .8477 .06197 .04687 .21143 .8731 4.7553 .7321 .7235 .09224 .08196 .32683 2.0042 3.5699 .8863 .6527 .06804 .02704 .31820 2.0074 21.9988 .8184 .9490 .06625 .02559 .31407 2.0293 23.7208 .8180 .9529 .06830 .02725 .31908 2.0115 21.8371 .8191 .9486 .06642 .02581 .31458 2.0319 23.5069 .8186 .9525 .23727 .24782 .39440 .9772 1.4982 .7695 .249C .20773 .19144 .42387 1.2426 2.6840 .8003 5448 .07007 .02983 .32449 2.0109 19.1769 .8241 .941C .07007 .02983 .32449 2.0109 19.1769 .8241 .9410 .07972 .04864 .32126 1.6460 9.0404 .8105 .8664 .06994 .08349 .12449 3.4834 .7089 .9784 -.2002 .05601 .02921 .25382 3.5145 14.1605 .9585 .9140 .05522 .02584 .25669 3.5505 17.4252 .9568 .9314 .05579 .02947 .25217 3.5174 13.7287 .9591 .9110 .05504 .02607 .25548 3.5517 17.0853 .9572 .9299 .21364 .26277 .13977 2.5310 .2754 .9293 -.6536 .19864 .23742 .17568 2.6802 .4293 .9454 -.4713 .05559 .03078 .24885 3.5107 12.4491 .9603 .9009 .05559 .03078 .24885 3.5107 12.4491 .9603 .9009 .06487 .04303 .26154 2.9696 9.8629 .9359 .8767 .06639 .08293 .01118 5.1234 .0090 .9874 -.9896 .02565 .03027 .04399 5.1416 .6303 .9974 -.2523 .02284 .02631 .04670 5.1853 .8690 .9976 -.0782 .02609 .03090 .04343 5.1396 .5996 .9973 -.2783 .02300 .02659 .04617 5.1865 .8369 .9976 -.0990 .21449 .26810 .00866 4.3435 .0019 .9416 -.9983 .19608 .24505 .01118 4.4516 .0037 .9585 -.9964 .02659 .03158 .04314 5.1285 .5702 .9973 -.3041 .02659 .03158 .04314 5.1285 5702 .9973 -.3041 .03563 .03920 .08192 45 364 1.3445 .9940 .1669 .06587 .08234 .00000 5.9044 .0000 .9874 1.0000 .02357 .02946 .00000 6.7514 .0000 .9984-1.000C .02095 .02619 .00000 6.8288 .0000 .9987 1.0000 .02386 .02982 .00000 6.7425 .0000 .9983 1.0000 .02117 .02646 .00000 6.8205 .0000 .9987 1.0000 .21411 .26764 .00000 4.9873 .0000 .9412-1.0000 .19446 .24308 .00000 5.2322 .0000 .9583 1.0000 .02405 .03006 .00000 6.7372 .0000 .9983 1.0000 .02405 .03006 .00000 6.7372 .0000 .9983 1.0000 .03195 .03992 .00500 6.2250 .0079 .9975 -.9897 130 Table 10.2.1.1.3.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P*I,p = 5, mNz- .5,pNz = A N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OLEX Rej Fp Fn Tp Tn .2496 .15977 .04978 .67525 .32475 .95022 .0321 .03400 .00572 .92358 .07643 .99428 .0308 .03296 .00542 .92573 .07427 .99458 .0328 .03467 .00583 .92208 .07792 .99417 .0316 .03364 .00555 .92423 .07578 .99445 .4994 .27457 .21530 .63653 .36348 .78470 .1618 .12597 .07605 .79915 .20085 .92395 .0461 .04432 .00828 .90163 .09838 .99172 .0461 .04432 .00828 .90163 .09838 .99172 .1032 .07101 .02110 .85413 .14588 .97890 .2423 .28735 .05037 .35718 .64283 .94963 .0337 .12573 .00607 .69477 .30523 .99393 .0293 .11958 .00507 .70865 .29135 .99493 .0344 .12694 .00623 .69200 .30800 .99377 .0301 .12084 .00522 .70572 .29428 .99478 .7088 .51233 .34825 .24155 .75845 .65175 .4369 .37738 .21455 .37838 .62163 .78545 .0414 .13688 .00747 .66900 .33100 .99253 .0414 .13688 .00747 .66900 .33100 .99253 .1222 .18465 .02518 37615 .42385 .97482 .2317 .41567 .04875 .03395 .96605 .95125 .0403 .34855 .00750 .13988 .86013 .99250 .0274 .33740 .00497 .16395 .83605 .99503 .0413 .34936 .00768 .13813 .86187 .99232 .0277 .33818 .00502 .16208 .83793 .99498 .7957 .64683 .41425 .00430 .99570 .58575 .7179 .61054 .35592 .00753 .99248 .64408 .0433 .35138 .00808 .13368 .86633 .99192 .0433 .35138 .00808 .13368 .86633 .99192 .0873 .34674 .01817 .16040 .83960 .98183 .2403 .42984 .04998 .00038 .99963 .95002 .0432 .40362 .00813 .00315 .99685 .99187 .0259 .40110 .00473 .00435 .99565 .99527 .0443 .40381 .00833 .00297 .99703 .99167 .0263 .40117 .00483 .00433 .99568 .99517 .7959 .65439 .42398 .00000 1.00000 .57602 .7261 .62316 .37193 .00000 1.00000 .62807 .0457 .40404 .00863 .00285 .99715 .99137 .0457 .40404 .00863 .00285 .99715 .99137 .0728 .40573 .01463 .00762 .99238 .98537 .2391 .43048 .05080 .00000 1.00000 .94920 .0449 .40514 .00857 .00000 1.00000 .99143 .0284 .40309 .00515 .00000 1.00000 .99485 .0457 .40523 .00872 .00000 1.00000 .99128 .0290 .40317 .00528 .00000 1.00000 .99472 .7983 .65469 .42448 .00000 1.00000 .57552 .7317 .62250 .37083 .00000 1.00000 .62917 .0462 .40530 .00883 .00000 1.00000 .99117 .0462 .40530 .00883 .00000 1.00000 .99117 .0753 .409401 .01567 .00000 1.00000 .98433 s:Rej s:FpTn s:FnTp .11988 .06255 .05951 .06307 .06009 .28055 .24248 .07163 .07163 .10210 .12291 .11083 .10356 .11118 .10395 .26240 .31401 .11362 .11362 .11925 .06914 .08325 .08430 .08284 .08380 .19686 .20288 .08182 .08182 .07755 .05880 .02644 .02211 .02651 .02231 .19877 .20268 .02687 .02687 .03821 .06076 .02517 .01875 .02541 .01905 .19784 .20118 .02561 .02561 .03644 .09474 .03246 .03111 .03274 .03144 .28423 .21250 .03917 .03917 .06892 .09935 .03403 .02966 .03472 .03024 .32141 .31158 .03764 .03764 .07470 .09882 .03865 .03078 .03908 .03091 .32611 .33119 .04012 .04012 .06462 .09804 .04035 .03018 .04079 .03062 .33129 .33779 .04161 .04161 .05659 .10126 .04195 .03125 .04235 .03175 .32974 .33530 .04269 .04269 .06073 a" .25960 1.1925 .14609 1.0996 .14006 1.1034 .14726 1.1029 .14152 1.1055 .35553 .4390 .33793 .5936 .16593 1.1053 .16593 1.1053 .20998 .9773 .26963 2.0073 .26919 1.9988 .25450 2.0218 .26966 1.9971 .25523 2.0202 .30436 1.0914 .42306 1.1005 .27481 1.9968 .27481 1.9968 .26279 1.7648 .09867 3.4828 .20024 3.5133 .20763 3.5565 .19886 3.5124 .20641 3.5606 .03751 2.8442 .06423 2.8006 .19568 3.5143 .19568 3.5143 .16344 3.0860 .00968 5.0156 .02899 5.1346 .03419 5.2183 .02824 5.1445 .03410 5.2131 .00000 4.4566 .00000 4.5916 .02747 5.1455 .02747 5.1455 .04328 4.6063 .00000 5.9020 .00000 6.6487 .00000 6.8305 .00000 6.6423 .00000 6.8216 .00000 4.4553 .00000 4.5945 .00000 6.6374 .00000 6.6374 .00000 6.4177 P A' B" 3.5007 .7840 .6451 8.8159 .7491 .8509 9.0493 .7491 .8547 8.7881 .7494 .8506 8.9919 .7493 .8539 1.2836 .6491 .1559 1.9619 .6891 .3910 7.6728 .7517 .8305 7.6728 .7517 .8305 4.5169 .7457 .7156 3.5965 .8864 .6552 20.4070 .8203 .9447 23.4462 .8176 .9523 20.0027 .8209 .9435 22.9514 .8182 .9512 .8438 .7926 -.1067 1.3030 .7933 .1652 17.5773 .8259 .9352 17.5773 .8259 .9352 6.6603 .8374 .8173 .7456 .9785 -.1715 10.7411 .9626 .8835 17.1985 .9573 .9304 10.4254 .9630 .8796 16.9187 .9578 .9291 .0324 .8942 -.9653 .0557 .9074 -.9369 9.7492 .9640 .8705 9.7492 .9640 .8705 5.4622 .9537 .7661 .0132 .9874 -.9843 .4299 .9972 -.4396 .9274 .9977 -.0420 .3997 .9972 -.4717 .9056 .9977 -.0553 .0000 .8940 •1.0000 .0000 .9070-1.0000 .3727 .9971 -.5015 .3727 .9971 -.5015 .5668 .9944 -.3117 .0000 .9873 -1.0000 .0000 .9979 -1.0000 .0000 .9987 -1.0000 .0000 .9978 -1.000C .0000 .9987 -1.0000 .0000 .8939 -1.0000 .0000 .9073 -1.0000 .0000 .9978 -1.0000 .0000 .9978 -1.0000 .0000 .9961 -1.0000 Table 10.2.1.1.3.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I, p = 5, mNz = .5, pNz = .6 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn Tp Tn .1609 .21461 .04905 .67502 .32498 .95095 .0225 .05028 .00618 .92032 .07968 .99383 .0214 .04713 .00573 .92527 .07473 .99428 .0229 .05115 .00630 .91895 .08105 .99370 .0219 .04797 .00585 .92395 .07605 .99415 .5505 .43369 .32615 .49462 .50538 .67385 .2150 .23357 .14330 .70625 .29375 .85670 .0303 .06395 .00840 .89902 .10098 .99160 .0303 .06395 .00840 .89902 .10098 .99160 .0992 .14015 .03335 .78865 .21135 .96665 .1574 .40587 .04960 .35662 .64338 .95040 .0232 .19398 .00658 .68108 .31892 .99343 .0175 .17534 .00470 .71090 .28910 .99530 .0236 .19573 .00680 .67832 .32168 .99320 .0181 .17704 .00485 .70817 .29183 .99515 .7684 .70457 .49203 .15373 .84627 .50798 .5520 57298 .36315 .28713 .71287 .63685 .0284 .20951 .00818 .65627 .34373 .99183 .0284 .20951 .00818 .65627 .34373 .99183 .1206 .32448 .04085 .48643 .51357 .95915 .1506 .59886 .04818 .03402 .96598 .95183 .0371 .52855 .01148 .12673 .87327 .98853 .0172 .50225 .00473 .16607 .83393 .99528 .0377 .52972 .01170 .12493 .87507 .98830 .0174 .50338 .00480 .16423 .83577 .99520 .8400 .82267 .55998 .00220 .99780 .44003 .7936 .80702 .52530 .00517 .99483 .47470 .0400 .53235 .01245 .12105 .87895 .98755 .0400 .53235 .01245 .12105 .87895 .98755 .0895 .54754 .03090 .10803 .89197 .96910 .1520 .61940 .04898 .00032 .99968 .95103 .0437 .60409 .01380 .00238 .99762 .98620 .0180 .59937 .00508 .00443 .99557 .99492 .0444 .60424 .01410 .00233 .99767 .98590 .0183 .59942 .00518 .00442 .99558 .99483 .8457 .82655 .56638 .00000 1.00000 .43363 .8079 .81426 .53565 .00000 1.00000 .46435 .0448 .60431 .01420 .00228 .99772 .98580 .0448 .60431 .01420 .00228 .99772 .98580 .0803 .60809 .02833 .00540 .99460 .97168 .1597 .62071 .05178 .00000 1.00000 .94823 .0444 .60562 .01405 .00000 1.00000 .98595 .0186 .60213 .00533 .00000 1.00000 .99468 .0453 .60573 .01433 .00000 1.00000 .98568 .0191 .60218 .00545 .00000 1.00000 .99455 .8535 .82771 .56928 .00000 1.00000 .43073 .8111 .81453 .53633 .00000 1.00000 .46367 .0460 .60581 .01453 .00000 1.00000 .98548 .0460 .60581 .01453 .00000 1.00000 .98548 .0805 .61165].02913 .00000 1.00000 .97088 CLEX Rej s:Rej s:FpTn s.FnTp A' B" d' P .16638 .12246 .26725 1.2003 3.5435 .7848 .6493 .08881 .04293 .14453 1.0948 8.4981 .7491 .8456 .07991 .03983 .13072 1.0872 8.6556 .7482 .8479 .08963 .04342 .14593 1.0968 8.4566 .7494 .8449 .08069 .04020 .13201 1.0889 8.6051 .7484 .8471 .34546 .36505 .39575 .4641 1.1067 .6552 .0643 .35265 .31012 .42194 .5231 1.5229 .6719 .2565 .09982 .05042 .16239 1.1151 7.7255 .7525 .8319 .09982 .05042 .16239 1.1151 7.7255 .7525 .8319 .16306 .11226 .24410 1.0319 3.8955 .7566 .6759 .17856 .12774 .28569 2.0163 3.6388 .8869 .6591 .17931 .04515 .29567 2.0090 19.3679 .8234 .9416 .15844 .03627 .26333 2.0411 24.9787 .8174 .9555 .17979 .04681 .29619 2.0047 18.8684 .8240 .9400 .15921 .03676 .26454 2.0383 24.3959 .8179 .9544 .25876 .37033 .27152 1.0405 .5942 .7790 -.3153 .37152 .39631 .42461 .9118 .9080 .7599 -.061C .18249 .05126 .30008 1.9987 16.4693 .8286 .9306 .18249 .05126 .30008 1.9987 16.4693 .8286 .9306 .18276 .12388 .28176 1.7749 4.5485 .8533 .7288 .07586 .12762 .09385 3.4876 .7539 .9786 -.1651 .12477 .06490 .20172 3.4162 6.9178 .9647 .8141 .12881 .03770 .21366 3.5652 18.1304 .9569 .9343 .12382 .06558 .19995 3.4175 6.7346 .9651 .8087 .12798 .03827 .21223 3.5671 17.7495 .9573 .9327 .14481 .35791 .02540 2.6971 .0175 .8584 -.9823 .15843 .37533 .06215 2.5010 .0374 .8653 -.9596 .12187 .06770 .19631 3.4127 6.2417 .9658 .7928 .12187 .06770 .19631 3.4127 6.2417 .9658 .7928 .09641 .11106 .13625 3.1048 2.6619 .9635 .5258 .05181 .12902 .00726 5.0718 .0115 .9877 -.9865 .03144 .07166 .02088 5.0253 .2109 .9959 -.7026 .02338 .03944 .02862 5.1878 .8864 .9976 -.0672 .03185 .07298 .02055 5.0237 .2031 .9959 -.7131 .02349 .03990 .02858 5.1823 .8682 .9976 -.0787 1.0000 .14270 .35674 .00000 4.0977 .0000 .8584 .14894 .37235 .00000 4.1754 .0000 .8661 • 1.0000 .03185 .07314 .02035 5.0278 .1979 .9959 -.7201 .03185 .07314 .02035 5.0278 .1979 .9959 -.7201 .04731 .10784 .02970 4.4551 .2387 .9915 -.6734 .05346 .13366 .00000 5.8928 .0000 .9871 1.0000 .02974 .07435 .00000 6.4608 .0000 .9965 1.0000 1.0000 .01639 .04096 .00000 6.8189 .0000 .9987 .03000 .07501 .00000 6.4532 .0000 .9964-1.000C .01653 .04133 .00000 6.8108 .0000 .9986-1.0000 .14101 .35254 .00000 4.0904 .0000 .8577 1.0000 .14809 .37022 .00000 4.1737 .0000 .8659 1.0000 .03016 .07539 .00000 6.4477 .0000 .9964-1.0000 .03016 .07539 .00000 6.4477 .0000 .9964-1.0000 .04474 .11184 .00000 6.1587 .0000 .9927 1.0000 132 Table 10.2.1.1.3.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * 1,^ = 5, mNz = .5,pNz = .8 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Tp Tn .0874 .26936 .04730 .67513 .32488 .95270 .0107 .06562 .00575 .91941 .08059 .99425 .0089 .06058 .00475 .92546 .07454 .99525 .0108 .06683 .00580 .91791 .08209 .99420 .0093 .06175 .00495 .92405 .07595 .99505 .5781 .57126 .43315 .39421 .60579 .56685 .2765 .32957 .22480 .64424 .35576 .77520 .0153 .08405 .00825 .89700 .10300 .99175 .0153 .08405 .00825 .89700 .10300 .99175 .0578 .20596 .03165 .75046 .24954 .96835 .0896 .52463 .04880 .35641 .64359 .95120 .0193 .26816 .01095 .66754 .33246 .98905 .0097 .23247 .00500 .71066 .28934 .99500 .0197 .27067 .01115 .66445 .33555 .98885 .0098 .23450 .00505 .70814 .29186 .99495 .8021 .87012 .64045 .07246 .92754 .35955 .6593 .76468 33440 .17775 .82225 .46560 .0219 .28877 .01235 .64213 .35788 .98765 .0219 .28877 .01235 .64213 .35788 .98765 .0751 .47140 .04115 .42104 .57896 .95885 .0884 .78272 .04795 .03359 .96641 .95205 .0323 .72334 .01795 .10031 .89969 .98205 .0103 .66915 .00530 .16489 .83511 .99470 .0331 .72463 .01835 .09880 .90120 .98165 .0106 .67073 .00550 .16296 .83704 .99450 .8421 .93750 .68930 .00045 .99955 .31070 .8196 .93515 .67800 .00056 .99944 .32200 .0340 .72673 .01890 .09631 .90369 .98110 .0340 .72673 .01890 .09631 .90369 .98110 .0702 .75177 .03970 .07021 .92979 .96030 .0898 .80969 .04965 .00030 .99970 .95035 .0458 .80436 .02675 .00124 .99876 .97325 .0102 .79757 .00530 .00436 .99564 .99470 .0466 .80447 .02720 .00121 .99879 .97280 .0103 .79759 .00535 .00435 .99565 .99465 .8450 .93793 .68965 .00000 1.00000 .31035 .8237 .93580 .67900 .00000 1.00000 .32100 .0468 .80452 .02730 .00118 .99883 .97270 .0468 .80452 .02730 .00118 .99883 .97270 .0612 .80623 .03560 .00111 .99889 .96440 .0896 .80985 .04925 .00000 1.00000 .95075 .0459 .80536 .02680 .00000 1.00000 .97320 .0097 .80101 .00505 .00000 1.00000 .99495 .0466 .80543 .02715 .00000 1.00000 .97285 .0100 .80104 .00520 .00000 1.00000 .99480 .8434 .93768 .68840 .00000 1.00000 .31160 .8239 .93573 .67865 .00000 1.00000 .32135 .0468 .80545 .02725 .00000 1.00000 .97275 .0468 .80545 .02725 .00000 1.00000 .97275 .0611 .80700 .03500 .00000 1.00000 .96500 CtEX Rej Fp Fn A' B" s.Rej s.FpTn s.FnTp d' .18638 .15816 .22721 1.2175 3.6475 .7864 .6591 .10481 .05694 .12805 1.1260 9.1303 .7510 .8567 .09122 .05150 .11240 1.1507 10.1993 .7516 .8717 .10565 .05715 .12912 1.1329 9.1874 .7515 .8578 .09209 .05245 .11342 1.1465 9.9729 .7515 .8688 .33796 .41602 .35491 .4367 .9784 .6474 -.0139 .41157 .38527 .43561 .3863 1.2429 .6343 .1361 .11967 .06825 .14577 1.1330 7.9625 .7539 .8373 .11967 .06825 .14577 1.1330 7.9625 .7539 .8373 .20239 .13257 .24217 1.1811 4.4634 .7745 .7187 .19439 .16131 .24154 2.0247 3.6856 .8874 .6634 .21840 .08158 .26876 1.8590 12.5920 .8230 .9069 .17999 .05124 .22410 2.0205 23.6475 .8171 .9528 .21885 .08217 .26932 1.8606 12.4405 .8237 .9058 .18080 .05148 .22510 2.0244 23.5344 .8177 .9525 .16266 .38661 .15896 1.0980 .3687 .7770 -.5481 .33640 .43172 .35098 .8376 .6550 .7421 -.2599 .22082 .08616 .27157 1.8819 11.6592 .8288 .8992 .22082 .08616 .27157 1.8819 11.6592 .8288 .8992 .19536 .14995 .23946 1.9367 4.4353 .8725 .7214 .06541 .15946 .07945 3.4956 .7489 .9788 -.1689 .12997 .10224 .16083 3.3778 3.9829 .9695 .6732 .14362 .05265 .17991 3.5302 16.2924 .9570 .9263 .12893 .10315 .15953 3.3775 3.8656 .9697 .6635 .14275 .05404 .17885 3.5251 15.6449 .9574 .9229 .07453 .37119 .00847 2.8262 .0046 .8272 -.9958 .07771 .38410 .01316 2.7951 .0055 .8300 -.9949 .12709 .10485 .15718 3.3799 3.6998 .9702 .6487 .12709 .10485 .15718 3.3799 3.6998 .9702 .6487 .07664 .15125 .09249 3.2284 1.5713 .9710 .2626 .03300 .16466 .00612 5.0799 .0108 .9875 -.9874 .02770 .12850 .01323 4.9572 .0662 .9930 -.9094 .02171 .05312 .02509 5.1783 .8407 .9976 -.0966 .02780 .12947 .01311 4.9562 .0641 .9929 -.9125 .02162 .05335 .02506 5.1760 .8316 .9976 -.1026 .07386 .36929 .00000 3.7700 .0000 .8276-1.0000 1.OOO0 .07632 .38161 .00000 3.8000 .0000 .8302 .02774 .12964 .01282 4.9641 .0621 .9929 -.9153 .02774 .12964 .01282 4.9641 .0621 .9929 -.9153 .03027 .14675 .01174 4.8626 .0474 .9908 -.9373 1.0000 .03265 .16325 .00000 5.9171 .0000 .9877 .02572 .12858 .00000 6.1949 .0000 .9933 1.0000 1.0000 .01039 .05196 .00000 6.8373 .0000 .9987 .02584 .12919 .00000 6.1893 .0000 .9932 1.0000 1.0OO0 .01053 .05266 .00000 6.8271 .0000 .9987 .07403 .37016 .00000 3.7736 .0000 .8279 l.OOOO .07628 .38141 .00000 3.8010 .0000 .8303 1.0000 .02587 .12936 .00000 6.1877 .0000 .9932 1.0000 .02587 .12936 .00000 6.1877 .0000 .9932-1.000C .02879 .14397 .00000 6.0768 .0000 .9913-1.000C 133 Table 10.2.1.2.1.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P*\,p = 10, mNz = .l,pNz = .2 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn .8449 .05427 .05022 .92956 .0378 .00132 .00107 .99768 .0378 .00132 .00107 .99769 .0384 .00135 .00109 .99759 .0383 .00135 .00108 .99759 .3951 .06235 .06110 .93264 .0629 .00444 .00388 .99334 .0552 .00191 .00159 .99679 .0552 .00191 .00159 .99679 .0607 .00264 .00231 .99607 .8398 .05857 .04974 .90610 .0386 .00168 .00109 .99599 .0385 .00168 .00109 .99599 .0393 .00171 .00111 .99591 .0392 .00171 .00111 .99591 .4548 .07980 .07770 .91180 .0581 .00451 .00354 .99161 .0480 .00203 .00136 .99531 .0480 .00203 .00136 .99531 .0724 .00346 .00258 .99302 .8412 .07352 .05024 .83339 .0385 .00307 .00109 .98900 .0382 .00306 .00108 .98902 .0394 .00316 .00112 .98867 .0393 .00315 .00111 .98871 .5951 .13628 .12920 .83539 .0730 .01231 .00841 .97209 .0411 .00333 .00117 .98802 .0411 .00333 .00117 .98802 .1242 .00887 .00469 .97440 .8370 .09820 .04990 .70858 .0426 .00733 .00121 .96816 .0423 .00730 .00119 .96829 .0436 .00747 .00123 .96760 .0435 .00743 .00123 .96779 .7674 .23810 .21436 .66694 .1645 .04708 .03229 .89377 .0450 .00766 .00127 .96679 .0450 .00766 .00127 .96679 .2055 .02257 .00776 .91822 .8415 .14361 .04983 .48126 .0407 .02174 .00115 .89588 .0400 .02151 .00113 .89697 .0414 .02202 .00117 .89456 .0406 .02177 .00114 .89574 .9354 .40941 .34450 .33096 .4740 .18579 .13030 .59226 .0421 .02222 .00119 .89364 .0421 .02222 .00119 .89364 .2871 .05757 .01081 .75538 Tp Tn .07044 .94978 .00232 .99893 .00231 .99893 .00241 .99891 .00241 .99892 .06736 .93890 .00666 .99612 .00321 .99841 .00321 .99841 .00393 .99769 .09390 .95026 .00401 .99891 .00401 .99891 .00409 .99889 .00409 .99889 .08820 .92230 .00839 .99646 .00469 .99864 .00469 .99864 .00698 .99742 .16661 .94976 .01100 .99891 .01098 .99892 .01133 .99888 .01129 .99889 .16461 .87080 .02791 .99159 .01198 .99883 .01198 .99883 .02560 .99531 .29142 .95010 .03184 .99879 .03171 .99881 .03240 .99877 .03221 .99877 .33306 .78564 .10623 .96771 .03321 .99873 .03321 .99873 .08178 .99224 .51874 .95018 .10412 .99885 .10303 .99887 .10544 .99883 .10426 .99886 .66904 .65550 .40774 .86970 .10636 .99881 .10636 .99881 .24462 .98919 s.Rej s:FpTn s.FnTpd' P A' B" .03427 .03650 .08902 .1702 1.3036 .5771 .1571 .00538 .00544 .01635 .2395 2.0273 .6352 .3688 .00537 .00544 .01632 .2380 2.0185 .6346 .3668 .00546 .00550 .01664 .2462 2.0631 .6374 .3772 .00544 .00548 .01664 .2477 2.0728 .6380 .3794 .13031 .12969 .15125 .0498 1.0787 .5249 .0454 .02891 .02777 .04371 .1867 1.6154 .6048 .2617 .00652 .00669 .01920 .2246 1.8914 .6267 .3372 .00652 .00669 .01920 .2246 1.8914 .6267 .3372 .01069 .01045 .02392 .1741 1.6127 .6033 .2585 .03561 .03632 .10217 .3303 1.6316 .6292 .2857 .00607 .00552 .02102 .4122 3.2470 .6825 .5703 .00606 .00551 .02102 .4130 3.2546 .6827 .5711 .00613 .00556 .02121 .4134 3.2504 .6826 .5708 .00611 .00556 .02121 .4142 3.2579 .6828 .5716 .14467 .14492 .16695 .0688 1.1000 .5326 .0576 .02956 .02709 .04882 .3019 2.1544 .6458 .4049 .00663 .00612 .02261 .3995 3.0579 .6783 .5489 .00663 .00612 .02261 .3995 3.0579 .6783 .5489 .01173 .01032 .03171 .3384 2.4331 .6586 .4583 .04042 .03697 .13579 .6748 2.4126 .7052 .4885 .00841 .00549 .03545 .7745 7.9547 .7277 .8182 .00837 .00547 .03539 .7760 7.9969 .7279 .8191 .00853 .00557 .03590 .7783 7.9776 .7279 .8189 .00848 .00554 .03581 .7783 7.9868 .7279 .8191 .19232 .19142 .23096 .1545 1.1767 .5640 .1000 .06306 .05536 .10941 .4781 2.7975 .6796 .5298 .00876 .00570 .03698 .7863 8.0407 .7284 .8207 .00876 .00570 .03698 .7863 8.0407 .7284 .8207 .02013 .01431 .06492 .6484 4.3683 .7095 .6849 .04527 .03665 .17102 1.0966 3.3319 .7707 .6266 .01309 .00577 .06075 1.1799 17.8877 .7482 .9248 .01298 .00573 .06028 1.1809 17.9776 .7482 .9252 .01319 .00583 .06126 1.1808 17.7706 .7483 .9244 .01310 .00582 .06081 1.1788 17.7216 .7482 .9241 .23857 .23709 .30427 .3599 1.2461 .6269 .1376 .13328 .11592 .23434 .6013 2.5358 .6931 .5048 .01338 .00591 .06214 1.1824 17.6287 .7484 .9239 .01338 .00591 .06214 1.1824 17.6287 .7484 .9239 .03042 .01748 .11618 1.0266 7.0800 .7449 .8139 .04791 .03641 .19073 1.6936 3.8747 .8494 .6812 .02279 .00563 .11145 1.7901 47.2206 .7730 .9757 .02241 .00557 .10965 1.7899 47.7069 .7728 .9759 .02292 .00567 .11201 1.7923 46.9278 .7733 .9755 .02251 .00560 .11014 1.7923 47.4745 .7731 .9758 .24508 .25886 .28908 .8375 .9846 .7451 -.0098 .23824 .21475 .40430 .8916 1.8323 .7499 .3612 .02298 .00572 .11229 1.7923 46.5129 .7735 .9753 .02298 .00572 .11229 1.7923 46.5129 .7735 .9753 .04055 .01973 .17165 1.6056 11.0157 .7980 .8906 Table 10.2.1.2.1.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I,p = 10, mNz = .l,pNz = .4 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OLEX Rej Fp Fn .7512 .05828 .04996 .92923 .0285 .00156 .00107 .99770 .0285 .00156 .00107 .99770 .0296 .00161 .00111 .99764 .0294 .00160 .00110 .99764 .4038 .07881 .07627 .91739 .0545 .00592 .00481 .99243 .0405 .00217 .00153 .99686 .0405 .00217 .00153 .99686 .0586 .00385 .00278 .99454 .7520 .06766 .05017 .90609 .0292 .00223 .00109 .99607 .0292 .00223 .00109 .99607 .0297 .00228 .00111 .99596 .0296 .00228 .00111 .99597 5192 .12228 .11694 .86973 .0600 .00991 .00721 .98604 .0355 .00270 .00133 .99523 .0355 .00270 .00133 .99523 .0903 .00724 .00420 .98820 .7534 .09725 .05078 .83305 .0286 .00507 .00108 .98893 .0285 .00505 .00107 .98898 .0291 .00520 .00110 .98864 .0290 .00517 .00109 .98871 .7464 .26960 .24715 .69672 .1152 .04202 .03051 .94071 .0308 .00552 .00116 .98794 .0308 .00552 .00116 .98794 .1924 .02492 .00945 .95188 .7431 .14718 .04995 .70697 .0324 .01366 .00122 .96767 .0316 .01352 .00119 .96799 .0330 .01388 .00124 .96716 .0325 .01376 .00122 .96744 .9353 51333 .45039 .39227 .3318 .18756 .14383 .74684 .0337 .01419 .00127 .96644 .0337 .01419 .00127 .96644 .3339 .07031 .01730 .85018 .7439 .23760 .05011 .48116 .0320 .04286 .00120 .89465 .0304 .04186 .00114 .89707 .0328 .04343 .00123 .89327 .0309 .04241 .00116 .89571 .9975 .74774 .64184 .09339 .8033 57515 .46509 .25976 .0331 .04381 .00125 .89236 .0331 .04381 .00125 .89236 .4464 .16366 .02393 .62673 Tp Tn .07077 .95004 .00230 .99893 .00230 .99893 .00236 .99889 .00236 .99890 .08261 .92373 .00757 .99519 .00314 .99847 .00314 .99847 .00546 .99722 .09391 .94983 .00393 .99891 .00393 .99891 .00404 .99889 .00403 .99889 .13027 .88306 .01396 .99279 .00477 .99867 .00477 .99867 .01180 .99580 .16695 .94922 .01107 .99892 .01102 .99893 .01136 .99890 .01129 .99891 .30328 .75285 .05929 .96949 .01206 .99884 .01206 .99884 .04812 .99055 .29303 .95005 .03233 .99878 .03201 .99881 .03284 .99876 .03256 .99878 .60773 .54961 .25316 .85617 .03356 .99873 .03356 .99873 .14982 .98270 51884 .94989 .10535 .99880 .10293 .99886 .10673 .99877 .10429 .99884 .90661 .35816 .74024 .53491 .10764 .99875 .10764 .99875 .37327 .97607 s:Rej s:FpTn s.FnTp d' P A' B" .03615 .04194 .06452 .1752 1.3137 .5790 .1616 .00587 .00626 .01142 .2372 2.0145 .6344 .3658 .00587 .00626 .01142 .2372 2.0145 .6344 .3658 .00598 .00637 .01156 .2344 1.9933 .6331 .3609 .00594 .00635 .01155 .2357 2.0013 .6336 .3628 .15040 .15082 .15983 .0429 1.0623 5209 .0365 .03875 .03602 .04650 .1597 1.4928 .5916 .2209 .00698 .00749 .01337 .2296 1.9227 .6289 .3451 .00698 .00749 .01337 .2296 1.9227 .6289 .3451 .01330 .01247 .02197 .2277 1.8323 .6235 .3245 .03952 .04204 .07586 .3262 1.6206 .6280 .2821 .00709 .00633 .01511 .4061 3.1955 .6813 5642 .00709 .00633 .01511 .4061 3.1955 .6813 .5642 .00717 .00638 .01530 .4100 3.2222 .6820 .5675 .00715 .00637 .01527 .4105 3.2281 .6821 .5682 .18774 .18752 .20089 .0653 1.0785 .5293 .0463 .05828 .05160 .07304 .2480 1.7791 .6225 .3156 .00779 .00697 .01662 .4123 3.1702 .6811 .5626 .00779 .00697 .01662 .4123 3.1702 .6811 .5626 .01839 .01466 .03361 .3720 2.4872 .6629 .4720 .04863 .04270 .10217 .6711 2.3956 .7046 .4853 .01094 .00633 .02546 .7798 8.0722 .7282 .8209 .01087 .00632 .02530 .7793 8.0694 .7281 .8208 .01106 .00639 .02573 .7848 8.1306 .7284 .8223 .01096 .00638 .02552 .7833 8.1108 .7284 .8219 .26537 .26391 .28912 .1685 1.1063 .5649 .0635 .14363 .12572 .17915 .3126 1.7105 .6288 .3069 .01142 .00655 .02658 .7909 8.1366 .7287 .8229 .01142 .00655 .02658 .7909 8.1366 .7287 .8229 .03627 .02207 .07285 .6840 3.9423 .7106 .6606 .05819 .04263 .13022 1.1008 3.3377 .7713 .6272 .01868 .00674 .04531 1.1825 17.8854 .7483 .9249 .01839 .00663 .04469 1.1865 18.1954 .7484 .9262 .01880 .00680 .04560 1.1841 17.8221 .7484 .9247 .01852 .00672 .04500 1.1856 17.9893 .7485 .9254 .27930 .29066 .29711 .3981 .9708 .6363 -.0188 .29823 .26460 .36738 .3987 1.4112 .6399 .2111 .01899 .00686 .04610 1.1876 17.8053 .7486 .9247 .01899 .00686 .04610 1.1876 17.8053 .7486 .9247 .05610 .02876 .12163 1.0758 5.4440 .7548 .7645 .06395 .04271 .14626 1.6911 3.8572 .8492 .6797 .03408 .00664 .08461 1.7840 45.8029 .7732 .9749 .03285 .00645 .08163 1.7869 47.3175 .7727 .9757 .03430 .00672 .08513 1.7842 45.2143 .7735 .9746 .03304 .00650 .08209 1.7895 47.0671 .7730 .9756 .17897 .23884 .12901 .9568 .4469 .7578 -.4616 .32810 .32658 .37238 .7317 .8158 .7215 -.1281 .03448 .00681 .08554 1.7846 44.8760 .7737 .9744 .03448 .00681 .08554 1.7846 44.8760 .7737 .9744 .06409 .03254 .14558 1.6554 6.7209 .8234 .8184 Table 10.2.1.2.1.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P*I,p = 10, mNz = ,l,pNz = .6 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OEX Rej Fp Fn .5971 .06201 .04956 .92969 .0187 .00176 .00105 .99777 .0187 .00176 .00105 .99777 .0192 .00181 .00108 .99770 .0191 .00180 .00107 .99771 .4034 .09925 .09469 .89771 .0459 .00777 .00611 .99112 .0264 .00246 .00149 .99689 .0264 .00246 .00149 .99689 .0535 .00567 .00357 .99293 .6040 .07674 .05031 .90563 .0192 .00278 .00108 .99608 .0192 .00277 .00108 .99610 .0193 .00284 .00108 .99599 .0193 .00283 .00108 .99601 .5449 .17081 .16024 .82214 .0585 .01727 .01240 .97948 .0231 .00335 .00129 .99529 .0231 .00335 .00129 .99529 .0896 .01276 .00598 .98273 3989 .12044 .05041 .83287 .0187 .00713 .00106 .98883 .0187 .00708 .00106 .98890 .0190 .00731 .00108 .98853 .0189 .00724 .00107 .98866 .8334 .42006 .37919 .55269 .1693 .09562 .07404 .89000 .0200 .00777 .00113 .98780 .0200 .00777 .00113 .98780 .2187 .04904 .01542 .92855 .5958 .19574 .05028 .70729 .0223 .01980 .00126 .96784 .0219 .01951 .00123 .96830 .0228 .02013 .00128 .96730 .0223 .01984 .00126 .96776 .9814 .72023 .63628 .22381 .4848 .36833 .30357 .58850 .0234 .02057 .00132 .96659 .0234 .02057 .00132 .96659 .3713 .13439 .02821 .79483 .5908 .33160 .04991 .48061 .0217 .06484 .00123 .89275 .0202 .06252 .00113 .89656 .0221 .06570 .00126 .89133 .0208 .06332 .00117 .89525 .9999 .89478 .79484 .03860 .9145 .80569 .69573 .12101 .0223 .06624 .00127 .89044 .0223 .06625 .00127 .89043 .4713 .28821 .03801 .54499 Tp Tn .07031 .95044 .00223 .99895 .00223 .99895 .00230 .99892 .00229 .99893 .10229 .90531 .00888 .99389 .00311 .99851 .00311 .99851 .00707 .99643 .09437 .94969 .00392 .99892 .00390 .99892 .00401 .99892 .00399 .99892 .17786 .83976 .02052 .98760 .00471 .99871 .00471 .99871 .01727 .99402 .16713 .94959 .01117 .99894 .01110 .99894 .01147 .99892 .01134 .99893 .44731 .62081 .11000 .92596 .01220 .99887 .01220 .99887 .07145 .98458 .29271 .94972 .03216 .99874 .03170 .99877 .03270 .99872 .03224 .99874 .77619 .36372 .41150 .69643 .03341 .99868 .03341 .99868 .20517 .97179 .51939 .95009 .10725 .99877 .10344 .99887 .10867 .99874 .10475 .99883 .96140 .20516 .87899 .30427 .10956 .99873 .10957 .99873 .45501 .96199 s:Rej s:FpTn s.FnTp d' P A' B" .03837 .05137 .05383 .1757 1.3156 .5792 .1624 .00628 .00765 .00919 .2320 1.9871 .6326 .3592 .00628 .00765 .00919 .2320 1.9871 .6326 .3592 .00640 .00774 .00931 .2341 1.9955 .6332 .3613 .00636 .00772 .00930 .2346 1.9991 .6334 .3622 .17445 .17551 .18064 .0438 1.0581 .5207 .0344 .04926 .04622 .05346 .1355 1.3914 .5788 .1839 .00752 .00916 .01087 .2327 1.9419 .6302 .3498 .00752 .00916 .01087 .2327 1.9419 .6302 .3498 .01723 .01639 .02291 .2369 1.8391 .6248 .3278 .04423 .05168 .06521 .3275 1.6227 .6283 .2828 .00804 .00774 .01244 .4089 3.2249 .6819 .5676 .00799 .00774 .01235 .4073 3.2112 .6816 .5660 .00813 .00776 .01262 .4156 3.2803 .6833 .5740 .00807 .00776 .01250 .4134 3.2613 .6828 .5719 .22936 .22940 .23789 .0699 1.0693 .5300 .0415 .08839 .07710 .09864 .2013 1.5398 .6010 .2427 .00881 .00845 .01365 .4167 3.2173 .6822 .5680 .00881 .00845 .01365 .4167 3.2173 .6822 .5680 .02723 .02066 .03892 .3995 2.5199 .6662 .4811 .05749 .05251 .08845 .6753 2.4113 .7053 .4882 .01317 .00777 .02129 .7880 8.2551 .7288 .8249 .01303 .00771 .02109 .7873 8.2518 .7287 .8248 .01337 .00782 .02163 .7934 8.3252 .7291 .8265 .01317 .00780 .02129 .7908 8.2872 .7289 .8257 .30457 .30740 .31601 .1752 1.0393 .5655 .0245 .23582 .20844 .25903 .2198 1.3415 .5914 .1763 .01380 .00801 .02234 .8020 8.3875 .7295 .8282 .01380 .00801 .02234 .8020 8.3875 .7295 .8282 .05491 .03292 .08164 .6940 3.5170 .7103 .6275 .07273 .05256 .11587 1.0966 3.3184 .7709 .6251 .02337 .00837 .03837 1.1720 17.3754 .7480 .9225 .02290 .00830 .03762 1.1710 17.4530 .7479 .9228 .02355 .00846 .03865 1.1728 17.2654 .7481 .9221 .02305 .00837 .03787 1.1731 17.4100 .7480 .9227 .23392 .27173 .22902 .4109 .7964 .6412 -.1424 .40176 .36752 .43442 .2905 1.1131 .6043 .0678 .02384 .00856 .03914 1.1747 17.1670 .7482 .9217 .02384 .00856 .03914 1.1747 17.1670 .7482 .9217 .08129 .04310 .12533 1.0846 4.3975 .7612 .7122 .08056 .05262 .12952 1.6943 3.8690 .8495 .6807 .04529 .00837 .07521 1.7861 45.2485 .7737 .9746 .04277 .00793 .07111 1.7907 47.6301 .7729 .9759 .04560 .00844 .07572 1.7883 44.9384 .7740 .9744 .04300 .00804 .07148 1.7892 46.8052 .7731 .9755 .09835 .18220 .06107 .9438 .2945 .7463 -.6292 .26688 .28958 .27386 .6578 .5751 .7027 -.3312 .04589 .00848 .07619 1.7905 44.8417 .7742 .9744 .04589 .00848 .07619 1.7905 44.8427 .7742 .9744 .08454 .04883 .13329 1.6612 4.7951 .8375 .7430 Table 10.2.1.2.1.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * \,p = 10, mNz = ,\>pNz - .8 N:p OEX Rej Fp Fn . 3 5 2 8 . 0 6 6 1 5 . 0 4 8 73 .92949 NT 2 MB 2 .0107 .00202 .00120 .99777 RB 2 .0107 .00202 .00120 .99777 MD 2 .0110 .00208 .00123 .99771 RD 2 .0110 .00207 .00123 .99772 OR 2 .3567 .12203 .11598 .87646 OS 2 .0341 .01122 .00887 .98819 OF 2 .0142 .00277 .00159 .99694 CF 2 .0142 .00277 .00159 .99694 CS 2 .0396 .00824 .00504 .99096 NT 4 .3637 .08541 .05008 .90576 MB 4 .0108 .00329 .00120 .99619 RB 4 .0107 .00328 .00119 .99620 MD 4 .0110 .00337 .00122 .99610 RD 4 .0110 .00336 .00122 .99611 OR 4 .5353 .22676 .21091 .76927 OS 4 .0569 .02844 .02111 .96972 OF 4 .0131 .00394 .00146 .99543 CF 4 .0131 .00394 .00146 .99543 CS 4 .0651 .01998 .00821 .97708 NT 10 .3655 .14397 .05014 .83257 MB 10 .0105 .00926 .00120 .98873 RB 10 .0104 .00917 .00119 .98883 MD 10 .0108 .00946 .00123 .98848 RD 10 .0107 .00936 .00122 .98860 OR 10 .8617 .55677 50034 .42912 OS 10 .2203 .16527 .13283 .82662 OF 10 .0111 .01000 .00127 .98782 CF 10 .0111 .01000 .00127 .98782 CS 10 .1706 .07954 .02212 .90610 NT 20 .3687 .24415 .05064 .70748 MB 20 .0110 .02597 .00126 .96785 RB 20 .0107 .02543 .00122 .96852 MD 20 .0111 .02640 .00127 .96732 RD 20 .0109 .02588 .00124 .96797 OR 20 .9892 .84567 .75982 .13286 OS 20 5978 .52712 .45566 .45501 OF 20 .0112 .02695 .00128 .96663 CF 20 .0112 .02695 .00128 .96663 CS 20 .2900 .20879 .03960 .74891 NT 40 .3666 .42540 .05079 .48094 MB 40 .0107 .08718 .00121 .89133 RB 40 .0100 .08280 .00112 .89678 MD 40 .0110 .08831 .00124 .88993 RD 40 .0103 .08389 .00116 .89543 OR 40 1.0000 .95792 .87333 .02094 OS 40 .9574 .91370 .82189 .06334 OF 40 .0110 .08904 .00124 .88901 CF 40 .0110 .08904 .00124 .88901 CS 40 .3792 .42553 .05492 .48182 Tp Tn .07051 .95127 .00223 .99880 .00223 .99880 .00229 .99877 .00228 .99877 .12354 .88402 .01181 .99113 .00306 .99841 .00306 .99841 .00904 .99496 .09424 .94992 .00381 .99880 .00380 .99881 .00390 .99878 .00389 .99878 .23073 .78909 .03028 .97889 .00457 .99854 .00457 .99854 .02292 .99179 .16743 .94986 .01128 .99880 .01117 .99881 .01152 .99877 .01140 .99878 57088 .49966 .17338 .86717 .01218 .99873 .01218 .99873 .09390 .97788 .29253 .94936 .03215 .99874 .03148 .99878 .03268 .99873 .03203 .99876 .86714 .24018 .54499 .54434 .03337 .99872 .03337 .99872 .25109 .96040 .51906 .94921 .10868 .99879 .10322 .99888 .11008 .99876 .10457 .99884 .97906 .12667 .93666 .17811 .11099 .99876 .11099 .99876 .51818 .94508 s.Rej s.FpTns.FnTp .04115 .07389 .00682 .01159 .00682 .01159 .00694 .01175 .00689 .01175 .20221 .21244 .06803 .06525 .00798 .01329 .00798 .01329 .02255 .02679 .04936 .07427 .00880 .01149 .00876 .01143 .00889 .01159 .00886 .01159 .26666 .27608 .12549 .11237 .00957 .01263 .00957 .01263 .03728 .03308 .06686 .07406 .01556 .01180 .01532 .01175 .01573 .01196 .01547 .01191 .30762 .32991 .31699 .28574 .01620 .01211 .01620 .01211 .07444 .05271 .08706 .07438 .02779 .01206 .02698 .01191 .02803 .01211 .02721 .01201 .16826 .23811 .43812 .41427 .02837 .01216 .02837 .01216 .10618 .06878 .09725 .07493 .05611 .01175 .05161 .01122 .05653 .01190 .05191 .01138 .05082 .15316 .20017 .23885 .05675 .01190 .05675 .01190 .10397 .08040 .04807 .00804 .00804 .00815 .00812 .20447 .07024 .00940 .00940 .02550 .05884 .01061 .01057 .01072 .01068 .27064 .13059 .01155 .01155 .04348 .08121 .01917 .01889 .01938 .01907 .31085 .32779 .01997 .01997 .08873 .10703 .03458 .03358 .03487 .03386 .16253 .44883 .03530 .03530 .12825 .11997 .07008 .06447 .07059 .06485 .03595 .20088 .07086 .07086 .12637 A' B" d' P .1852 1.3362 5829 .1714 .1917 1.7569 .6156 .2994 .1917 1.7569 .6156 .2994 .1928 1.7598 .6159 .3003 .1909 1.7502 .6151 .2975 .0379 1.0456 .5174 .0273 .1078 1.2839 .5630 .1408 .2091 1.8132 .6207 .3163 .2091 1.8132 .6207 .3163 .2087 1.6741 .6115 .2818 .3290 1.6272 .6288 .2843 .3673 2.8507 .6719 5201 .3691 2.8676 .6724 .5224 .3698 2.8636 .6724 .5221 .3688 2.8563 .6722 .5210 .0668 1.0528 .5278 .0322 .1546 1.3526 5780 .1738 .3699 2.8092 .6711 .5155 .3699 2.8092 .6711 .5155 .4026 2.4227 .6642 .4667 .6791 2.4241 .7060 .4907 .7547 7.4353 .7259 .8058 .7539 7.4378 .7259 .8058 .7546 7.3875 .7258 .8048 .7533 7.3803 .7257 .8045 .1778 .9842 5662 -.0101 .1722 1.1935 .5702 .1088 .7679 7.5656 .7267 .8097 .7679 7.5656 .7267 .8097 .6947 3.1779 .7095 .5946 1.0926 3.2986 .7704 .623C 1.1719 17.3712 .7480 .9225 1.1707 17.4987 .7479 .9230 1.1765 17.4649 .7482 .9230 1.1730 17.4606 .7480 .9229 .4072 .6905 .6426 -.2260 .2244 .9998 .5820 -.0001 1.1833 17.6288 .7485 .9239 1.1833 17.6288 .7485 .9239 1.0843 3.7264 .7656 .6635 1.6850 3.8158 .8489 .6763 1.7993 46.4477 .7741 .9753 1.7924 47.9861 .7728 .9761 1.7986 45.7264 .7744 .9749 1.7911 47.1597 .7731 .9756 .8925 .2423 .7357 -.6873 .6047 .4768 .6917 -.4232 1.8034 45.9990 .7747 .9751 1.8034 45.9990 .7747 .9751 1.6445 35865 .8460 .6558 Table 10.2.1.2.2.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * l,p = 10, mNz ~ 3,pNz = .2 N:p N 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 N 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 N 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 N 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 N 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn Tp Tn . 2 5 601 .94978 .8329 .09138 .05022 .74399 .0381 .00484 .00108 .98013 .01987 .99892 .0381 .00480 .00108 .98030 .01970 .99892 .0392 .00494 .00111 .97976 .02024 .99889 .0389 .00490 .00110 .97990 .02010 .99890 .6936 .21114 .19063 .70682 .29318 .80937 .1487 .04270 .03052 .90857 .09143 .96948 .0554 .00646 .00159 .97409 .02591 .99841 .0554 .00646 .00159 .97409 .02591 .99841 .1941 .02037 .00818 .93089 .06911 .99182 .8246 .13496 .04990 .52478 .47522 .95010 .0399 .01700 .00114 .91957 .08043 .99886 .0394 .01678 .00113 .92058 .07942 .99888 .0408 .01726 .00117 .91834 .08166 .99883 .0401 .01703 .00114 .91942 .08058 .99886 .8894 .37573 .31955 .39952 .60048 .68045 .3941 .16285 .11691 .65338 .34662 .88309 .0479 .01913 .00138 .90986 .09014 .99863 .0479 .01913 .00138 .90986 .09014 .99863 .2771 .05218 .01118 .78383 .21617 .98882 .8283 .21323 .05026 .13488 .86512 .94974 .0425 .08549 .00122 .57743 .42257 .99878 .0380 .08341 .00108 .58727 .41273 .99892 .0439 .08602 .00126 .57496 .42504 .99874 .0395 .08406 .00112 .58420 .41580 .99888 .9905 34538 .44036 .03453 .96547 .55964 .9342 .45374 .33521 .07212 .92788 .66479 .0465 .08764 .00134 .56716 .43284 .99866 .0465 .08764 .00134 .56714 .43286 .99866 .2497 .12817 .00960 .39757 .60243 .99040 .8221 .23796 .04960 .00862 .99138 .95040 .0486 .17508 .00139 .13018 .86982 .99861 .0402 .17295 .00114 .13981 .86019 .99886 .0502 .17539 .00144 .12882 .87118 .99856 .0414 .17332 .00118 .13810 .86190 .99882 .9948 .56643 .45819 .00059 .99941 .54181 .9894 .49616 .37036 .00068 .99932 .62964 .0517 .17580 .00149 .12696 .87304 .99851 .0517 .17580 .00149 .12696 .87304 .99851 .1168 .17393 .00429 .14753 .85247 .99571 .8254 .23983 .04979 .00000 1.00000 .95021 .0490 .20089 .00141 .00118 .99882 .99859 .0389 .20061 .00112 .00143 .99857 .99888 .0502 .20093 .00144 .00112 .99888 .99856 .0400 .20064 .00115 .00140 .99860 .99885 .9946 .56441 .45552 .00000 1.00000 .54448 .9892 .49444 .36805 .00000 1.00000 .63195 .0508 .20095 .00146 .00110 .99890 .99854 .0508 .20095 .00146 .00110 .99890 .99854 .0663 .19613 .00257 .02964 .97036 .99743 OEX Rej A' B" s.Rej s.FpTn s.FnTp d' . 9 8 7 0 . 5 995 . 0 3 8 0 2 . 1 9 3 7 6 3 . 1 0 9 0 . 7 5 5 1 .04956 .01120 .00549 .05121 1.0106 13.3176 .7411 .8950 .01107 .00549 .05054 1.0072 13.2227 .7409 .8941 .01135 .00556 .05188 1.0101 13.1906 .7411 .8940 .01122 .00554 .05128 1.0094 13.2015 .7410 .8941 .24574 .24156 .31913 .3315 1.2653 .6191 .1464 .13637 .11997 .23484 .5413 2.3808 .6823 .4748 .01302 .00671 .05916 1.0043 11.6733 .7407 .8813 .01302 .00671 .05916 1.0043 11.6733 .7407 .8813 .03429 .02005 .12370 .9183 5.9477 .7358 .7760 .05555 .03833 .23018 1.5837 3.8668 .8357 .6805 .02257 .00566 .11038 1.6492 39.3497 .7663 .9697 .02215 .00563 .10842 1.6461 39.4156 .7661 .9697 .02278 .00574 .11140 1.6501 38.9313 .7666 .9694 .02230 .00567 .10916 1.6487 39.2278 .7663 .9696 .26335 .26869 .33896 .7235 1.0806 .7202 .0491 .24301 .21835 .41253 .7961 1.8794 .7307 .3738 .02408 .00623 .11763 1.6545 36.0730 .7684 .9671 .02408 .00623 .11763 1.6545 36.0730 .7684 .9671 .04679 .02172 .19663 1.4990 9.9788 .7889 .8775 .04242 .03860 .14457 2.7460 2.0952 .9500 .4194 .04654 .00587 .23123 2.8362 97.1204 .8547 .9901 .04532 .00550 .22558 2.8466107.7040 .8524 .9911 .04659 .00597 .23139 2.8323 94.2883 .8553 .9898 .04536 .00560 .22580 2.8432 104.2106 .8531 .9908 .21439 .26078 .09400 1.9680 .1937 .8705 -.7617 .22607 .25872 .20773 1.8857 .3770 .8826 -.5381 .04678 .00616 .23215 2.8340 89.5751 .8572 .9892 .04678 .00616 .23215 2.8340 89.5756 .8573 .9892 .04256 .01944 .18975 2.6012 14.9939 .8957 .9236 .03147 .03861 .03340 4.0302 .2284 .9852 -.6930 .02975 .00626 .14686 4.1163 46.4664 .9670 .9758 .03019 .00562 .14953 4.1326 58.6188 .9647 .9813 .02964 .00637 .14617 4.1125 44.7462 .9673 .9747 .03005 .00573 .14865 4.1302 56.3690 .9651 .9804 .20682 .25838 .00880 3.3492 .0052 .8851 -.9953 .20040 .25032 .00960 3.5348 .0062 .9071 -.9942 .02950 .00648 .14532 4.1115 43.0083 .9678 .9736 .02950 .00648 .14532 4.1115 43.0083 .9678 .9736 .02546 .01343 .10743 3.6753 18.2760 .9617 .9343 .03064 .03829 .00000 5.9118 .0000 .9876-1.000C .00560 .00631 .01222 6.0284 .8492 .9994 -.0890 .00521 .00564 .01340 6.0390 1.2565 .9994 .1240 .00562 .00637 .01197 6.0357 .7953 .9994 -.1244 .00526 .00573 .01326 6.0374 1.1971 .9994 .0979 .20371 .25464 .00000 4.3766 .0000 .8861 1.0000 .19783 .24729 .00000 4.6019 .0000 .9080-l.oood .00564 .00640 .01187 6.0382 .7727 .9994 -.1399 .00564 .00640 .01187 6.0382 .7727 .9994 -.1399 .01429 .01117 .05036 4.6842 8.4686 .9919 .8364| 138 Table 10.2.1.2.2.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * \,p = 10, mNz - .3,pNz - .4 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .7234 .13163 .0270 .00874 .0266 .00866 .0283 .00891 .0279 .00882 .8525 .42885 .2597 .14823 .0396 .01165 .0396 .01165 .2949 .05953 .7212 .22075 .0292 .03310 .0284 .03222 .0298 .03357 .0288 .03265 .9839 .69702 .6551 .46728 .0357 .03720 .0357 .03720 .4185 .14803 .7263 .37663 .0340 .17483 .0285 .16601 .0352 .17611 .0294 .16720 1.0000 .82275 .9954 .78891 .0376 .17931 .0376 .17931 .3459 .30074 .7170 .42640 .0455 .35455 .0299 .34486 .0464 .35512 .0308 .34554 1.0000 .82876 1.0000 .79945 .0478 .35592 .0478 .35592 .1448 .36907 .7208 .43009 .0497 .40079 .0297 .40007 .0508 .40083 .0305 .40011 1.0000 .82734 .9999 .79831 .0514 .40084 .0514 .40084 .0808 .39647 Fp .04934 .00100 .00099 .00105 .00104 .37566 .11430 .00149 .00149 .01677 .04991 .00109 .00106 .00112 .00108 .60040 .37696 .00135 .00135 .02459 .05073 .00130 .00109 .00136 .00112 .70791 .65490 .00144 .00144 .01879 .04971 .00176 .00113 .00180 .00116 .71464 .66580 .00185 .00185 .00734 .05016 .00196 .00115 .00201 .00118 .71223 .66384 .00204 .00204 .00430 Fn Tp Tn .74495 .25505 .95066 .97967 .02033 .99900 .97984 .02016 .99901 .97930 .02070 .99895 .97951 .02049 .99896 .49137 .50863 .62434 .80087 .19913 .88570 .97310 .02690 .99851 .97310 .02690 .99851 .87634 .12366 .98323 .52298 .47702 .95009 .91888 .08112 .99891 .92105 .07895 .99894 .91775 .08225 .99888 .91999 .08001 .99892 .15806 .84194 .39960 .39724 .60276 .62304 .90902 .09098 .99865 .90902 .09098 .99865 .66681 .33319 .97541 .13452 .86548 .94927 .56488 .43512 .99870 58661 .41339 .99891 .56177 .43823 .99864 .58368 .41632 .99888 .00500 .99500 .29209 .01008 .98992 .34510 .55390 .44610 .99856 55390 .44610 .99856 .27632 .72368 .98121 .00856 .99144 .95029 .11627 .88373 .99824 .13954 .86046 .99887 .11491 .88509 .99820 .13791 .86209 .99884 .00006 .99994 .28536 .00007 .99993 .33420 .11298 .88702 .99815 .11298 .88702 .99815 .08834 .91166 .99266 .00001 .99999 .94984 .00096 .99904 .99804 .00154 .99846 .99885 .00096 .99904 .99799 .00150 .99850 .99882 .00000 1.00000 .28777 .00000 1.00000 .33616 .00094 .99906 .99796 .00094 .99906 .99796 .01528 .98472 .99570 s.FnTp s.Rej s.FpTn .06875 .01601 .01576 .01619 .01591 .31131 .28418 .01880 .01880 .06664 .08175 .03614 .03452 .03650 .03479 .23088 .37417 .03873 .03873 .08397 .05492 .07978 .07499 .07987 .07520 .13052 .15123 .07989 .07989 .05633 .02896 .04625 .04899 .04592 .04871 .12641 .13770 .04540 .04540 .02876 .02781 .00610 .00555 .00615 .00556 .12593 .13662 .00616 .00616 .01532 .04471 .00604 .00599 .00617 .00613 .30850 .25158 .00739 .00739 .03184 .04569 .00633 .00625 .00642 .00629 .26980 .35225 .00710 .00710 .03695 .04629 .00706 .00642 .00723 .00651 .21393 .23644 .00743 .00743 .03164 .04592 .00826 .00647 .00833 .00659 .21064 .22946 .00845 .00845 .02073 .04635 .00884 .00669 .00898 .00680 .20988 .22770 .00903 .00903 .01683 .15730 .03890 .03832 .03935 .03867 .34833 .35158 .04559 .04559 .14219 .19227 .08974 .08577 .09059 .08644 .22179 .44277 .09612 .09612 .19120 .11874 .19889 .18720 .19907 .18769 .02033 .06843 .19916 .19916 .13067 .02486 .11504 .12219 .11421 .12144 .00184 .00200 .11294 .11294 .06098 .00056 .00774 .00975 .00772 .00963 .00000 .00000 .00768 .00768 .02539 d' P .9926 3.1468 1.0422 14.5334 1.0431 14.6272 1.0357 14.1343 1.0358 14.1989 .3385 1.0512 .3592 1.4448 1.0416 12.8208 1.0416 12.8208 .9687 4.9028 15881 3.8674 1.6662 41.1368 1.6599 41.3321 1.6667 40.6855 1.6629 41.2253 .7481 .6249 .5740 1.0153 1.6648 36.8898 1.6648 36.8898 1.5359 6.3073 2.7431 2.0758 2.8472 91.6965 2.8470107.3227 2.8433 88.5968 2.8455 104.5564 .0421 2.0285 .0728 1.9249 2.8438 83.8468 2.8438 83.8468 2.6732 7.2843 4.0320 .2265 4.1116 34.6034 4.1372 59.1599 4.1127 33.7342 4.1349 56.9714 3.2746 .0007 .0008 3.3721 4.1132 32.4194 4.1132 32.4194 3.7910 7.8776 6.0377 .0002 5.9860 .5208 6.0077 1.3113 .5060 5.9795 6.0081 1.2438 3.7050 .0000 .0000 3.8419 5.9795 .4956 5.9795 .4956 4.7899 3.0425 A' B" .7557 .6040 .7425 .9042 .7425 .9047 .7422 .9014 .7422 .9019 .6186 .0318 .6304 .2234 .7425 .8925 .7425 .8925 .7433 .7359 .8362 .6806 .7667 .9711 .7661 .9712 .7669 .9708 .7664 .9712 .7228 -.2865 .6843 .0097 .7687 .9679 .7687 .9679 .8106 .8051 .9499 .4148 .8578 .9895 .8525 .9911 .8586 .9891 .8532 .9908 .8179 -.9530 .8273 -.9154 .8605 .9884 .8605 .9884 .9231 .8312 .9852 -.6955 .9704 .9663 .9647 .9814 .9707 .9653 .9651 .9806 .8213 -.9994 .8335 -.9994 .9712 .9638 .9712 .9638 .9757 .8340 .9875 -.9998 .9993 -.3422 .9993 .1470 .9993 -.3561 .9993 .1186 .8219-1.0000 .8340-1.0000 .9993 -.3660 .9993 -.3660 .9951 .5567 Table 10.2.1.2.2.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P*I,p = 10, mNz = ,3,pNz = .6 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn Rej Tp Tn . 0 4 9 3 0 . 7 4 5 9 0 . 2 5 410 .95070 .5605 .17218 .0176 .01248 .00101 .97987 .02013 .99899 .0174 .01229 .00099 .98018 .01982 .99901 .0183 .01274 .00104 .97946 .02054 .99896 .0180 .01252 .00103 .97981 .02019 .99897 .9052 .60987 33948 .34320 .65680 .46052 .3659 .28227 .23223 .68438 .31562 .76777 .0277 .01665 .00159 .97331 .02669 .99841 .0277 .01665 .00159 .97331 .02669 .99841 .3129 .11520 .02731 .82620 .17380 .97269 3593 .30555 .04952 .52376 .47624 .95048 .0202 .04998 .00113 .91745 .08255 .99887 .0188 .04779 .00105 .92104 .07896 .99895 .0209 .05068 .00117 .91631 .08369 .99883 .0193 .04847 .00108 .91994 .08006 .99892 .9941 .85773 .76151 .07813 .92187 .23849 .7747 .67654 38265 .26086 .73914 .41735 .0244 .05589 .00137 .90776 .09224 .99863 .0244 .05589 .00137 .90776 .09224 .99863 .4425 .26318 .03915 .58747 .41253 .96085 3551 33911 .04938 .13440 .86560 .95062 .0269 .27060 .00159 35005 .44995 .99841 .0191 .24827 .00110 38695 .41305 .99890 .0276 .27259 .00163 34676 .45324 .99837 .0196 .25005 .00113 .58400 .41600 .99887 .9999 .93360 .83692 .00195 .99805 .16308 .9981 .92248 .81212 .00394 .99606 .18788 .0299 .27746 .00176 33874 .46126 .99824 .0299 .27746 .00176 .53874 .46126 .99824 .3516 .48554 .02953 .21045 .78955 .97047 .5564 .61505 .05007 .00829 .99171 .94993 .0438 34126 .00263 .09966 .90034 .99737 .0193 .51688 .00111 .13928 .86072 .99889 .0451 .54213 .00271 .09825 .90175 .99729 .0196 .51782 .00113 .13771 .86229 .99887 1.0000 .93586 .83967 .00002 .99998 .16033 1.0000 .92701 .81757 .00003 .99997 .18243 .0462 .54307 .00277 .09673 .90327 .99723 .0462 .54307 .00277 .09673 .90327 .99723 .1505 .56958 .01196 .05868 .94132 .98804 3580 .61998 .04995 .00001 .99999 .95005 .0461 .60072 .00283 .00068 .99932 .99717 .0201 39953 .00117 .00156 .99844 .99883 .0475 .60077 .00291 .00066 .99934 .99709 .0204 .59957 .00119 .00151 .99849 .99881 1.0000 .93533 .83833 .00000 1.00000 .16167 1.0000 .92679 .81698 .00000 1.00000 .18302 .0484 .60080 .00297 .00065 .99935 .99703 .0484 .60080 .00297 .00065 .99935 .99703 .0867 .59830 .00736 .00774 .99226 .99264 a£X A' B" s.FnTp s:Rej s.FpTn d' P .09261 .05697 .14932 .9900 3.1428 .7553 .6035 .02033 .00765 .03354 1.0374 14.3833 .7423 .9031 .01981 .00761 .03266 1.0345 14.3444 .7422 .9027 .02058 .00779 .03393 1.0347 14.1356 .7422 .9014 .02003 .00773 .03302 1.0322 14.1350 .7421 .9013 .32068 .32745 .33363 .3046 .9263 .6083 -.0486 .39223 .35457 .42631 .2515 1.1646 .5932 .0956 .02430 .00964 .03998 1.0171 11.9669 .7414 .8845 .02430 .00964 .03998 1.0171 11.9669 .7414 .8845 .10715 .05058 .16364 .9826 4.0783 .7484 .6878 .11391 .05754 .18577 1.5899 3.8910 .8362 .6825 .05204 .00787 .08647 1.6662 40.4908 .7670 .9707 .04821 .00761 .08020 1.6636 41.8021 .7662 .9716 .05252 .00806 .08727 1.6621 39.4878 .7672 .9699 .04860 .00770 .08085 1.6633 41.2455 .7664 .9712 .16119 .22145 .14663 .7066 .4713 .7116 -.4321 .38177 .37194 .40421 .4320 .8324 .6467 -.1155 .05577 .00868 .09269 1.6692 36.8987 .7690 .9679 .05577 .00868 .09269 1.6692 36.8987 .7690 .9679 .12400 .05687 .19644 1.5396 43974 .8234 .7313 .07130 .05777 .11220 2.7567 2.1198 .9504 .4250 .12047 .00994 .20031 2.8242 76.9632 .8614 .9873 .10880 .00801 .18117 2.8421 105.9766 .8524 .9910 .12058 .01004 .20049 2.8250 75.3619 .8622 .9870 .10897 .00811 .18146 2.8422103.7517 .8532 .9908 .06438 .15764 .00990 1.9046 .0251 .7874 -.9719 .08171 .17881 .04352 1.7714 .0434 .7909 -.9498 .12075 .01041 .20075 2.8208 70.3014 .8641 .9860 .12075 .01041 .20075 2.8208 70.3014 .8641 .986C .06875 .04959 .10895 2.6926 4.2965 .9364 .7058 .02652 .05889 .02129 4.0400 .2191 .9852 -.7052 .06224 .01275 .10323 4.0744 21.5632 .9743 .9432 .06987 .00803 .11632 4.1439 60.0798 .9648 .9817 .06173 .01293 .10237 4.0731 20.7846 .9746 .9409 .06944 .00814 .11559 4.1450 583422 .9652 .9812 .06227 .15565 .00091 3.0900 .0004 .7900 -.9997 .06942 .17351 .00098 3.1410 .0004 .7956 -.9997 .06108 .01308 .10126 4.0740 20.1019 .9750 .9387 .06108 .01308 .10126 4.0740 20.1019 .9750 .9387 .03002 .03362 .04190 3.8245 3.7599 .9820 .6476 .02340 .05849 .00052 5.9768 .0003 .9875 -.9997 .00623 .01348 .00529 5.9711 .2714 .9991 -.6126 .00594 .00834 .00822 5.9992 1.3063 .9993 .1450 .00628 .01366 .00522 5.9695 .2577 .9991 -.6300 .00592 .00845 .00810 6.0032 1.2478 .9993 .1204 .06275 .15687 .00000 3.2773 .0000 .7904-1.000C 1.0000 .06942 .17356 .00000 3.3610 .0000 .7958 .00632 .01382 .00518 5.9676 .2490 .9991 -.6413 .00632 .01382 .00518 5.9676 .2490 .9991 -.6413 .01540 .02826 .01524 4.8602 1.0464 .9962 .0256 Table 10.2.1.2.2.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I,p = 10, mNz = .3,pNz = .8 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp .3173 .21289 .04826 .0100 .01634 .00113 .0100 .01599 .00113 .0103 .01669 .00117 .0102 .01631 .00116 .9177 .73039 .65617 .4393 .39096 .33701 .0146 .02182 .00166 .0146 .02182 .00166 .2332 .18078 .03830 .3256 .39096 .04954 .0103 .06729 .00114 .0098 .06329 .00109 .0108 .06827 .00120 .0101 .06417 .00112 .9966 .93420 .84846 .8373 .79475 .71102 .0123 .07532 .00137 .0123 .07532 .00137 .3353 .39246 .05571 .3247 .70271 .04946 .0171 .37743 .00202 .0105 .33109 .00119 .0176 .38014 .00209 .0106 .33349 .00120 .9999 .97904 .89944 .9993 .97686 .89132 .0187 .38649 .00222 .0187 .38649 .00222 .2803 .68180 .04576 .3249 .80313 .04946 .0391 .74077 .00481 .0094 .68907 .00109 .0398 .74174 .00492 .0094 .69039 .00110 1.0000 .98045 .90230 1.0000 .97898 .89500 .0408 .74264 .00503 .0408 .74264 .00503 .1287 .77469 .02087 .3271 .81015 .05078 .0462 .80094 .00588 .0093 .79886 .00109 .0473 .80098 .00600 .0095 .79890 .00112 .9999 .98017 .90083 .9999 .97869 .89346 .0475 .80098 .00602 .0475 .80098 .00602 .0846 .80128 .01537 OLEX Rej Fn Tp Tn .74595 .25405 .95174 .97986 .02014 .99887 .98030 .01970 .99887 .97943 .02058 .99883 .97990 .02010 .99884 .25106 .74894 .34383 59555 .40445 .66299 .97313 .02687 .99834 .97313 .02687 .99834 .78360 .21640 .96170 .52368 .47632 .95046 .91617 .08383 .99886 .92116 .07884 .99891 .91497 .08503 .99880 .92007 .07993 .99888 .04437 .95563 .15154 .18431 .81569 .28898 .90619 .09381 .99863 .90619 .09381 .99863 .52336 .47664 .94429 .13398 .86602 .95054 52872 .47128 .99798 .58643 .41357 .99881 52534 .47466 .99791 .58343 .41657 .99880 .00106 .99894 .10056 .00175 .99825 .10868 .51745 .48255 .99778 .51745 .48255 .99778 .15919 .84081 .95424 .00845 .99155 .95054 .07524 .92476 .99519 .13893 .86107 .99891 .07406 .92594 .99508 .13729 .86271 .99890 .00002 .99998 .09770 .00002 .99998 .10500 .07296 .92704 .99497 .07296 .92704 .99497 .03686 .96314 .97913 .00001 .99999 .94922 .00030 .99970 .99412 .00170 .99830 .99891 .00028 .99972 .99400 .00165 .99835 .99888 .00000 1.00000 .09917 .00000 1.00000 .10654 .00028 .99972 .99398 .00028 .99972 .99398 .00224 .99776 .98463 A' s.Rej s.FpTn s:FnTp B" d' P . 7 5 6 6 .6099 .11745 .08467 .14482 1.0002 3.1968 .02522 .01149 .03142 1.0019 12.8946 .7407 .8915 .02423 .01149 .03016 .9929 12.6572 .7403 .8893 .02553 .01165 .03180 1.0021 12.7883 .7407 .8907 .02452 .01160 .03053 .9953 12.6465 .7404 .8893 .29976 .32549 .30272 .2691 .8655 .5984 -.0909 .44701 .41597 .45871 .1788 1.0610 .5671 .0376 .03046 .01382 .03791 1.0083 11.6267 .7409 .8811 .03046 .01382 .03791 1.0083 11.6267 .7409 .8811 .15081 .08536 .18123 .9864 3.5260 .7521 .6431 .14604 .08505 .18121 1.5899 3.8896 .8363 .6824 .06829 .01122 .08528 1.6702 40.4137 .7673 .9707 .06135 .01095 .07662 1.6519 40.3859 .7660 .9705 .06888 .01149 .08602 1.6637 39.1138 .7675 .9697 .06185 .01111 .07724 1.6503 39.6962 .7662 .9700 .10641 .18813 .09906 .6723 .3992 .7048 -.5040 .35480 .35673 .36094 .3427 .7793 .6226 -.1549 .07296 .01225 .09113 1.6786 37.3601 .7695 .9684 .07296 .01225 .09113 1.6786 37.3601 .7695 .9684 .16413 .09780 .19986 1.5333 3.5440 .8322 .6517 .08822 .08484 .10820 2.7579 2.1126 .9505 .4233 .16570 .01605 .20690 2.8026 62.1332 .8665 .9839 .14048 .01165 .17556 2.8201 98.7238 .8525 .9903 .16576 .01635 .20696 2.8009 60.3657 .8673 .9834 .14075 .01170 .17588 2.8250 98.0482 .8532 .9902 .02814 .13570 .00620 1.7950 .0201 .7723 -.9769 .03868 .14778 .02531 1.6859 .0302 .7727 -.9645 .16559 .01685 .20673 2.8010 57.1330 .8692 .9824 .16559 .01685 .20673 2.8010 57.1330 .8692 .9824 .07675 .08917 .09261 2.6853 2.5243 .9447 5081 .02322 .08477 .02000 4.0389 .2250 .9853 -.6974 .07274 .02520 .09048 4.0269 10.1566 .9798 .8712 .08958 .01139 .11197 4.1500 60.8196 .9649 .9820 .07204 .02566 .08958 4.0275 9.8320 .9801 .8667 .08899 .01155 .11122 4.1544 59.7698 .9653 .9816 .02628 .13132 .00068 2.8546 .0004 .7744 -.9996 .02841 .14196 .00073 2.8604 .0005 .7762 -.9996 .07123 .02588 .08858 4.0277 95285 .9803 .8621 .07123 .02588 .08858 4.0277 9.5285 .9803 .8621 .02813 .06435 .02931 3.8246 1.6061 .9852 .2694 .01748 .08739 .00039 6.0317 .0002 .9873 -.9998 .00615 .02857 .00295 5.9510 .0663 .9985 -.9024 .00644 .01150 .00760 5.9944 1.5001 .9993 .2181 .00614 .02878 .00283 5.9619 .0611 .9984 -.9102 .00638 .01176 .00748 5.9941 1.4222 .9993 .1901 1.0000 .02681 .13405 .00000 2.9786 .0000 .7748 .02878 .14389 .00000 3.0198 .0000 .7766-1.0000 .00615 .02882 .00283 5.9606 .0609 .9984 -.9105 .00615 .02882 .00283 5.9606 .0609 .9984 -.9105 .01391 .06047 .00759 5.0021 .1821 .9956 -.7422 141 Table 10.2.1.2.3.1 Table of empirical results for tests of correlations with no preliminary omnibus test when P * L p = 10, mNz = .S,pNz - .2 N:p CtEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 M D 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn Tp Tn .8044 .16903 .05024 .35579 .64421 .94976 .0402 .3050 .00118 .85221 .14779 .99882 .0391 .02983 .00114 .85541 .14459 .99886 .0411 .03091 .00121 .85027 .14973 .99879 .0399 .03021 .00117 .85361 .14639 .99883 .9403 .45475 .37358 .22056 .77944 .62643 .6060 .27964 .20415 .41842 .58158 .79585 .0568 .03713 .00168 .82107 .17893 .99832 .0568 .03713 .00168 .82107 .17893 .99832 .3100 .08226 .01393 .64442 .35558 .98607 .7957 .22404 .04956 .07800 .92200 .95044 .0424 .10830 .00126 .46354 .53646 .99874 .0383 .10583 .00113 .47537 .52463 .99887 .0426 .10888 .00127 .46068 33932 .99873 .0390 .10645 .00116 .47238 .52762 .99884 .9888 .55337 .44576 .01620 .98380 .55424 .9518 .47350 .35156 .03876 .96124 .64844 .0501 .11347 .00151 .43868 36132 .99849 .0501 .11347 .00151 .43868 36132 .99849 .2213 .14463 .00927 .31391 .68609 .99073 .7934 .23964 .04962 .00028 .99972 .95038 .0467 .19847 .00140 .01326 .98674 .99860 .0378 .19787 .00112 .01514 .98486 .99888 .0482 .19856 .00144 .01300 .98700 .99856 .0390 .19794 .00116 .01493 .98507 .99884 .9896 35815 .44769 .00002 .99998 35231 .9818 .49504 .36881 .00002 .99998 .63119 .0513 .19875 .00154 .01240 .98760 .99846 .0513 .19875 .00154 .01240 .98760 .99846 .0992 .19124 .00455 .06201 .93799 .99545 .7906 .23966 .04958 .00000 1.00000 .95042 .0464 .20112 .00139 .00000 1.00000 .99861 .0383 .20091 .00114 .00000 1.00000 .99886 .0475 .20114 .00143 .00000 1.00000 .99857 .0395 .20094 .00118 .00000 1.00000 .99882 .9912 .56394 .45493 .00000 1.00000 .54507 .9835 .49747 .37184 .00000 1.00000 .62816 .0492 .20119 .00148 .00000 1.00000 .99852 .0492 .20119 .00148 .00000 1.00000 .99852 .0910 .20300 .00422 .00187 .99813 .99578 .7916 .23982 .04977 .00000 1.00000 .95023 .0473 .20114 .00142 .00000 1.00000 .99858 .0389 .20093 .00116 .00000 1.00000 .99884 .0483 .20116 .00145 .00000 1.00000 .99855 .0395 .20094 .00118 .00000 1.00000 .99883 .9908 .56369 .45461 .00000 1.00000 34539 .9840 .49789 .37236 .00000 1.00000 .62764 .0491 .20118 .00148 .00000 1.00000 .99852 .0491 .20118 .00148 .00000 1.00000 .99852 .0906 .20324 .00405 .00000 1.00000 .99595 s:Rej s:FpTn S.FnTp .06001 .03508 .03398 .03533 .03424 .25977 .28184 .03853 .03853 .05746 .04124 .05408 .05297 .05404 .05301 .21905 .22535 .05380 .05380 .04335 .03385 .01055 .01084 .01049 .01080 .21705 .21251 .01036 .01036 .02133 .03388 .00527 .00474 .00533 .00481 .21695 .21223 .00544 .00544 .01318 .03360 .00529 .00474 .00534 .00477 .21646 .21093 .00540 .00540 .01242 d' P A' B" .04196 .24812 2.0123 3.5990 .8869 .6554 .00593 .17298 1.9953 59.0123 .7847 .9815 .00581 .16779 1.9915 59.9670 .7839 .9818 .00600 .17417 1.9967 58.2732 .7852 .9813 .00590 .16901 1.9921 59.1481 .7844 .9815 .28300 .29250 1.0927 .7829 .7922 -.1530 .26863 .44557 1.0328 1.3780 .7808 .1993 .00708 .18958 2.0132 48.3013 .7920 .9774 .00708 .18958 2.0132 48.3013 .7920 .9774 .02625 .24371 1.8289 10.4812 .8268 .8869 .04150 .12530 3.0678 1.4242 .9660 .2085 .00620 .26896 3.1122 95.3939 .8834 .9899 .00583 .26393 3.1154 105.6717 .8805 .9910 .00624 .26878 3.1167 943666 .8841 .9898 .00591 .26408 3.1163103.5246 .8812 .9908 .27080 .06408 2.2758 .1024 .8794 -.8788 .26781 .16201 2.1464 .2264 .8936 -.7191 .00681 .26731 3.1204 80.3806 .8895 .9878 .00681 .26731 3.1204 80.3806 .8895 .9878 .02130 .18895 2.8395 14.2212 .9174 .9182 .04229 .00577 5.1010 .0100 .9875 -.9883 .00658 .04631 5.2075 7.4300 .9963 .8069 .00583 .04953 5.2229 10.2255 .9959 .8605 .00667 .04578 5.2055 7.1005 .9964 .7979 .00592 .04914 5.2189 9.8133 .9960 .8545 .27130 .00157 4.2146 .0002 .8881 -.9998 .26563 .00157 4.4181 .0003 .9078 -.9998 .00686 .04447 5.2049 6.4439 .9965 .7774 .00686 .04447 5.2049 6.4439 .9965 .7774 .01687 .06991 4.1464 9.1950 .9832 .8555 .04234 .00000 5.9139 .0000 .9876-1.0000 1.0000 .00659 .00000 7.2550 .0000 .9997 .00592 .00000 7.3156 .0000 .9997 1.000C .00666 .00000 7.2478 .0000 .9996-1.0000 .00601 .00000 7.3062 .0000 .9997 • 1.0000 .27119 .00000 4.3781 .0000 .8863 -1.0000 .26529 .00000 4.5919 .0000 .9070-1.0000 .00680 .00000 7.2361 .0000 .9996-1.0000 .00680 .00000 7.2361 .0000 .9996-1.0000 .01596 .01428 5.5341 .4795 .9985 -.3853 .04200 .00000 5.9120 .0000 .9876-1.0000 .00661 .00000 7.2490 .0000 .9996-1.0000 .00593 .00000 7.3112 .0000 .9997-1.0000 .00668 .00000 7.2424 .0000 .9996-1.0000 1.0000 .00597 .00000 7.3069 .0000 .9997 1.0000 .27057 .00000 4.3789 .0000 .8863 1.0000 .26366 .00000 4.5905 .0000 .9069 1.0000 .00675 .00000 7.2366 .0000 .9996 .00675 .00000 7.2366 .0000 .9996-1.0000 .01553 .00000 6.9125 .0000 .9990-1.0000 Table 10.2.1.2.3.2 Table of empirical results for tests of correlations with no preliminary omnibus test when P * !,/> = 10, mNz - .5,pNz = .4 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 CtEX Rej .6747 .28707 .0277 .06116 .0262 .05855 .0284 .06198 .0266 .05927 .9951 .76198 .8353 .62608 .0406 .07377 .0406 .07377 .4098 .21427 .6688 .39860 .0354 .22258 .0274 .21088 .0361 .22375 .0281 .21216 .9994 .81964 .9972 .78867 .0421 .23224 .0421 .23224 .2809 .32740 .6713 .43009 .0439 .39683 .0280 .39462 .0450 .39692 .0285 .39472 .9996 .82315 .9993 .79451 .0475 .39717 .0475 .39717 .1164 .39138 .6633 .42981 .0436 .40108 .0274 .40067 .0448 .40112 .0286 .40070 .9996 .82164 .9996 .79354 .0469 .40117 .0469 .40117 .1082 .40416 .6607 .43015 .0461 .40120 .0287 .40072 .0475 .40123 .0288 .40073 .9996 .82304 .9994 .79476 .0479 .40124 .0479 .40124 .1137 .40462 Fp .04951 .00106 .00100 .00109 .00102 .64538 50441 .00161 .00161 .02722 .04969 .00142 .00107 .00145 .00110 .70069 .65038 .00172 .00172 .01773 .05035 .00184 .00111 .00189 .00113 .70525 .65752 .00201 .00201 .00796 .04969 .00181 .00112 .00187 .00116 .70274 .65590 .00195 .00195 .00743 .05026 .00199 .00121 .00204 .00121 .70507 .65794 .00206 .00206 .00769 Fn Tp Tn .35658 .64342 .95049 .84869 .15131 .99894 .85512 .14488 .99900 .84669 .15331 .99891 .85336 .14664 .99898 .06313 .93687 .35462 .19143 .80857 .49559 .81800 .18200 .99839 .81800 .18200 .99839 .50516 .49484 .97278 .07802 .92198 .95031 .44569 .55431 .99858 .47439 52561 .99893 .44281 55719 .99855 .47124 .52876 .99890 .00193 .99807 .29931 .00390 .99610 .34962 .42197 57803 .99828 .42197 57803 .99828 .20811 .79189 .98227 .00029 .99971 .94965 .01068 .98932 .99816 .01512 .98488 .99889 .01053 .98947 .99811 .01489 .98511 .99887 .00000 1.00000 .29475 .00000 1.00000 .34248 .01008 .98992 .99799 .01008 .98992 .99799 .03350 .96650 .99204 .00000 1.00000 .95031 .00000 1.00000 .99819 .00000 1.00000 .99888 .00000 1.00000 .99813 .00000 1.00000 .99884 .00000 1.00000 .29726 .00000 1.00000 .34410 .00000 1.00000 .99805 .00000 1.00000 .99805 .00075 .99925 .99257 .00000 1.00000 .94974 .00000 1.00000 .99801 .00000 1.00000 .99879 .00000 1.00000 .99796 .00000 1.00000 .99879 .00000 1.00000 .29493 .00000 1.00000 .34206 .00000 1.00000 .99794 .00000 1.00000 .99794 .00000 1.00000 .99231 s.FnTp d' s:Rej s.FpTn .09198 .05143 .21752 2.0172 .06058 .00639 .15091 2.0423 .05678 .00624 .14154 2.0305 .06101 .00649 .15196 2.0425 .05714 .00628 .14244 2.0338 .18600 .25630 .14108 1.1562 .32036 .33305 .36096 .8616 .06659 .00808 .16580 2.0379 .06659 .00808 .16580 2.0379 .09403 .04335 .21612 1.9104 .05173 .05222 .10642 3.0664 .09609 .00773 .23962 3.1206 .09183 .00655 .22932 3.1342 .09596 .00783 .23929 3.1216 .09189 .00662 .22946 3.1350 .13677 .22671 .01371 2.3634 .15120 .24602 .04297 2.2743 .09497 .00858 .23670 3.1225 .09497 .00858 .23670 3.1225 .05727 .03644 .12917 2.9161 .03169 .05284 .00433 5.0781 .01409 .00923 .03296 5.2051 .01633 .00676 .03999 5.2257 .01401 .00933 .03266 5.2031 .01618 .00681 .03958 5.2269 .13684 .22807 .00000 3.7253 .14812 .24687 .00000 3.8592 .01385 .00970 .03188 5.1996 .01385 .00970 .03188 5.1996 .02428 .02756 .03620 4.2425 .03161 .05268 .00000 5.9128 .00543 .00904 .00000 7.1748 .00425 .00708 .00000 7.3217 .00554 .00923 .00000 7.1647 .00431 .00718 .00000 7.3100 .13753 .22922 .00000 3.7326 .14811 .24685 .00000 3.8636 .00563 .00939 .00000 7.1513 .00563 .00939 .00000 7.1513 .01637 .02681 .00641 5.6103 .03226 .05377 .00000 5.9073 .00588 .00981 .00000 7.1442 .00449 .00748 .00000 7.2987 .00594 .00989 .00000 7.1361 .00451 .00751 .00000 7.2969 .13702 .22837 .00000 3.7259 .14659 .24431 .00000 3.8580 .00595 .00992 .00000 7.1338 .00595 .00992 .00000 7.1338 .01574 .02623 .00000 6.6881 A' B" P 3.6439 .8870 .6596 66.0658 .7859 .9837 67.4237 .7843 .9839 64.9796 .7863 .9834 67.0657 .7847 .9839 .3331 .7833 -5893 .6834 .7475 -.2352 50.7321 .7930 .9786 50.7321 .7930 .9786 6.3569 .8564 .8084 1.4215 .9660 .2075 85.0299 .8878 .9886 111.0879 .8808 .9915 83.3590 .8885 .9883 108.6185 .8815 .9912 .0177 .8229 -.9818 .0313 .8340 -.9664 70.8403 .8936 .9860 70.8403 .8936 .9860 6.5598 .9414 .8088 .0105 .9873 -.9878 4.7930 .9969 .7033 10.2828 .9959 .8613 4.6308 .9969 .6935 9.9937 .9960 .8571 .0000 .8237 -1.000C .0000 .8356 -1.000C 4.2130 .9970 .6652 4.2130 .9970 .6652 3.4158 .9895 .6079 .0000 .9876 -1.0000 .0000 .9995 -1.0000 .0000 .9997 1.0000 .0000 .9995 -1.0000 .0000 .9997 1.000C .0000 .8243 1.0000 .0000 .8360 1.0000 .0000 .9995 1.0000 .0000 .9995 1.0000 .1258 .9980 -.8156 .0000 .9874-1.0000 .0000 .9995 • 1.0000 .0000 .9997 • 1.0000 .0000 .9995 • 1.0000 .0000 .9997 1.000C 1.0000 .0000 .8237 .0000 .8355 1.0000 .0000 .9995-1.0000 .0000 .9995 1.0000 .0000 .9981 1.0000 143 Table 10.2.1.2.3.3 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I,p = 10, mNz = £,pNz N:p OEX Rej Fp = .6 Fn Tp Tn NT 2 .4983 .40612 .04918 .35592 .64408 .95082 MB 2 .0187 .09331 .00116 .84525 .15475 .99884 RB 2 .0173 .08650 .00107 .85654 .14346 .99893 MD 2 .0195 .09457 .00121 .84319 .15681 .99879 RD 2 .0174 .08758 .00107 .85474 .14526 .99893 .9983 .89964 .79392 .02989 .97011 .20608 OR 2 OS 2 .8902 .80221 .68963 .12274 .87726 .31037 OF 2 .0289 .11236 .00185 .81397 .18603 .99815 CF 2 .0289 .11236 .00185 .81397 .18603 .99815 CS 2 .4009 .36774 .04262 .41551 .58449 .95738 NT 4 .5007 .57281 .04978 .07850 .92150 .95022 MB 4 .0276 .34483 .00179 .42649 37351 .99821 RB 4 .0178 .31500 .00109 .47574 32426 .99891 MD 4 .0280 .34683 .00182 .42316 .57684 .99818 RD 4 .0181 .31685 .00111 .47266 .52734 .99889 OR 4 1.0000 .93214 .83162 .00084 .99916 .16838 OS 4 .9984 .92113 .80653 .00247 .99753 .19347 OF 4 .0327 .35880 .00212 .40341 39659 .99788 CF 4 .0327 .35880 .00212 .40341 .59659 .99788 CS 4 .2807 .52080 .02858 .15106 .84894 .97142 NT 10 .4901 .61957 .04932 .00026 .99974 .95068 MB 10 .0412 .59635 .00276 .00791 .99209 .99724 RB 10 .0182 39138 .00112 .01511 .98489 .99888 MD 10 .0419 .59646 .00282 .00777 .99223 .99718 RD 10 .0185 .59155 .00113 .01484 .98516 .99887 OR 10 1.0000 .93187 .82967 .00000 1.00000 .17033 OS 10 .9999 .92260 .80651 .00000 1.00000 .19349 OF 10 .0433 .59671 .00289 .00742 .99258 .99711 CF 10 .0433 .59671 .00289 .00742 .99258 .99711 CS 10 .1200 .59355 .01348 .01974 .98026 .98652 NT 20 .4859 .61979 .04947 .00000 1.00000 .95053 MB 20 .0432 .60115 .00287 .00000 1.00000 .99713 RB 20 .0192 .60049 .00122 .00000 1.00000 .99878 MD 20 .0440 .60118 .00294 .00000 1.00000 .99706 RD 20 .0199 .60050 .00126 .00000 1.00000 .99874 OR 20 .9999 .93262 .83156 .00000 1.00000 .16844 OS 20 .9999 .92331 .80827 .00000 1.00000 .19173 OF 20 .0448 .60121 .00302 .00000 1.00000 .99698 CF 20 .0448 .60121 .00302 .00000 1.00000 .99698 CS 20 .1198 .60537 .01369 .00017 .99983 .98631 NT 40 .4915 .61992 .04980 .00000 1.00000 .95020 MB 40 .0415 .60115 .00288 .00000 1.00000 .99712 RB 40 .0178 .60047 .00118 .00000 1.00000 .99882 MD 40 .0424 .60118 .00294 .00000 1.00000 .99706 RD 40 .0178 .60047 .00118 .00000 1.00000 .99882 OR 40 1.0000 .93226 .83066 .00000 1.00000 .16934 OS 40 .9999 .92330 .80825 .00000 1.00000 .19175 OF 40 .0426 .60118 .00296 .00000 1.00000 .99704 CF 40 .0426 .60118 .00296 .00000 1.00000 .99704 CS 40 .1181 .60544 .01359 .00000 1.00000 .98641 A' B" s.Rej s.FpTn s:FnTpd' P .13450 .06783 .21954 2.0223 3.6612 .8873 .6612 .09244 .00885 .15381 2.0308 61.9172 .7866 .9825 .08198 .00847 .13647 2.0061 63.3432 .7838 .9828 .09323 .00903 .15510 2.0266 60.0841 .7870 .9820 .08256 .00849 .13743 2.0125 63.5731 .7842 .9829 .11568 .20284 .09170 1.0624 .2380 .7592 -.6989 .29629 .31637 .30953 .6666 .5758 .7046 -.3306 .10174 .01147 .16910 2.0101 45.3478 .7936 .9759 .10174 .01147 .16910 2.0101 45.3478 .7936 .9759 .13607 .07088 .21554 1.9345 4.2987 .8733 .7123 .06792 .06834 .10510 3.0622 1.4259 .9658 .2093 .14988 .01171 .24935 3.0975 68.2577 .8924 .9855 .13894 .00856 .23153 3.1242 108.8686 .8804 .9913 .14968 .01176 .24903 3.1022 67.3893 .8932 .9853 .13902 .00861 .23166 3.1274 107.3172 .8812 .9911 .06922 .17156 .00778 2.1795 .0115 .7907 -.9880 .08506 .19269 .04070 1.9452 .0280 .7947 -.9689 .14786 .01273 .24593 3.1039 57.8690 .8980 .9826 .14786 .01273 .24593 3.1039 57.8690 .8980 .9826 .07000 .05965 .10827 2.9339 3.5839 .9527 .6440 .02759 .06892 .00326 5.1225 .0095 .9876 -.9890 .01596 .01480 .02475 5.1884 2.5624 .9973 .4815 .02220 .00866 .03664 5.2245 10.2294 .9959 .8606 .01588 .01502 .02454 5.1879 2.4718 .9973 .4660 .02185 .00871 .03605 5.2273 9.9340 .9960 .8562 .06848 .17120 .00000 3.3120 .0000 .7926 -1.0000 .07583 .18958 .00000 3.3998 .0000 .7984•1.0000 .01552 .01515 .02383 5.1958 2.3154 .9974 .4368 .01552 .01515 .02383 5.1958 2.3154 .9974 .4368 .02557 .04630 .02338 4.2711 1.3860 .9916 .1853 .02794 .06985 .00000 5.9149 .0000 .9876 -1.0000 .00594 .01485 .00000 7.0269 .0000 .9993 -1.0000 .00376 .00940 .00000 7.2964 .0000 .9997 1.0000 .00603 .01508 .00000 7.0194 .0000 .9993 -1.0000 .00380 .00951 .00000 7.2869 .0000 .9997 • 1.OOO0 .06822 .17056 .00000 3.3046 .0000 .7921 1.0000 .07563 .18908 .00000 3.3934 .0000 .7979-1.0000 .00617 .01542 .00000 7.0103 .0000 .9992 1.0000 .00617 .01542 .00000 7.0103 .0000 .9992-1.0000 .01874 .04662 .00253 5.7826 .0190 .9965 -.9745 .02816 .07039 .00000 5.9117 .0000 .9875 1.0000 .00629 .01573 .00000 7.0263 .0000 .9993 1.0000 .00381 .00952 .00000 7.3062 .0000 .9997 1.0000 .00635 .01588 .00000 7.0194 .0000 .9993 1.0000 .00383 .00957 .00000 7.3048 .0000 .9997 1.000C .06893 .17234 .00000 3.3081 .0000 .7923 1.0000 .07629 .19073 .00000 3.3934 .0000 .7979 1.0000 .00637 .01593 .00000 7.0176 .0000 .9993 1.0000 .00637 .01593 .00000 7.0176 .0000 .9993-1.0000 .01867 .04667 .00000 6.4738 .0000 .9966 1.0000 Table 10.2.1.2.3.4 Table of empirical results for tests of correlations with no preliminary omnibus test when P * I,/? = 10, mNz = J5,pNz = .8 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn Tp Tn . 0 4 8 4 0 . 3 5 6 3 2 . 6 4 368 .95160 .2657 .52462 .0118 .12842 .00150 .83985 .16015 .99850 .0103 .11470 .00127 .85694 .14306 .99873 .0122 .13016 .00156 .83769 .16231 .99844 .0104 .11621 .00128 .85506 .14494 .99872 .9986 .95948 .87027 .01822 .98178 .12973 .9212 .89043 .79829 .08654 .91346 .20171 .0154 .15398 .00209 .80804 .19196 .99791 .0154 .15399 .00209 .80804 .19196 .99791 .2992 53556 .06430 .34662 .65338 .93570 .2727 .74755 .04953 .07795 .92205 .95047 .0184 .48328 .00258 .39654 .60346 .99742 .0081 .42021 .00099 .47498 52502 .99901 .0194 .48606 .00270 .39310 .60690 .99730 .0082 .42271 .00100 .47186 52814 .99900 .9999 .97913 .89732 .00042 .99958 .10268 .9992 .97680 .88843 .00111 .99889 .11157 .0229 .50118 .00312 .37431 .62569 .99688 .0229 .50118 .00312 .37431 .62569 .99688 .2186 .72375 .04752 .10719 .89281 .95248 .2710 .80968 .04949 .00028 .99972 .95051 .0416 .79748 .00594 .00463 .99537 .99406 .0086 .78804 .00108 .01521 .98479 .99892 .0423 .79762 .00607 .00450 .99550 .99393 .0088 .78827 .00111 .01494 .98506 .99889 .9997 .97932 .89659 .00000 1.00000 .10341 .9997 .97766 .88828 .00000 1.00000 .11172 .0437 .79778 .00622 .00433 .99567 .99378 .0437 .79778 .00622 .00433 .99567 .99378 .1208 .79828 .02869 .00932 .99068 .97131 .2647 .80981 .04904 .00000 1.00000 .95096 .0402 .80119 .00596 .00000 1.00000 .99404 .0076 .80021 .00104 .00000 1.00000 .99896 .0409 .80122 .00610 .00000 1.00000 .99390 .0079 .80022 .00108 .00000 1.00000 .99892 1.0000 .97932 .89662 .00000 1.00000 .10338 .9998 .97765 .88823 .00000 1.00000 .11177 .0415 .80124 .00619 .00000 1.00000 .99381 .0415 .80124 .00619 .00000 1.00000 .99381 .1110 .80559 .02808 .00003 .99997 .97192 .7916 .23982 .04977 .00000 1.00000 .95023 .0399 .80120 .00601 .00000 1.00000 .99399 .0080 .80023 .00113 .00000 1.00000 .99887 .0404 .80122 .00609 .00000 1.00000 .99391 .0081 .80023 .00114 .00000 1.00000 .99886 .9999 .97984 .89922 .00000 1.00000 .10078 .9999 .97812 .89059 .00000 1.00000 .10941 .0408 .80123 .00613 .00000 1.00000 .99387 .0408 .80123 .00613 .00000 1.00000 .99387 .1151 .80591 .02956 .00000 1.00000 .97044 OLEX Rej s:FnTp d' A' B" s.Rej s:FpTn P .17536 .10299 .21709 2.0289 3.7096 .8876 .6655 .12800 .01552 .15984 1.9739 49.8932 .7874 .9780 .10647 .01364 .13302 1.9526 54.0152 .7833 .9796 .12905 .01579 .16115 1.9715 48.6889 .7879 .9774 .10732 .01368 .13409 1.9583 54.0602 .7838 .9796 .07655 .17356 .06862 .9643 .2117 .7433 -.7265 .26353 .28906 .26725 .5269 .5605 .6743 -.3415 .14077 .01900 .17573 1.9937 41.4041 .7948 .9735 .14077 .01900 .17573 1.9937 41.4045 .7948 .9735 .17675 .13206 .21350 1.9141 2.9355 .8828 .5802 .08416 .10397 .10182 3.0684 1.4240 .9661 .2084 .20737 .02174 .25880 3.0595 48.3091 .8995 .9787 .18332 .01134 .22915 3.1563119.4687 .8807 .9921 .20695 .02209 .25826 3.0534 46.2190 .9003 .9777 .18344 .01140 .22930 3.1608118.1880 .8814 .9920 .02953 .14484 .00442 2.0714 .0085 .7745 -.9909 .04096 .15823 .02665 1.8398 .0196 .7751 -.9778 .20340 .02323 .25382 3.0551 39.9585 .9049 .9738 .20340 .02323 .25382 3.0551 39.9585 .9049 .9738 .07808 .11767 .09003 2.9110 1.8638 .9586 .3578 .02088 .10368 .00323 5.1023 .0101 .9876 -.9883 .01561 .03312 .01754 5.1175 .8012 .9973 -.1233 .02820 .01218 .03512 5.2324 10.6289 .9959 .8659 .01538 .03349 .01718 5.1205 .7662 .9973 -.1478 .02781 .01242 .03462 5.2306 10.1739 .9960 .8597 .02898 .14488 .00000 3.0025 .0000 .7759-1.0000 .03129 .15644 .00000 3.0475 .0000 .7779 -1.0000 .01518 .03372 .01688 5.1242 .7248 .9973 -.1780 .01518 .03372 .01688 5.1242 .7248 .9973 -.1780 .02443 .09690 .01499 4.2532 .3821 .9904 -.5024 .02110 .10549 .00000 5.9191 .0000 .9877 -1.0000 .00698 .03488 .00000 6.7797 .0000 .9985 1.0000 .00270 .01348 .00000 7.3422 .0000 .9997 -1.0000 .00710 .03550 .00000 6.7712 .0000 .9985 -1.000C .00272 .01361 .00000 7.3328 .0000 .9997 1.0000 .02869 .14343 .00000 3.0024 .0000 .7758 1.000C 1.0000 .03119 .15595 .00000 3.0477 .0000 .7779 .00715 .03573 .00000 6.7661 .0000 .9985 1.0000 .00715 .03573 .00000 6.7661 .0000 .9985 1.0000 .01990 .09940 .00088 5.9408 .0018 .9930 -.9980 .03360 .04200 .00000 5.9120 .0000 .9876-1.0000 .00709 .03545 .00000 6.7764 .0000 .9985 1.000C .00278 .01392 .00000 7.3178 .0000 .9997 1.0000 .00714 .03570 .00000 6.7718 .0000 .9985 1.000C .00279 .01396 .00000 7.3148 .0000 .9997-1.0000 .02864 .14320 .00000 2.9878 .0000 .7752 1.0000 .03115 .15577 .00000 3.0352 .0000 .7774-1.000C .00715 .03576 .00000 6.7693 .0000 .9985 1.0000 .00715 .03576 .00000 6.7693 .0000 .9985 1.0000 .02060 .10298 .00000 6.1523 .0000 .9926-1.0000 145 Table 10.2.2.1.1.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz - .2 N:p Ct£X NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn .0526 .01127 .01114 .98820 .0203 .00273 .00263 .99685 .0202 .00270 .00259 .99685 .0210 .00281 .00273 .99685 .0209 .00277 .00268 .99685 .0560 .02353 .02335 .97575 .0276 .00728 .00706 .99185 .0253 .00341 .00331 .99620 .0253 .00341 .00331 .99620 .0481 .00700 .00690 .99260 .0544 .01189 .01131 .98580 .0213 .00298 .00275 .99610 .0213 .00297 .00274 .99610 .0216 .00301 .00279 .99610 .0215 .00299 .00276 .99610 .0585 .02341 .02300 .97495 .0268 .00742 .00704 .99105 .0232 .00331 .00301 .99550 .0232 .00331 .00301 .99550 .0466 .00705 .00661 .99120 .0637 .01548 .01263 .97310 .0212 .00406 .00275 .99070 .0211 .00401 .00271 .99080 .0213 .00412 .00276 .99045 .0212 .00407 .00274 .99060 .0719 .02995 .02799 .96220 .0280 .00943 .00779 .98400 .0219 .00429 .00286 .99000 .0219 .00429 .00286 .99000 .0495 .00895 .00683 .98255 .0919 .02427 .01673 .94555 .0255 .00673 .00329 .97950 .0252 .00663 .00324 .97980 .0258 .00678 .00333 .97940 .0255 .00671 .00328 .97955 .1094 .04699 .04283 .93635 .0423 .01565 .01141 .96740 .0264 .00696 .00340 .97880 .0264 .00696 .00340 .97880 .0636 .01426 .00883 .96400 .1440 .04427 .02413 .87515 .0313 .01451 .00401 .94350 .0304 .01433 .00386 .94380 .0314 .01460 .00403 .94310 .0305 .01443 .00389 .94340 .1857 .08294 .07062 .86780 .0757 .03343 .02079 .91600 .0317 .01477 .00406 .94240 .0317 .01477 .00406 .94240 .0794 .02598 .01100 .91410 Tp Tn .01180 .98886 .00315 .99738 .00315 .99741 .00315 .99728 .00315 .99733 .02425 .97665 .00815 .99294 .00380 .99669 .00380 .99669 .00740 .99310 .01420 .98869 .00390 .99725 .00390 .99726 .00390 .99721 .00390 .99724 .02505 .97700 .00895 .99296 .00450 .99699 .00450 .99699 .00880 .99339 .02690 .98738 .00930 .99725 .00920 .99729 .00955 .99724 .00940 .99726 .03780 .97201 .01600 .99221 .01000 .99714 .01000 .99714 .01745 .99318 .05445 .98328 .02050 .99671 .02020 .99676 .02060 .99668 .02045 .99673 .06365 .95718 .03260 .98859 .02120 .99660 .02120 .99660 .03600 .99118 .12485 .97588 .05650 .99599 .05620 .99614 .05690 .99598 .05660 .99611 .13220 .92938 .08400 .97921 .05760 .99594 .05760 .99594 .08590 .98900 s.Rej s:FpTn S.FnTp .04944 .05178 .08100 .01702 .01852 .04019 .01676 .01823 .04019 .01730 .01893 .04019 .01695 .01852 .04019 .11098 .11121 .13356 .05506 .05485 .07489 .01938 .02099 .04400 .01938 .02099 .04400 .03113 .03283 .06241 .05079 .05186 .08860 .01786 .01892 .04455 .01778 .01880 .04455 .01800 .01904 .04455 .01784 .01888 .04455 .10611 .10632 .13285 .05553 .05505 .07743 .01903 .01989 .04827 .01903 .01989 .04827 .02969 .03165 .06688 .05776 .05259 .12561 .02153 .01909 .07010 .02114 .01872 .06976 .02175 .01913 .07095 .02137 .01888 .07044 .11809 .11557 .16209 .06113 .05756 .10293 .02233 .01960 .07246 .02233 .01960 .07246 .03303 .03123 .09445 .07126 .05768 .18443 .02724 .02062 .10189 .02667 .02040 .10071 .02735 .02073 .10211 .02698 .02051 .10153 .14517 .14174 .20863 .07669 .06757 .14980 .02774 .02094 .10369 .02774 .02094 .10369 .04148 .03540 .13400 .09176 .06476 .27748 .03912 .02259 .16796 .03849 .02199 .16702 .03928 .02263 .16857 .03869 .02213 .16763 .18607 .17839 .29348 .11189 .09473 .24176 .03953 .02272 .16966 .03953 .02272 .16966 .05375 .03946 .20031 d' .0221 .0595 .0642 .0474 .0534 .0160 .0519 .0458 .0458 .0252 .0880 .1156 .1171 .1112 .1141 .0363 .0877 .1344 .1344 .1037 .3091 .4227 .4232 .4311 .4282 .1344 .2743 .4368 .4368 .3568 .5235 .6741 .6731 .6724 .6744 .1940 .4325 .6769 .6769 .5738 .8241 1.0662 1.0764 1.0686 1.0777 .3551 .6591 1.0716 1.0716 .9239 A' B" 1.0515 .5142 1.1787 .5418 1.1941 3448 1.1396 .5338 1.1588 .5378 1.0323 .5095 1.1344 .5336 1.1312 .5322 1.1312 .5322 1.0637 3170 1.2174 .5516 1.3692 .5740 1.3748 .5748 1.3525 .5716 1.3636 .5732 1.0744 .5210 1.2354 .5539 1.4333 .5830 1.4333 .5830 1.2861 .5627 1.9039 .6363 2.9572 .6777 2.9659 .6779 3.0143 .6794 2.9973 .6789 1.2813 .5674 1.8699 .6304 3.0389 .6802 3.0389 .6802 2.2623 .6549 2.6545 .6828 4.9767 .7142 4.9831 .7142 4.9466 .7140 4.9835 .7143 1.3697 .5872 2.4374 .6678 4.9676 .7144 4.9676 .7144 3.3097 .6956 3.6260 .7275 9.5646 .7454 9.8552 .7460 9.5911 .7456 9.8533 .7461 1.5831 .6330 3.0829 .7043 9.6038 .7458 9.6038 .7458 5.4157 .7369 .0285 .0906 .0978 .0721 .0813 .0184 .0709 .0683 .0683 .0347 .1117 .1724 .1746 .1658 .1702 .0416 .1187 .1973 .1973 .1409 .3548 .5412 .5423 .5489 .5466 .1442 .3416 .5524 .5524 .4333 .5158 .7194 .7196 .7178 .7198 .1850 .4730 .7193 .7193 3974 .6455 .8605 .8647 .8610 .8648 .2722 .5816 .8613 .8613 .7566 P 146 Table 10.2.2.1.1.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz - .4 N.p OLEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn .0484 .01193 .01153 .98748 .0166 .00308 .00282 .99653 .0165 .00306 .00280 .99655 .0172 .00315 .00292 .99650 .0172 .00313 .00292 .99655 .0567 .02558 .02507 .97365 .0263 .00829 .00805 .99135 .0217 .00383 .00370 .99598 .0217 .00383 .00370 .99598 .0400 .00759 .00742 .99215 .0530 .01459 .01253 .98233 .0177 .00364 .00302 .99543 .0176 .00358 .00297 .99550 .0179 .00367 .00305 .99540 .0179 .00361 .00302 .99550 .0659 .02932 .02798 .96868 .0261 .00927 .00805 .98890 .0194 .00416 .00337 .99465 .0194 .00416 .00337 .99465 .0400 .00880 .00740 .98910 .0713 .02405 .01582 .96360 .0167 .00611 .00282 .98895 .0162 .00599 .00272 .98910 .0171 .00622 .00288 .98878 .0166 .00611 .00278 .98890 .0984 .04778 .04280 .94475 .0336 .01579 .01122 .97735 .0175 .00643 .00297 .98838 .0175 .00643 .00297 .98838 .0468 .01412 .00852 .97748 .1107 .04568 .02302 .92033 .0226 .01257 .00387 .97438 .0221 .01233 .00378 .97485 .0226 .01273 .00387 .97398 .0221 .01248 .00378 .97448 .1761 .09211 .07800 .88673 .0607 .03397 .02143 .94723 .0235 .01308 .00403 .97335 .0235 .01308 .00403 .97335 .0657 .02716 .01200 .95010 .1736 .09958 .03448 .80278 .0274 .03214 .00467 .92665 .0261 .03148 .00442 .92793 .0282 .03248 .00480 .92600 .0269 .03185 .00457 .92723 .3439 .19798 .15523 .73790 .1321 .09065 .05065 .84935 .0285 .03275 .00485 .92540 .0285 .03275 .00485 .92540 .0897 .06054 .01645 .87333 Tp Tn .01253 .98847 .00348 .99718 .00345 .99720 .00350 .99708 .00345 .99708 .02635 .97493 .00865 .99195 .00403 .99630 .00403 .99630 .00785 .99258 .01768 .98747 .00458 .99698 .00450 .99703 .00460 .99695 .00450 .99698 .03133 .97202 .01110 .99195 .00535 .99663 .00535 .99663 .01090 .99260 .03640 .98418 .01105 .99718 .01090 .99728 .01123 .99712 .01110 .99722 .05525 .95720 .02265 .98878 .01163 .99703 .01163 .99703 .02253 .99148 .07968 .97698 .02563 .99613 .02515 .99622 .02603 .99613 .02553 .99622 .11328 .92200 .05278 .97857 .02665 .99597 .02665 .99597 .04990 .98800 .19723 .96552 .07335 .99533 .07208 .99558 .07400 .99520 .07278 .99543 .26210 .84477 .15065 .94935 .07460 .99515 .07460 .99515 .12668 .98355 s:FnTp cT A' s.Rej s.FpTn P . 0 3 1 7 1 . 0 7 4 1 . 5 2 00 . 0 5 5 8 3 . 0 6 6 7 1 .05043 .01796 .02187 .02969 .0692 1.2081 .5475 .01780 .02181 .02959 .0687 1.2068 .5473 .01820 .02223 .02980 .0602 1.1783 .5418 .01803 .02223 .02959 .0554 1.1631 .5388 .11517 .11731 .12642 .0215 1.0427 .5125 .05850 .06027 .06755 .0264 1.0652 .5175 .02041 .02512 .03225 .0283 1.0784 .5203 .02041 .02512 .03225 .0283 1.0784 .5203 .03255 .03812 .04745 .0206 1.0513 .5139 .05616 .05796 .08105 .1360 1.3439 .5740 .02017 .02271 .03533 .1396 1.4528 .5855 .01973 .02229 .03471 .1394 1.4533 .5856 .02029 .02283 .03541 .1378 1.4455 .5846 .01985 .02247 .03471 .1339 1.4315 .5828 .11992 .12087 .13487 .0496 1.0981 .5275 .06249 .06101 .07707 .1197 1.3244 .5695 .02179 .02438 .03820 .1574 1.5131 .5932 .02179 .02438 .03820 .1574 1.5131 .5932 .03380 .03809 .05574 .1434 1.4039 .5812 .07237 .06180 .12202 .3549 2.0133 .6466 .02634 .02174 .05514 .4797 3.3637 .6883 .02564 .02124 .05449 .4863 3.4342 .6897 .02660 .02199 .05562 .4781 3.3388 .6879 .02596 .02149 .05502 .4853 3.4134 .6894 .15136 .14862 .17784 .1231 1.2264 .5596 .08345 .07416 .11369 .2811 1.8262 .6291 .02709 .02241 .05643 .4821 3.3547 .6884 .02709 .02241 .05643 .4821 3.3547 .6884 .04302 .03996 .08031 .3818 2.3119 .6590 .09650 .07052 .18079 .5878 2.7183 .6923 .03749 .02575 .08307 .7141 5.1917 .7177 .03654 .02549 .08158 .7134 5.2118 .7178 .03775 .02575 .08377 .7208 5.2594 .7184 .03675 .02549 .08212 .7198 5.2768 .7184 .20506 .19735 .24991 .2094 1.3167 .5874 .12296 .10399 .17567 .4065 2.0970 .6565 .03838 .02636 .08487 .7168 5.1658 .7178 .03838 .02636 .08487 .7168 5.1658 .7178 .05845 .04727 .11816 .6113 3.2968 .6995 .12928 .08140 .26592 .9671 3.6369 .7484 .05959 .02810 .13959 1.1483 10.2352 .7513 .05794 .02718 .13648 1.1579 10.6075 .7517 .05988 .02847 .14023 1.1433 10.0488 .7512 .05828 .02771 .13725 1.1516 10.3724 .7514 .27173 .26031 .34647 .3774 1.3655 .6336 .19679 .16071 .29279 .6049 2.2440 .6923 .06021 .02860 .14102 1.1440 10.0180 .7513 .06021 .02860 .14102 1.1440 10.0180 .7513 .08267 .05482 .17854 .9911 5.0689 .7456 B" .0407 .1043 .1037 .0906 .0835 .0243 .0356 .0419 .0419 .0282 .1677 .2045 .2046 .2019 .1966 .0546 .1578 .2266 .2266 .1896 .3852 .5911 .5983 .5885 .5963 .1205 .3324 .5905 .5905 .4456 .5306 .7327 .7335 .7362 .7368 .1655 .4089 .7318 .7318 .5999 .6525 .8721 .8766 .8697 .8738 .1919 .4537 .8693 .8693 .7448 147 Table 10.2.2.1.1.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz = .6 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn .0354 .01284 .01155 .98630 .0103 .00337 .00260 .99612 .0103 .00336 .00260 .99613 .0105 .00343 .00265 .99605 .0105 .00341 .00265 .99608 .0515 .02777 .02680 .97158 .0206 .00970 .00905 .98987 .0140 .00422 .00358 .99535 .0140 .00422 .00358 .99535 .0273 .00808 .00738 .99145 .0438 .01843 .01365 .97838 .0112 .00409 .00288 .99510 .0112 .00405 .00285 .99515 .0113 .00413 .00290 .99505 .0113 .00407 .00288 .99513 .0717 .03813 .03568 .96023 .0226 .01204 .00972 .98642 .0127 .00482 .00333 .99418 .0127 .00482 .00333 .99418 .0297 .01117 .00810 .98678 .0640 .03390 .01845 .95580 .0132 .00860 .00333 .98788 .0131 .00847 .00328 .98807 .0132 .00871 .00333 .98770 .0131 .00861 .00328 .98783 .1176 .07013 .05960 .92285 .0366 .02539 .01675 .96885 .0133 .00909 .00335 .98708 .0133 .00909 .00335 .98708 .0378 .02103 .01003 .97163 .1007 .07303 .02820 .89708 .0159 .01866 .00410 .97163 .0154 .01815 .00398 .97240 .0160 .01891 .00413 .97123 .0156 .01836 .00403 .97208 .2410 .15602 .12680 .82450 .0755 .06062 .03665 .92340 .0168 .01944 .00433 .97048 .0168 .01944 .00433 .97048 .0599 .04611 .01598 .93380 .1467 .16190 .04083 .75738 .0195 .04997 .00500 .92005 .0186 .04835 .00475 .92258 .0195 .05066 .00503 .91892 .0187 .04899 .00480 .92155 .4804 .34850 .26988 .59908 .2000 .18237 .10968 .76917 .0198 .05113 .00510 .91818 .0198 .05113 .00510 .91818 .0859 .10953 .02330 .83298 OLEX Rej Tn Tp .01370 .98845 .00388 .99740 .00387 .99740 .00395 .99735 .00392 .99735 .02842 .97320 .01013 .99095 .00465 .99643 .00465 .99643 .00855 .99263 .02162 .98635 .00490 .99713 .00485 .99715 .00495 .99710 .00487 .99713 .03977 .96433 .01358 .99028 .00582 .99668 .00582 .99668 .01322 .99190 .04420 .98155 .01212 .99668 .01193 .99673 .01230 .99668 .01217 .99673 .07715 .94040 .03115 .98325 .01292 .99665 .01292 .99665 .02837 .98998 .10292 .97180 .02837 .99590 .02760 .99603 .02877 .99588 .02792 .99598 .17550 .87320 .07660 .96335 .02952 .99568 .02952 .99568 .06620 .98403 .24262 .95918 .07995 .99500 .07742 .99525 .08108 .99498 .07845 .99520 .40092 .73013 .23083 .89033 .08182 .99490 .08182 .99490 .16702 .97670 A' s.Rej s.FpTn s:FnTp d' P .05293 .06465 .06326 .0660 1.1592 .5398 .01875 .02561 .02601 .1323 1.4348 .5829 .01867 .02561 .02596 .1309 1.4293 .5822 .01895 .02585 .02632 .1319 1.4320 .5826 .01880 .02585 .02611 .1290 1.4212 .5812 .12090 .12819 .12637 .0255 1.0500 .5146 .06510 .07198 .06963 .0422 1.1039 .5270 .02168 .03031 .02883 .0890 1.2653 .5581 .02168 .03031 .02883 .0890 1.2653 .5581 .03404 .04585 .04271 .0539 1.1388 .5347 .06389 .06848 .08180 .1858 1.4810 .5942 .02131 .02735 .03016 .1789 1.6130 .6038 .02104 .02701 .02975 .1782 1.6109 .6036 .02149 .02746 .03047 .1796 1.6150 .6040 .02108 .02712 .02979 .1765 1.6032 .6028 .13928 .14452 .14747 .0498 1.0927 .5268 .07690 .07466 .08604 .1277 1.3367 .5720 .02372 .02992 .03354 .1908 1.6481 .6077 .02372 .02992 .03354 .1908 1.6481 .6077 .03945 .04832 .05287 .1846 1.5323 .5981 .08663 .07448 .12008 .3830 2.0665 .6522 .03162 .02886 .04843 .4605 3.1382 .6836 .03097 .02843 .04752 .4596 3.1395 .6836 .03186 .02886 .04881 .4663 3.1793 .6847 .03135 .02843 .04821 .4671 3.1926 .6849 .18676 .18341 .20609 .1336 1.2205 .5615 .11374 .09929 .13532 .2619 1.6862 .6192 .03266 .02896 .05008 .4827 3.2946 .6876 .03266 .02896 .05008 .4827 3.2946 .6876 .05529 .05190 .08029 .4201 2.4317 .6663 .12119 .08856 .17763 .6428 2.7728 .7007 .04675 .03272 .07368 .7384 5.3625 .7200 .04499 .03226 .07114 .7369 5.3889 .7199 .04730 .03282 .07462 .7424 5.3959 .7203 .04531 .03244 .07162 .7376 5.3798 .7200 .26584 .25827 .29554 .2090 1.2420 .5833 .17804 .14869 .21399 .3626 1.7927 .6408 .04804 .03354 .07556 .7376 5.2840 .7197 .04804 .03354 .07556 .7376 5.2840 .7197 .08130 .06539 .12128 .6403 3.2172 .7024 .15535 .10352 .23820 1.0433 3.5692 .7605 .07628 .03588 .12383 1.1704 10.2765 .7532 .07268 .03486 .11834 1.1709 10.4979 .7529 .07688 .03614 .12482 1.1763 10.3403 .7536 .07313 .03521 .11902 1.1743 10.5055 .7532 .33516 .33639 .37596 .3622 1.1694 .6266 .29929 .25431 .35644 .4922 1.6216 .6652 .07721 .03639 .12536 1.1760 10.2744 .7537 .07721 .03639 .12536 1.1760 10.2744 .7537 .11620 .07903 .17915 1.0239 4.5416 .7519 B" .0841 .1973 .1953 .1963 .1923 .0284 .0559 .1302 .1302 .0732 .2220 .2595 .2588 .2602 .2563 .0521 .1636 .2714 .2714 .2376 .3999 .5663 .5664 .5714 .5728 .1191 .2939 .5850 .585C .4705 .5422 .7419 .7429 .7436 .7426 .1330 .3341 .7386 .7386 .5945 .6487 .8733 .8758 .8742 .8760 .0987 .2903 .8735 .8735 .7188 148 Table 10.2.2.1.1.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .1, pNz = .8 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 OEX .0212 .0050 .0050 .0053 .0053 .0411 .0137 .0064 .0064 .0140 .0309 .0067 .0066 .0069 .0068 .0610 .0190 .0080 .0080 .0176 .0410 .0063 .0061 .0063 .0062 .1161 .0325 .0069 .0069 .0226 .0630 .0095 .0093 .0096 .0095 .2589 .0796 .0097 .0097 .0367 .0888 .0095 .0089 .0100 .0091 .5170 .2369 .0103 .0103 .0587 Fp Fn Rej .01349 .01175 .98608 .00338 .00250 .99640 .00336 .00250 .99643 .00347 .00265 .99633 .00344 .00265 .99636 .02899 .02750 .97064 .00968 .00865 .99006 .00417 .00320 .99559 .00417 .00320 .99559 .00859 .00735 .99110 .02131 .01640 .97746 .00474 .00335 .99491 .00468 .00330 .99498 .00482 .00345 .99484 .00474 .00340 .99492 .04460 .04135 .95459 .01458 .01220 .98483 .00558 .00400 .99403 .00558 .00400 .99403 .01318 .00890 .98575 .04418 .02130 .95010 .01082 .00315 .98726 .01063 .00305 .98748 .01100 .00315 .98704 .01084 .00310 .98723 .09438 .07895 .90176 .03485 .02165 .96185 .01162 .00345 .98634 .01162 .00345 .98634 .02747 .01145 .96853 .09935 .03265 .88398 .02510 .00485 .96984 .02438 .00475 .97071 .02555 .00490 .96929 .02475 .00485 .97028 .22289 .18050 .76651 .09019 .05550 .90114 .02630 .00495 .96836 .02630 .00495 .96836 .06639 .01875 .92170 .22274 .04580 .73303 .06775 .00480 .91651 .06475 .00450 .92019 .06867 .00505 .91543 .06562 .00460 .91913 .49661 .38100 .47449 .27941 .17810 .69526 .06936 .00520 .91460 .06936 .00520 .91460 .16678 .03025 .79909 Tn Tp .01393 .98825 .00360 .99750 .00358 .99750 .00368 .99735 .00364 .99735 .02936 .97250 .00994 .99135 .00441 .99680 .00441 .99680 .00890 .99265 .02254 .98360 .00509 .99665 .00503 .99670 .00516 .99655 .00508 .99660 .04541 .95865 .01518 .98780 .00598 .99600 .00598 .99600 .01425 .99110 .04990 .97870 .01274 .99685 .01253 .99695 .01296 .99685 .01278 .99690 .09824 .92105 .03815 .97835 .01366 .99655 .01366 .99655 .03148 .98855 .11603 .96735 .03016 .99515 .02929 .99525 .03071 .99510 .02973 .99515 .23349 .81950 .09886 .94450 .03164 .99505 .03164 .99505 .07830 .98125 .26698 .95420 .08349 .99520 .07981 .99550 .08458 .99495 .08088 .99540 .52551 .61900 .30474 .82190 .08540 .99480 .08540 .99480 .20091 .96975 s.Rej s.FpTn s.FnTp d' A' P .05460 .08299 .05863 .0658 1.1583 .5396 .01883 .03527 .02186 .1196 1.3889 .5767 .01862 .03527 .02165 .1173 1.3803 5754 .01916 .03631 .02213 .1077 1.3423 .5700 .01893 .03631 .02196 .1042 1.3300 .5681 .12470 .14123 .12743 .0286 1.0560 .5163 .06610 .07779 .06815 .0516 1.1291 .5327 .02153 .03987 .02470 .1078 1.3339 .5690 .02153 .03987 .02470 .1078 1.3339 .5690 .03577 .06302 .03932 .0699 1.1831 .5439 .06959 .09424 .07618 .1306 1.3102 .5696 .02314 .04079 .02677 .1416 1.4533 .5858 .02272 .04049 .02634 .1423 1.4570 .5863 .02338 .04139 .02704 .1369 1.4340 .5834 .02284 .04109 .02644 .1358 1.4309 .5829 .15261 .17268 .15597 .0442 1.0786 .5234 .08562 .09302 .08882 .0853 1.2072 .5498 .02543 .04454 .02951 .1385 1.4299 .5831 .02543 .04454 .02951 .1385 1.4299 .5831 .04486 .06687 .05136 .1794 1.5054 .5952 .09775 .10487 .11276 .3818 2.0162 .6506 .03565 .03956 .04303 .4976 3.4403 .6906 .03482 .03893 .04216 .5017 3.4905 .6915 .03596 .03956 .04344 .5044 3.4928 .6917 .03525 .03925 .04268 .5040 3.4991 .6918 .21612 .23270 .22454 .1205 1.1769 .5543 .13577 .12553 .14587 .2482 1.6013 .6123 .03708 .04139 .04472 .4948 3.3680 .6894 .03708 .04139 .04472 .4948 3.3680 .6894 .06421 .07579 .07515 .4156 2.3610 .6641 .13719 .12810 .16264 .6481 2.6766 .7012 .05430 .05002 .06620 .7079 4.8568 .7162 .05208 .04952 .06359 .7021 4.8284 .7156 .05500 .05026 .06708 .7124 4.8852 .7166 .05252 .05002 .06409 .7015 4.7983 .7155 .30914 .32974 .32206 .1861 1.1649 .5729 .22418 .20093 .24006 .3057 1.5533 .6211 .05593 .05050 .06822 .7220 4.9615 .7176 .05593 .05050 .06822 .7220 4.9615 .7176 .09915 .09708 .11808 .6637 3.1912 .7053 .16677 .14901 .20189 1.0650 3.4198 .7651 .08846 .04927 .10955 1.2079 11.0108 .7554 .08255 .04775 .10245 1.2057 11.2743 .7548 .08916 .05050 .11036 1.1974 10.6266 .7551 .08311 .04826 .10313 1.2053 11.1663 .7549 .34845 .40970 .36249 .3668 1.0448 .6271 .36792 .34204 .39115 .4118 1.3433 .6424 .08952 .05122 .11080 1.1926 10.4289 .7549 .08952 .05122 .11080 1.1926 10.4289 .7549 .14275 .12293 .17253 1.0388 4.0975 .7564 B" .0836 .1798 .1764 .1616 .1566 .0318 .0686 .1587 .1587 .0946 .1546 .2051 .2064 .1980 .1968 .0447 .1072 .1970 .1970 .2285 .3892 .6004 .6053 .6059 .6064 .0984 .2680 .5935 .5935 .4585 .5291 .7167 .7148 .7185 .7133 .095C .2591 .7230 .7230 .5937 .6349 .8825 .8850 .8781 .8840 .0278 .1828 .8758 .8758 .6910 149 Table 10.2.2.1.2.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .2 N:p OLEX NT 2 MB 2 . RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .0741 .01872 .0240 .00472 .0236 .00466 .0246 .00488 .0242 .00482 .0868 .03977 .0398 .01412 .0308 .00594 .0308 .00594 .0579 .01175 .1250 .03686 .0298 .01116 .0296 .01105 .0302 .01132 .0298 .01115 .1572 .07302 .0650 .02882 .0335 .01219 .0335 .01219 .0786 .02234 .2304 .10159 .0366 .04500 .0348 .04423 .0373 .04533 .0354 .04456 .3525 .17673 .1765 .10264 .0397 .04657 .0397 .04657 .0921 .06054 .3111 .19640 .0432 .12188 .0378 .11966 .0444 .12249 .0382 .12015 .5675 .32127 .3761 .24458 .0466 .12344 .0466 .12344 .0869 .12243 .3364 .23851 .0486 .19348 .0391 .19161 .0498 .19377 .0404 .19200 .6508 .38166 .5084 .32560 .0503 .19390 .0503 .19390 .0599 .18362 Fp .01454 .00308 .00301 .00315 .00310 .03754 .01169 .00403 .00403 .00860 .02218 .00386 .00381 .00393 .00384 .06414 .01973 .00438 .00438 .01135 .03718 .00476 .00449 .00485 .00456 .13179 .05546 .00521 .00521 .01324 .04800 .00565 .00493 .00581 .00498 .20634 .11989 .00611 .00611 .01215 .05060 .00641 .00505 .00656 .00524 .22970 .16021 .00663 .00663 .00856 Fn .96455 .98870 .98875 .98820 .98830 .95130 .97615 .98640 .98640 .97565 .90440 .95965 .96000 .95910 .95960 .89145 .93480 .95655 .95655 .93370 .64075 .79405 .79680 .79275 .79545 .64350 .70865 .78800 .78800 .75025 .21000 .41320 .42140 .41080 .41915 .21900 .25665 .40725 .40725 .43645 .00985 .05825 .06215 .05740 .06095 .01050 .01285 .05700 .05700 .11615 Tp Tn .03545 .98546 .01130 .99693 .01125 .99699 .01180 .99685 .01170 .99690 .04870 .96246 .02385 .98831 .01360 .99598 .01360 .99598 .02435 .99140 .09560 .97782 .04035 .99614 .04000 .99619 .04090 .99608 .04040 .99616 .10855 .93586 .06520 .98028 .04345 .99563 .04345 .99563 .06630 .98865 .35925 .96283 .20595 .99524 .20320 .99551 .20725 .99515 .20455 .99544 .35650 .86821 .29135 .94454 .21200 .99479 .21200 .99479 .24975 .98676 .79000 .95200 .58680 .99435 37860 .99508 .58920 .99419 .58085 .99503 .78100 .79366 .74335 .88011 .59275 .99389 .59275 .99389 .56355 .98785 .99015 .94940 .94175 .99359 .93785 .99495 .94260 .99344 .93905 .99476 .98950 .77030 .98715 .83979 .94300 .99338 .94300 .99338 .88385 .99144 s:Rej s.FpTn s:FnTp .06467 .02249 .02210 .02285 .02247 .14362 .08044 .02574 .02574 .04158 .08645 .03487 .03439 .03527 .03457 .18195 .10896 .03675 .03675 .05304 .12038 .06534 .06385 .06566 .06416 .23848 .18259 .06685 .06685 .07374 .10254 .07682 .07515 .07703 .07520 .23504 .21053 .07720 .07720 .07256 .06501 .04076 .03976 .04077 .03970 .20653 .18756 .04075 .04075 .05259 .05682 .01984 .01957 .02007 .01992 .14214 .07575 .02304 .02304 .03785 .06492 .02241 .02214 .02264 .02221 .17515 .09577 .02397 .02397 .04168 .07537 .02502 .02405 .02522 .02423 .22720 .15597 .02633 .02633 .04446 .07931 .02715 .02533 .02754 .02544 .25583 .21277 .02826 .02826 .04156 .07884 .02907 .02543 .02935 .02597 .25638 .23125 .02947 .02947 .03615 d' .14856 .3764 .07598 .4595 .07582 .4646 .07753 .4682 .07723 .4702 .18610 .1224 .12887 .2872 .08316 .4413 .08316 .4413 .11416 .4112 .24580 .7037 .14352 .9172 .14213 .9176 .14450 .9181 .14290 .9200 .27009 .2867 .21413 .5469 .14963 .9097 .14963 .9097 .18113 .7745 .42146 1.4240 .30335 1.7721 .29886 1.7828 .30460 1.7703 .29999 1.7819 .42418 .7502 .40995 1.0446 .30792 1.7619 .30792 1.7619 .30440 1.5439 .35540 2.4710 .36281 2.7526 .35928 2.7794 .36291 2.7488 .35945 2.7816 .36035 1.5948 .39271 1.8293 .36268 2.7402 .36268 2.7402 .31176 2.4123 .07667 3.9711 .16744 4.0582 .17180 4.1094 .16631 4.0573 .17033 4.1066 .07931 3.0478 .09144 3.2243 .16584 4.0574 .16584 4.0574 .21387 3.5785 P A' 2.1182 .6528 3.1690 .6840 3.2157 .6851 3.2198 .6854 3.2425 .6859 1.2342 .5602 1.8403 .6306 2.9216 .6784 2.9216 .6784 2.4476 .6657 3.2137 .7108 7.5587 .7352 7.5935 .7353 7.5331 .7353 7.6105 .7354 1.4843 .6141 2.6560 .6860 7.1788 .7346 7.1788 .7346 4.3265 .7211 4.6051 .8078 20.5768 .7948 21.5256 .7944 20.3211' .7950 21.2933 .7946 1.7459 .7223 3.0635 .7648 19.3133 .7958 19.3133 .7958 9.3405 .7967 2.8870 .9297 24.1602 .8937 27.4193 .8920 23.5275 .8942 27.1422 .8925 1.0354 .8650 1.6117 .8868 22.4535 .8950 22.4535 .8950 12.4753 .8842 .2526 .9846 6.4538 .9836 8.3934 .9830 6.2505 .9838 8.0026 .9833 .0917 .9386 .1361 .9556 6.1643 .9839 6.1643 .9839 8.4014 .9683 B" .4095 .5694 .5748 .5757 .5782 .1237 .3368 .5398 .5398 .4718 .5990 .8192 .8200 .8187 .8205 .2343 .5183 .8103 .8103 .6931 .7309 .9437 .9463 .9429 .9457 .3344 .5952 .9398 .9398 .8697 .5681 .9547 .9606 .9534 .9601 .0217 .2878 .9509 .9509 .9069 -.6625 .7919 .8413 .7849 .8331 -.8891 -.8277 .7818 .7818 .8472 150 Table 10.2.2.1.2.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .4 N:p CLEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn .0865 .03231 .01980 .94893 .0246 .00796 .00420 .98640 .0244 .00785 .00417 .98663 .0253 .00816 .00433 .98610 .0251 .00803 .00428 .98635 .1276 .06922 .06050 .91770 .0530 .02814 .02070 .96070 .0310 .01020 .00542 .98263 .0310 .01020 .00542 .98263 .0629 .02057 .01243 .96723 .1508 .08265 .03162 .84080 .0270 .02431 .00470 .94628 .0262 .02357 .00448 .94780 .0274 .02463 .00477 .94558 .0267 .02393 .00457 .94703 .2781 .16646 .13213 .78205 .1090 .07839 .04712 .87470 .0315 .02726 .00555 .94018 .0315 .02726 .00555 .94018 .0904 .05165 .01747 .89708 .2281 .23777 .04545 .47375 .0302 .10178 .00523 .75340 .0257 .09735 .00438 .76320 .0303 .10270 .00525 .75113 .0260 .09820 .00443 .76115 .6432 .44338 .30738 .35263 .3695 .30403 .16865 .49290 .0321 .10569 .00557 .74412 .0321 .10569 .00557 .74412 .1135 .15556 .02212 .64427 .2508 .37505 .04902 .13590 .0383 .25398 .00670 .37510 .0286 .24237 .00497 .40153 .0391 .25528 .00688 .37213 .0291 .24366 .00505 .39843 .8039 .62678 .40995 .04797 .6920 .57131 .33643 .07637 .0402 .25700 .00710 .36815 .0402 .25700 .00710 .36815 .1029 .28203 .01968 .32445 .2580 .42667 .05028 .00875 .0453 .38503 .00817 .04968 .0295 .37830 .00522 .06207 .0463 .38559 .00835 .04855 .0300 .37878 .00532 .06103 .8267 .65286 .42207 .00095 .7590 .61963 .36688 .00125 .0468 .38577 .00843 .04822 .0468 .38577 .00843 .04822 .0660 .37660 .01275 .07763 Tp Tn .05108 .98020 .01360 .99580 .01338 .99583 .01390 .99567 .01365 .99572 .08230 .93950 .03930 .97930 .01738 .99458 .01738 .99458 .03278 .98757 .15920 .96838 .05373 .99530 .05220 .99552 .05443 .99523 .05298 .99543 .21795 .86787 .12530 .95288 .05983 .99445 .05983 .99445 .10293 .98253 52625 .95455 .24660 .99477 .23680 .99562 .24888 .99475 .23885 .99557 .64738 .69262 .50710 .83135 .25588 .99443 .25588 .99443 .35573 .97788 .86410 .95098 .62490 .99330 59848 .99503 .62788 .99312 .60158 .99495 .95203 .59005 .92363 .66357 .63185 .99290 .63185 .99290 .67555 .98032 .99125 .94972 .95033 .99183 .93793 .99478 .95145 .99165 .93898 .99468 .99905 57793 .99875 .63312 .95178 .99157 .95178 .99157 .92237 .98725 s:Rej s.FpTn s.FnTp d' P A' B" .08554 .07030 .15008 .4234 2.1851 .6611 .4281 .02995 .02675 .06146 .4269 2.8124 .6752 .5247 .02938 .02665 .06033 .4231 2.7920 .6744 .5216 .03036 .02725 .06221 .4248 2.7869 .6745 5212 .02971 .02700 .06094 .4217 2.7722 .6739 5189 .18985 .18310 .22515 .1608 1.2667 .5720 .1412 .12127 .10904 .15952 .2806 1.7040 .6231 .3013 .03458 .03093 .07003 .4368 2.7663 .6751 5203 .03458 .03093 .07003 .4368 2.7663 .6751 .5203 .05741 .05197 .10251 .4020 2.2728 .6603 .4416 .12684 .08211 .25478 .8598 3.4126 .7333 .6277 .05417 .02887 .12548 .9874 7.9800 .7404 .8315 .05184 .02768 .12068 .9894 8.1354 .7405 .8345 .05460 .02905 .12652 .9889 7.9615 .7406 .8312 .05250 .02791 .12228 .9904 8.0969 .7406 .8338 .26634 .25268 .33591 .3372 1.3766 .6232 .1956 .19486 .16420 .27941 .5246 2.0965 .6765 .4188 .05780 .03162 .13315 .9833 7.4908 .7405 .8213 .05780 .03162 .13315 .9833 7.4908 .7405 .8213 .08518 .05954 .17401 .8441 4.1538 .7293 .6865 .14658 .09104 .32484 1.7565 4.1662 .8543 .7036 .10238 .03028 .24866 1.8748 20.9487 .8054 .9455 .09651 .02739 .23642 1.9044 24.0010 .8037 .9528 .10284 .03033 .24985 1.8809 20.9922 .8060 .9456 .09690 .02754 .23724 1.9071 23.8716 .8042 .9526 .29259 .31346 .37206 .8815 1.0567 .7540 .0349 .30720 .27891 .43723 .9773 1.5844 .7686 .2813 .10402 .03122 .25237 1.8824 20.2208 .8075 .9435 .10402 .03122 .25237 1.8824 20.2208 .8075 .9435 .11230 .06661 .25228 1.6420 7.0667 .8197 .8275 .09358 .09178 .19078 2.7534 2.1486 .9501 .4317 .11380 .03439 .27816 2.7913 20.2282 .9029 .9448 .10965 .02949 .27100 2.8275 26.9048 .8970 .9597 .11358 .03504 .27742 2.7895 19.7016 .9036 .9431 .10925 .02971 .26992 2.8298 26.4552 .8978 .9589 .21046 .31906 .14153 1.8925 .2567 .8720 -.6823 .23790 .32571 .21809 1.8521 .3933 .8802 -5198 .11324 .03566 .27638 2.7889 19.1031 .9045 .9412 .11324 .03566 .27638 2.7889 19.1031 .9045 .9412 .09502 .06155 .21558 2.5156 7.5296 .9100 .8382 .05840 .09269 .04781 4.0181 .2289 .9850 -.6926 .05255 .03895 .11679 4.0494 4.5967 .9853 .7071 .05388 .03070 .12735 4.0987 8.1466 .9830 .8363 .05203 .03937 .11525 4.0523 4.4262 .9856 .696C .05359 .03104 .12643 4.1007 7.9041 .9832 .8310 .18914 .31456 .01578 3.3020 .0082 .8940 -.9922 .19262 .31995 .02058 3.3635 .0110 .9077 -.9893 .05183 .03953 .11465 4.0519 4.3643 .9856 .6918 .05183 .03953 .11465 4.0519 4.3643 .9856 .6918 .05954 .05144 .12473 3.6550 4.4143 .9769 .7010 151 Table 10.2.2.1.2.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p - 5, mNz = .3, pNz = .6 N:p ct£X NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp Fn .0735 .05250 .02308 .92788 .0167 .01159 .00435 .98358 .0166 .01134 .00433 .98398 .0172 .01189 .00448 .98317 .0171 .01161 .00445 .98362 .1721 .12091 .09872 .86430 .0603 .05114 .03430 .93763 .0218 .01532 .00580 .97833 .0218 .01532 .00580 .97833 .0544 .03575 .01605 .95112 .1196 .13427 .03548 .79987 .0181 .03798 .00470 .93983 .0174 .03623 .00450 .94262 .0183 .03855 .00475 .93892 .0177 .03672 .00458 .94185 .3818 .29427 .22680 .66075 .1616 .15660 .09610 .80307 .0210 .04306 .00548 .93188 .0210 .04306 .00548 .93188 .0823 .09617 .02440 .85598 .1632 .35092 .04683 .44635 .0210 .15812 .00555 .74017 .0169 .14588 .00435 .75977 .0214 .15978 .00565 .73747 .0170 .14726 .00438 .75748 .7497 .65688 .46693 .21648 .5132 30965 .32008 .36397 .0231 .16423 .00615 .73038 .0231 .16423 .00615 .73038 .1165 .27527 .03405 .56392 .1719 .53822 .04925 .13580 .0323 .39075 .00885 .35465 .0193 .36075 .00508 .40213 .0328 .39256 .00898 .35172 .0194 .36247 .00510 .39928 .8705 .81556 .56713 .01882 .8101 .78562 .52093 .03792 .0336 .39499 .00918 .34780 .0336 .39499 .00918 .34780 .0990 .46087 .02948 .25153 .1760 .61530 .05060 .00823 .0442 .58080 .01228 .04018 .0213 .56533 .00563 .06153 .0448 .58120 .01245 .03963 .0217 .56604 .00575 .06043 .8846 .83130 .57855 .00020 .8462 .81803 .54555 .00032 .0453 .58145 .01258 .03930 .0453 .58145 .01258 .03930 .0700 .57913 .02115 .04888 Tp Tn .07212 .97693 .01642 .99565 .01602 .99568 .01683 .99553 .01638 .99555 .13570 .90128 .06237 .96570 .02167 .99420 .02167 .99420 .04888 .98395 .20013 .96453 .06017 .99530 .05738 .99550 .06108 .99525 .05815 .99543 .33925 .77320 .19693 .90390 .06812 .99453 .06812 .99453 .14402 .97560 .55365 .95318 .25983 .99445 .24023 .99565 .26253 .99435 .24252 .99563 .78352 .53308 .63603 .67993 .26962 .99385 .26962 .99385 .43608 .96595 .86420 .95075 .64535 .99115 .59787 .99492 .64828 .99103 .60072 .99490 .98118 .43288 .96208 .47908 .65220 .99083 .65220 .99083 .74847 .97053 .99177 .94940 .95982 .98773 .93847 .99438 .96037 .98755 .93957 .99425 .99980 .42145 .99968 .45445 .96070 .98743 .96070 .98743 .95112 .97885 s.FnTp s.Rej s.FpTn d' P A' B" .10986 .08846 .16034 .5338 2.5142 .6826 .4960 .03764 .03400 .05841 .4895 3.2042 .6868 .5770 .03627 .03391 .05619 .4816 3.1534 .6854 3708 .03808 .03444 .05912 .4899 3.1919 .6866 .5758 .03661 .03436 .05668 .4809 3.1344 .6851 .5687 .25412 .24518 .28311 .1890 1.2532 .5784 .1372 .17606 .15553 .20712 .2858 1.6155 .6198 .2768 .04415 .03989 .06793 3036 3.1403 .6871 .5723 .04415 .03989 .06793 .5036 3.1403 .6871 .5723 .08102 .07189 .11663 .4874 23238 .6763 .4929 .16389 .10325 .25275 .9647 3.5848 .7484 .6478 .07355 .03522 .11990 1.0438 8.7243 .7444 .8472 .06850 .03435 .11175 1.0349 8.7386 .7437 .8470 .07448 .03539 .12145 1.0478 8.7452 .7447 .8477 .06915 .03461 .11282 1.0359 8.7009 .7438 .8465 .34663 .33380 .39302 .3349 1.2152 .6192 .1221 .29643 .24846 .35596 .4515 1.6272 .6559 .2909 .07935 .03810 .12937 1.0543 8.3872 .7457 .842C .07935 .03810 .12937 1.0543 8.3872 .7457 .8420 .13017 .08842 .19737 .9079 3.9619 .7383 .6763 .18028 .11359 .28750 1.8113 4.0396 .8618 .6940 .15029 .03930 .24769 1.8957 20.4369 .8086 .9442 .13504 .03382 .22381 1.9181 24.3565 .8047 .9536 .15097 .03960 .24873 1.8977 20.2225 .8092 .9436 .13540 .03391 .22441 1.9234 24.3569 .8053 .9537 .28913 .36142 .31398 .8671 .7379 .7495 -.1894 .37843 .37451 .43745 .8154 1.0500 .7404 .0309 .15238 .04168 .25101 1.8894 19.0120 .8106 .9398 .15238 .04168 .25101 1.8894 19.0120 .8106 .9398 .16352 .10095 .25744 1.6634 5.2131 .8345 .7641 .11557 .11549 .17561 2.7516 2.1394 .9500 .4296 .16783 .05059 .27548 2.7446 15.5384 .9071 .9262 .15862 .03699 .26318 2.8185 26.4031 .8968 .9589 .16750 .05088 .27491 2.7473 15.3030 .9078 .9249 .15809 .03707 .26232 2.8242 26.2391 .8975 .9586 .15182 .33954 .08007 1.9098 .1169 .8446 -.8601 .19697 .36136 .16230 1.7229 .2071 .8448 -.7449 .16672 .05133 .27363 2.7497 14.9493 .9087 .9229 .16672 .05133 .27363 2.7497 14.9493 .9087 .9229 .12303 .09586 .19082 2.5582 4.7546 .9254 .7362 .05299 .11754 .03992 4.0375 .2159 .9851 -.7094 .06383 .06026 .09684 3.9970 2.7154 .9866 .5217 .07168 .03905 .11637 4.0769 7.5668 .9830 .8234 .06336 .06069 .09593 3.9979 2.6524 .9867 .5117 .07110 .03959 .11532 4.0783 7.3167 .9833 .8171 .13308 .33229 .00577 3.3419 .0019 .8552 -.9984 .14051 .35016 .01166 3.3025 .0029 .8634 -.9974 .06322 .06092 .09557 3.9980 2.6116 .9868 3050 .06322 .06092 .09557 3.9980 2.6116 .9868 .5050 .06378 .08376 .08655 3.6863 1.9953 .9820 .3838 152 Table 10.2.2.1.2.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .3, pNz = .8 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn OEX Rej .0480 .06789 .02595 .92163 .0097 .01497 .00495 .98253 .0094 .01461 .00480 .98294 .0099 .01533 .00515 .98213 .0095 .01488 .00485 .98261 .1845 .16190 .13440 .83123 .0607 .06880 .04655 .92564 .0116 .01961 .00610 .97701 .0116 .01961 .00610 .97701 .0357 .04917 .01940 .94339 .0749 .18098 .03950 .78365 .0090 .04997 .00450 .93866 .0084 .04728 .00420 .94195 .0097 .05081 .00485 .93770 .0088 .04795 .00440 .94116 .4212 .40619 .31580 .57121 .1924 .22666 .15060 .75433 .0112 .05693 .00570 .93026 .0112 .05693 .00570 .93026 .0541 .14081 .02875 .83118 .0891 .46146 .04680 .43488 .0136 .21798 .00715 .72931 .0093 .19468 .00470 .75783 .0138 .22061 .00725 .72605 .0097 .19656 .00490 .75553 .7762 .83253 .60745 .11120 .6128 .71018 .48655 .23391 .0149 .22706 .00785 .71814 .0149 .22706 .00785 .71814 .0782 .40898 .04150 .49915 .0957 .70150 .05010 .13565 .0263 .54470 .01385 .32259 .0103 .47914 .00525 .40239 .0268 .54719 .01410 .31954 .0104 .48144 .00530 .39953 .8558 .93381 .69240 .00584 .8332 .92775 .67900 .01006 .0272 .55010 .01435 .31596 .0272 .55010 .01435 .31596 .0800 .65106 .04300 .19693 .0966 .80350 .05060 .00828 .0429 .78283 .02295 .02720 .0098 .75151 .00495 .06185 .0436 .78334 .02330 .02665 .0101 .75242 .00510 .06075 .8656 .94006 .70055 .00006 .8472 .93820 .69135 .00009 .0436 .78354 .02330 .02640 .0436 .78354 .02330 .02640 .0562 .78479 .03050 .02664 Tp Tn .07838 .97405 .01748 .99505 .01706 .99520 .01788 .99485 .01739 .99515 .16878 .86560 .07436 .95345 .02299 .99390 .02299 .99390 .05661 .98060 .21635 .96050 .06134 .99550 .05805 .99580 .06230 .99515 .05884 .99560 .42879 .68420 .24568 .84940 .06974 .99430 .06974 .99430 .16883 .97125 .56513 .95320 .27069 .99285 .24218 .99530 .27395 .99275 .24448 .99510 .88880 .39255 .76609 .51345 .28186 .99215 .28186 .99215 .50085 .95850 .86435 .94990 .67741 .98615 .59761 .99475 .68046 .98590 .60048 .99470 .99416 .30760 .98994 .32100 .68404 .98565 .68404 .98565 .80308 .95700 .99173 .94940 .97280 .97705 .93815 .99505 .97335 .97670 .93925 .99490 .99994 .29945 .99991 .30865 .97360 .97670 .97360 .97670 .97336 .96950 s.FnTp d' A' B" s:Rej s:FpTn P .12395 .11939 .14509 .5279 2.4275 .6807 .4816 .04277 .05050 .05155 .4704 3.0120 .6823 .5542 .04111 .04977 .04961 .4713 3.0332 .6828 .5566 .04343 .05243 .05220 .4658 2.9643 .6812 .5482 .04143 .05002 .05003 .4754 3.0541 .6834 .5595 .29142 .30215 .30199 .1468 1.1637 .5608 .0933 .20793 .19316 .21968 .2352 1.4439 .6008 .2160 .05066 .05756 .06069 .5107 3.1567 .6879 .5749 .05066 .05756 .06069 .5107 3.1567 .6879 .5749 .10119 .10428 .11700 .4824 2.4121 .6738 .4747 .18275 .14227 .22018 .9719 3.4380 .7504 .6343 .08532 .04722 .10521 1.0684 9.2073 .7459 .8556 .07802 .04564 .09629 1.0642 9.3792 .7454 .8579 .08631 .04901 .10635 1.0506 8.7174 .7450 .8474 .07877 .04670 .09721 1.0551 9.0929 .7450 .8534 .37783 .40421 .39514 .3000 1.1039 .6072 .0626 .35930 .32715 .38135 .3457 1.3467 .6247 .1832 .09239 .05402 .11338 1.0524 8.2405 .7457 .8393 .09239 .05402 .11338 1.0524 8.2405 .7457 .8393 .16284 .12349 .19439 .9407 3.8358 .7435 .6681 .18799 .15317 .23174 1.8407 4.0238 .8652 .6927 .17993 .06224 .22272 1.8389 16.6742 .8098 .9306 .15317 .04877 .19078 1.8978 22.8300 .8048 .9503 .18096 .06263 .22397 1.8437 16.5695 .8105 .9302 .15354 .04976 .19122 1.8908 22.1100 .8052 .9486 .20650 .39255 .20620 .9475 .4930 .7583 -.4139 .36243 .43210 .37888 .7597 .7688 .7273 -.1646 .18281 .06530 .22614 1.8385 15.6651 .8121 .9259 .18281 .06530 .22614 1.8385 15.6651 .8121 .9259 .18615 .14639 .22730 1.7357 4.4932 .8491 .7255 .12142 .15746 .14835 2.7440 2.1088 .9498 .4226 .19752 .08622 .24436 2.6620 10.1483 .9131 .8823 .17945 .05194 .22408 2.8061 25.6225 .8967 .9575 .19680 .08690 .24346 2.6635 9.9535 .9138 .8798 .17907 .05217 .22359 2.8102 25.3602 .8974 .9570 .07890 .36235 .02993 2.0192 .0472 .8211 -.9469 .10196 .37533 .07155 1.8591 .0748 .8207 -.9126 .19569 .08786 .24206 2.6666 9.7574 .9146 .8772 .19569 .08786 .24206 2.6666 9.7574 .9146 .8772 .12120 .15051 .14686 2.5695 3.0354 .9352 .5870 .04089 .15825 .03447 4.0356 .2169 .9851 -.7082 .06111 .11158 .07039 3.9199 1.1531 .9871 .0826 .08008 .05001 .09975 4.1187 8.5121 .9831 .8435 .06057 .11229 .06957 3.9224 1.1192 .9872 .0654 .07934 .05074 .09881 4.1175 8.1733 .9834 .8367 .07143 .35675 .00279 3.3101 .0007 .8248 -.9994 .07372 .36807 .00331 3.2529 .0010 .8271 -.9992 .06029 .11229 .06922 3.9265 1.1105 .9873 .0608 .06029 .11229 .06922 3.9265 1.1105 .9873 .0608 .05140 .12931 .05586 3.8062 .8935 .9853 -.0656 153 Table 10.2.2.1.3.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .2 N:p aEX Rej Fp Fn Tp Tn .1438 .04528 .02546 .87545 .12455 .97454 NT 2 .0337 .01406 .00438 .94720 .05280 .99563 MB 2 .0331 .01385 .00430 .94795 .05205 .99570 RB 2 MD 2 .0343 .01429 .00448 .94645 .05355 .99553 .0338 .01405 .00439 .94730 .05270 .99561 RD 2 .1896 .09236 .08085 .86160 .13840 .91915 OR 2 OS 2 .0910 .04269 .02960 .90495 .09505 .97040 .0452 .01733 .00594 .93710 .06290 .99406 OF 2 .0452 .01733 .00594 .93710 .06290 .99406 CF 2 CS 2 .0869 .02822 .01313 .91140 .08860 .98687 .2554 .12005 .04176 .56680 .43320 .95824 NT 4 .0392 .05441 .00520 .74875 .25125 .99480 MB 4 .0361 .05337 .00473 .75205 .24795 .99528 RB 4 .0397 .05487 .00528 .74675 .25325 .99473 MD 4 RD 4 .0366 .05375 .00480 .75045 .24955 .99520 OR 4 .4059 .21314 .15860 .56870 .43130 .84140 OS 4 .2217 .13646 .07771 .62855 .37145 .92229 OF 4 .0473 .05869 .00631 .73180 .26820 .99369 CF 4 .0473 .05869 .00631 .73180 .26820 .99369 CS 4 .1045 .07297 .01536 .69660 .30340 .98464 .3146 .23184 .05013 .04130 .95870 .94988 NT 10 MB 10 .0444 .17283 .00601 .15990 .84010 .99399 RB 10 .0375 .17037 .00494 .16790 .83210 .99506 M D 10 .0455 .17345 .00615 .15735 .84265 .99385 .0380 .17074 .00501 .16635 .83365 .99499 RD 10 .6121 .37097 .22440 .04275 .95725 .77560 OR 10 OS 10 .4578 .31183 .15279 .05200 .94800 .84721 OF 10 .0492 .17495 .00669 .15200 .84800 .99331 .0492 .17495 .00669 .15200 .84800 .99331 CF 10 CS 10 .0833 .16212 .01205 .23760 .76240 .98795 NT 20 .3171 .24013 .05023 .00025 .99975 .94978 M B 20 .0481 .20446 .00655 .00390 .99610 .99345 RB 20 .0388 .20324 .00515 .00440 .99560 .99485 MD 20 .0492 .20464 .00675 .00380 .99620 .99325 RD 20 .0395 .20334 .00525 .00430 .99570 .99475 OR 20 .6246 .38657 .23325 .00015 .99985 .76675 OS 20 .4914 .33260 .16581 .00025 .99975 .83419 OF 20 .0513 .20489 .00705 .00375 .99625 .99295 CF 20 .0513 .20489 .00705 .00375 .99625 .99295 CS 20 .0668 .20509 .00981 .01380 .98620 .99019 NT 40 .3221 .24044 .05055 .00000 1.00000 .94945 MB 40 .0483 .20516 .00645 .00000 1.00000 .99355 RB 40 .0393 .20414 .00518 .00000 1.00000 .99483 MD 40 .0495 .20529 .00661 .00000 1.00000 .99339 RD 40 .0403 .20424 .00530 .00000 1.00000 .99470 OR 40 .6266 .38802 .23503 .00000 1.00000 .76498 OS 40 .4933 .33335 .16669 .00000 1.00000 .83331 .0502 .20537 .00671 .00000 1.00000 .99329 OF 40 .0502 .20537 .00671 .00000 1.00000 .99329 CF 40 CS 40 .0661 .20782 .00979 .00005 .99995 .99021 s.FnTp d' A' B" s:Rej s.FpTn P .09562 .06911 .27896 .7996 3.4598 .7243 .6292 .03922 .02384 .16222 1.0034 8.3909 .7414 .8398 .03838 .02366 .15990 1.0023 8.4253 .7413 .8403 .03964 .02421 .16374 1.0026 8.3156 .7414 .8384 .03867 .02387 .16086 1.0015 8.3569 .7413 .8391 .20791 .20083 .30357 .3118 1.4737 .6196 .2321 .13832 .12301 .25835 .5764 2.5128 .6890 .4993 .04495 .02791 .17999 .9850 7.3370 .7407 .8180 .04495 .02791 .17999 .9850 7.3370 .7407 .8180 .06197 .04687 .21143 .8731 4.7553 .7321 .7235 .12420 .07966 .43578 1.5624 4.4075 .8280 .7197 .06923 .02642 .32142 1.8917 21.2787 .8067 .9465 .06733 .02493 .31687 1.9144 23.0121 .8063 .9508 .06959 .02664 .32272 1.8930 21.0973 .8071 .9460 .06759 .02516 .31774 1.9140 22.7690 .8067 .9503 .25732 .24969 .43909 .8272 1.6246 .7391 .2953 .21249 .19148 .43740 1.0926 2.5994 .7773 .5302 .07223 .02914 .33163 1.8759 18.5289 .8100 .9381 .07223 .02914 .33163 1.8759 18.5289 .8100 .9381 .07972 .04864 .32126 1.6460 9.0404 .8105 .8664 .07342 .08348 .15603 3.3794 .8558 .9761 -.0919 .05686 .02921 .25834 3.5063 14.2778 .9580 .9148 .05607 .02584 .26109 3.5427 17.5570 .9563 .9321 .05668 .02947 .25685 3.5088 13.8478 .9586 .9119 .05591 .02607 .26002 3.5437 17.2195 .9567 .9306 .21536 .26294 .15922 2.4770 .3037 .9276 -.6193 .19935 .23742 .18493 2.6503 .4508 .9444 -.4484 .05653 .03078 .25387 3.5015 12.5677 .9598 .9020 .05653 .03078 .25387 3.5015 12.5677 .9598 .902C .06487 .04303 .26154 2.9696 9.8629 .9359 .8767 .06639 .08293 .01118 5.1234 .0090 .9874 -.9896 .02565 .03027 .04399 5.1416 .6303 .9974 -.2523 .02284 .02631 .04670 5.1853 .8690 .9976 -.0782 .02609 .03090 .04343 5.1396 .5996 .9973 -.2783 .02300 .02659 .04617 5.1865 .8369 .9976 -.099C .21449 .26810 .00866 4.3435 .0019 .9416 -.9983 .19608 .24505 .01118 4.4516 .0037 .9585 -.9964 .02659 .03158 .04314 5.1285 .5702 .9973 -.3041 .02659 .03158 .04314 5.1285 .5702 .9973 -.3041 .03563 .03920 .08192 4.5364 1.3445 .9940 .1669 .06587 .08234 .00000 5.9044 .0000 .9874-1.000C .02357 .02946 .00000 6.7514 .0000 .9984-1.0000 .02095 .02619 .00000 6.8288 .0000 .9987 1.0000 .02386 .02982 .00000 6.7425 .0000 .9983 1.0000 .02117 .02646 .00000 6.8205 .0000 .9987 1.0000 .21411 .26764 .00000 4.9873 .0000 .9412-1.0000 .19446 .24308 .00000 5.2322 .0000 .9583 1.0000 .02405 .03006 .00000 6.7372 .0000 .9983 1.0000 1.0000 .02405 .03006 .00000 6.7372 .0000 .9983 .03195 .03992 .00500 6.2250 .0079 .9975 -.9897 154 Table 10.2.2.1.3.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .4 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn Tp Tn .1713 .11088 .03623 .77715 .22285 .96377 .0291 .03069 .00522 .93110 .06890 .99478 .0278 .02965 .00492 .93325 .06675 .99508 .0296 .03124 .00530 .92985 .07015 .99470 .0284 .03021 .00502 .93200 .06800 .99498 .3514 .22556 .17585 .69988 .30013 .82415 .1556 .12015 .07502 .81215 .18785 .92498 .0405 .03902 .00735 .91348 .08653 .99265 .0405 .03902 .00735 .91348 .08653 .99265 .1032 .07101 .02110 .85413 .14588 .97890 .2309 .27388 .04838 .38788 .61213 .95162 .0335 .12434 .00603 .69820 .30180 .99397 .0291 .11819 .00503 .71208 .28793 .99497 .0342 .12552 .00620 .69550 .30450 .99380 .0299 .11942 .00518 .70923 .29078 .99482 .6686 .49698 .33690 .26290 .73710 .66310 .4364 .37537 .21447 .38328 .61673 .78553 .0411 .13514 .00742 .67328 .32673 .99258 .0411 .13514 .00742 .67328 .32673 .99258 .1222 .18465 .02518 .57615 .42385 .97482 .2317 .41567 .04875 .03395 .96605 .95125 .0403 .34855 .00750 .13988 .86013 .99250 .0274 .33740 .00497 .16395 .83605 .99503 .0413 .34936 .00768 .13813 .86187 .99232 .0277 .33818 .00502 .16208 .83793 .99498 .7957 .64683 .41425 .00430 .99570 .58575 .7179 .61054 .35592 .00753 .99248 .64408 .0433 .35138 .00808 .13368 .86633 .99192 .0433 .35138 .00808 .13368 .86633 .99192 .0873 .34674 .01817 .16040 .83960 .98183 .2403 .42984 .04998 .00038 .99963 .95002 .0432 .40362 .00813 .00315 .99685 .99187 .0259 .40110 .00473 .00435 .99565 .99527 .0443 .40381 .00833 .00297 .99703 .99167 .0263 .40117 .00483 .00433 .99568 .99517 .7959 .65439 .42398 .00000 1.00000 .57602 .7261 .62316 .37193 .00000 1.00000 .62807 .0457 .40404 .00863 .00285 .99715 .99137 .0457 .40404 .00863 .00285 .99715 .99137 .0728 .40573 .01463 .00762 .99238 .98537 .2391 .43048 .05080 .00000 1.00000 .94920 .0449 .40514 .00857 .00000 1.00000 .99143 .0284 .40309 .00515 .00000 1.00000 .99485 .0457 .40523 .00872 .00000 1.00000 .99128 .0290 .40317 .00528 .00000 1.00000 .99472 .7983 .65469 .42448 .00000 1.00000 .57552 .7317 .62250 .37083 .00000 1.00000 .62917 .0462 .40530 .00883 .00000 1.00000 .99117 .0462 .40530 .00883 .00000 1.00000 .99117 .0753 .40940 .01567 .00000 1.00000 .98433 s.Rej s:FpTn s:FnTp .14303 .08784 .06161 .03124 .05847 .02983 .06213 .03144 .05905 .03009 .30112 .28704 .24400 .21246 .07101 .03733 .07101 .03733 .10210 .06892 .14333 .09853 .11177 .03395 .10448 .02958 .11215 .03465 .10490 .03016 .28313 .32681 .31599 .31161 .11493 .03754 .11493 .03754 .11925 .07470 .06914 .09882 .08325 .03865 .08430 .03078 .08284 .03908 .08380 .03091 .19686 .32611 .20288 .33119 .08182 .04012 .08182 .04012 .07755 .06462 .05880 .09804 .02644 .04035 .02211 .03018 .02651 .04079 .02231 .03062 .19877 .33129 .20268 .33779 .02687 .04161 .02687 .04161 .03821 .05659 .06076 .10126 .02517 .04195 .01875 .03125 .02541 .04235 .01905 .03175 .19784 .32974 .20118 .33530 .02561 .04269 .02561 .04269 .03644 .06073 A' B" P .29690 1.0336 3.7522 .7578 .6644 .14337 1.0771 8.8331 .7471 .8503 .13710 1.0812 9.0862 .7471 .8544 .14455 1.0809 8.8309 .7474 .8504 .13857 1.0838 9.0536 .7474 .8540 .38026 .4073 1.3449 .6412 .1835 .33983 .5536 1.9033 .6807 .3747 .16358 1.0772 7.7514 .7487 .8310 .16358 1.0772 7.7514 .7487 .8310 .20998 .9773 4.5169 .7457 .7156 .31555 1.9456 3.8131 .8783 .6752 .27145 1.9910 20.4038 .8194 .9447 .25671 2.0141 23.4538 .8167 .9523 .27201 1.9891 19.9966 .8199 .9435 .25751 2.0122 22.9547 .8173 .9512 .34115 1.0554 .8935 .7866 -.0710 .42834 1.0879 1.3084 .7911 .1677 .27797 1.9874 17.5893 .8247 .9352 .27797 1.9874 17.5893 .8247 .9352 .26279 1.7648 6.6603 .8374 .8173 .09867 3.4828 .7456 .9785 -.1715 .20024 3.5133 10.7411 .9626 .8835 .20763 3.5565 17.1985 .9573 .9304 .19886 3.5124 10.4254 .9630 .8796 .20641 3.5606 16.9187 .9578 .9291 .03751 2.8442 .0324 .8942 -.9653 .06423 2.8006 .0557 .9074 -.9369 .19568 3.5143 9.7492 .9640 .8705 .19568 3.5143 9.7492 .9640 .8705 .16344 3.0860 5.4622 .9537 .7661 .00968 5.0156 .0132 .9874 -.9843 .02899 5.1346 .4299 .9972 -.4396 .03419 5.2183 .9274 .9977 -.0420 .02824 5.1445 .3997 .9972 -.4717 .03410 5.2131 .9056 .9977 -.0553 .00000 4.4566 .0000 .8940-1.0000 .00000 4.5916 .0000 .9070-1.000C .02747 5.1455 .3727 .9971 -.5015 .02747 5.1455 .3727 .9971 -.5015 .04328 4.6063 .5668 .9944 -.3117 .00000 5.9020 .0000 .9873 1.0000 .00000 6.6487 .0000 .9979 1.0000 .00000 6.8305 .0000 .9987 1.000C .00000 6.6423 .0000 .9978 1.0000 1.0000 .00000 6.8216 .0000 .9987 .00000 4.4553 .0000 .8939-1.0000 .00000 4.5945 .0000 .9073 l.oood .00000 6.6374 .0000 .9978 l.oood .00000 6.6374 .0000 .9978 l.oood .00000 6.4177 .0000 .9961 -l.ooool d' 155 Table 10.2.2.1.3.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 5, mNz = .5, pNz = .6 N:p aEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Fp Fn Tp Tn .1196 .18302 .03840 .72057 .27943 .96160 .0208 .04879 .00575 .92252 .07748 .99425 .0197 .04564 .00530 .92747 .07253 .99470 .0211 .04957 .00585 .92128 .07872 .99415 .0201 .04639 .00540 .92628 .07372 .99460 .4512 .39263 .28990 .53888 .46112 .71010 .2121 .23080 .14258 .71038 .28962 .85743 .0275 .06140 .00770 .90280 .09720 .99230 .0275 .06140 .00770 .90280 .09720 .99230 .0992 .14015 .03335 .78865 .21135 .96665 .1522 .39823 .04830 .36848 .63152 .95170 .0231 .19363 .00655 .68165 .31835 .99345 .0174 .17499 .00468 .71147 .28853 .99533 .0235 .19537 .00678 .67890 .32110 .99323 .0180 .17668 .00483 .70875 .29125 .99518 .7428 .69292 .48193 .16642 .83358 .51808 .5518 .57242 .36310 .28803 .71197 .63690 .0283 .20901 .00815 .65708 .34292 .99185 .0283 .20901 .00815 .65708 .34292 .99185 .1206 .32448 .04085 .48643 .51357 .95915 .1506 .59884 .04818 .03405 .96595 .95183 .0371 .52855 .01148 .12673 .87327 .98853 .0172 .50225 .00473 .16607 .83393 .99528 .0377 .52972 .01170 . .12493 .87507 .98830 .0174 .50338 .00480 .16423 .83577 .99520 .8400 .82267 .55998 .00220 .99780 .44003 .7936 .80702 .52530 .00517 .99483 .47470 .0400 .53235 .01245 .12105 .87895 .98755 .0400 .53235 .01245 .12105 .87895 .98755 .0895 .54754 .03090 .10803 .89197 .96910 .1520 .61940 .04898 .00032 .99968 .95103 .0437 .60409 .01380 .00238 .99762 .98620 .0180 .59937 .00508 .00443 .99557 .99492 .0444 .60424 .01410 .00233 .99767 .98590 .0183 .59942 .00518 .00442 .99558 .99483 .8457 .82655 .56638 .00000 1.00000 .43363 .8079 .81426 .53565 .00000 1.00000 .46435 .0448 .60431 .01420 .00228 .99772 .98580 .0448 .60431 .01420 .00228 .99772 .98580 .0803 .60809 .02833 .00540 .99460 .97168 .1597 .62071 .05178 .00000 1.00000 .94823 .0444 .60562 .01405 .00000 1.00000 .98595 .0186 .60213 .00533 .00000 1.00000 .99468 .0453 .60573 .01433 .00000 1.00000 .98568 .0191 .60218 .00545 .00000 1.00000 .99455 .8535 .82771 .56928 .00000 1.00000 .43073 .8111 .81453 .53633 .00000 1.00000 .46367 .0460 .60581 .01453 .00000 1.00000 .98548 .0460 .60581 .01453 .00000 1.00000 .98548 .0805 .61165 .02913 .00000 1.00000 .97088 s.Rej s.FpTn s.FnTp d' P A' B" .18906 .11446 29141 1.1850 4.0343 .7783 .6901 .08880 .04174 .14446 1.1049 8.8627 .7495 .8519 .07985 .03853 .13056 1.0984 9.0603 .7486 .8547 .08964 .04217 .14588 1.1073 8.8333 .7497 .8515 .08063 .03884 .13186 1.1004 9.0220 .7489 .8542 .37484 .37323 .42850 .4561 1.1601 .6531 .0938 .35401 .31016 .42385 .5143 1.5181 .6698 .2546 .10015 .04877 .16275 1.1252 8.1099 .7527 .8398 .10015 .04877 .16275 1.1252 8.1099 .7527 .8398 .16306 .11226 .24410 1.0319 3.8955 .7566 .6759 .19208 .12696 .30639 1.9974 3.7584 .8841 .6701 .17959 .04508 .29612 2.0087 19.4187 .8233 .9418 .15871 .03619 .26378 2.0413 25.0747 .8172 .9557 .18009 .04675 .29665 2.0044 18.9156 .8238 .9401 .15950 .03668 .26499 2.0384 24.4857 .8178 .9545 .28197 .37677 .30070 1.0137 .6263 .7752 -.2857 .37229 .39634 .42590 .9093 .9094 .7594 -.0600 .18292 .05120 .30079 1.9976 16.4989 .8284 .9307 .18292 .05120 .30079 1.9976 16.4989 .8284 .9307 .18276 .12388 .28176 1.7749 4.5485 .8533 .7288 .07601 .12762 .09416 3.4872 .7545 .9786 -.1646 .12477 .06490 .20172 3.4162 6.9178 .9647 .8141 .12881 .03770 .21366 3.5652 18.1304 .9569 .9343 .12382 .06558 .19995 3.4175 6.7346 .9651 .8087 .12798 .03827 .21223 3.5671 17.7495 .9573 .9327 .14481 .35791 .02540 2.6971 .0175 .8584 -.9823 .15843 .37533 .06215 2.5010 .0374 .8653 -.9596 .12187 .06770 .19631 3.4127 6.2417 .9658 .7928 .12187 .06770 .19631 3.4127 6.2417 .9658 .7928 .09641 .11106 .13625 3.1048 2.6619 .9635 .5258 .05181 .12902 .00726 5.0718 .0115 .9877 -.9865 .03144 .07166 .02088 5.0253 .2109 .9959 -.7026 .02338 .03944 .02862 5.1878 .8864 .9976 -.0672 .03185 .07298 .02055 5.0237 .2031 .9959 -.7131 .02349 .03990 .02858 5.1823 .8682 .9976 -.0787 .14270 .35674 .00000 4.0977 .0000 .8584-1.0000 .14894 .37235 .00000 4.1754 .0000 .8661 1.0000 .03185 .07314 .02035 5.0278 .1979 .9959 -.7201 .03185 .07314 .02035 5.0278 .1979 .9959 -.7201 .04731 .10784 .02970 4.4551 .2387 .9915 -.6734 .05346 .13366 .00000 5.8928 .0000 .9871 1.0000 .02974 .07435 .00000 6.4608 .0000 .9965 1.0000 1.000C .01639 .04096 .00000 6.8189 .0000 .9987 .03000 .07501 .00000 6.4532 .0000 .9964-1.0000 .01653 .04133 .00000 6.8108 .0000 .9986-1.0000 .14101 .35254 .00000 4.0904 .0000 .8577 1.0000 1.0000 .14809 .37022 .00000 4.1737 .0000 .8659 .03016 .07539 .00000 6.4477 .0000 .9964-1.0000 .03016 .07539 .00000 6.4477 .0000 .9964-1.000C .04474 .11184 .00000 6.1587 .0000 .9927 1.0000 156 Table 10.2.2.1.3.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p ~ 5, mNz = .5, pNz = .8 N:p OEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .0688 .24623 .0106 .06463 .0088 .05959 .0107 .06579 .0092 .06071 .5174 .53126 .2758 .32777 .0149 .08244 .0149 .08244 .0578 .20596 .0882 .52303 .0193 .26812 .0097 .23243 .0197 .27062 .0098 .23445 .7967 .86660 .6593 .76461 .0219 .28871 .0219 .28871 .0751 .47140 .0884 .78272 .0323 .72334 .0103 .66915 .0331 .72463 .0106 .67073 .8421 .93750 .8196 .93515 .0340 .72673 .0340 .72673 .0702 .75177 .0898 .80969 .0458 .80436 .0102 .79757 .0466 .80447 .0103 .79759 .8450 .93793 .8237 .93580 .0468 .80452 .0468 .80452 .0612 .80623 .0896 .80985 .0459 .80536 .0097 .80101 .0466 .80543 .0100 .80104 .8434 .93768 .8239 .93573 .0468 .80545 .0468 .80545 .0611 .80700 Fp .03790 .00570 .00470 .00575 .00490 .39705 .22445 .00805 .00805 .03165 .04810 .01095 .00500 .01115 .00505 .63680 .53440 .01235 .01235 .04115 .04795 .01795 .00530 .01835 .00550 .68930 .67800 .01890 .01890 .03970 .04965 .02675 .00530 .02720 .00535 .68965 .67900 .02730 .02730 .03560 .04925 .02680 .00505 .02715 .00520 .68840 .67865 .02725 .02725 .03500 Fn Tp Tn .70169 .29831 .96210 .92064 .07936 .99430 .92669 .07331 .99530 .91920 .08080 .99425 .92534 .07466 .99510 .43519 .56481 .60295 .64640 .35360 .77555 .89896 .10104 .99195 .89896 .10104 .99195 .75046 .24954 .96835 .35824 .64176 .95190 .66759 .33241 .98905 .71071 .28929 .99500 .66451 .33549 .98885 .70820 .29180 .99495 .07595 .92405 .36320 .17784 .82216 .46560 .64220 .35780 .98765 .64220 .35780 .98765 .42104 .57896 .95885 .03359 .96641 .95205 .10031 .89969 .98205 .16489 .83511 .99470 .09880 .90120 .98165 .16296 .83704 .99450 .00045 .99955 .31070 .00056 .99944 .32200 .09631 .90369 .98110 .09631 .90369 .98110 .07021 .92979 .96030 .00030 .99970 .95035 .00124 .99876 .97325 .00436 .99564 .99470 .00121 .99879 .97280 .00435 .99565 .99465 .00000 1.00000 .31035 .00000 1.00000 .32100 .00118 .99883 .97270 .00118 .99883 .97270 .00111 .99889 .96440 .00000 1.00000 .95075 .00000 1.00000 .97320 .00000 1.00000 .99495 .00000 1.00000 .97285 .00000 1.00000 .99480 .00000 1.00000 .31160 .00000 1.00000 .32135 .00000 i.ooooo .97275 .00000 1.00000 .97275 .00000 1.00000 .96500 s.Rej s.FpTn s:FnTp d' .20827 .10495 .09133 .10581 .09222 .38050 .41274 .12011 .12011 .20239 .19802 .21844 .18003 .21890 .18085 .17652 .33654 .22088 .22088 .19536 .06541 .12997 .14362 .12893 .14275 .07453 .07771 .12709 .12709 .07664 .03300 .02770 .02171 .02780 .02162 .07386 .07632 .02774 .02774 .03027 .03265 .02572 .01039 .02584 .01053 .07403 .07628 .02587 .02587 .02879 .14497 .05673 .05127 .05694 .05221 .42337 .38524 .06753 .06753 .13257 ,16043 .08158 .05124 .08217 .05148 .38904 .43172 .08616 .08616 .14995 .15946 .10224 .05265 .10315 .05404 .37119 .38410 .10485 .10485 .15125 .16466 .12850 .05312 .12947 .05335 .36929 .38161 .12964 .12964 .14675 .16325 .12858 .05196 .12919 .05266 .37016 .38141 .12936 .12936 .14397 .25213 1.2463 .12822 1.1208 .11253 1.1456 .12931 1.1274 .11357 1.1409 .39935 .4242 .43701 .3816 .14630 1.1310 .14630 1.1310 .24217 1.1811 .24565 2.0267 .26881 1.8588 .22415 2.0204 .26938 1.8604 .22516 2.0243 .17419 1.0829 .35117 .8373 .27165 1.8817 .27165 1.8817 .23946 1.9367 .07945 3.4956 .16083 3.3778 .17991 3.5302 .15953 3.3775 .17885 3.5251 .00847 2.8262 .01316 2.7951 .15718 3.3799 .15718 3.3799 .09249 3.2284 .00612 5.0799 .01323 4.9572 .02509 5.1783 .01311 4.9562 .02506 5.1760 .00000 3.7700 .00000 3.8000 .01282 4.9641 .01282 4.9641 .01174 4.8626 .00000 5.9171 .00000 6.1949 .00000 6.8373 .00000 6.1893 .00000 6.8271 .00000 3.7736 .00000 3.8010 .00000 6.1877 .00000 6.1877 .00000 6.0768 A' 4.2051 9.0954 10.1665 9.1485 9.9332 1.0210 1.2413 8.0229 8.0229 4.4634 3.7351 12.5912 23.6456 12.4395 23.5320 .3809 .6552 11.6584 11.6584 4.4353 .7489 3.9829 16.2924 3.8656 15.6449 .0046 .0055 3.6998 3.6998 1.5713 .0108 .0662 .8407 .0641 .8316 .0000 .0000 .0621 .0621 .0474 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 B" .7859 .7033 .7506 .8560 .7512 .8712 .7511 .8571 .7511 .8682 .6438 .0132 .6329 .1353 .7535 .8384 .7535 .8384 .7745 .7187 .8872 .6678 .8230 .9069 .8171 .9527 .8237 .9057 .8177 .9525 .7754 -.5344 .7420 -.2597 .8288 .8992 .8288 .8992 .8725 .7214 .9788 -.1689 .9695 .6732 .9570 .9263 .9697 .6635 .9574 .9229 .8272 -.9958 .8300 -.9949 .9702 .6487 .9702 .6487 .9710 .2626 .9875 -.9874 .9930 -.9094 .9976 -.0966 .9929 -.9125 .9976 -.1026 .8276 1.0000 .8302 1.0000 .9929 -.9153 .9929 -.9153 .9908 -.9373 .9877 -1.0000 .9933 -1.0000 .9987 •1.0000 .9932-1.0000 .9987 -1.0000 .8279 -1.0000 .8303-1.0000 .9932 -1.0000 .9932 -1.0000 .9913-1.0000 Table 10.2.2.2.1.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz - .2 N:p CLEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .0787 .00887 .0142 .00053 .0142 .00052 .0143 .00054 .0142 .00053 .0787 .02796 .0231 .00284 .0178 .00069 .0178 .00069 .0607 .00264 .1051 .01191 .0145 .00065 .0144 .00065 .0148 .00066 .0147 .00066 .1061 .03803 .0245 .00298 .0179 .00078 .0179 .00078 .0724 .00346 .2246 .02758 .0231 .00193 .0228 .00192 .0237 .00198 .0236 .00197 .2295 .08820 .0522 .01070 .0245 .00206 .0245 .00206 .1242 .00887 .4568 .06358 .0336 .00621 .0333 .00618 .0344 .00632 .0343 .00628 .4863 .19898 .1503 .04544 .0357 .00647 .0357 .00647 .2055 .02257 .7598 .13377 .0399 .02124 .0392 .02100 .0406 .02150 .0398 .02125 .8595 .39784 .4705 .18492 .0413 .02170 .0413 .02170 .2871 .05757 Fp .00799 .00041 .00041 .00042 .00041 .02720 .00251 .00055 .00055 .00231 .00975 .00041 .00041 .00042 .00042 .03691 .00247 .00051 .00051 .00258 .01816 .00066 .00065 .00068 .00067 .08276 .00776 .00071 .00071 .00469 .03165 .00096 .00094 .00098 .00098 .17827 .03184 .00101 .00101 .00776 .04655 .00113 .00111 .00115 .00112 .33483 .13018 .00117 .00117 .01081 Fn .98763 .99902 .99903 .99899 .99899 .96899 .99583 .99876 .99876 .99607 .97946 .99838 .99838 .99836 .99836 .95752 .99498 .99814 .99814 .99302 .93473 .99298 .99299 .99280 .99284 .89003 .97753 .99252 .99252 .97440 .80872 .97276 .97289 .97233 .97252 .71819 .90018 .97171 .97171 .91822 .51737 .89832 .89941 .89707 .89824 .35014 .59609 .89617 .89617 .75538 Tp Tn .01237 .99201 .00098 .99959 .00097 .99959 .00101 .99958 .00101 .99959 .03101 .97280 .00417 .99749 .00124 .99945 .00124 .99945 .00393 .99769 .02054 .99025 .00162 .99959 .00162 .99959 .00164 .99958 .00164 .99958 .04248 .96309 .00502 .99753 .00186 .99949 .00186 .99949 .00698 .99742 .06527 .98184 .00702 .99934 .00701 .99935 .00720 .99932 .00716 .99933 .10997 .91724 .02247 .99224 .00748 .99929 .00748 .99929 .02560 .99531 .19128 .96835 .02724 .99904 .02711 .99906 .02767 .99902 .02748 .99903 .28181 .82173 .09982 .96816 .02829 .99899 .02829 .99899 .08178 .99224 .48263 .95345 .10168 .99887 .10059 .99889 .10293 .99885 .10176 .99888 .64986 .66517 .40391 .86982 .10383 .99883 .10383 .99883 .24462 .98919 S.FnTp s.Rej s.FpTn .03159 .00357 .00354 .00363 .00359 .11245 .02806 .00429 .00429 .01069 .03616 .00400 .00398 .00404 .00402 .12821 .02888 .00436 .00436 .01173 .05292 .00712 .00708 .00723 .00717 .19169 .06289 .00740 .00740 .02013 .06927 .01269 .01257 .01280 .01269 .25560 .13360 .01300 .01300 .03042 .06435 .02302 .02263 .02316 .02274 .25870 .23882 .02322 .02322 .04055 .02954 .00352 .00352 .00357 .00352 .11077 .02685 .00425 .00425 .01045 .03106 .00342 .00341 .00346 .00344 .12673 .02649 .00381 .00381 .01032 .03825 .00436 .00433 .00444 .00440 .18707 .05524 .00453 .00453 .01431 .04174 .00518 .00514 .00524 .00523 .24784 .11598 .00533 .00533 .01748 .03849 .00558 .00552 .00562 .00555 .26660 .21481 .00567 .00567 .01973 .05530 .01096 .01090 .01112 .01112 .12745 .04005 .01251 .01251 .02392 .07262 .01378 .01378 .01387 .01387 .14483 .04463 .01467 .01467 .03171 .13723 .02961 .02959 .02994 .02983 .23116 .10696 .03071 .03071 .06492 .22297 .05868 .05819 .05919 .05871 .33194 .23481 .06008 .06008 .11618 .24475 .11246 .11066 .11308 .11119 .32131 .40713 .11337 .11337 .17165 d' P .1638 1.4639 .2464 2.2112 .2430 2.1882 .2527 2.2523 .2564 2.2803 .0575 1.1151 .1670 1.5754 .2389 2.1196 .2389 2.1196 .1741 1.6127 .2932 1.9001 .4016 3.5356 .4035 3.5579 .4003 3.5136 .4021 3.5354 .0650 1.1209 .2363 1.8895 .3841 3.2821 .3841 3.2821 .3384 2.4331 .5814 2.8523 .7550 8.4939 .7581 8.5819 .7556 8.4542 .7558 8.4720 .1600 1.2326 .4149 2.5047 .7589 8.4550 .7589 8.4550 .6484 4.3683 .9838 3.8308 1.1808 19.4467 1.1821 19.5764 1.1807 19.2862 1.1785 19.2265 .3445 1.2947 .5718 2.4520 1.1796 19.0036 1.1796 19.0036 1.0266 7.0800 1.6357 4.0918 1.7823 47.2496 1.7821 47.7431 1.7842 46.9325 1.7842 47.4903 .8115 1.0171 .8823 1.8291 1.7841 46.5086 1.7841 46.5086 1.6056 11.0157 A' B" .5895 .6443 .6431 .6464 .6478 .5317 .5996 .6397 .6397 .6033 .6341 .6869 .6874 .6865 .6870 .5342 .6276 .6818 .6818 .6586 .6925 .7281 .7283 .7280 .7280 .5693 .6674 .7281 .7281 .7095 .7498 .7478 .7478 .7478 .7478 .6234 .6878 .7479 .7479 .7449 .8402 .7724 .7722 .7727 .7724 .7396 .7481 .7729 .7729 .7980 .2128 .4050 .4002 .4133 .4189 .0635 .2466 .3867 .3867 .2585 .3515 .5952 .5974 .5931 .5953 .0672 .3391 .5695 .5695 .4583 .5478 .8269 .8287 .8262 .8266 .1264 .4810 .8265 .8265 .6849 .6692 .9305 .9309 .9299 .9297 .1602 .4891 .9289 .9289 .8139 .6981 .9756 .9759 .9755 .9758 .0107 .3603 .9753 .9753 .8906 158 Table 10.2.2.2.1.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz - A N:p ctEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB. 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .1036 .01251 .0117 .00073 .0117 .00073 .0121 .00075 .0119 .00074 .1079 .04099 .0255 .00428 .0145 .00093 .0145 .00093 .0586 .00385 .1726 .02233 .0133 .00124 .0133 .00124 .0134 .00127 .0134 .00127 .1862 .07606 .0364 .00829 .0156 .00148 .0156 .00148 .0903 .00724 .4095 .06297 .0238 .00428 .0237 .00426 .0240 .00437 .0239 .00434 .4736 .22480 .1063 .04079 .0252 .00462 .0252 .00462 .1924 .02492 .6611 .13604 .0313 .01338 .0305 .01324 .0319 .01360 .0314 .01347 .8418 .49581 .3297 .18716 .0326 .01390 .0326 .01390 .3339 .07031 .7427 .23737 .0320 .04285 .0304 .04185 .0328 .04342 .0309 .04240 .9956 .74722 .8033 .57514 .0331 .04380 .0331 .04380 .4464 .16366 Fp .00990 .00044 .00044 .00046 .00045 .03937 .00352 .00056 .00056 .00278 .01508 .00050 .00050 .00050 .00050 .07168 .00623 .00058 .00058 .00420 .03114 .00090 .00089 .00090 .00090 .20397 .03014 .00095 .00095 .00945 .04575 .00118 .00115 .00120 .00118 .43411 .14375 .00123 .00123 .01730 .05006 .00120 .00114 .00123 .00116 .64133 .46509 .00125 .00125 .02393 Fn .98358 .99884 .99884 .99881 .99882 .95658 .99458 .99852 .99852 .99454 .96679 .99765 .99765 .99758 .99758 .91737 .98862 .99718 .99718 .98820 .88928 .99064 .99069 .99043 .99049 .74394 .94324 .98988 .98988 .95188 .72853 .96831 .96863 .96781 .96809 .41165 .74771 .96711 .96711 .85018 .48166 .89467 .89708 .89329 .89573 .09394 .25979 .89237 .89237 .62673 Tp Tn .01642 .99010 .00116 .99956 .00116 .99956 .00119 .99954 .00118 .99955 .04342 .96063 .00542 .99648 .00148 .99944 .00148 .99944 .00546 .99722 .03321 .98492 .00235 .99950 .00235 .99950 .00242 .99950 .00242 .99950 .08263 .92832 .01138 .99377 .00282 .99942 .00282 .99942 .01180 .99580 .11072 .96886 .00936 .99910 .00931 .99911 .00957 .99910 .00951 .99910 .25606 .79603 .05676 .96986 .01012 .99905 .01012 .99905 .04812 .99055 .27147 .95425 .03169 .99882 .03137 .99885 .03219 .99880 .03191 .99882 .58835 .56589 .25229 .85625 .03289 .99877 .03289 .99877 .14982 .98270 .51834 .94994 .10533 .99880 .10292 .99886 .10671 .99877 .10427 .99884 .90606 .35867 .74021 .53491 .10763 .99875 .10763 .99875 .37327 .97607 s.Rej s.FpTn s.FnTp .03743 .00419 .00419 .00429 .00424 .13739 .03815 .00493 .00493 .01330 .04855 .00556 .00556 .00562 .00561 .18359 .05798 .00612 .00612 .01839 .07151 .01045 .01037 .01055 .01045 .28526 .14381 .01091 .01091 .03627 .07472 .01871 .01842 .01883 .01855 .30152 .29845 .01903 .01903 .05610 .06463 .03409 .03285 .03431 .03305 .18058 .32811 .03449 .03449 .06409 .03273 .00413 .00413 .00420 .00417 .13565 .03522 .00474 .00474 .01247 .03755 .00429 .00429 .00431 .00431 .17983 .05128 .00463 .00463 .01466 .04533 .00579 .00578 .00581 .00580 .27712 .12574 .00594 .00594 .02207 .04431 .00663 .00653 .00669 .00661 .30614 .26464 .00676 .00676 .02876 .04274 .00664 .00645 .00672 .00650 .23968 .32658 .00681 .00681 .03254 .05372 .00831 .00831 .00842 .00840 .14500 .04506 .00961 .00961 .02197 .07706 .01216 .01216 .01234 .01230 .19654 .07199 .01338 .01338 .03361 .13259 .02422 .02406 .02447 .02424 .31324 .17932 .02530 .02530 .07285 .15914 .04536 .04473 .04566 .04505 .32584 .36787 .04617 .04617 .12163 .14767 .08463 .08165 .08515 .08211 .13217 .37243 .08556 .08556 .14558 a" .1961 .2765 .2765 .2759 .2790 .0458 .1471 .2847 .2847 .2277 .3324 .4657 .4657 .4733 .4726 .0758 .2215 .4801 .4801 .3720 .6416 .7714 .7708 .7775 .7761 .1720 .2961 .7842 .7842 .6840 1.0791 1.1838 1.1880 1.1852 1.1869 .3892 .3963 1.1885 1.1885 1.0758 1.6902 1.7839 1.7868 1.7841 1.7895 .9548 .7316 1.7845 1.7845 1.6554 P 1.5493 2.4126 2.4126 2.4021 2.4281 1.0826 1.4704 2.4271 2.4271 1.8323 1.9452 4.1577 4.1577 4.2439 4.2351 1.1142 1.6973 4.2373 4.2373 2.4872 2.6923 8.2579 8.2549 8.3614 8.3419 1.1360 1.6693 8.4006 8.4006 3.9423 3.4514 18.1476 18.4755 18.0728 18.2500 .9889 1.4092 18.0436 18.0436 5.4440 3.8602 45.7977 47.3119 45.2092 47.0617 .4486 .8158 44.8710 44.8710 6.7209 A' .6009 .6540 .6540 .6536 .6547 .5244 .5881 .6550 .6550 .6235 .6411 .6977 .6977 .6989 .6988 .5361 .6145 .6991 .6991 .6629 .7002 .7282 .7281 .7286 .7285 .5672 .6241 .7289 .7289 .7106 .7670 .7483 .7484 .7484 .7484 .6337 .6392 .7486 .7486 .7548 .8491 .7732 .7727 .7735 .7730 .7576 .7215 .7737 .7737 .8234 B" .2447 .4442 .4442 .4424 .4470 .0468 .2115 .4479 .4479 .3245 .3674 .6507 .6507 .6572 .6566 .0651 .2902 .6577 .6577 .4720 .5309 .8238 .8237 .8261 .8256 .0797 .2936 .8273 .8273 .6606 .6383 .9259 .9273 .9257 .9264 -.0071 .2103 .9257 .9257 .7645 .6800 .9749 .9757 .9746 .9756 -.4598 -.1281 .9744 .9744 .8184 159 Table 10.2.2.2.1.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .1, pNz = .6 N:p aEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 M D 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 M D 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 M D 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .1176 .01707 .0082 .00094 .0082 .00094 .0084 .00096 .0083 .00095 .1419 .05932 .0263 .00612 .0110 .00126 .0110 .00126 .0535 .00567 .2055 .03438 .0103 .00192 .0103 .00191 .0104 .00197 .0104 .00196 .2681 .12412 .0448 .01587 .0125 .00228 .0125 .00228 .0896 .01276 .4476 .09836 .0169 .00665 .0169 .00661 .0170 .00680 .0169 .00673 .6635 .38544 .1657 .09487 .0177 .00722 .0177 .00722 .2187 .04904 .5811 .19273 .0222 .01973 .0218 .01944 .0227 .02006 .0222 .01977 .9537 .71423 .4846 .36822 .0233 .02049 .0233 .02049 .3713 .13439 .5908 .33159 .0217 .06484 .0202 .06252 .0221 .06570 .0208 .06332 .9998 .89477 .9145 .80569 .0223 .06624 .0223 .06625 .4713 .28821 Fp .01219 .00047 .00047 .00048 .00047 .05566 .00489 .00064 .00064 .00357 .01984 .00057 .00057 .00058 .00058 .11489 .01160 .00069 .00069 .00598 .03929 .00096 .00096 .00097 .00096 .34591 .07381 .00101 .00101 .01542 .04931 .00125 .00123 .00128 .00125 .63048 .30356 .00131 .00131 .02821 .04991 .00123 .00113 .00126 .00117 .79483 .69573 .00127 .00127 .03801 Fn .97968 .99874 .99874 .99872 .99873 .93825 .99307 .99833 .99833 .99293 .95592 .99717 .99719 .99711 .99713 .86972 .98129 .99667 .99667 .98273 .86227 .98956 .98962 .98931 .98943 .58821 .89109 .98864 .98864 .92855 .71165 .96795 .96841 .96742 .96788 .22994 .58867 .96672 .96672 .79483 .48062 .89275 .89656 .89133 .89525 .03860 .12101 .89044 .89043 .54499 Tp .02032 .00126 .00126 .00128 .00127 .06175 .00693 .00167 .00167 .00707 .04408 .00283 .00281 .00289 .00287 .13028 .01871 .00333 .00333 .01727 .13773 .01044 .01038 .01069 .01057 .41179 .10891 .01136 .01136 .07145 .28835 .03205 .03159 .03258 .03212 .77006 .41133 .03328 .03328 .20517 .51938 .10725 .10344 .10867 .10475 .96140 .87899 .10956 .10957 .45501 Tn .98781 .99953 .99953 .99952 .99953 .94434 .99511 .99936 .99936 .99643 .98016 .99943 .99943 .99942 .99942 .88511 .98840 .99931 .99931 .99402 .96071 .99904 .99904 .99903 .99904 .65409 .92619 .99899 .99899 .98458 .95069 .99875 .99877 .99872 .99875 .36952 .69644 .99869 .99869 .97179 .95009 .99877 .99887 .99874 .99883 .20517 .30427 .99873 .99873 .96199 s.FnTp d' s.Rej s.FpTn .04340 .00480 .00480 .00488 .00485 .16631 .04877 .00578 .00578 .01723 .05923 .00699 .00693 .00707 .00702 .23484 .08835 .00767 .00767 .02723 .07961 .01301 .01287 .01321 .01300 .33341 .23606 .01364 .01364 .05491 .07873 .02339 .02293 .02358 .02308 .24821 .40185 .02387 .02387 .08129 .08058 .04529 .04277 .04560 .04300 .09841 .26688 .04589 .04589 .08454 .03785 .00519 .00519 .00525 .00522 .16300 .04545 .00618 .00618 .01639 .04484 .00561 .00561 .00564 .00564 .22904 .07690 .00617 .00617 .02066 .05277 .00741 .00735 .00743 .00741 .32840 .20848 .00757 .00757 .03292 .05280 .00835 .00828 .00844 .00835 .28135 .36753 .00854 .00854 .04310 .05262 .00837 .00793 .00844 .00804 .18224 .28958 .00848 .00848 .04883 .05408 .00714 .00714 .00719 .00718 .17235 .05268 .00841 .00841 .02291 .07879 .01097 .01087 .01111 .01101 .24412 .09847 .01200 .01200 .03892 .11568 .02102 .02082 .02135 .02101 .34742 .25937 .02207 .02207 :08164 .12409 .03841 .03766 .03870 .03791 .24538 .43456 .03918 .03918 .12533 .12956 .07521 .07111 .07572 .07148 .06113 .27386 .07619 .07619 .13329 P .2040 1.5502 .2879 2.4879 .2879 2.4879 .2866 2.4736 .2890 2.4939 .0520 1.0849 .1223 1.3613 .2857 2.4096 .2857 2.4096 .2369 1.8391 .3519 1.9385 .4851 4.3057 .4829 4.2802 .4899 4.3581 .4878 4.3329 .0759 1.0922 .1891 1.5089 .4839 4.1784 .4839 4.1784 .3995 2.5199 .6684 2.5916 .7920 8.5264 .7913 8.5242 .7992 8.6566 .7967 8.6185 .1734 1.0552 .2157 1.3351 .8103 8.7973 .8103 8.7973 .6940 3.5170 1.0934 3.3472 1.1718 17.3962 1.1708 17.4745 1.1725 17.2838 1.1729 17.4294 .4059 .8045 .2901 1.1130 1.1743 17.1799 1.1743 17.1799 1.0846 4.3975 1.6943 3.8691 1.7861 45.2485 1.7907 47.6301 1.7883 44.9384 1.7892 46.8052 .9438 .2945 .6578 .5751 1.7905 44.8417 1.7905 44.8427 1.6612 4.7951 A' B" .6021 .6573 .6573 .6567 .6575 .5263 .5739 .6544 .6544 .6248 .6437 .6999 .6996 .7006 .7003 .5339 .5968 .6986 .6986 .6662 .7043 .7294 .7293 .7298 .7297 .5652 .5901 .7305 .7305 .7103 .7701 .7480 .7479 .7480 .7480 .6397 .6042 .7482 .7482 .7612 .8495 .7737 .7729 .7740 .7731 .7463 .7027 .7742 .7742 .8375 .2463 .4578 .4578 .4554 .4589 .0486 .1711 .4454 .4454 .3278 .3685 .6626 .6607 .6664 .6646 .0540 .2313 .6544 .6544 .4811 .5176 .8300 .8299 .8327 .8319 .0341 .1734 .8357 .8357 .6275 .6281 .9226 .9229 .9222 .9228 -.1363 .0677 .9218 .9218 .7122 .6807 .9746 .9759 .9744 .9755 -.6292 -.3312 .9744 .9744 .7430 160 Table 10.2 2.2.1.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz - .1, pNz = .8 N:p NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 OLEX Rej .0927 .02247 .0054 .00124 .0054 .00124 .0055 .00127 .0055 .00126 .1693 .08169 .0240 .00962 .0067 .00161 .0067 .00161 .0396 .00824 .1650 .04742 .0063 .00254 .0062 .00253 .0065 .00260 .0065 .00259 .3348 .18012 .0500 .02722 .0077 .00298 .0077 .00298 .0651 .01998 .3165 .13056 .0101 .00902 .0100 .00893 .0104 .00921 .0103 .00911 .7718 .53267 .2198 .16493 .0107 .00973 .0107 .00973 .1706 .07954 .3666 .24332 .0110 .02596 .0107 .02542 .0111 .02638 .0109 .02586 .9824 .84348 .5978 .52711 .0112 .02694 .0112 .02694 .2900 .20879 .3666 .42540 .0107 .08718 .0100 .08280 .0110 .08831 .0103 .08389 1.0000 .95792 .9574 .91370 .0110 .08904 .0110 .08904 .3792 .42553 Fp .01474 .00061 .00061 .00062 .00062 .07692 .00768 .00076 .00076 .00504 .02418 .00070 .00069 .00072 .00072 .16489 .02034 .00086 .00086 .00821 .04406 .00116 .00114 .00119 .00118 .47704 .13278 .00122 .00122 .02212 .05041 .00126 .00122 .00127 .00124 .75776 .45566 .00128 .00128 .03960 .05079 .00121 .00112 .00124 .00116 .87333 .82189 .00124 .00124 .05492 Fn .97559 .99861 .99861 .99857 .99858 .91712 .98989 .99818 .99818 .99096 .94678 .99699 .99701 .99693 .99694 .81608 .97107 .99649 .99649 .97708 .84782 .98902 .98912 .98878 .98890 .45342 .82703 .98814 .98814 .90610 .70846 .96786 .96853 .96734 .96798 .13509 .45502 .96664 .96664 .74891 .48094 .89133 .89678 .88993 .89543 .02094 .06334 .88901 .88901 .48182 Tp .02441 .00139 .00139 .00143 .00142 .08288 .01011 .00182 .00182 .00904 .05322 .00301 .00299 .00307 .00306 .18392 .02893 .00351 .00351 .02292 .15218 .01098 .01088 .01122 .01110 .54658 .17297 .01186 .01186 .09390 .29154 .03214 .03147 .03266 .03202 .86491 .54498 .03336 .03336 .25109 .51906 .10868 .10322 .11008 .10457 .97906 .93666 .11099 .11099 .51818 Tn .98526 .99939 .99939 .99938 .99938 .92308 .99232 .99924 .99924 .99496 .97582 .99930 .99931 .99928 .99928 .83511 .97966 .99914 .99914 .99179 .95594 .99884 .99886 .99881 .99882 .52296 .86722 .99878 .99878 .97788 .94959 .99874 .99878 .99873 .99876 .24224 .54434 .99872 .99872 .96040 .94921 .99879 .99888 .99876 .99884 .12667 .17811 .99876 .99876 .94508 s:Rejs.FpTn s.FnTp d' .04980 .00557 .00557 .00566 .00563 .19945 .06776 .00649 .00649 .02255 .06801 .00804 .00799 .00811 .00809 .28014 .12558 .00871 .00871 .03728 .08403 .01553 .01528 .01569 .01544 .33741 .31715 .01617 .01617 .07444 .08894 .02780 .02699 .02803 .02722 .17681 .43814 .02838 .02838 .10618 .09725 .05611 .05161 .05653 .05191 .05082 .20017 .05675 .05675 .10397 .05129 .00837 .00837 .00844 .00844 .20151 .06425 .00927 .00927 .02679 .06048 .00879 .00872 .00893 .00893 .27972 .11213 .00971 .00971 .03308 .07233 .01160 .01154 .01175 .01170 .35011 .28576 .01191 .01191 .05271 .07436 .01206 .01191 .01211 .01201 .24279 .41427 .01216 .01216 .06878 .07493 .01175 .01122 .01190 .01138 .15316 .23885 .01190 .01190 .08040 .05525 .00662 .00662 .00671 .00667 .20168 .06991 .00769 .00769 .02550 .07760 .00974 .00969 .00983 .00980 .28452 .13067 .01055 .01055 .04348 .10009 .01913 .01884 .01933 .01902 .34169 .32798 .01994 .01994 .08873 .10920 .03459 .03359 .03488 .03387 .17178 .44884 .03531 .03531 .12825 .11997 .07008 .06447 .07059 .06485 .03595 .20088 .07086 .07086 .12637 .2067 .2429 .2429 .2456 .2438 .0401 .1016 .2651 .2651 .2087 .3599 .4475 .4509 .4454 .4445 .0740 .1500 .4405 .4405 .4026 .6783 .7560 .7553 .7555 .7543 .1746 .1709 .7684 .7684 .6947 1.0920 1.1717 1.1705 1.1763 1.1728 .4036 .2244 1.1831 1.1831 1.0843 1.6850 1.7993 1.7924 1.7986 1.7911 .8925 .6047 1.8034 1.8034 1.6445 P 1.5350 2.1299 2.1299 2.1443 2.1329 1.0580 1.2726 2.2390 2.2390 1.6741 1.9074 3.7788 3.8222 3.7417 3.7326 1.0719 1.3441 3.6127 3.6127 2.4227 2.5264 7.5216 7.5266 7.4651 7.4598 .9948 1.1920 7.6388 7.6388 3.1779 3.3056 17.3650 17.4923 17.4587 17.4543 .6952 .9998 17.6227 17.6227 3.7264 3.8158 46.4477 47.9861 45.7264 47.1597 .2423 .4768 45.9990 45.9990 3.5865 A' .6014 .6404 .6404 .6413 .6406 .5196 .5607 .6466 .6466 .6115 .6439 .6924 .6931 .6918 .6916 .5316 .5764 .6897 .6897 .6642 .7059 .7261 .7261 .7260 .7259 .5650 .5697 .7269 .7269 .7095 .7703 .7480 .7479 .7482 .7480 .6416 .5820 .7485 .7485 .7656 .8489 .7741 .7728 .7744 .7731 .7357 .6917 .7747 .7747 .8460 B" .2421 .3894 .3894 .3926 .3901 .0341 .1355 .4134 .4134 .2818 .3622 .6215 .6252 .6183 .6175 .0431 .1700 .6073 .6073 .4667 .5078 .8079 .8079 .8066 .8064 -.0033 .1081 .8113 .8113 .5946 .6237 .9225 .9230 .9230 .9229 -.2221 -.0001 .9239 .9239 .6635 .6763 .9753 .9761 .9749 .9756 -.6873 -.4232 .9751 .9751 .6558 161 Table 10.2.2.2.2.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz - .3, pNz - .2 N:p NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .3695 .05327 .0294 .00399 .0294 .00396 .0299 .00406 .0296 .00403 .3937 .17133 .1285 .04095 .0400 .00519 .0400 .00519 .1941 .02037 .6585 .11759 .0372 .01637 .0367 .01616 .0380 .01662 .0373 .01639 .7514 .35514 .3885 .16177 .0444 .01839 .0444 .01839 .2771 .05218 .8270 .21301 .0425 .08548 .0380 .08340 .0439 .08601 .0395 .08405 .9890 .54518 .9342 .45374 .0465 .08763 .0465 .08763 .2497 .12817 .8221 .23796 .0486 .17508 .0402 .17295 .0502 .17539 .0414 .17332 .9948 .56643 .9894 .49616 .0517 .17580 .0517 .17580 .1168 .17393 .8254 .23983 .0490 .20089 .0389 .20061 .0502 .20093 .0400 .20064 .9946 .56441 .9892 .49444 .0508 .20095 .0508 .20095 .0663 .19613 aEX Fp .02715 .00084 .00084 .00085 .00084 .15353 .02986 .00117 .00117 .00818 .04236 .00106 .00105 .00109 .00107 .30172 .11674 .00128 .00128 .01118 .05021 .00122 .00108 .00126 .00112 .44024 .33521 .00134 .00134 .00960 .04960 .00139 .00114 .00144 .00118 .45819 .37036 .00149 .00149 .00429 .04979 .00141 .00112 .00144 .00115 .45552 .36805 .00146 .00146 .00257 Fn Tp Tn .84227 .15773 .97285 .98339 .01661 .99916 .98354 .01646 .99916 .98310 .01690 .99915 .98324 .01676 .99916 .75746 .24254 .84647 .91469 .08531 .97014 .97872 .02128 .99883 .97872 .02128 .99883 .93089 .06911 .99182 .58149 .41851 .95764 .92240 .07760 .99894 .92340 .07660 .99895 .92123 .07877 .99891 .92231 .07769 .99893 .43116 .56884 .69828 .65809 .34191 .88326 .91314 .08686 .99872 .91314 .08686 .99872 .78383 .21617 .98882 .13578 .86422 .94979 .57747 .42253 .99878 .58730 .41270 .99892 .57500 .42500 .99874 .58424 .41576 .99888 .03509 .96491 .55976 .07217 .92783 .66479 .56720 .43280 .99866 .56719 .43281 .99866 .39757 .60243 .99040 .00862 .99138 .95040 .13018 .86982 .99861 .13981 .86019 .99886 .12882 .87118 .99856 .13810 .86190 .99882 .00059 .99941 .54181 .00068 .99932 .62964 .12696 .87304 .99851 .12696 .87304 .99851 .14753 .85247 .99571 .00000 1.00000 .95021 .00118 .99882 .99859 .00143 .99857 .99888 .00112 .99888 .99856 .00140 .99860 .99885 .00000 1.00000 .54448 .00000 1.00000 .63195 .00110 .99890 .99854 .00110 .99890 .99854 .02964 .97036 .99743 s.FnTp d' A' B" s.Rej s:FpTn B .07052 .04234 .22860 .9206 3.8490 .7405 .6683 .01066 .00488 .04873 1.0126 14.4262 .7413 .9024 .01053 .00488 .04802 1.0089 14.3105 .7412 .9015 .01080 .00492 .04939 1.0148 14.4211 .7414 .9024 .01066 .00490 .04875 1.0142 14.4455 .7414 .9026 .25820 .24893 .33857 .3233 1.3204 .6180 .1714 .13664 .12003 .23483 .5127 2.3024 .6768 .4586 .01245 .00586 .05649 1.0161 13.1571 .7413 .8940 .01245 .00586 .05649 1.0161 13.1571 .7413 .8940 .03429 .02005 .12370 .9183 5.9477 .7358 .7760 .07734 .04171 .29270 1.5183 4.3271 .8229 .7143 .02270 .00549 .11093 1.6504 40.7625 .7657 .9707 .02229 .00546 .10897 1.6474 40.8520 .7655 .9708 .02293 .00556 .11198 1.6514 40.3578 .7660 .9705 .02244 .00549 .10973 1.6502 40.7005 .7657 .9707 .28248 .28074 .37712 .6929 1.1274 .7130 .0758 .24363 .21843 .41529 .7842 1.8717 .7284 .3715 .02430 .00603 .11850 1.6563 37.5203 .7677 .9683 .02430 .00603 .11850 1.6563 37.5203 .7677 .9683 .04679 .02172 .19663 1.4990 9.9788 .7889 .8775 .04326 .03864 .14862 2.7424 2.1066 .9497 .4221 .04655 .00587 .23129 2.8361 97.1188 .8547 .9901 .04533 .00550 .22563 2.8465 107.7020 .8524 .9911 .04660 .00597 .23146 2.8322 94.2863 .8553 .9898 .04538 .00560 .22587 2.8431 104.2081 .8531 .9908 .21486 .26095 .09855 1.9611 .1963 .8703 -.7584 .22609 .25872 .20792 1.8854 .3772 .8825 -.5379 .04679 .00616 .23223 2.8339 89.5734 .8572 .9892 .04679 .00616 .23222 2.8339 89.5738 .8572 .9892 .04256 .01944 .18975 2.6012 14.9939 .8957 .9236 .03147 .03861 .03340 4.0302 .2284 .9852 -.6930 .02975 .00626 .14686 4.1163 46.4664 .9670 .9758 .03019 .00562 .14953 4.1326 58.6188 .9647 .9813 .02964 .00637 .14617 4.1125 44.7462 .9673 .9747 .03005 .00573 .14865 4.1302 56.3690 .9651 .9804 .20682 .25838 .00880 3.3492 .0052 .8851 -.9953 .20040 .25032 .00960 3.5348 .0062 .9071 -.9942 .02950 .00648 .14532 4.1115 43.0083 .9678 .9736 .02950 .00648 .14532 4.1115 43.0083 .9678 .9736 .02546 .01343 .10743 3.6753 18.2760 .9617 .9343 .03064 .03829 .00000 5.9118 .0000 .9876 •1.0000 .00560 .00631 .01222 6.0284 .8492 .9994 -.0890 .00521 .00564 .01340 6.0390 1.2565 .9994 .1240 .00562 .00637 .01197 6.0357 .7953 .9994 -.1244 .00526 .00573 .01326 6.0374 1.1971 .9994 .0979 .20371 .25464 .00000 4.3766 .0000 .8861 -1.0000 .19783 .24729 .00000 4.6019 .0000 .9080-1.0000 .00564 .00640 .01187 6.0382 .7727 .9994 -.1399 .00564 .00640 .01187 6.0382 .7727 .9994 -.1399 .01429 .01117 .05036 4.6842 8.4686 .9919 .8364 162 Table 10.2.2.2.2.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .3, pNz = .4 N:p <XEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej Fp .5371 .11108 .03912 .0247 .00833 .00092 .0243 .00826 .00090 .0259 .00850 .00096 .0255 .00841 .00095 .6824 .40063 .34970 .2543 .14741 .11409 .0363 .01105 .00137 .0363 .01105 .00137 .2949 .05953 .01677 .7018 .21811 .04893 .0291 .03306 .00109 .0283 .03217 .00106 .0297 .03353 .00111 .0287 .03261 .00107 .9614 .69290 .59667 .6548 .46720 .37695 .0355 .03714 .00134 .0355 .03714 .00134 .4185 .14803 .02459 .7263 .37663 .05073 .0340 .17483 .00130 .0285 .16601 .00109 .0352 .17611 .00136 .0294 .16720 .00112 1.0000 .82275 .70791 .9954 .78891 .65490 .0376 .17931 .00144 .0376 .17931 .00144 .3459 .30074 .01879 .7170 .42640 .04971 .0455 .35455 .00176 .0299 .34486 .00113 .0464 .35512 .00180 .0308 .34554 .00116 1.0000 .82876 .71464 1.0000 .79945 .66580 .0478 .35592 .00185 .0478 .35592 .00185 .1448 .36907 .00734 .7208 .43009 .05016 .0497 .40079 .00196 .0297 .40007 .00115 .0508 .40083 .00201 .0305 .40011 .00118 1.0000 .82734 .71223 .9999 .79831 .66384 .0514 .40084 .00204 .0514 .40084 .00204 .0808 .39647 .00430 Fn Tp Tn .78099 .21901 .96088 .98054 .01946 .99908 .98072 .01928 .99910 .98019 .01981 .99904 .98039 .01961 .99905 .52298 .47702 .65030 .80261 .19739 .88591 .97443 .02557 .99863 .97443 .02557 .99863 .87634 .12366 .98323 .52812 .47188 .95107 .91899 .08101 .99891 .92116 .07884 .99894 .91786 .08214 .99889 .92009 .07991 .99893 .16277 .83723 .40333 .39743 .60257 .62305 .90916 .09084 .99866 .90916 .09084 .99866 .66681 .33319 .97541 .13452 .86548 .94927 .56488 .43512 .99870 .58661 .41339 .99891 .56177 .43823 .99864 .58368 .41632 .99888 .00500 .99500 .29209 .01008 .98992 .34510 .55390 .44610 .99856 .55390 .44610 .99856 .27632 .72368 .98121 .00856 .99144 .95029 .11627 .88373 .99824 .13954 .86046 .99887 .11491 .88509 .99820 .13791 .86209 .99884 .00006 .99994 .28536 .00007 .99993 .33420 .11298 .88702 .99815 .11298 .88702 .99815 .08834 .91166 .99266 .00001 .99999 .94984 .00096 .99904 .99804 .00154 .99846 .99885 .00096 .99904 .99799 .00150 .99850 .99882 .00000 1.00000 .28777 .00000 1.00000 .33616 .00094 .99906 .99796 .00094 .99906 .99796 .01528 .98472 .99570 s.FnTp d' A' B" s:Rej s.FpTn P .08969 .04709 .18988 .9855 3.4898 .7522 .6397 .01594 .00578 .03871 1.0503 15.1871 .7428 .9082 .01569 .00574 .03813 1.0514 15.2997 .7429 .9089 .01613 .00592 .03917 1.0438 14.7674 .7426 .9056 .01584 .00587 .03848 1.0440 14.8471 .7426 .9061 .33682 .32690 .37989 .3285 1.0756 .6157 .0463 .28455 .25166 .35233 .3541 1.4390 .6290 .2210 .01880 .00710 .04553 1.0460 13.2892 .7427 .8961 .01880 .00710 .04553 1.0460 13.2892 .7427 .8961 .06664 .03184 .14219 .9687 4.9028 .7433 .7359 .08722 .04612 .20221 1.5848 3.9260 .8353 .6853 .03617 .00632 .08980 1.6665 41.2245 .7666 .9712 .03455 .00624 .08583 1.6602 41.4226 .7661 .9713 .03652 .00641 .09065 1.6670 40.7699 .7669 .9709 .03482 .00628 .08650 1.6632 41.3144 .7664 .9712 .24066 .27588 .23621 .7384 .6355 .7209 -.2769 .37427 .35226 .44300 .5735 1.0155 .6841 .0097 .03877 .00708 .09621 1.6657 37.0335 .7687 .9680 .03877 .00708 .09621 1.6657 37.0335 .7687 .9680 .08397 .03695 .19120 1.5359 6.3073 .8106 .8051 .05492 .04629 .11874 2.7431 2.0758 .9499 .4148 .07978 .00706 .19889 2.8472 91.6965 .8578 .9895 .07499 .00642 .18720 2.8470 107.3227 .8525 .9911 .07987 .00723 .19907 2.8433 88.5968 .8586 .9891 .07520 .00651 .18769 2.8455 104.5564 .8532 .9908 .13052 .21393 .02033 2.0285 .0421 .8179 -.9530 .15123 .23644 .06843 1.9249 .0728 .8273 -.9154 .07989 .00743 .19916 2.8438 83.8468 .8605 .9884 .07989 .00743 .19916 2.8438 83.8468 .8605 .9884 .05633 .03164 .13067 2.6732 7.2843 .9231 .8312 .02896 .04592 .02486 4.0320 .2265 .9852 -.6955 .04625 .00826 .11504 4.1116 34.6034 .9704 .9663 .04899 .00647 .12219 4.1372 59.1599 .9647 .9814 .04592 .00833 .11421 4.1127 33.7342 .9707 .9653 .04871 .00659 .12144 4.1349 56.9714 .9651 .9806 .12641 .21064 .00184 3.2746 .0007 .8213 -.9994 .13770 .22946 .00200 3.3721 .0008 .8335 -.9994 .04540 .00845 .11294 4.1132 32.4194 .9712 .9638 .04540 .00845 .11294 4.1132 32.4194 .9712 .9638 .02876 .02073 .06098 3.7910 7.8776 .9757 .8340 .02781 .04635 .00056 6.0377 .0002 .9875 -.9998 .00610 .00884 .00774 5.9860 .5208 .9993 -.3422 .00555 .00669 .00975 6.0077 1.3113 .9993 .1470 .00615 .00898 .00772 5.9795 .5060 .9993 -.3561 .00556 .00680 .00963 6.0081 1.2438 .9993 .1186 .12593 .20988 .00000 3.7050 .0000 .8219-1.0000 .13662 .22770 .00000 3.8419 .0000 .8340-1.0000 .00616 .00903 .00768 5.9795 .4956 .9993 -.3660 .00616 .00903 .00768 5.9795 .4956 .9993 -.3660 .01532 .01683 .02539 4.7899 3.0425 .9951 .5567 163 Table 10.2.2.2.2.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .3, pNz = .6 Fp Fn N:p aEX Rej Tp Tn NT 2 .4800 .16043 .04353 .76164 .23836 .95647 MB 2 .0165 .01227 .00094 .98018 .01982 .99906 RB 2 .0163 .01208 .00093 .98048 .01952 .99907 MD 2 .0172 .01253 .00098 .97977 .02023 .99902 RD 2 .0169 .01231 .00097 .98013 .01987 .99903 OR 2 .8108 .59082 .52158 .36302 .63698 .47842 OS 2 .3638 .28188 .23211 .68494 .31506 .76789 OF 2 .0260 .01635 .00150 .97375 .02625 .99850 CF 2 .0260 .01635 .00150 .97375 .02625 .99850 CS 2 .3129 .11520 .02731 .82620 .17380 .97269 NT 4 .5555 .30489 .04925 .52468 .47532 .95075 MB 4 .0202 .04997 .00113 .91746 .08254 .99887 RB 4 .0188 .04778 .00105 .92106 .07894 .99895 MD 4 .0209 .05068 .00117 .91632 .08368 .99883 RD 4 .0193 .04846 .00108 .91995 .08005 .99892 OR 4 .9889 .85654 .76044 .07939 .92061 .23956 OS 4 .7747 .67653 .58265 .26089 .73911 .41735 OF 4 .0244 .05588 .00137 .90777 .09223 .99863 CF 4 .0244 .05588 .00137 .90777 .09223 .99863 CS 4 .4425 .26318 .03915 .58747 .41253 .96085 NT 10 .5551 .53911 .04938 .13440 .86560 .95062 MB 10 .0269 .27060 .00159 .55005 .44995 .99841 RB 10 .0191 .24827 .00110 .58695 .41305 .99890 MD 10 .0276 .27259 .00163 .54676 .45324 .99837 RD 10 .0196 .25005 .00113 .58400 .41600 .99887 OR 10 .9999 .93360 .83692 .00195 .99805 .16308 OS 10 .9981 .92248 .81212 .00394 .99606 .18788 OF 10 .0299 .27746 .00176 .53874 .46126 .99824 CF 10 .0299 .27746 .00176 .53874 .46126 .99824 CS 10 .3516 .48554 .02953 .21045 .78955 .97047 NT 20 .5564 .61505 .05007 .00829 .99171 .94993 MB 20 .0438 .54126 .00263 .09966 .90034 .99737 RB 20 .0193 .51688 .00111 .13928 .86072 .99889 MD 20 .0451 .54213 .00271 .09825 .90175 .99729 RD 20 .0196 .51782 .00113 .13771 .86229 .99887 OR 20 1.0000 .93586 .83967 .00002 .99998 .16033 OS 20 1.0000 .92701 .81757 .00003 .99997 .18243 OF 20 .0462 .54307 .00277 .09673 .90327 .99723 CF 20 .0462 .54307 .00277 .09673 .90327 .99723 CS 20 .1505 .56958 .01196 .05868 .94132 .98804 NT 40 .5580 .61998 .04995 .00001 .99999 .95005 MB 40 .0461 .60072 .00283 .00068 .99932 .99717 RB 40 .0201 .59953 .00117 .00156 .99844 .99883 MD 40 .0475 .60077 .00291 .00066 .99934 .99709 RD 40 .0204 .59957 .00119 .00151 .99849 .99881 OR 40 1.0000 .93533 .83833 .00000 1.00000 .16167 OS 40 1.0000 .92679 .81698 .00000 1.00000 .18302 OF 40 .0484 .60080 .00297 .00065 .99935 .99703 CF 40 .0484 .60080 .00297 .00065 .99935 .99703 CS 40 .0867 .59830 .00736 .00774 .99226 .99264 A' B" s.Rej s.FpTn s.FnTp d' .10746 .05739 .16806 .9995 3.3559 .7553 .6269 .02035 .00744 .03356 1.0497 15.0402 .7428 .9074 .01982 .00740 .03268 1.0469 15.0066 .7427 .9071 .02060 .00757 .03395 1.0461 14.7432 .7427 .9055 .02004 .00752 .03304 1.0439 14.7528 .7426 .9055 .34715 .34683 .36276 .2963 .9418 .6056 -.0381 .39249 .35464 .42669 .2504 1.1640 .5928 .0953 .02437 .00937 .04007 1.0287 12.4758 .7419 .8893 .02437 .00937 .04007 1.0287 12.4758 .7419 .8893 .10715 .05058 .16364 .9826 4.0783 .7484 .6878 .11543 .05757 .18780 1.5903 3.9076 .8361 .6838 .05204 .00787 .08648 1.6661 40.4854 .7670 .9707 .04822 .00761 .08021 1.6635 41.7961 .7662 .9716 .05252 .00806 .08728 1.6620 39.4826 .7671 .9699 .04860 .00770 .08086 1.6632 41.2398 .7664 .9712 .16631 .22425 .15312 .7015 .4759 .7106 -.4274 .38180 .37194 .40426 .4319 .8324 .6466 -.1155 .05578 .00868 .09271 1.6691 36.8943 .7690 .9679 .05578 .00868 .09271 1.6691 36.8943 .7690 .9679 .12400 .05687 .19644 1.5396 4.5974 .8234 .7313 .07130 .05777 .11220 2.7567 2.1198 .9504 .4250 .12047 .00994 .20031 2.8242 76.9632 .8614 .9873 .10880 .00801 .18117 2.8421 105.9766 .8524 .9910 .12058 .01004 .20049 2.8250 75.3619 .8622 .9870 .10897 .00811 .18146 2.8422 103.7517 .8532 .9908 .06438 .15764 .00990 1.9046 .0251 .7874 -.9719 .08171 .17881 .04352 1.7714 .0434 .7909 -.9498 .12075 .01041 .20075 2.8208 70.3014 .8641 .9860 .12075 .01041 .20075 2.8208 70.3014 .8641 .9860 .06875 .04959 .10895 2.6926 4.2965 .9364 .7058 .02652 .05889 .02129 4.0400 .2191 .9852 -.7052 .06224 .01275 .10323 4.0744 21.5632 .9743 .9432 .06987 .00803 .11632 4.1439 60.0798 .9648 .9817 .06173 .01293 .10237 4.0731 20.7846 .9746 .9409 .06944 .00814 .11559 4.1450 58.5422 .9652 .9812 .06227 .15565 .00091 3.0900 .0004 .7900 -.9997 .06942 .17351 .00098 3.1410 .0004 .7956 -.9997 .06108 .01308 .10126 4.0740 20.1019 .9750 .9387 .06108 .01308 .10126 4.0740 20.1019 .9750 .9387 .03002 .03362 .04190 3.8245 3.7599 .9820 .6476 .02340 .05849 .00052 5.9768 .0003 .9875 -.9997 .00623 .01348 .00529 5.9711 .2714 .9991 -.6126 .00594 .00834 .00822 5.9992 1.3063 .9993 .1450 .00628 .01366 .00522 5.9695 .2577 .9991 -.6300 .00592 .00845 .00810 6.0032 1.2478 .9993 .1204 .06275 .15687 .00000 3.2773 .0000 .7904-1.0000 .06942 .17356 .00000 3.3610 .0000 .7958 1.0000 .00632 .01382 .00518 5.9676 .2490 .9991 -.6413 .00632 .01382 .00518 5.9676 .2490 .9991 -.6413 .01540 .02826 .01524 4.8602 1.0464 .9962 .0256 164 Table 10.2.2.2.2.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p - 10, mNz = .3, pNz = .8 N:p OEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 M D 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 N T 40 MB 40 RB 40 M D 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .2886 .20510 .0097 .01622 .0097 .01588 .0100 .01657 .0099 .01619 .8676 .71627 .4387 .39073 .0143 .02164 .0143 .02164 .2332 .18078 .3245 .39067 .0103 .06729 .0098 .06329 .0108 .06826 .0101 .06417 .9944 .93354 .8372 .79474 .0123 .07532 .0123 .07532 .3353 .39246 .3247 .70271 .0171 .37743 .0105 .33109 .0176 .38014 .0106 .33349 .9999 .97904 .9993 .97686 .0187 .38649 .0187 .38649 .2803 .68180 .3249 .80313 .0391 .74077 .0094 .68907 .0398 .74174 .0094 .69039 1.0000 .98045 1.0000 .97898 .0408 .74264 .0408 .74264 .1287 .77469 .3271 .81015 .0462 .80094 .0093 .79886 .0473 .80098 .0095 .79890 .9999 .98017 .9999 .97869 .0475 .80098 .0475 .80098 .0846 .80128 Fp .04464 .00110 .00110 .00113 .00112 .64291 .33694 .00162 .00162 .03830 .04942 .00114 .00109 .00120 .00112 .84792 .71101 .00137 .00137 .05571 .04946 .00202 .00119 .00209 .00120 .89944 .89132 .00222 .00222 .04576 .04946 .00481 .00109 .00492 .00110 .90230 .89500 .00503 .00503 .02087 .05078 .00588 .00109 .00600 .00112 .90083 .89346 .00602 .00602 .01537 Fn Tp Tn .75479 .24521 .95536 .97999 .02001 .99890 .98043 .01957 .99890 .97957 .02043 .99887 .98005 .01995 .99888 .26539 .73461 .35709 .59583 .40417 .66306 .97336 .02664 .99838 .97336 .02664 .99838 .78360 .21640 .96170 .52402 .47598 .95058 .91618 .08383 .99886 .92116 .07884 .99891 .91497 .08503 .99880 .92007 .07993 .99888 .04506 .95494 .15208 .18432 .81568 .28899 .90620 .09380 .99863 .90620 .09380 .99863 .52336 .47664 .94429 .13398 .86602 .95054 .52872 .47128 .99798 .58643 .41357 .99881 .52534 .47466 .99791 .58343 .41657 .99880 .00106 .99894 .10056 .00175 .99825 .10868 .51745 .48255 .99778 .51745 .48255 .99778 .15919 .84081 .95424 .00845 .99155 .95054 .07524 .92476 .99519 .13893 .86107 .99891 .07406 .92594 .99508 .13729 .86271 .99890 .00002 .99998 .09770 .00002 .99998 .10500 .07296 .92704 .99497 .07296 .92704 .99497 .03686 .96314 .97913 .00001 .99999 .94922 .00030 .99970 .99412 .00170 .99830 .99891 .00028 .99972 .99400 .00165 .99835 .99888 .00000 1.00000 .09917 .00000 1.00000 .10654 .00028 .99972 .99398 .00028 .99972 .99398 .00224 .99776 .98463 s.Rej s.FpTn s.FnTp d' .12816 .02525 .02425 .02556 .02454 .32581 .44721 .03052 .03052 .15081 .14674 .06829 .06135 .06888 .06185 .11101 .35482 .07296 .07296 .16413 .08822 .16570 .14048 .16576 .14075 .02814 .03868 .16559 .16559 .07675 .02322 .07274 .08958 .07204 .08899 .02628 .02841 .07123 .07123 .02813 .01748 .00615 .00644 .00614 .00638 .02681 .02878 .00615 .00615 .01391 .08370 .01133 .01133 .01149 .01144 .34358 .41601 .01369 .01369 .08536 .08504 .01122 .01095 .01149 .01111 .19008 .35675 .01225 .01225 .09780 .08484 .01605 .01165 .01635 .01170 .13570 .14778 .01685 .01685 .08917 .08477 .02520 .01139 .02566 .01155 .13132 .14196 .02588 .02588 .06435 .08739 .02857 .01150 .02878 .01176 .13405 .14389 .02882 .02882 .06047 .15673 1.0095 .03144 1.0082 .03019 .9991 .03183 1.0079 .03056 1.0011 .32972 .2606 .45894 .1783 .03798 1.0110 .03798 1.0110 .18123 .9864 .18200 1.5902 .08529 1.6701 .07663 1.6519 .08603 1.6637 .07724 1.6503 .10426 .6672 .36096 .3427 .09114 1.6785 .09114 1.6785 .19986 1.5333 .10820 2.7579 .20690 2.8026 .17556 2.8201 .20696 2.8009 .17588 2.8250 .00620 1.7950 .02531 1.6859 .20673 2.8010 .20673 2.8010 .09261 2.6853 .02000 4.0389 .09048 4.0269 .11197 4.1500 .08958 4.0275 .11122 4.1544 .00068 2.8546 .00073 2.8604 .08858 4.0277 .08858 4.0277 .02931 3.8246 .00039 6.0317 .00295 5.9510 .00760 5.9944 .00283 5.9619 .00748 5.9941 .00000 2.9786 .00000 3.0198 .00283 5.9606 .00283 5.9606 .00759 5.0021 P 3.3393 13.1793 12.9350 13.0543 12.9110 .8786 1.0609 11.7623 11.7623 3.5260 3.8971 40.4127 40.3848 39.1129 39.6951 .4032 .7793 37.3584 37.3584 3.5440 2.1126 62.1332 98.7238 60.3657 98.0482 .0201 .0302 57.1330 57.1330 2.5243 .2250 10.1566 60.8196 9.8320 59.7698 .0004 .0005 9.5285 9.5285 1.6061 .0002 .0663 1.5001 .0611 1.4222 .0000 .0000 .0609 .0609 .1821 A' B" .7570 .6254 .7410 .8939 .7406 .8917 .7410 .8929 .7407 .8916 .5954 -.0815 .5669 .0375 J410 .8824 .7410 .8824 .7521 .6431 .8362 .6830 .7673 .9707 .7660 .9705 .7675 .9697 .7662 .9700 .7040 -.4996 .6226 -.1549 .7695 .9684 .7695 .9684 .8322 .6517 .9505 .4233 .8665 .9839 .8525 .9903 .8673 .9834 .8532 .9902 .7723 -.9769 .7727 -.9645 .8692 .9824 .8692 .9824 .9447 .5081 .9853 -.6974 .9798 .8712 .9649 .9820 .9801 .8667 .9653 .9816 .7744 -.9996 .7762 -.9996 .9803 .8621 .9803 .8621 .9852 .2694 .9873 -.9998 .9985 -.9024 .9993 .2181 .9984 -.9102 .9993 .1901 .7748 -1.0000 .7766-1.0000 .9984 -.9105 .9984 -.9105 .9956 -.7422 165 Table 10.2.2.2.3.1 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .5, pNz = .2 N:p aEX Rej NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT 4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 .7364 .16087 .0389 .03029 .0378 .02962 .0398 .03069 .0386 .02999 .8776 .44542 .6030 .27916 .0549 .03683 .0549 .03683 .3100 .08226 .7952 .22398 .0424 .10830 .0383 .10583 .0426 .10888 .0390 .10645 .9885 .55332 .9518 .47349 .0501 .11347 .0501 .11347 .2213 .14463 .7934 .23964 .0467 .19847 .0378 .19787 .0482 .19856 .0390 .19794 .9896 .55815 .9818 .49504 .0513 .19875 .0513 .19875 .0992 .19124 .7906 .23966 .0464 .20112 .0383 .20091 .0475 .20114 .0395 .20094 .9912 .56394 .9835 .49747 .0492 .20119 .0492 .20119 .0910 .20300 .7916 .23982 .0473 .20114 .0389 .20093 .0483 .20116 .0395 .20094 .9908 .56369 .9840 .49789 .0491 .20118 .0491 .20118 .0906 .20324 Fp .04726 .00114 .00110 .00117 .00113 .36581 .20406 .00163 .00163 .01393 .04954 .00126 .00113 .00127 .00116 .44574 .35156 .00151 .00151 .00927 .04962 .00140 .00112 .00144 .00116 .44769 .36881 .00154 .00154 .00455 .04958 .00139 .00114 .00143 .00118 .45493 .37184 .00148 .00148 .00422 .04977 .00142 .00116 .00145 .00118 .45461 .37236 .00148 .00148 .00405 Fn .38467 .85312 .85632 .85121 .85456 .23612 .42043 .82236 .82236 .64442 .07827 .46354 .47537 .46068 .47238 .01633 .03877 .43869 .43869 .31391 .00028 .01326 .01514 .01300 .01493 .00002 .00002 .01240 .01240 .06201 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00187 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 Tp .61533 .14688 .14368 .14879 .14544 .76388 .57957 .17764 .17764 .35558 .92173 .53646 .52463 .53932 .52762 .98367 .96123 .56131 .56131 .68609 .99972 .98674 .98486 .98700 .98507 .99998 .99998 .98760 .98760 .93799 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 .99813 1.00000 1,00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 Tn .95274 .99886 .99890 .99883 .99887 .63419 .79594 .99837 .99837 .98607 .95046 .99874 .99887 .99873 .99884 .55426 .64844 .99849 .99849 .99073 .95038 .99860 .99888 .99856 .99884 55231 .63119 .99846 .99846 .99545 .95042 .99861 .99886 .99857 .99882 .54507 .62816 .99852 .99852 .99578 .95023 .99858 .99884 .99855 .99883 .54539 .62764 .99852 .99852 .99595 s.FnTp s.Rej s.FpTn .07449 .03519 .03410 .03546 .03436 .27177 .28227 .03873 .03873 .05746 .04151 .05408 .05297 .05404 .05301 .21916 .22535 .05381 .05381 .04335 .03385 .01055 .01084 .01049 .01080 .21705 .21251 .01036 .01036 .02133 .03388 .00527 .00474 .00533 .00481 .21695 .21223 .00544 .00544 .01318 .03360 .00529 .00474 .00534 .00477 .21646 .21093 .00540 .00540 .01242 .04346 .00585 .00573 .00593 .00582 .28961 .26870 .00699 .00699 .02625 .04151 .00620 .00583 .00624 .00591 .27083 .26781 .00681 .00681 .02130 .04229 .00658 .00583 .00667 .00592 .27130 .26563 .00686 .00686 .01687 .04234 .00659 .00592 .00666 .00601 .27119 .26529 .00680 .00680 .01596 .04200 .00661 .00593 .00668 .00597 .27057 .26366 .00675 .00675 .01553 .29593 .17346 .16827 .17467 .16951 .32341 .44767 .19040 .19040 .24371 .12683 .26896 .26393 .26878 .26408 .06583 .16208 .26733 .26733 .18895 .00577 .04631 .04953 .04578 .04914 .00157 .00157 .04447 .04447 .06991 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .01428 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 d' P 1.9653 3.8762 2.0007 60.4684 1.9971 61.5012 2.0018 59.6658 1.9974 60.6121 1.0618 .8191 1.0280 1.3799 2.0181 49.4997 2.0181 49.4997 1.8289 10.4812 3.0662 1.4283 3.1122 95.3939 3.1154105.6717 3.1167 94.5666 3.1163 103.5246 2.2726 .1031 .2264 2.1463 3.1203 80.3809 3.1203 80.3809 2.8395 14.2212 .0100 5.1010 5.2075 7.4300 5.2229 10.2255 5.2055 7.1005 5.2189 9.8133 .0002 4.2146 4.4181 .0003 5.2049 6.4439 5.2049 6.4439 4.1464 9.1950 .0000 5.9139 7.2550 .0000 7.3156 .0000 7.2478 .0000 .0000 7.3062 4.3781 .0000 .0000 4.5919 7.2361 .0000 .0000 7.2361 5.5341 .4795 .0000 5.9120 7.2490 .0000 .0000 7.3112 .0000 7.2424 .0000 7.3069 .0000 4.3789 .0000 4.5905 7.2366 .0000 7.2366 .0000 .0000 6.9125 A' B" .8799 .6804 .7845 .9820 .7838 .9823 .7850 .9817 .7842 .9820 .7872 -.1252 .7799 .2001 .7918 .978C .7918 .9780 .8268 .8869 .9660 .2102 .8834 .9899 .8805 .9910 .8841 .9898 .8812 .9908 .8793 -.8779 .8936 -.7190 .8895 .9878 .8895 .9878 .9174 .9182 .9875 -.9883 .9963 .8069 .9959 .8605 .9964 .7979 .9960 .8545 .8881 -.9998 .9078 -.9998 .9965 .7774 .9965 .7774 .9832 .8555 .9876 -1.0000 .9997 -1.0000 .9997 -1.0000 .9996 1.0000 .9997 1.0000 .8863 1.000C .9070 1.0000 .9996 1.0000 .9996 1.0000 .9985 -.3853 .9876- 1.0000 .9996- 1.0000 .9997 -1.0000 .9996- 1.0000 .9997 -1.0000 .8863 -1.0000 .9069 -1.0000 .9996- 1.0000 .9996- 1.0000 .9990 -1.0000 166 Table 10.2.2.2.3.2 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * l,p = 10, mNz = .5,pNz = A OEX NT 2 MB 2 RB 2 MD 2 RD 2 OR 2 OS 2 OF 2 CF 2 CS 2 NT .4 MB 4 RB 4 MD 4 RD 4 OR 4 OS 4 OF 4 CF 4 CS 4 NT 10 MB 10 RB 10 MD 10 RD 10 OR 10 OS 10 OF 10 CF 10 CS 10 NT 20 MB 20 RB 20 MD 20 RD 20 OR 20 OS 20 OF 20 CF 20 CS 20 NT 40 MB 40 RB 40 MD 40 RD 40 OR 40 OS 40 OF 40 CF 40 CS 40 Rej .6698 .28626 .0277 .06114 .0262 .05854 .0284 .06196 .0266 .05924 .9877 .76021 .8352 .62604 .0405 .07374 .0405 .07374 .4098 .21427 .6688 .39860 .0354 .22258 .0274 .21088 .0361 .22375 .0281 .21216 .9994 .81964 .9972 .78867 .0421 .23224 .0421 .23224 .2809 .32740 .6713 .43009 .0439 .39683 .0280 .39462 .0450 .39692 .0285 .39472 .9996 .82315 .9993 .79451 .0475 .39717 .0475 .39717 .1164 .39138 .6633 .42981 .0436 .40108 .0274 .40067 .0448 .40112 .0286 .40070 .9996 .82164 .9996 .79354 .0469 .40117 .0469 .40117 .1082 .40416 .6607 .43015 .0461 .40120 .0287 .40072 .0475 .40123 .0288 .40073 .9996 .82304 .9994 .79476 .0479 .40124 .0479 .40124 .1137 .40462 Fp .04924 .00106 .00100 .00109 .00102 .64380 .50441 .00161 .00161 .02722 .04969 .00142 .00107 .00145 .00110 .70069 .65038 .00172 .00172 .01773 .05035 .00184 .00111 .00189 .00113 .70525 .65752 .00201 .00201 .00796 .04969 .00181 .00112 .00187 .00116 .70274 .65590 .00195 .00195 .00743 .05026 .00199 .00121 .00204 .00121 .70507 .65794 .00206 .00206 .00769 Fn .35822 .84873 .85517 .84674 .85342 .06517 .19152 .81807 .81807 .50516 .07802 .44569 .47439 .44281 .47124 .00193 .00390 .42197 .42197 .20811 .00029 .01068 .01512 .01053 .01489 .00000 .00000 .01008 .01008 .03350 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00075 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 Tp .64178 .15127 .14483 .15326 .14658 .93483 .80848 .18193 .18193 .49484 .92198 .55431 32561 .55719 .52876 .99807 .99610 .57803 .57803 .79189 .99971 .98932 .98488 .98947 .98511 1.00000 1.00000 .98992 .98992 .96650 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 .99925 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 Tn .95076 .99894 .99900 .99891 .99898 .35620 .49559 .99839 .99839 .97278 .95031 .99858 .99893 .99855 .99890 .29931 .34962 .99828 .99828 .98227 .94965 .99816 .99889 .99811 .99887 .29475 .34248 .99799 .99799 .99204 .95031 .99819 .99888 .99813 .99884 .29726 .34410 .99805 .99805 .99257 .94974 .99801 .99879 .99796 .99879 .29493 .34206 .99794 .99794 .99231 s:Rej s:FpTn s.FnTp a" .09411 .06060 .05679 .06102 .05716 .19164 .32043 .06662 .06662 .09403 .05173 .09609 .09183 .09596 .09189 .13677 .15120 .09497 .09497 .05727 .03169 .01409 .01633 .01401 .01618 .13684 .14812 .01385 .01385 .02428 .03161 .00543 .00425 .00554 .00431 .13753 .14811 .00563 .00563 .01637 .03226 .00588 .00449 .00594 .00451 .13702 .14659 .00595 .00595 .01574 .05151 .00639 .00624 .00649 .00628 .25911 .33305 .00807 .00807 .04335 .05222 .00773 .00655 .00783 .00662 .22671 .24602 .00858 .00858 .03644 .05284 .00923 .00676 .00933 .00681 .22807 .24687 .00970 .00970 .02756 .05268 .00904 .00708 .00923 .00718 .22922 .24685 .00939 .00939 .02681 .05377 .00981 .00748 .00989 .00751 .22837 .24431 .00992 .00992 .02623 .22158 .15095 .14157 .15200 .14248 .15195 .36113 .16586 .16586 .21612 .10642 .23962 .22932 .23929 .22946 .01371 .04297 .23670 .23670 .12917 .00433 .03296 .03999 .03266 .03958 .00000 .00000 .03188 .03188 .03620 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00641 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 2.0154 3.6655 2.0421 66.0529 2.0303 67.4098 2.0423 64.9640 2.0336 67.0487 .3408 1.1442 .6836 .8613 2.0384 50.8270 2.0384 50.8270 1.9104 6.3569 3.0664 1.4215 3.1206 85.0299 3.1342111.0879 3.1216 83.3590 3.1350108.6185 .0177 2.3634 2.2743 .0313 3.1225 70.8403 3.1225 70.8403 2.9161 6.5598 5.0781 .0105 5.2051 4.7930 5.2257 10.2828 5.2031 4.6308 5.2269 9.9937 3.7253 .0000 .0000 3.8592 5.1996 4.2130 5.1996 4.2130 4.2425 3.4158 5.9128 .0000 7.1748 .0000 7.3217 .0000 7.1647 .0000 7.3100 .0000 3.7326 .0000 3.8636 .0000 .0000 7.1513 7.1513 .0000 .1258 5.6103 .0000 5.9073 7.1442 .0000 7.2987 .0000 7.1361 .0000 .0000 7.2969 .0000 3.7259 3.8580 .0000 .0000 7.1338 .0000 7.1338 6.6881 .0000 A' B" .8866 .6616 .7858 .9837 .7843 .9839 .7863 .9834 .7847 .9839 .7821 -.5802 .7474 -.2350 .7929 .9787 .7929 .9787 .8564 .8084 .9660 .2075 .8878 .9886 .8808 .9915 .8885 .9883 .8815 .9912 .8229 -.9818 .8340 -.9664 .8936 .9860 .8936 .9860 .9414 .8088 .9873 -.9878 .9969 .7033 .9959 .8613 .9969 .6935 .9960 .8571 .8237 -1.0000 .8356 -1.0000 .9970 .6652 .9970 .6652 .9895 .6079 .9876 -1.0000 .9995 -1.0000 .9997 -l.OOOOj .9995 -l.ooooj .9997 -i.oooo| .8243 -l.OOOOj .8360 -1.0000 .9995 -1.0000 .9995 -1.0OO0 .9980 -.8156 .9874 -1.0000 .9995 -1.0000 .9997-1.0000 .9995 -1.0000 .9997 -1.0000 .8237 -1.0000 .8355 -1.0000 .9995 -1.0000 .9995 -1.0000 .9981 -1.0000 167 Table 10.2 2.23.3 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10, mNz = .5, pNz = .6 N:p OEX Rej Fp Fn Tp Tn NT 2 .4967 .40592 .04906 .35616 .64384 .95094 MB 2 .0187 .09331 .00116 .84526 .15474 .99884 RB 2 .0173 .08650 .00107 .85654 .14346 .99893 MD 2 .0195 .09457 .00121 .84319 .15681 .99879 RD 2 .0174 .08758 .00107 .85474 .14526 .99893 OR 2 .9966 .89927 .79353 .03023 .96977 .20647 OS 2 .8902 .80221 .68963 .12274 .87726 .31037 OF 2 .0289 .11236 .00185 .81397 .18603 .99815 CF 2 .0289 .11236 .00185 .81397 .18603 .99815 CS 2 .4009 .36774 .04262 .41551 .58449 .95738 .5007 .57281 .04978 .07850 .92150 .95022 NT 4 MB 4 .0276 .34483 .00179 .42649 .57351 .99821 RB 4 .0178 .31500 .00109 .47574 .52426 .99891 MD 4 .0280 .34683 .00182 .42316 .57684 .99818 RD 4 .0181 .31685 .00111 .47266 .52734 .99889 OR 4 1.0000 .93214 .83162 .00084 .99916 .16838 OS 4 .9984 .92113 .80653 .00247 .99753 .19347 OF 4 .0327 .35880 .00212 .40341 .59659 .99788 CF 4 .0327 .35880 .00212 .40341 .59659 .99788 CS 4 .2807 .52080 .02858 .15106 .84894 .97142 NT 10 .4901 .61957 .04932 .00026 .99974 .95068 MB 10 .0412 .59635 .00276 .00791 .99209 .99724 RB 10 .0182 .59138 .00112 .01511 .98489 .99888 M D 10 .0419 .59646 .00282 .00777 .99223 .99718 RD 10 .0185 .59155 .00113 .01484 .98516 .99887 OR 10 1.0000 .93187 .82967 .00000 1.00000 .17033 OS 10 .9999 .92260 .80651 .00000 1.00000 .19349 OF 10 .0433 .59671 .00289 .00742 .99258 .99711 CF 10 .0433 .59671 .00289 .00742 .99258 .99711 CS 10 .1200 .59355 .01348 .01974 .98026 .98652 NT 20 .4859 .61979 .04947 .00000 1.00000 .95053 MB 20 .0432 .60115 .00287 .00000 1.00000 .99713 RB 20 .0192 .60049 .00122 .00000 1.00000 .99878 M D 20 .0440 .60118 .00294 .00000 1.00000 .99706 RD 20 .0199 .60050 .00126 .00000 1.00000 .99874 OR 20 .9999 .93262 .83156 .00000 1.00000 .16844 OS 20 .9999 .92331 .80827 .00000 1.00000 .19173 OF 20 .0448 .60121 .00302 .00000 1.00000 .99698 CF 20 .0448 .60121 .00302 .00000 1.00000 .99698 CS 20 .1198 .60537 .01369 .00017 .99983 .98631 NT 40 .4915 .61992 .04980 .00000 1.00000 .95020 MB 40 .0415 .60115 .00288 .00000 1.00000 .99712 RB 40 .0178 .60047 .00118 .00000 1.00000 .99882 MD 40 .0424 .60118 .00294 .00000 1.00000 .99706 RD 40 .0178 .60047 .00118 .00000 1.00000 .99882 OR 40 1.0000 .93226 .83066 .00000 1.00000 .16934 OS 40 .9999 .92330 .80825 .00000 1.00000 .19175 OF 40 .0426 .60118 .00296 .00000 1.00000 .99704 CF 40 .0426 .60118 .00296 .00000 1.00000 .99704 CS 40 .1181 .60544 .01359 .00000 1.00000 .98641 A' B" s.Rej s:FpTn s.FnTp a" P .13502 .06783 .22019 2.0228 3.6693 .8873 .6619 .09244 .00885 .15382 2.0307 61.9162 .7866 .9825 .08198 .00847 .13648 2.0061 63.3421 .7838 .9828 .09323 .00903 .15511 2.0266 60.0832 .7870 .9820 .08256 .00849 .13744 2.0125 63.5720 .7842 .9829 .11806 .20407 .09481 1.0587 .2400 .7588 -.6965 .29630 .31637 .30954 .6666 .5758 .7046 -.3306 .10174 .01147 .16910 2.0100 45.3473 .7936 .9759 .10174 .01147 .16910 2.0100 45.3473 .7936 .9759 .13607 .07088 .21554 1.9345 4.2987 .8733 .7123 .06792 .06834 .10510 3.0622 1.4259 .9658 .2093 .14988 .01171 .24935 3.0975 68.2577 .8924 .9855 .13894 .00856 .23153 3.1242 108.8686 .8804 .9913 .14968 .01176 .24903 3.1022 67.3893 .8932 .9853 .13902 .00861 .23166 3.1274 107.3172 .8812 .9911 .06922 .17156 .00778 2.1795 .0115 .7907 -.9880 .08506 .19269 .04070 1.9452 .0280 .7947 -.9689 .14786 .01273 .24593 3.1039 57.8690 .8980 .9826 .14786 .01273 .24593 3.1039 57.8690 .8980 .9826 .07000 .05965 .10827 2.9339 3.5839 .9527 .6440 .02759 .06892 .00326 5.1225 .0095 .9876 -.9890 .01596 .01480 .02475 5.1884 2.5624 .9973 .4815 .02220 .00866 .03664 5.2245 10.2294 .9959 .8606 .01588 .01502 .02454 5.1879 2.4718 .9973 .4660 .02185 .00871 .03605 5.2273 9.9340 .9960 .8562 .06848 .17120 .00000 3.3120 .0000 .7926 1.0000 .07583 .18958 .00000 3.3998 .0000 .7984 1.0000 .01552 .01515 .02383 5.1958 2.3154 .9974 .4368 .01552 .01515 .02383 5.1958 2.3154 .9974 .4368 .02557 .04630 .02338 4.2711 1.3860 .9916 .1853 .02794 .06985 .00000 5.9149 .0000 .9876 1.0000 .00594 .01485 .00000 7.0269 .0000 .9993 1.0000 .00376 .00940 .00000 7.2964 .0000 .9997 1.0000 .00603 .01508 .00000 7.0194 .0000 .9993-1.0000 .00380 .00951 .00000 7.2869 .0000 .9997 1.0000 .06822 .17056 .00000 3.3046 .0000 .7921 -1.0000 .07563 .18908 .00000 3.3934 .0000 .7979 -1.0000 .00617 .01542 .00000 7.0103 .0000 .9992 1.0000 .00617 .01542 .00000 7.0103 .0000 .9992 -1.0000 .01874 .04662 .00253 5.7826 .0190 .9965 -.9745 .02816 .07039 .00000 5.9117 .0000 .9875 -1.0000 .00629 .01573 .00000 7.0263 .0000 .9993 -1.0000 .00381 .00952 .00000 7.3062 .0000 .9997 -1.0000 .00635 .01588 .00000 7.0194 .0000 .9993 -1.0000 .00383 .00957 .00000 7.3048 .0000 .9997 -1.0000 .06893 .17234 .00000 3.3081 .0000 .7923-1.0000 .07629 .19073 .00000 3.3934 .0000 .7979 -1.0000 .00637 .01593 .00000 7.0176 .0000 .9993 -1.000C .00637 .01593 .00000 7.0176 .0000 .9993 -l.OOOC .01867 .04667 .00000 6.4738 .0000 .9966-1.0000 168 Table 10.2.2.2.3.4 Table of empirical results for a preliminary omnibus test and follow-up tests of correlations when P * I, p = 10; mNz = .5, pNz = .8 N:p OEX Rej Fp Fn Tp Tn s.Rej s.FpTns.FnTp d' P A' B" NT 2 .2654 .52452 .04837 .35645 .64355 .95163 .17566 .10299 .21744 2.0289 3.7121 .8876 .6658 .0118 .12842 .00150 .83985 .16015 .99850 .12800 .01552 .15985 1.9739 49.8926 .7874 .9780 MB 2 .0103 .11470 .00127 .85695 .14305 .99873 .10647 .01364 .13303 1.9526 54.0145 .7833 .9796 RB 2 .0122 .13016 .00156 .83769 .16231 .99844 .12906 .01579 .16115 1.9715 48.6883 .7879 .9774 MD 2 .0104 .11620 .00128 .85506 .14494 .99872 .10732 .01368 .13409 1.9583 54.0595 .7838 .9796 RD 2 .9977 .95927 .87011 .01844 .98156 .12989 .07887 .17424 .07140 .9601 .2138 .7429 -.7239 OR 2 OS 2 .9212 .89043 .79829 .08654 .91346 .20171 .26353 .28906 .26726 .5269 .5605 .6743 -.3415 OF 2 .0154 .15398 .00209 .80804 .19196 .99791 .14077 .01900 .17574 1.9937 41.4038 .7948 .9735 .0154 .15398 .00209 .80804 .19196 .99791 .14077 .01900 .17573 1.9937 41.4041 .7948 .9735 CF 2 .2992 .53556 .06430 .34662 .65338 .93570 .17675 .13206 .21350 1.9141 2.9355 .8828 5802 CS 2 .2727 .74755 .04953 .07795 .92205 .95047 .08416 .10397 .10182 3.0684 1.4240 .9661 .2084 NT 4 .0184 .48328 .00258 .39654 .60346 .99742 .20737 .02174 .25880 3.0595 48.3091 .8995 .9787 MB 4 .0081 .42021 .00099 .47498 .52502 .99901 .18332 .01134 .22915 3.1563119.4687 .8807 .9921 RB 4 MD 4 .0194 .48606 .00270 .39310 .60690 .99730 .20695 .02209 .25826 3.0534 46.2190 :9003 .9777 RD 4 .0082 .42271 .00100 .47186 .52814 .99900 .18344 .01140 .22930 3.1608118.1880 .8814 .9920 OR 4 .9999 .97913 .89732 .00042 .99958 .10268 .02953 .14484 .00442 2.0714 .0085 .7745 -.9909 .9992 .97680 .88843 .00111 .99889 .11157 .04096 .15823 .02665 1.8398 OS 4 .0196 .7751 -.9778 OF 4 .0229 .50118 .00312 .37431 .62569 .99688 .20340 .02323 .25382 3.0551 39.9585 .9049 .9738 CF 4 .0229 .50118 .00312 .37431 .62569 .99688 .20340 .02323 .25382 3.0551 39.9585 .9049 .9738 CS 4 .2186 .72375 .04752 .10719 .89281 .95248 .07808 .11767 .09003 2.9110 1.8638 .9586 .3578 NT 10 .2710 .80968 .04949 .00028 .99972 .95051 .02088 .10368 .00323 5.1023 .0101 .9876 -.9883 MB 10 .0416 .79748 .00594 .00463 .99537 .99406 .01561 .03312 .01754 5.1175 .8012 .9973 -.1233 RB 10 .0086 .78804 .00108 .01521 .98479 .99892 .02820 .01218 .03512 5.2324 10.6289 .9959 .8659 MD 10 .0423 .79762 .00607 .00450 .99550 .99393 .01538 .03349 .01718 5.1205 .7662 .9973 -.1478 RD 10 .0088 .78827 .00111 .01494 .98506 .99889 .02781 .01242 .03462 5.2306 10.1739 .9960 .8597 OR 10 .9997 .97932 .89659 .00000 1.00000 .10341 .02898 .14488 .00000 3.0025 .0000 .7759 -1.000C OS 10 .9997 .97766 .88828 .00000 1.00000 .11172 .03129 .15644 .00000 3.0475 .0000 .7779 -1.0000 OF 10 .0437 .79778 .00622 .00433 .99567 .99378 .01518 .03372 .01688 5.1242 .7248 .9973 -.1780 CF 10 .0437 .79778 .00622 .00433 .99567 .99378 .01518 .03372 .01688 5.1242 .7248 .9973 -.1780 CS 10 .1208 .79828 .02869 .00932 .99068 .97131 .02443 .09690 .01499 4.2532 .3821 .9904 -.5024 NT 20 .2647 .80981 .04904 .00000 1.00000 .95096 .02110 .10549 .00000 5.9191 .0000 .9877 -1.0000 MB 20 .0402 .80119 .00596 .00000 1.00000 .99404 .00698 .03488 .00000 6.7797 .0000 .9985 1.0000 RB 20 .0076 .80021 .00104 .00000 1.00000 .99896 .00270 .01348 .00000 7.3422 .0000 .9997- 1.0000 MD 20 .0409 .80122 .00610 .00000 1.00000 .99390 .00710 .03550 .00000 6.7712 .0000 .9985 -1.0000 RD 20 .0079 .80022 .00108 .00000 1.00000 .99892 .00272 .01361 .00000 7.3328 .0000 .9997- 1.0000 OR 20 1.0000 .97932 .89662 .00000 1.00000 .10338 .02869 .14343 .00000 3.0024 .0000 .7758 -1.0000 OS 20 .9998 .97765 .88823 .00000 1.00000 .11177 .03119 .15595 .00000 3.0477 .0000 .7779 -1.0000 OF 20 .0415 .80124 .00619 .00000 1.00000 .99381 .00715 .03573 .00000 6.7661 .0000 .9985 -1.0000 CF 20 .0415 .80124 .00619 .00000 1.00000 .99381 .00715 .03573 .00000 6.7661 .0000 .9985 -1.000C CS 20 .1110 .80559 .02808 .00003 .99997 .97192 .01990 .09940 .00088 5.9408 .0018 .9930 -.9980 NT 40 .2683 .81014 .05070 .00000 1.00000 .94930 .02168 .10840 .00000 5.9030 .0000 .9873 -1.0000 MB 40 .0399 .80120 .00601 .00000 1.00000 .99399 .00709 .03545 .00000 6.7764 .0000 .9985 -1.0000 RB 40 .0080 .80023 .00113 .00000 1.00000 .99887 .00278 .01392 .00000 7.3178 .0000 .9997 -1.0000 MD 40 .0404 .80122 .00609 .00000 1.00000 .99391 .00714 .03570 .00000 6.7718 .0000 .9985 -1.0000 RD 40 .0081 .80023 .00114 .00000 1.00000 .99886 .00279 .01396 .00000 7.3148 .0000 .9997 -1.000C OR 40 .9999 .97984 .89922 .00000 1.00000 .10078 .02864 .14320 .00000 2.9878 .0000 .7752 -1.0000 OS 40 .9999 .97812 .89059 .00000 1.00000 .10941 .03115 .15577 .00000 3.0352 .0000 .7774 -1.0000 OF 40 .0408 .80123 .00613 .00000 1.00000 .99387 .00715 .03576 .00000 6.7693 .0000 .9985 -1.0000 CF 40 .0408 .80123 .00613 .00000 1.00000 .99387 .00715 .03576 .00000 6.7693 .0000 .9985 -1.0000 CS 40 .1151 .80591 .02956 .00000 1.00000 .97044 .02060 .10298 .00000 6.1523 .0000 .9926-1.0000 Table 11.1 S E X f ° tests on correlations when P = I,across N:pxOxp r Scrit .01 = -00562, S N:p NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 n-5 .3612 .0007 .0007 .0025 .0025 .179 .0237 .0237 .0237 .0042 .3578 -.002 -.002 -.0008 -.0008 .1829 .0097 .0097 .0097 .0032 .3493 -.004 -.004 -.0031 -.0031 .1818 -.0004 -.0004 -.0004 -.0009 .3449 -.0025 -.0025 -.0017 -.0017 .1837 .0002 .0002 .0002 .0007 .3577 -.0012 -.0012 1E-04 1E-04 .1855 .0009 .0009 .0009 -.0014 C R I T . 01 = 00717) 0 n-10 y-5 .8544 .0041 -.0004 -.0261 -.0004 -.0261 .0007 -.0257 .0007 -.0257 .3341 .0042 .0211 -.0205 .0211 -.0205 .0211 -.0205 .005 .0042 .8542 .003 -.0015 -.0249 -.0015 -.0249 -.0002 -.0245 -.0002 -.0245 .3354 .0032 .0091 -.022 .0091 -.022 .0091 -.022 -.0022 .0032 .8535 -.0012 -.0026 -.0275 -.0026 -.0275 -.0013 -.0273 -.0013 -.0273 .3384 -.001 .0016 -.0267 .0016 -.0267 .0016 -.0267 .0034 -.0009 .8484 .0003 .0017 -.0261 .0017 -.0261 .0032 -.0259 .0032 -.0259 .3412 .0007 .0051 -.0253 .0051 -.0253 .0051 -.0253 .0018 .0007 .8547 -.0017 -.0008 -.0272 -.0008 -.0272 0 -.0266 0 -.0266 .3375 -.0016 .0007 -.0263 .0007 -.0263 .0007 -.0263 -.0008 -.0014 V-10 .005 -.0346 -.0346 -.0345 -.0345 .005 -.0307 -.0307 -.0307 .005 -.0022 -.0381 -.0381 -.038 -.038 -.0022 -.0367 -.0367 -.0367 -.0022 .0034 -.0357 -.0357 -.0355 -.0355 .0034 -.0349 -.0349 -.0349 .0034 .0018 -.0351 -.0351 -.0348 -.0348 .0018 -.0345 -.0345 -.0345 .0018 -.0008 -.0363 -.0363 -.0362 -.0362 -.0008 -.036 -.036 -.036 -.0008 (5 £t 05 = .00427, cr 170 Table 11.2 SEX f o r t e s t s o n correlations with no preliminary omnibus test when P * I, p = 5 across N:p x mNz x pNz (8 crit N.p NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 .05 = -00427, «5 c r i t . 1 = .00562, <5crit .001 = - 0 ° 7 1 7 0 ) 1-2 1-4 1-6 1-8 3-2 3-4 3-6 3-8 5-2 5-4 5-6 5-8 .2901 -.0084 -.0085 -.0068 -.0069 .1597 .0122 .0095 .0095 -.0019 .2886 -.0101 -.0101 -.0094 -.0095 .1726 .0021 -.0022 -.0022 -.0034 .2785 -.0126 -.0128 -.0122 -.0124 .1975 -.0028 -.01 -.01 -.0005 .2804 -.0119 -.0122 -.0109 -.0112 .247 .0091 -.0094 -.0094 .0136 .2955 -.01 -.0109 -.0096 -.0105 .3237 .0374 -.0091 -.0091 .0294 .2203 -.0184 -.0185 -.0174 -.0174 .1331 .0022 -.0029 -.0029 -.01 .2144 -.0189 -.019 -.0181 -.0181 .1575 -.0068 -.0141 -.0141 -.01 .2086 -.0218 -.0224 -.0214 -.022 .2049 -.003 -.0201 -.0201 -.0032 .2087 -.0207 -.0212 -.0204 -.0209 .3025 .02 -.0188 -.0188 .0157 .2156 -.0198 -.0211 -.019 -.0203 .4534 .0857 -.0187 -.0187 .0397 .1351 -.0297 -.0297 -.0292 -.0292 .0974 -.0122 -.0193 -.0193 -.0227 .0477 -.0411 -.0411 -.0407 -.0407 .0473 -.0284 -.0361 -.0361 -.036 .051 -.0406 -.0407 -.0404 -.0405 .0766 -.0264 -.0379 -.0379 -.0324 .0449 -.0405 -.0407 -.0402 -.0403 .1511 -.0137 -.0394 -.0394 -.0274 .0467 -.0394 -.0396 -.0392 -.0393 .2927 .0314 -.039 -.039 -.0133 .0486 -.0404 -.041 -.0399 -.0408 .5164 .187 -.0396 -.0396 .0087 .2841 -.0096 -.01 -.0079 -.0083 .2113 .0192 .009 .009 .0079 .2855 -.0098 -.01 -.009 -.0094 .2949 .0302 -.0027 -.0027 .0286 .2746 -.0109 -.0127 -.0102 -.0121 .4433 .1315 -.0076 -.0076 .0421 .2761 -.0065 -.0119 -.0053 -.0115 .5512 .3274 -.0031 -.0031 .0369 .2866 -.0014 -.0109 -.0002 -.0096 .6011 .4584 .0003 .0003 .0099 .2095 -.0166 -.0168 -.0151 -.0153 .237 .0188 -.0042 -.0042 .0129 .2114 -.0182 -.019 -.0177 -.0184 .3861 .0663 -.0123 -.0123 .0404 .1971 -.0194 -.0239 -.0193 -.0236 .651 .3199 -.0175 -.0175 .0635 .2011 -.0117 -.0214 -.0109 -.0209 .7565 .642 -.0098 -.0098 .0529 .208 -.0047 -.0205 -.0037 -.02 .7767 .709 -.0032 -.0032 .016 .1246 -.0279 -.028 -.0272 -.0273 .2513 .0197 -.0196 -.0196 .0044 .1272 -.0302 -.0309 -.0298 -.0304 .4432 .1143 -.0265 -.0265 .0323 .1209 -.029 -.0331 -.0286 -.033 .7279 .4632 -.0269 -.0269 .0665 .1219 -.0177 -.0307 -.0172 -.0306 .8215 .7601 -.0164 -.0164 .049 .126 -.0058 -.0287 -.0052 -.0283 .8346 .7962 -.0047 -.0047 .02 .0428 -.0382 -.0385 -.0379 -.0383 .2178 .0143 -.0349 -.0349 -.0143 .0485 -.0404 -.041 -.0397 -.0406 .2749 -.0088 -.0094 -.0075 -.008 .3177 .0565 .0077 .0077 .0369 .2753 -.0085 -.0116 -.008 -.0111 .4698 .1771 .0002 .0002 .0545 .2652 -.0056 -.0125 -.0045 -.012 .5644 .4078 -.0008 -.0008 .0333 .2671 -.0019 -.0112 -.0008 -.0105 .5746 .4414 .0013 .0013 .0168 .2721 -.0017 -.0107 -.0005 -.0097 .5766 .4433 .0002 .0002 .0161 .1996 -.0179 -.0192 -.0172 -.0184 .4494 .1118 -.0039 -.0039 .0532 .1109 -.0275 -.0286 -.0271 -.0281 .5005 .165 -.0197 -.0197 .0492 .1074 -.0268 -.0325 -.0264 -.0319 .7184 .502 -.0216 -.0216 .0706 .1006 -.0129 -.0328 -.0123 -.0326 .79 .7436 -.01 -.01 .0395 .102 -.0063 -.032 -.0056 -.0317 .7957 .7579 -.0052 -.0052 .0303 .1097 -.0056 -.0314 -.0047 -.0309 .8035 .7611 -.004 -.004 .0305 .0374 -.0393 -.0411 -.0392 -.0407 .5281 .2265 -.0347 -.0347 .0078 .0396 -.0307 -.0403 -.0303 -.0402 .7521 .6093 -.0281 -.0281 .0251 .0384 -.0177 -.0397 -.0169 -.0394 .7921 .7696 -.016 -.016 .0202 .0398 -.0042 -.0398 -.0034 -.0397 .795 .7737 -.0032 -.0032 .0112 .0396 -.0041 -.0403 -.0034 -.04 .7934 .7739 -.0032 -.0032 .0111 .1345 -.0306 -.0306 -.0301 -.0301 .1318 -.0161 -.0266 -.0266 -.0203 .1323 -.031 -.0311 -.0309 -.031 .1985 -.0062 -.03 -.03 -.0122 .131 -.0306 -.0311 -.0304 -.0308 .3261 .0299 -.0294 -.0294 .0099 .1357 -.0296 -.0305 -.0295 -.0303 .5347 .1513 -.0292 -.0292 .0359 ^/]/)/) .1431 -.0382 -.0382 .0041 .0405 -.0363 -.0406 -.0361 -.0402 .7323 .5629 -.035 -.035 .0282 .0457 -.0237 -.0397 -.0232 -.0396 .8058 .7832 -.0228 -.0228 .03 .0466 -.0071 -.0402 -.0064 -.0399 .8156 .7972 -.0064 -.0064 .0062 .1923 -.0163 -.0207 -.0156 -.0199 .6588 .3869 -.0086 -.0086 .0722 .1817 -.0097 -.0226 -.0087 -.0223 .7457 .6679 -.0067 -.0067 .0373 .1903 -.0068 -.0241 -.0057 -.0237 .7459 .6761 -.0043 -.0043 .0228 .1891 -.0051 -.0216 -.0043 -.021 .7483 .6817 -.0038 -.0038 .0253 171 Table 11.3 SEX f ° rt e s t s o n correlations with a preliminary omnibus test when P * I, p = 5 across N:p x mNz x pNz (8 crit NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS N:p 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 1-2 .005 -.0346 -.0346 -.0345 -.0345 .005 -.0307 -.0307 -.0307 .005 -.0022 -.0381 -.0381 -.038 -.038 -.0022 -.0367 -.0367 -.0367 -.0022 .0034 -.0357 -.0357 -.0355 -.0355 .0034 -.0349 -.0349 -.0349 .0034 .0018 -.0351 -.0351 -.0348 -.0348 .0018 -.0345 -.0345 -.0345 .0018 -.0008 -.0363 -.0363 -.0362 -.0362 -.0008 -.036 -.036 -.036 -.0008 , 5 = -00427, S 0 1-4 -.0016 -.0334 -.0335 -.0328 -.0328 .0067 -.0237 -.0283 -.0283 -.01 .003 -.0323 -.0324 -.0321 -.0321 .0159 -.0239 -.0306 -.0306 -.01 .0213 -.0333 -.0338 -.0329 -.0334 .0484 -.0164 -.0325 -.0325 -.0032 .0607 -.0274 -.0279 -.0274 -.0279 .1261 .0107 -.0265 -.0265 .0157 .1236 -.0226 -.0239 -.0218 -.0231 .2939 .0821 -.0215 -.0215 .0397 1-6 -.0146 -.0397 -.0397 -.0395 -.0395 .0015 -.0294 -.036 -.036 -.0227 -.0062 -.0388 -.0388 -.0387 -.0387 .0217 -.0274 -.0373 -.0373 -.0203 .014 -.0368 -.0369 -.0368 -.0369 .0676 -.0134 -.0367 -.0367 -.0122 .0507 -.0341 -.0346 -.034 -.0344 .191 .0255 -.0332 -.0332 .0099 .0967 -.0305 -.0314 -.0305 -.0313 .4304 .15 -.0302 -.0302 .0359 1-8 -.0288 -.045 -.045 -.0447 -.0447 -.0089 -.0363 -.0436 -.0436 -.036 -.0191 -.0433 -.0434 -.0431 -.0432 .011 -.031 -.042 -.042 -.0324 -.009 -.0437 -.0439 -.0437 -.0438 .0661 -.0175 -.0431 -.0431 -.0274 .013 -.0405 -.0407 -.0404 -.0405 .2089 .0296 -.0403 -.0403 -.0133 .0388 -.0405 -.0411 -.04 -.0409 .467 .1869 -.0397 -.0397 .0087 CRII M 3-2 .0241 -.026 -.0264 -.0254 -.0258 .0368 -.0102 -.0192 -.0192 .0079 .075 -.0202 -.0204 -.0198 -.0202 .1072 .015 -.0165 -.0165 .0286 .1804 -.0134 -.0152 -.0127 -.0146 .3025 .1265 -.0103 -.0103 .0421 .2611 -.0068 -.0122 -.0056 -.0118 .5175 .3261 -.0034 -.0034 .0369 .2864 -.0014 -.0109 -.0002 -.0096 .6008 .4584 .0003 .0003 .0099 = .00562, S 3-4 .0365 -.0254 -.0256 -.0247 -.0249 .0776 .003 -.019 -.019 .0129 .1008 -.023 -.0238 -.0226 -.0233 .2281 .059 -.0185 -.0185 .0404 .1781 -.0198 -.0243 -.0197 -.024 .5932 .3195 -.0179 -.0179 .0635 .2008 -.0117 -.0214 -.0109 -.0209 .7539 .642 -.0098 -.0098 .0529 .208 -.0047 -.0205 -.0037 -.02 .7767 .709 -.0032 -.0032 .016 CRIT M L 3-6 3-8 .0235 -.0333 -.0334 -.0328 -.0329 .1221 .0103 -.0282 -.0282 .0044 .0696 -.0319 -.0326 -.0317 -.0323 .3318 .1116 -.029 -.029 .0323 .1132 -.029 -.0331 -.0286 -.033 .6997 .4632 -.0269 -.0269 .0665 .1219 -.0177 -.0307 -.0172 -.0306 .8205 .7601 -.0164 -.0164 .049 .126 -.0058 -.0287 -.0052 -.0283 .8346 .7962 -.0047 -.0047 .02 -.002 -.0403 -.0406 -.0401 -.0405 .1345 .0107 -.0384 -.0384 -.0143 .0249 -.041 -.0416 -.0403 -.0412 .3712 .1424 -.0388 -.0388 .0041 .0391 -.0364 -.0407 -.0362 -.0403 .7262 .5628 -.0351 -.0351 .0282 .0457 -.0237 -.0397 -.0232 -.0396 .8058 .7832 -.0228 -.0228 .03 .0466 -.0071 -.0402 -.0064 -.0399 .8156 .7972 -.0064 -.0064 .0062 = .00717) 5-2 .0938 -.0163 -.0169 -.0157 -.0162 .1396 .041 -.0048 -.0048 .0369 .2054 -.0108 -.0139 -.0103 -.0134 .3559 .1717 -.0027 -.0027 .0545 .2646 -.0056 -.0125 -.0045 -.012 .5621 .4078 -.0008 -.0008 .0333 .2671 -.0019 -.0112 -.0008 -.0105 .5746 .4414 .0013 .0013 .0168 .2721 -.0017 -.0107 -.0005 -.0097 .5766 .4433 .0002 .0002 .0161 5-4 .1213 -.0209 -.0222 -.0204 -.0216 .3014 .1056 -.0095 -.0095 .0532 .1809 -.0165 -.0209 -.0158 -.0201 .6186 .3864 -.0089 -.0089 .0722 .1817 -.0097 -.0226 -.0087 -.0223 .7457 .6679 -.0067 -.0067 .0373 .1903 -.0068 -.0241 -.0057 -.0237 .7459 .6761 -.0043 -.0043 .0228 .1891 -.0051 -.0216 -.0043 -.021 .7483 .6817 -.0038 -.0038 .0253 5-6 5-8 .0696 -.0292 -.0303 -.0289 -.0299 .4012 .1621 -.0225 -.0225 .0492 .1022 -.0269 -.0326 -.0265 -.032 .6928 .5018 -.0217 -.0217 .0706 .1006 -.0129 -.0328 -.0123 -.0326 .79 .7436 -.01 -.01 .0395 .102 -.0063 -.032 -.0056 -.0317 .7957 .7579 -.0052 -.0052 .0303 .1097 -.0056 -.0314 -.0047 -.0309 .8035 .7611 -.004 -.004 .0305 .0188 -.0394 -.0412 -.0393 -.0408 .4674 .2258 -.0351 -.0351 .0078 .0382 -.0307 -.0403 -.0303 -.0402 .7467 .6093 -.0281 -.0281 .0251 .0384 -.0177 -.0397 -.0169 -.0394 .7921 .7696 -.016 -.016 .0202 .0398 -.0042 -.0398 -.0034 -.0397 .795 .7737 -.0032 -.0032 .0112 .0396 -.0041 -.0403 -.0034 -.04 .7934 .7739 -.0032 -.0032 .0111 172 Table 11.4 SEX f ° r t e s t s o n correlations with no preliminary omnibus test when P * I, p = 10 across N:p x mNz x pNz(8 [ cr N:p NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 1-2 .7949 -.0122 -.0122 -.0116 -.0117 .3451 .0129 .0052 .0052 .0107 .7898 -.0114 -.0115 -.0107 -.0108 .4048 .0081 -.002 -.002 .0224 .7912 -.0115 -.0118 -.0106 -.0107 .5451 .023 -.0089 -.0089 .0742 .787 -.0074 -.0077 -.0064 -.0065 .7174 .1145 -.005 -.005 .1555 .7915 -.0093 -.01 -.0086 -.0094 .8854 .424 -.0079 -.0079 .2371 1-4 .7012 -.0215 -.0215 -.0204 -.0206 .3538 .0045 -.0095 -.0095 .0086 .702 -.0208 -.0208 -.0203 -.0204 .4692 .01 -.0145 -.0145 .0403 .7034 -.0214 -.0215 -.0209 -.021 .6964 .0652 -.0192 -.0192 .1424 .6931 -.0176 -.0184 -.017 -.0175 .8853 .2818 -.0163 -.0163 .2839 .6939 -.018 -.0196 -.0172 -.0191 .9475 .7533 -.0169 -.0169 .3964 t .05 = .00427, t 5 j . o i = .00562, t 5 j .001 = -00717) c r 1-6 .5471 -.0313 -.0313 -.0308 -.0309 .3534 -.0041 -.0236 -.0236 .0035 .554 -.0308 -.0308 -.0307 -.0307 .4949 .0085 -.0269 -.0269 .0396 5489 -.0313 -.0313 -.031 -.0311 .7834 .1193 -.03 -.03 .1687 .5458 -.0277 -.0281 -.0272 -.0277 .9314 .4348 -.0266 -.0266 .3213 .5408 -.0283 -.0298 -.0279 -.0292 .9499 .8645 -.0277 -.0277 .4213 1-8 .3028 -.0393 -.0393 -.039 -.039 .3067 -.0159 -.0358 -.0358 -.0104 .3137 -.0392 -.0393 -.039 -.039 .4853 .0069 -.0369 -.0369 .0151 .3155 -.0395 -.0396 -.0392 -.0393 .8117 .1703 -.0389 -.0389 .1206 .3187 -.039 -.0393 -.0389 -.0391 .9392 .5478 -.0388 -.0388 .24 .3166 -.0393 -.04 -.039 -.0397 .95 .9074 -.039 -.039 .3292 3-2 .7829 -.0119 -.0119 -.0108 -.0111 .6436 .0987 .0054 .0054 .1441 .7746 -.0101 -.0106 -.0092 -.0099 .8394 .3441 -.0021 -.0021 .2271 .7783 -.0075 -.012 -.0061 -.0105 .9405 .8842 -.0035 -.0035 .1997 .7721 -.0014 -.0098 .0002 -.0086 .9448 .9394 .0017 .0017 .0668 .7754 -.001 -.0111 .0002 -.01 .9446 .9392 .0008 .0008 .0163 t 3-4 .6734 -.023 -.0234 -.0217 -.0221 .8025 .2097 -.0104 -.0104 .2449 .6712 -.0208 -.0216 -.0202 -.0212 .9339 .6051 -.0143 -.0143 .3685 .6763 -.016 -.0215 -.0148 -.0206 .95 .9454 -.0124 -.0124 .2959 .667 -.0045 -.0201 -.0036 -.0192 .95 .95 -.0022 -.0022 .0948 .6708 -.0003 -.0203 .0008 -.0195 .95 .9499 .0014 .0014 .0308 c r 3-6 .5105 -.0324 -.0326 -.0317 -.032 .8552 .3159 -.0223 -.0223 .2629 .5093 -.0298 -.0312 -.0291 -.0307 .9441 .7247 -.0256 -.0256 .3925 .5051 -.0231 -.0309 -.0224 -.0304 .9499 .9481 -.0201 -.0201 .3016 .5064 -.0062 -.0307 -.0049 -.0304 .95 .95 -.0038 -.0038 .1005 .508 -.0039 -.0299 -.0025 -.0296 .95 .95 -.0016 -.0016 .0367 3-8 .2673 -.04 -.04 -.0397 -.0398 .8677 .3893 -.0354 -.0354 .1832 .2756 -.0397 -.0402 -.0392 -.0399 .9466 .7873 -.0377 -.0377 .2853 .2747 -.0329 -.0395 -.0324 -.0394 .9499 .9493 -.0313 -.0313 .2303 .2749 -.0109 -.0406 -.0102 -.0406 .95 .95 -.0092 -.0092 .0787 .2771 -.0038 -.0407 -.0027 -.0405 .9499 .9499 -.0025 -.0025 .0346 t 5-2 .7544 -.0098 -.0109, -.0089 -.0101 .8903 .556 .0068 .0068 .26 .7457 -.0076 -.0117 -.0074 -.011 .9388 .9018 1E-04 1E-04 .1713 .7434 -.0033 -.0122 -.0018 -.011 .9396 .9318 .0013 .0013 .0492 .7406 -.0036 -.0117 -.0025 -.0105 .9412 .9335 -.0008 -.0008 .041 .7416 -.0027 -.0111 -.0017 -.0105 .9408 .934 -.0009 -.0009 .0406 5-4 .6247 -.0223 -.0238 -.0216 -.0234 .9451 .7853 -.0094 -.0094 .3598 .6188 -.0146 -.0226 -.0139 -.0219 .9494 .9472 -.0079 -.0079 .2309 .6213 -.0061 -.022 -.005 -.0215 .9496 .9493 -.0025 -.0025 .0664 .6133 -.0064 -.0226 -.0052 -.0214 .9496 .9496 -.0031 -.0031 .0582 .6107 -.0039 -.0213 -.0025 -.0212 .9496 .9494 -.0021 -.0021 .0637 5-6 .4483 -.0313 -.0327 -.0305 -.0326 .9483 .8402 -.0211 -.0211 .3509 .4507 -.0224 -.0322 -.022 -.0319 .95 .9484 -.0173 -.0173 .2307 .4401 -.0088 -.0318 -.0081 -.0315 .95 .9499 -.0067 -.0067 .07 .4359 -.0068 -.0308 -.006 -.0301 .9499 .9499 -.0052 -.0052 .0698 .4415 -.0085 -.0322 -.0076 -.0322 .95 .9499 -.0074 -.0074 .0681 5-8 .2157 -.0382 -.0397 -.0378 -.0396 .9486 .8712 -.0346 -.0346 .2492 .2227 -.0316 -.0419 -.0306 -.0418 .9499 .9492 -.0271 -.0271 .1686 .221 -.0084 -.0414 -.0077 -.0412 .9497 .9497 -.0063 -.0063 .0708 .2147 -.0098 -.0424 -.0091 -.0421 .95 .9498 -.0085 -.0085 .061 .7416 -.0101 -.042 -.0096 -.0419 .9499 .9499 -.0092 -.0092 .0651 173 Table 11.5 SEX f ° tests on correlations with a preliminary omnibus test when P * to I, p = 10 r across N:p x mNz x pNz (5 j 05 = .00427, <5 j .01 = .00562, i5 j .001 = -00717) cr N:p NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS NT MB RB MD RD OR OS OF CF CS 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 4040 40 40 40 1-2 .0287 -.0358 -.0358 -.0357 -.0358 .0287 -.0269 -.0322 -.0322 .0107 .0551 -.0355 -.0356 -.0352 -.0353 .0561 -.0255 -.0321 -.0321 .0224 .1746 -.0269 -.0272 -.0263 -.0264 .1795 .0022 -.0255 -.0255 .0742 .4068 -.0164 -.0167 -.0156 -.0157 .4363 .1003 -.0143 -.0143 .1555 .7098 -.0101 -.0108 -.0094 -.0102 .8095 .4205 -.0087 -.0087 .2371 1-4 .0536 -.0383 -.0383 -.0379 -.0381 .0579 -.0245 -.0355 -.0355 .0086 .1226 -.0367 -.0367 -.0366 -.0366 .1362 -.0136 -.0344 -.0344 .0403 .3595 -.0262 -.0263 -.026 -.0261 .4236 .0563 -.0248 -.0248 .1424 .6111 -.0187 -.0195 -.0181 -.0186 .7918 .2797 -.0174 -.0174 .2839 .6927 -.018 -.0196 -.0172 -.0191 .9456 .7533 -.0169 -.0169 .3964 t 1-6 .0676 -.0418 -.0418 -.0416 -.0417 .0919 -.0237 -.039 -.039 .0035 .1555 -.0397 -.0397 -.0396 -.0396 .2181 -.0052 -.0375 -.0375 .0396 .3976 -.0331 -.0331 -.033 -.0331 .6135 .1157 -.0323 -.0323 .1687 .5311 -.0278 -.0282 -.0273 -.0278 .9037 .4346 -.0267 -.0267 .3213 .5408 -.0283 -.0298 -.0279 -.0292 .9498 .8645 -.0277 -.0277 .4213 cr t cr t 1-8 3-2 3-4 3-6 3-8 5-2 5-4 5-6 5-8 .0427 -.0446 -.0446 -.0445 -.0445 .1193 -.026 -.0433 -.0433 -.0104 .115 -.0437 -.0438 -.0435 -.0435 .2848 0 -.0423 -.0423 .0151 .2665 -.0399 -.04 -.0396 -.0397 .7218 .1698 -.0393 -.0393 .1206 .3166 -.039 -.0393 -.0389 -.0391 .9324 .5478 -.0388 -.0388 .24 .3166 -.0393 -.04 -.039 -.0397 .95 .9074 -.039 -.039 .3292 .3195 -.0206 -.0206 -.0201 -.0204 .3437 .0785 -.01 -.01 .1441 .6085 -.0128 -.0133 -.012 -.0127 .7014 .3385 -.0056 -.0056 .2271 .777 -.0075 -.012 -.0061 -.0105 .939 .8842 -.0035 -.0035 .1997 .7721 -.0014 -.0098 .0002 -.0086 .9448 .9394 .0017 .0017 .0668 .7754 -.001 -.0111 .0002 -.01 .9446 .9392 .0008 .0008 .0163 .4871 -.0253 -.0257 -.0241 -.0245 .6324 .2043 -.0137 -.0137 .2449 .6518 -.0209 -.0217 -.0203 -.0213 .9114 .6048 -.0145 -.0145 .3685 .6763 -.016 -.0215 -.0148 -.0206 .95 .9454 -.0124 -.0124 .2959 .667 -.0045 -.0201 -.0036 -.0192 .95 .95 -.0022 -.0022 .0948 .6708 -.0003 -.0203 .0008 -.0195 .95 .9499 .0014 .0014 .0308 .43 -.0335 -.0337 -.0328 -.0331 .7608 .3138 -.024 -.024 .2629 .5055 -.0298 -.0312 -.0291 -.0307 .9389 .7247 -.0256 -.0256 .3925 .5051 -.0231 -.0309 -.0224 -.0304 .9499 .9481 -.0201 -.0201 .3016 .5064 -.0062 -.0307 -.0049 -.0304 .95 .95 -.0038 -.0038 .1005 .508 -.0039 -.0299 -.0025 -.0296 .95 .95 -.0016 -.0016 .0367 .2386 -.0403 -.0403 -.04 -.0401 .8176 .3887 -.0357 -.0357 .1832 .2745 -.0397 -.0402 -.0392 -.0399 .9444 .7872 -.0377 -.0377 .2853 .2747 -.0329 -.0395 -.0324 -.0394 .9499 .9493 -.0313 -.0313 .2303 .2749 -.0109 -.0406 -.0102 -.0406 .95 .95 -.0092 -.0092 .0787 .2771 -.0038 -.0407 -.0027 -.0405 .9499 .9499 -.0025 -.0025 .0346 .6864 -.0111 -.0122 -.0102 -.0114 .8276 .553 .0049 .0049 .26 .7452 -.0076 -.0117 -.0074 -.011 .9385 .9018 1E-04 1E-04 .1713 .7434 -.0033 -.0122 -.0018 -.011 .9396 .9318 .0013 .0013 .0492 .7406 -.0036 -.0117 -.0025 -.0105 .9412 .9335 -.0008 -.0008 .041 .7416 -.0027 -.0111 -.0017 -.0105 .9408 .934 -.0009 -.0009 .0406 .6198 -.0223 -.0238 -.0216 -.0234 .9377 .7852 -.0095 -.0095 .3598 .6188 -.0146 -.0226 -.0139 -.0219 .9494 .9472 -.0079 -.0079 .2309 .6213 -.0061 -.022 -.005 -.0215 .9496 .9493 -.0025 -.0025 .0664 .6133 -.0064 -.0226 -.0052 -.0214 .9496 .9496 -.0031 -.0031 .0582 .6107 -.0039 -.0213 -.0025 -.0212 .9496 .9494 -.0021 -.0021 .0637 .4467 -.0313 -.0327 -.0305 -.0326 .9466 .8402 -.0211 -.0211 .3509 .4507 -.0224 -.0322 -.022 -.0319 .95 .9484 -.0173 -.0173 .2307 .4401 -.0088 -.0318 -.0081 -.0315 .95 .9499 -.0067 -.0067 .07 .4359 -.0068 -.0308 -.006 -.0301 .9499 .9499 -.0052 -.0052 .0698 .4415 -.0085 -.0322 -.0076 -.0322 .95 .9499 -.0074 -.0074 .0681 .2154 -.0382 -.0397 -.0378 -.0396 .9477 .8712 -.0346 -.0346 .2492 .2227 -.0316 -.0419 -.0306 -.0418 .9499 .9492 -.0271 -.0271 .1686 .221 -.0084 -.0414 -.0077 -.0412 .9497 .9497 -.0063 -.0063 .0708 .2147 -.0098 -.0424 -.0091 -.0421 .95 .9498 -.0085 -.0085 .061 .2183 -.0101 -.042 -.0096 -.0419 .9499 .9499 -.0092 -.0092 .0651 174 Table 12.1 Table of parametri
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A comprehensive examination of procedures for testing the significance of a correlation matrix and its… Fouladi, Rachel Tanya 1991
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Title | A comprehensive examination of procedures for testing the significance of a correlation matrix and its elements |
Creator |
Fouladi, Rachel Tanya |
Publisher | University of British Columbia |
Date Issued | 1991 |
Description | Correlational techniques are important tools in multivariate behavioural and social science exploratory research. A wide array of procedures have been proposed for testing (a) whether any of the variables are related, and (b) which variables are related. In the current study, the performance of the procedures currently available for testing these distinct questions is assessed on the primary Neyman-Pearson criterion for an optimal test. According to this criterion, an optimal procedure is the most powerful procedure that controls experimentwise Type I error rate at or below the nominal level. The findings of the first part of this study addressing how to test complete multivariate independence suggest that the statistic traditionally used (QBA) is not the optimal test, and that one of several recently derived statistics (QSE> QSA> QF) should be used. Computational efficiency of the procedures is also considered with the resulting recommendation of the use of QSA- The second part of this study addresses how to test which variables are correlated; the findings suggest the use of a multi-stage order statistics approach with z-tests (CF). The conditions necessary to ensure maximal power when addressing these questions are also considered. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-01-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0098755 |
URI | http://hdl.handle.net/2429/30565 |
Degree |
Master of Arts - MA |
Program |
Psychology |
Affiliation |
Arts, Faculty of Psychology, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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