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Um, 3%" ’11:”) .r‘q» Q” o . >5 “E‘s. ‘ ~ . ‘ lg g 1% Vt as e; b llllllml‘lll Ml HlMH M 3 1293 006004 LIBRARY Michigan State University This is to certify that the dissertation entitled An Empirical Study of the Type I Error Rate and Power for Some Selected Normal-Theory and Nonparametric Tests of the Independence of Two Sets of Variables presented by Abdul Razak Habib has been accepted towards fulfillment of the requirements for Ph. D. Education degree in fiL/E £74”. )fl: /z.__. a 101' professor Date August 10th, 1988 MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 _._.._—————u >~ . PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE [JUL 2 319;), ‘ :70“ S, (972. i '¥ —. 12;) 4mm ‘ i j! W I .J- mewz‘er/J l ~ «I?! 3:39 1:3; T MSU Is An Affirmative Halon/Equal Opportunity Institution AN EMPIRICAL STUDY OF THE TYPE I ERROR RATE AND POWER FOR SOME SELECTED NORMAL-THEORY AND NONPARAMETRIC TESTS OF THE INDEPENDENCE OF TWO SETS OF VARIABLES By Abdul Razak Habib A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Counseling, Educational Psychology and Special Education 1988 /" “3(2) 51‘) {Cf ,/ J" \— ABSTRACT AN EMPIRICAL STUDY OF THE TYPE I ERROR RATE AND POWER FOR SOME SELECTED NORMAL-THEORY AND NONPARAMETRIC TESTS OF THE INDEPENDENCE OF TWO SETS OF VARIABLES By Abdul Razak Habib The present study empirically examined the effect of non- normality, sample size, number of variables, and degree of dependency on the Type I error and power properties of five normal-theory and nonparametric tests of' the independence of’ two sets of ‘variables. Simulated data representing light-, moderately heavy-, and heavy-tailed distributions, three sample sizes, three sets of correlations-among- variables, and three sets of numbers-of-variables were included. This study yielded the following results. The Type I error rates of the normal-theory Bartlett and Rao E tests increase subtantially for the moderately heavy- and heavy-tailed distributions, whereas the Type I error rates of the nonparametric rank-transfonm Rao E and the pure- and mixed-rank tests are not affected by the form of a parent distribution. for' moderately-small and. moderately-large samples. The Type I error rates of the Bartlett and Rao E tests increase with increases in the correlation among predictor and/or dependent variables, and with increases in the number-of-variables for heavy- tailed distributions. The Type I error rate of the rank- transform Rao E test is not affected by the within-set-correlation and the number-of- variables factors, while those of the pure- and mixed-rank tests are not affected by the within-set-correlation factor but decrease as the number of variables increases for all distributions. The power values of the normal-theory Bartlett and Rao E tests increase subtantially only for extremely heavy-tailed distributions. The power values of all three nonparametric tests increase with increases in the kurtosis values. The power values of all five tests increase with increases in the sample size and the correlation among the predictor variables, and decrease with increases in the correlation among the dependent variables for all distributions. The increments due to the sample size are higher for the three nonparametric tests. The power values of the Bartlett, Rao E, and the rank-transform Rao E tests are not affected by the number of variables, while those of the pure- and mixed-rank tests decrease as the number of variables increases for all distributions. However, the reduction in the power values tends to be compensated for by increases in the sample size. To Faisal and Hilary iv ACKNOWLEDGEMENTS I wish to express my appreciation to all the individuals who made the completion of this program possible. To the members of my doctoral committee: Dr. Betsy J. Becker (Chairperson), Dr. Dennis C. Gilliland, Dr. Michael R. Harwell, Dr. Perry E. Lanier, and Dr. Stephen W. Raudenbush for their constant encouragement and valuable suggestions and comments. I wish to extend my appreciation to the Vice Chancellor of the Malaysian National University (MNU) for granting the study leave; to members of the Faculty of Education at MNU for carrying extra work during my absence; to the Chairpersons (past and present) and faculty members of the Department of Counseling, Educational Psychology, and Special Education at Michigan State University (MSU) for hiring me as a teaching and research assistant, research consultant, and instructor; to the President of the Sage Foundation and the MSU Coordinator of the Sage Foundation Fund for awarding me a dissertation grant; and to all friends who supported me financially and emotionally during my struggle to complete this program. Finally, I wish to express my deep gratitude to members of my family: Pn. Rafeah, En. Habib, Pn. Embun, Pn. Fatimah, Pn. Maimun, Ad. Mahadzir, Ad. Yahya, Ad. Rohani, Ad. Bashah, Ad. Basuri, Ad. Nazri, Ad. Salmah, Ad. Rokiah, Ad. Habibah, Ad. Mohammad Nasir, An. Faisal, and An. Hilmy for their continued love. May God blesses everyone. TABLE OF CONTENT LIST OF TABLES LIST OF FIGURES CHAPTER I. STATEMENT OF THE PROBLEM . Role of Multivariate Analysis in Educational Research Normal-Theory and Nonparametric-Multivariate Tests in Educational Research Purpose of the Study . Factors Which Influence the Choice of a Normal-Theory or Nonparametric-Multivariate Test . . . . . . Research Questions and Hypotheses Role of Simulation in Distributional Studies . Significance of the Study Limitations of the Study . II. REVIEW OF THE LITERATURE . Comparing the Normal Distribution and Some Non-normal Distributions Methods of Dealing With Non-Normal Data Nonparametric Methods Rank-Transform Methods . Robustness and Power Results for Some Normal-Theory and Nonparametric-Multivariate Tests . Analytic Results - Univariate Case . Empirical Results - Univariate Case vi Page ix xi 10 13 13 15 16 16 20 20 22 23 24 25 Empirical Results - Normal-Theory-Mu1tivariate Case Analytic Results - Nonparametric-Mu1tivariate Case . Empirical Results - Nonparametric-Mu1tivariate Case Dependency Among Variables . Summary III . METHODOLOGY Multivariate Statistical Models Test Statistics and Their Assumptions Bartlett Statistic . Rao E Statistic Rank-Transform Rao E Statistic . Pure-Rank Statistic Mixed-Rank Statistic . Data Generation Simulation Conditions Presentation of Simulation Results . IV. RESULTS Characteristics of the Generated Data Type I Error and Power Conditions Overall Type I Error and Power Results . Main Effects of Simulation Conditions . Distribution . Sample Size Within-Set Correlation . Number-of-Variables vii 28 30 31 35 35 37 37 41 42 43 44 44 47 48 52 54 S7 57 62 63 66 66 69 72 74 Interaction Effects of Simulation Conditions . Distribution and Sample Size Distribution and Within-Set Correlation . Distribution and Number-of-Variables Sample Size, Within-Set Correlation, and Number-of- Variables . . . . . . . . . . . . V. SUMMARY Research Questions . Methodology Findings . Conclusions Implications for Data Analysis in Educational Research . Recommendations for Further Study . APPENDICES A. Definition of Terms B. Procedures and Algorithms C. Pure-Rank Statistic D. Computer Programming . E. Tables . BIBLIOGRAPHY . viii 76 76 81 85 89 92 92 92 94 96 98 . 100 . 103 . 103 . 105 . 113 . 117 . 127 . 150 Table 1. Skewness and Kurtosis Values of Some Univariate Distributions . 2. Multivariate Skewness and Kurtosis Values of Some Bivariate Distributions 3. Asymptotic Relative Efficiencies of Some Nonparametric and Normal-Thery Tests of Location . . . . . 4. Simulation Factors . 5. Average Mean, Variance, Skewness, Kurtosis, and Within- Set Correlations of the Simulated Data . 6. Sample Measures of Multivariate Skewness and Kurtosis For the Normal Distribution 7. Regression Coefficients Used for Power Simulations and the Resulting Between-Set Correlations . 8. Overall Empirical Type I Error and Power Values 9. Overall Number of Conservative and Liberal Type I Errors . 10. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Distribution . 11. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Sample Size 12. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Within-Set Correlation . . 13. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Number-of-Variables. 14. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Distribution and Sample Size 15. Average Type I Error and Power Values and Number of LIST OF TABLES Conservative and Liberal Error Rates by Distribution and Within-Set Correlation ix Page 18 19 24 52 59 61 62 64 66 67 70 72 74 78 82 16. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Distribution and Number-of-Variables . . . . . . . . . . . . . . . . . 86 17. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Sample Size and and Within-Set Variables For Rao E Test . . . . . . . . . 89 18. Average Type I Error and Power Values and Number of Conservative and Liberal Error Rates by Sample Size and Number-of-Variables For Rao E and Pure- and Mixed-Rank Tests . . . . . . . . . . . . . . . . . . . . .. . . . . . 90 LIST OF FIGURES Figure Page 1. Frequency-Distribution Curves of Simulated Data. . . . . . . 58 2. Type I Error and Power Curves by Alpha Level . . . . . . . . 64 3. Type I Error and Power Curves by Distribution . . . . . . . 68 4. Type I Error and Power Curves by Sample Size . . . . . . . . 71 5. Type I Error and Power Curves by Within-Set Correlation . . 73 6. Type I Error and Power Curves by Number-of-Variables . . . . 75 7. Type I Error Curves by Distribution and Sample Size . . . . 79 8. Power Curves by Distribution and Sample Size . . . . . . . . 80 9. Type I Error Curves by Distribution and Within-Set Correlation . . . . . . . . . . . . . . . 83 10. Power Curves by Distribution and Within-Set Correlation . . 84 11. Type I Error Curves by Distribution and Number-of-Variables 87 12. Power Curves by Distribution and Number-of-Variables. . . . 88 xi CHAPTER I' STATEMENT OF THE PROBLEM The present study used computer-simulated data to assess the distributional behavior (i.e., Type I error rate and power) of selected normal-theory and nonparametric tests of the independence of two sets of variables. The focus of the investigation was the behavior of the tests in, the 'presence of’ non-normal skewness and. kurtosis values. This chapter discusses the (a) role of multivariate analysis in educational research, (b) normal-theory and nonparametric-mu1tivariate tests in educational research, (c) purpose of the study, (d) factors which influence the choice of a normal-theory or nonparametric- multivariate test, (e) research questions and hypotheses, (f) role of simulation in distributional studies, (g) significance of the study, and (h) limitations of the study. The definitions of some statistical terms are given in Appendix A. va 1 Educ na esea Multivariate analysis refers to a collection of descriptive and inferential methods that have been developed for situations where one or more sets of correlated variables are treated as outcome measures, predictors, or both (Harris, 1975, p. 5). More specifically, multivariate methods allow researchers to simultaneously analyze the interrelationships among many variables. In contrast, univariate analyses are carried out separately for each outcome variable. One potential shortcoming of univariate methods is that they may lead to an 2 incomplete description of the data since they ignore interrelationships among predictor and outcome variables. Multivariate methods have found widespread use in educational research. A. primary reason for their popularity is the interest educational researchers show in testing theories that are multivariate in character, which implies the use of multiple variables. Because these variables are chosen to be consistent with the theory under test, they form a multidimensional system and are expected to be correlated (Takeuchi, Yanai, & Mukherjee, 1982, p. 54). Testing theories by collecting data on several variables leads quite naturally to multivariate data-analytic methods. Among the inferential multivariate methods used in educational research are multivariate analysis of variance (MANOVA), factor analysis, discriminant analysis, canonical-correlation analysis, and multivariate-multiple-regression. These methods have served as important explanatory tools for researchers attempting to summarize the information in a data set containing multiple (correlated) outcome variables. As noted above, many studies in education involve the analysis of relationships between multiple outcome and predictor variables. Canonical-correlation analysis and multivariate-multip1e-regression represent general data-analytic methods that may be used to study such relationships. The fundamental difference between the two approaches lies in the nature of the measured relationship. Canonical-correlation analysis assesses the degree of relationship among two sets of random variables (Takeuchi et a1., 1982, p. 225). Although researchers often 3 refer to one set of variables as predictors and the other as outcomes, the mathematical model underlying canonical correlation makes no such distinction (Gittins, 1985, p. 19). The multivariate-multiple-regression model, on the other hand, simultaneously assesses the degree of relationship between each of the random outcome variables and the set of fixed and known predictor variable values (Takeuchi et a1., 1982, p. 116). However, predictors are rarely fixed and known in practice and regression analysis is routinely performed for predictors that, in essence, are random variables. An important consequence of this practice is that data- analytic inferences are limited to predictor values appearing in the sample (Rogosa, 1980). As an example of the differing applications of canonical- correlation and multivariate-mu1tip1e-regression in educational research, consider a study of the relationship between school organizational climate and teacher job satisfaction. Two well known instruments in this area are the Teacher Job Satisfaction Questionaire (Lester, 1983), which measures nine identified factors of teacher job satisfaction, and the Organizational Climate Description Questionaire (Kottkamp, Mohlern, & Hay, 1985), which measures five dimensions of organizational climate. If the research question focuses on the interdependence between these two sets of (random) ‘variables, the relationship between the job satisfaction. factors and the organizational climate dimensions is most properly examined using canonical-correlation analysis. If one set of variables is conceptualized as outcomes and the other as predictors, then 4 multivariate-multiple-regression would be appropriate. It is important to emphasize that these models are conceptually different, yet are identical with respect to making, statistical inferences about the relationship between two sets of variables. Both of these procedures are important explanatory tools for educational researchers interested in testing theories that are multivariate in character. “MW Historically, researchers opting for multivariate methods have been confronted with the problem of fitting the observed data into the framework of multivariate-normal-theory procedures. Such methods have collectively been labelled parametric, and are identified by their reliance on the assumption that the population distribution of observations follows a multivariate-normal density function (Puri & Sen, 1971, p. 1). Yet in many data-analytic situations there is little doubt that the observations can be characterized as moderately or even distinctly non-normal (Puri & Sen, 1971, p. 1). Under the assumption of random sampling from a specified population, this casts doubt on the normality of the population distribution. For example, educationally- oriented variables such as the number of days absent from school are likely to produce (non-normal) data that are badly skewed or slightly or heavily kurtic. One approach for dealing with non-normality is to transform the original data to a form more closely resembling a normal distribution (see Box 6: Cox, 1964) and then employ normal-theory methods. This requires that the underlying distribution of the original variable(s) 5 be known before deciding which transformation is most appropriate. In many cases, the distribution of the original variable is not known, and transformations of this type may be problematic (Kendall & Stuart, 1969, V. 2, p. 487). Another issue is that the transformed variable may not be interpretable. A second approach is to transform the original data to ranks (or some other' monotonic transformation) and then. employ ‘nonparametric methods. These methods do not require that the form of the underlying distribution be known, and are characterized by their relaxation of the normality assumption. However, the underlying distribution must be continuous (Kendall & Stuart, 1969, V. 2, p. 487). In surveying the literature, a number of nonparametric alternatives to normal-theory, univariate methods are available (e.g., Conover, 1980; Gibbons, 1971; MarascuiLo & Mwaeeney, 1977). This is not true for the multivariate case, where nonparametric alternatives to normal-theory multivariate methods exist only in certain areas of statistical inference. A primary source of the development of nonparametric-multivariate methods is the work of Puri and Sen (1969, 1971, 1985). Of special importance are the tests these authors generated for hypotheses subsumed under the multivariate general linear model. One is the pure- rank procedure, in which values of all variables are ranked prior to any analysis, and the other is the mixed-rank procedure, in which some but not all variables are ranked. Another nonparametric approach that is closely related to classical nonparametric methods is the rank- transform procedure due to Iman and Conover (Conover & Iman, 1981; 6 Iman, 1974b; Iman & Conover, 1979). This procedure, which likewise does not require a normality assumption, involves transforming the original data to their corresponding ranks and then applying normal- theory procedures. The pure- and mixed-rank procedures as illustrated by Puri and Sen, and the rank-transform procedure of Conover and Iman, represent the primary nonparametric alternatives to normal-theory multivariate analysis. The purpose of the present study was to compare the Type I error and power properties of two normal-theory (Bartlett, Rao E) and three nonparametric (rank-transform Rao E, pure-and mixed-rank) tests of the independence of two sets of variables (i.e., tests of no canonical correlation or In) regression). The distributional properties of these tests hold exactly only for the asymptotic case (i.e., for very large samples and/or a parent normal distribution), and hence the focus was on the behavior of these tests for small samples under a variety of non-normal skewness and kurtosis conditions. Other factors examined included the sample size, the correlation within the set of predictors and within the set of dependent variables, the correlation among the sets of predictor and dependent variables, and the number of variables. Since the effects of such factors are difficult to evaluate analytically (Ito & Schull, 1964; Zwick, 1984, p. 2), a simulation study was performed to investigate the behavior of the tests. It is anticipated that the results of the present study will provide educational researchers with guidelines for choosing 7 between normal-theory tests of the hypothesis of no relationship among two sets of variables and their nonparametric counterparts under a variety of non-normal data and sample-size conditions. Wh e o N a - a - t s As a result of theoretical and computational advances a variety of normal-theory and nonparametric multivariate tests are available to educational researchers. The question arises of how best to choose among the two kinds of tests. The application of a normal-theory procedure to test a statistical hypothesis requires some statistical assumptions on the observations and the population distribution. For example, the omnibus test that all population regression coefficients equal zero in multivariate-multiple-regression assumes that (a) the population of outcomes, conditional on the predictors, is normally distributed, (b) the outcomes, conditional on the predictors, have a common covariance matrix, and (c) the residuals for a given outcome variable are independent. Violations of one or more of these assumptions have been shown to have adverse effects on the Type I error probability and power of normal-theory multivariate tests under a variety of data conditions (Ito 6: Schull, 1964; Mardia, 1971; Olson, 1976). Thus, the use of a normal-theory test when assumptions are violated may lead to an incorrect conclusion. For example, a true statistical hypothesis may be rejected, not because the statistical hypothesis is false but because one or more of the underlying statistical assumptions are violated (Conover, 1980, p. 84). Thus, the effect of violating underlying 8 statistical assumptions is an important factor in choosing a normal- theory or nonparametric-mu1tivariate test. It is important to emphasize that some violations of the assumptions underlying a statistical test: will always occur. This points to a need for criteria that define the ”best" test with respect to distributional properties when underlying assumptions are violated. Gibbons (1971, p. 16) defines the "best" test as the test which is most successful in correctly distinguishing between the conditions as stated in the null and alternative hypotheses. An equivalent and more technical definition of the "best" test is given by Ito (1980), who argues that it is the one which is robust (i.e., insensitive to the violation of test assumptions) with respect to the Type I error probability and also most powerful among its competitors. Unfortunately, the search for the "best" test is complicated by the variety of assumption violations that can affect the distributional properties of a test. Fundamental to the comparison of normal-theory and nonparametric tests is the assumption of normality; As noted earlier, the application of a nonparametric procedure in testing statistical hypotheses does not require a normality assumption, and hence the choice of a normal-theory or nonparametric test in general depends on the tenability of the normality assumption. It is important to emphasize that this is the fundamental difference ‘between. the normal-theory and nonparametric tests considered in the present study. While nonparametric methods may be less sensitive to other assumption violations than their normal-theory counterparts (e.g., heteroscedesti- city of variance), it is primarily the lack of a normality requirement 9 that distinguishes the two methodologies, and serves as an important factor influencing the choice of one kind of test rather than another. Factors other than the tenability of the normality assumption also influence the Type I error rate and power of tests, and hence should be considered in the choice of a normal-theory or nonparametric test. These include the number of variables, their degree of dependency, and the sample size . As noted earlier it would be desirable to analytically examine the effects of all of the above factors on the distributional properties of these tests. Such analyses are extremely difficult if not impossible in the multivariate case because the analytic methods rely on specific statistical assumptions about the underlying distributions and on the asymptotic distribution of sample statistics. Hence investigations of the effects of these factors on normal-theory and nonparametric-multivariate tests have primarily been empirical. The results of a number of studies suggest that the form of the underlying distribution plays a key role on the Type I error rate and power performance of multivariate tests (Arnold, 1964; Chase & Bulgren, 1971; Davis, 1982b; Harwell & Serlin, 1985; Mardia, 1970). The number of dependent variables has also been found to affect the distributional behavior of multivariate tests (Ito, 1980; Olson, 1974), in that the tests tend to become less robust as the number of outcome variables increases. This result may be explained by the fact that the degree of non-normality in the joint distribution of the dependent variables is likely to increase as their number increases (Puri & Sen, 1971, p. 2). 10 As implied by. Puri and Sen (1971, p. 176), the degree and pattern of dependency among variables would also be expected to affect the power of nonparametric-multivariate tests. For example, high correlations among variables would (other factors being equal) tend to produce less powerful tests than low correlations among variables. Sample size is also an important factor influencing the choice of a normal-theory or nonparametric test. This occurs because most of the normal-theory multivariate methods are based on sampling distributions of test statistics that are derived from large samples. However, the same is true for nonparametric tests, and hence both normal-theory and nonparametric tests are expected to be less robust for small samples. In addition to the form of the underlying distribution, the number of variables, their degree of dependency, and the sample size all have been cited as influencing the choice of a normal-theory or nonparametric test. Consequently, any investigation of normal-theory and nonparametric-multivariate tests. should consider these factors. Eesgggch Questions and flypotheses In comparing the Type I error and power properties of normal- theory and nonparametric tests of the independence of two sets of variables, special attention was given to the influence of skewness and kurtosis. Such attention is justified by previous research in both the univariate and multivariate cases, and has indicated the importance of these two characteristics. in the performance of a test (Chase & Bulgren, 1971; Harwell & Serlin, 1985; Mardia, 1970; Olson, 1976). The questions of particular interest and their related hypotheses are the 11 following: 1. Do the skewness and kurtosis values affect the Type I error and power values of normal-theory and nonparametric tests? Previous empirical results suggest that increasing skewness results in normal-theory tests that become liberal (i.e., rejecting a true null hypothesis more often than expected), while increasing kurtosis results in tests that become conservative (i.e., rejecting a true null hypothesis less often than expected) (Chase 6: Bulgren, 1971; Mardia, 1970; Olson, 1974). Increasing skewness or kurtosis may also reduce the power of normal-theory tests (Harwell & Serlin, 1985; Olson, 1974). Such effects would not be expected for nonparametric tests, since these procedures in general do not depend on the form of the underlying distribution. 2. Does sample size influence the effects .of skewness and kurtosis on the Type I error and power values of normal-theory and nonparametric tests? The sampling distributions of most normal- theory and nonparametric test statistics are derived for large samples. Thus, for small samples neither normal-theory or nonparametric tests would be expected to be robust with respect to Type I error rate and power. In particular, for small and moderate samples departures from normality would be expected to noticeably affect the Type I error rate and power of the normal-theory tests (Olson, 1974; Zwick, 1984, p. 2). Similarly, the nonparametric tests would not be expected to perform well for small samples, but would be expected to do well for moderate samples. The power values 12 of all tests would be expected to increase with increases in the sample size. Does the degree of dependency among variables influence the effects of skewness and kurtosis on the Type I error and power values of normal-theory and nonparametric tests? As implied by Puri and Sen (1971, p. 176), the degree of dependency among variables would be expected to affect the power of some multivariate tests. However, the results of the study by Harwell and Serlin (1985) suggested that in general the degree of dependency among variables would not affect the power of normal- theory tests, although for extremely skewed data a high degree of dependency among variables tended to slightly reduce the power of nonparametric tests. Thus, a high degree of dependency among variables might be expected to slightly reduce the power of the nonparametric tests for extremely skewed data. Does the number of variables influence the effects of skewness and kurtosis on the Type I error and power values of normal-theory and nonparametric tests? Previous empirical results suggest that normal-theory procedures become less robust with respect to Type I errors and power as the number of outcome variables increases (Ito, 1980; Olson, 1974). Puri and Sen (1971, p. 2) pointed out that as the number of variables increases, the degree of non- normality of their joint distribution might be expected to increase simply because of the increase in the dimensionality of the distribution. l3 WM It has been pointed out that analytic studies of the Type I error probability and power efficiency of multivariate tests for small samples are very difficult if not impossible to carry out. An alternative to analytic methods is the application of computer simulation in assessing the performance of various statistical tests. Hartley (1976) argues that computer simulation has become an important technique for verifying analytic results. For example, such techniques can be used to find the exact sampling distribution of most statistics using data that have been drawn from any parent distribution (Tracy & Conley, 1982, p. 262). Fawcett and Salter (1987) stressed that a distribution study should not be regarded as complete without the inclusion of computer simulation for finding the exact distribution of the statistic used. Although the present study will present analytic expressions when they exist, it will rely on simulated data to answer the research questions by examining the distributional behavior of the selected test statistics under various data conditions. §igni§ig§n§g of the Stgdy The significance of the present study is related to the multivariate character of educational research questions and the empirical tests of these questions. As noted earlier, studies in education often generate research questions that are multivariate in character and lead to the use of multiple (correlated) variables. The multivariate methods used in the analysis of this data typically assume normality of the underlying distribution. In many practical l4 situations, however, there is evidence that the underlying distributions are non-normal (Puri 6: Sen, 1971, p. 1). Because such non-normality can ‘have a deleterious effect on. the distributional properties of normal-theory tests, particularly for studies that employ small to moderate samples, the use of normal-theory methods may not be appropriate. Under these circumstances, educational researchers should consider a multivariate-nonparametric alternative. In studying normal-theory and nonparametric-multivariate tests, canonical-correlation analysis/multivariate—multiple-regression seem a natural starting point. Their importance as a general system of statistical inference has been demonstrated by several authors. For example, Knapp (1978) showed that many of the commonly used normal- theory tests can be treated as special cases of the canonical- correlation model. The same is true in the nonparametric case. It should be emphasized that the Bartlett, Rao E, and rank-transform Rao E tests were developed for an omnibus test involving canonical correlations, while the pure- and mixed-rank tests to be examined were developed for an omnibus test involving multivariate-multiple-regression. However, as shown in the next chapter the models underlying canonical-correlation and multivariate- multiple-regression. are identical, and. 'hence these tests are equivalent with respect to concluding whether two sets of variables are independent (Gittins, 1985, p. 19). The choice of the "best" test depends on a number of factors, including the form of the underlying distribution, the number of variables, their degree of dependency, and the sample size. These 15 factors were examined in the present simulation study. It is expected that the results will provide educational researchers with guidelines for performing either normal—theory or nonparametric canonical- correlation/multivariate-multiple-regression analysis for a variety of data conditions . t 0 ud The findings of the present study are valid only if the between- set correlation matrix containing zeros is a sufficient indication of the independence of two sets of non-normal variates. The generalizability of the results is also limited by the range of simulation conditions investigated. CHAPTER II REVIEW OF THE LITERATURE A review of the literature pertinent to the present study is presented in this chapter. The review includes the following areas (a) defining the normal distribution and some non-normal distributions on the basis of skewness and kurtosis, (b) methods of dealing with non- normal data, including transforming the original data to ranks, and (c) robustness and power results for some normal-theory and nonparametric- multivariate tests . a n t Norm st buti and Some Ngn-Egmal Distzibutigns Because of the critical role of skewness and kurtosis in the present study, these characteristics of a distribution are defined and illustrated for the normal and some non-normal distributions. In theory, the shape of a distribution is defined by its probability distribution function, which is an algebraic expression indicating the distribution of a variable across all of its possible values. In practice, an approximate distribution of a variable can be characterized by its first four central moments (i.e., mean, variance, skewness, and kurtosis) (Fleishman, 1978). In this scheme, the center and dispersion of a distribution are determined by the mean and variance, and its symmetry and tailedness by the skewness and kurtosis values. Assuming the observations are standardized to have a known mean and variance, this permits distributions to be classified according to their skewness and kurtosis values. 16 l7 Skewness and kurtosis are defined through central moments. For a continuous random variable X, the r(th) central moment (pr) is defined as (Kendall 6: Stuart, 1969, V. 1, p. 55): O "r - I 4). values that deviate subtantially from zero indicate a greater degree of non-normality, although it should be emphasized that the effects of both skewness and kurtosis must be considered. Another distribution that is important in simulation studies is the symmetric, extremely heavy-tailed Cauchy, since it reflects an extreme in non-normality that can occur in practice. Theoretically, the Cauchy distribution has an infinite variance, and hence does not possess a finite kurtosis value. However, in empirical studies a pseudo-Cauchy distribution can be generated using a zero skewness 19 value and a large positive kurtosis value (Harwell & Serlin, 1985). Although univariate measures of skewness and kurtosis are relatively unambiguous in their interpretation, such is not the case for their multivariate counterparts. Even so, these measures have been useful in identifying a particular member of a family of distributions, in developing a test of normality, and in investigating the robustness of normal-theory procedures (Mardia, 1970). Mardia (1970, 1974) developed measures of skewness and kurtosis for multivariate distributions, details of which appear in Chapter III. For illustrative purposes, multivariate measures of skewness and kurtosis for some bivariate distributions in which both variables are standardized are given in Table 2 (Mardia, 1970). Notice that the skewness values are zero for all symmetric distributions. In particular, Mardia (1974) showed. that the multivariate-normal distribution has skewness and kurtosis values of zero. Just as in the univariate case, any multivariate distribution is considered to be non-normal for non-zero skewness and/or kurtosis values. Table 2 Multivariate Skewness and Kurtosis Values of Some Bivariate Distributions (Mardia, 1970) Distribution Skewness Kurtosis normal 0.00 0.00 uniform 0.00 -2.24 double-exponential 0.00 6.00 exponential 8.00 12.00 20 e - at As noted earlier, the normality assumption underlying normal- theory tests is of prime importance in the correct use and interpretation. of these procedures. At the same time, there is a general recognition that data obtained in a ‘variety' of settings, including education, are frequently at least moderately non-normal and hence the use of normal-theory tests is problematic. This has led to the emergence of two approaches for fitting such data into the framework of statistical theory (a) transforming the data to an approximate normal form and then applying a normal-theory procedure (see Box & Cox, 1964; Kaskey, Kolman, Krishnaiah, & Steinberg, 1980), or (b) transforming the data to their corresponding 'ranks, which removes the normality requirement on the form of the underlying distribution, and employing nonparametric methods. The focus here is on (b), and in the following sections the applications of rank methods in hypothesis testing are discussed. WM Nonparametric methods have a long history in both theoretical and applied statistics (Noether, 1984). These methods require a transformation of the original scores such that the resulting transformed values have known distributional properties. These tests are often classified as distribution-free, since the methods are based on (sample) statistics whose sampling distributions do not depend on the form of the parent distribution from which the sample was drawn (Gibbons, 1971, p. 3). The valid use of these tests requires that the distributions underlying the data are continuous and that all 21 observations are independently and identically distributed (Puri & Sen, 1985, p. 307). The principle underlying nonparametric procedures is that under a ‘postulated statistical hypothesis the joint distribution. of the random variables is invariant under appropriate groups of transformations (Puri & Sen, 1985, p. 7). It is this invariance that produces what are called genuinely distribution-free tests. However, some distribution-free tests are not genuinely distribution-free, and are usually classified as asymptotically or permutationally distribution-free. In general, an asymptotically distribution-free test can be defined as one which is distribution-free given that the sample size is infinite (i.e., by virtue of the central limit theorem) (Conover & Iman, 1981; Hollander 6: Wolfe, 1973, p. 437), whereas a permutationally distribution-free test depends only on the set of permutations of the observations associated with testing a hypothesis, and not on the underlying distribution function (Puri & Sen, 1985, p. 149). Applications of nonparametric methods in testing univariate hypotheses are described in detail by Gibbons (1971), Conover (1980), and Marascuilo and McSweeney (1977). A primary source of nonparametric methods in testing multivariate hypotheses is the work of Puri and Sen (1969, 1971, 1985). Multivariate tests presented by these authors include the single-sample location problem (e.g., sign, signed-rank, extended signed-rank), the multi-sample location problem (e.g., median, rank sum), and a number of tests for hypotheses subsumed by the general linear model. 22 W A related set of nonparametric procedures that are used to test univariate and multivariate hypotheses are rank-transform methods. Transforming the original data to their corresponding ranks and applying the usual normal-theory procedures is an idea championed by Conover and Iman (1981). This approach generates a class of rank- transform methods that in many ways are comparable to well known univariate nonparametric procedures such as the Wilcoxon-Mann- Whitney, Kruskal-Wallis, Wilcoxon-signed ranks, and Friedman tests (Conover 6: Iman, 1976, 1980a, 1982, 1980b; Iman, 1974a, 1974b, 1976; Iman 6: Conover 1976, 1978, 1979, 1980a, 1980b). Other applications of the rank-transform approach include correlation and regression analysis (Boyer, Palachek, 6: Schucany; 1983; Hogg & Randles, 1975; Iman & Conover, 1979). An extension of this approach to tests based on the multivariate general linear model is possible by ranking each quantitative variable separately, and then applying the usual normal- theory methods. One advantage of this approach is that the resulting tests can be performed using existing statistical computer packages. Although the rank-transform approach seems promising, one important limitation should be noted. The rank-transform methods rely on the distributions of the normal-theory test statistics as approximations to the actual distributions of the rank transformation statistics. To date, the theoretical distributions of such statistics have not been established. Consequently their distributional properties can only be determined empirically through computer simulation studies and are always restricted by the conditions of a particular simulation 3 tudy . 23 In sum, the two primary nonparametric methods for handling non-normal data are the pure- and mixed-rank procedures illustrated in Puri and Sen (1985, pp. 307-328), and the rank-transform method of Conover and Iman (1981) . Both remove the underlying normality requirement. The major difference between them is that the pure- and mixed-rank procedures have a known theoretical substructure, which permits analytic statements about the distributional properties of a test, while the rank-transform procedure permits no such statements. 3 a d we N - e oa t'-utvaaeets In comparison to the univariate case, surprisingly little is known about the robustness and power of multivariate tests when underlying assumptions are violated (e.g., non-normality). A natural starting point is the robustness and power properties of univariate procedures, which have often been found to parallel those of their multivariate counterparts. A large number of studies comparing the distributional properties of univariate procedures are available. Some- of these studies have been analytic, involving asymptotic approximations, but most have been empirical. The following review will summarize the robustness and power of some nonparametric-univariate and -mu1tivariate tests as compared to their normal-theory counterparts. In reporting these results, the focus will be on the violation of the normality assumption. Analytic results, where available, will be presented first, followed by empirical results. The sparseness of information on multivariate tests clearly indicates the need for further work in this area. 24 u t - v at The power of a particular test relative to a competitor is usually reflected by its asymptotic relative efficiency (A.R.E.), which is the limiting ratio of the sample size required by one test relative to that required by a second test such that they have equal power for the same alternative hypothesis. A test is said to be more powerful and efficient than another test if the A.R.E. is greater than 1. Details of the computation of the A.R.E. appear in Appendix B. As an example, some A.R.E. results for nonparametric tests of location relative to the normal-theory E: and E tests are shown in Table 3 (Marascuilo & McSweeney, 1977, p. 87). The nonparametric "normal scores" tests in Table 3 refer to a rank test applied to normal scores, which are obtained by transforming the ranks to their standard normal scale counterparts. Table 3 Asymptotic Relative Efficiencies of Some Nonparametric and Normal-Theory Tests of Location. double- Test/Distribution normal uniform logistic exponential One-sample Sign 0.637 0.333 0.750 2.000 Wilcoxon 0.955 1.000 1.047 1.500 Two-sample Median 0.637 0.333 0.750 2.000 Mann-Whitney 0.955 1.000 1.047 1.500 Normal-scores 1.000 1.273 K-sample Median 0.637 0.333 0.750 2.000 Kruskal-Wallis 0.955 1.000 1.047 1.500 Normal-scores 1.000 1.273 25 As an example, consider the A.R.E. of the nonparametric two- sample Mann-Whitney test relative to the 3; test for a parent normal distribution. The A.R.E. of .955 implies that the Mann-Whitney test requires a sample size of 100 in order to have a power equal to that of the ; test using a sample size of approximately 95. In this case, the 5 test is said to be more efficient than the rank test since it requires a smaller sample size in order to have the same power. It should be noted that although all four distributions in Table 3 are symmetric, their kurtosis varies from moderate-negative to large- positive. The results indicate that for the normal, uniform, and logistic distributions most of the nonparametric tests are as efficient as their normal-theory counterparts, while for the double exponential distribution the nonparametric tests are more efficient than the corresponding ‘normal-theory tests. These findings suggest that these nonparametric alternatives are, in general, almost as powerful as the corresponding normal-theory procedures for normal and near-normal parent distributions, and certainly more powerful for leptokurtic distributions. — U v te A.R.E. is a large-sample property of a test which may not be valid for small to moderate samples (Gibbons, 1971, p. 19). As an alternative, simulation studies may be used to assess the performance of two or more tests by comparing their empirical Type I error and power values for various underlying distributions, alternative hypotheses, and sample sizes. This section summarizes the results of a number of empirical studies carried out to investigate the effects 26 of non-normality on the robustness and power of univariate normal- theory and nonparametric tests. The studies cover a variety of tests and were chosen because of their representativeness in comparing normal-theory and nonparametric tests for parent non-normal distributions. In general, empirical studies have suggested that univariate normal-theory tests are robust to moderate non-normality for large samples, especially when the underlying distribution is symmetric (Gaito, 1970; Glass, Peckham, & Sanders, 1972, Kendall & Stuart, V. 2, p. 484, Scheffe', 1959, p. 347). However, a number of studies have suggested that distinct departures from normality in combination with small samples affect univariate normal-theory tests. For example, Fair-Walsh and Toothaker (1974) compared the performance of the normal- theory E test and the nonparametric Kruskal-Wallis test when samples were drawn from an exponential (positively skewed) population. While the Type I error rates were somewhat conservative for both tests, the Kruskal-Wallis procedure was found to be more powerful then the E test. Srisukho (1974), in a similar study, found the power of the Kruskal- Wallis test to be greater than that of the E test when all samples were drawn from a double exponential (symmetric, leptokurtic) population, and less than the E test when all samples were drawn from a uniform (symmetric, platykurtic) population. The Type I error rates for the Kruskal-Wallis test tended to be closer to the nominal value than those of the E test for both the double-exponential and uniform populations. 27 Studies that examined the distributional behavior of normal- theory and rank-transform statistics have shown favorable results for the latter tests. Boyer, Palachek, and Schucany (1983) studied the distributional behavior of the Williams' (1959) test of the equality of dependent correlations (i.e., - p given that X and X H0: p yx1 yx2 l 2 are correlated), and its rank-transform alternative. The results indicated that although the power values of both procedures were similar, the Type I error rates of the rank-transform test were closer to the nominal alpha level than those of Williams' test for data that were drawn from a parent lognormal distribution. Williams' test, however, produced higher power values for data that were drawn from a parent normal distrution. Based on these results, the authors recommended the use of Williams' test when normality can be assumed, and the rank-transform version of Williams' test when the normality assumption is not tenable. Iman (1974b) examined the Type I error and power properties of the normal-theory E and rank-transform E tests for a two-way ANOVA problem. The results indicated that the Type I error rate of the rank- transform E test was similar to that of the E test, and that the rank- transform E was more powerful when the underlying distribution was non-normal. In sum, the smattering of empirical results for univariate tests presented above suggests that the Type I error rate of normal-theory tests is generally not affected by moderate departures from normality for large samples, particularly when the underlying distribution is symmetric and light-tailed (e.g., uniform). However, for distinctly 28 non-normal populations (e.g., exponential, double-exponential), nonparametric tests appear to be robust with respect to Type I error rate, and produce higher power values than their normal-theory counterparts. - o a - o - v Ca e Normal-theory-multivariate tests depend on the assumption that the observations are governed by the multivariate-normal density function. Since departures from this assumption are very difficult to investigate analytically (Ito & Schull, 1964), empirical methods are used. The following review summarizes some empirical studies that have examined the effects of non-normality on the Type I error and power properties of some normal-theory- multivariate tests. A number of studies for the one- and two-independent groups case have been carried out examining the effects of non—normal skewness and kurtosis and sample size on Hotelling's 12 statistic. The one-sample 12, like the univariate one-sample t statistic, is (a) not affected by small departures from normality, (b) more sensitive to non-normal skewness than. to non-normal kurtosis, (c) produces liberal. Type I error rates for a large skewness, and (d) produces conservative Type I error rates for large kurtosis (Chase & Bulgren, 1971; Davis, 1982a; Mardia, 1970). Similar results for the two-sample location problem were obtained by Davis (1980, 1982b) for Wilks's likelihood ratio and Roy's largest root tests. Other results have suggested that non-normal kurtosis has no substantial effect on the two-sample 12 statistic for large samples (Hopkins & Clay, 1963; Ito, 1980). 29 Olson (1974) empirically studied the effects of non-normality and heterogeneity of covariance matrices on six multi-sample, normal- theory' MANOVA tests (Roy, Hotelling-Lawleyy ‘Wilks, Pillai-Bartlett, Gnanadesikan, and Gnanadesikan-alternative) using small-to-large sample sizes (5, 10, 50). The empirical Type I error results indicated that moderate departures from a kurtosis of zero had mild effects on three tests (Hotelling-Lawley, Wilks, and Pillai-Bartlett), and severe effects on the remaining tests. The direction of the effect of positive kurtosis was generally toward conservatism. The results of this study also suggested that the Gnanadesikan and Gnanadesikan- alternative tests tended to produce liberal Type I error rates with increases in the number of outcome variables for non-zero kurtosis values, especially for small samples crossed with a large number of groups. The power results indicated that all six tests suffer under moderate departures from a kurtosis of zero, and that increases in the number of outcome variables tended to decrease the power of all of the tests. In general, the results of the studies that used large samples suggest that normal-theory tests are robust to non-normality (Zwick, 1984, p. 2). This result is expected for many tests because of the role of the multivariate analog of the central limit theorem (Morrison, 1976, p. 85). This theorem states that any statistic which can be represented as a.linear combination of the observations has a sampling distribution that can be approximated by the normal distribution for large samples. Since many test statistics are derived from some linear function of the observations, the sampling 30 distribution of the test statistic can be approximated by the normal distribution as sample size increases. Consequently the test would be robust for large samples; this is not necessarily the case for small or moderate samples. In sum, these studies suggest that (a) small-to-moderate departures from normality have only minor effects on normal-theory- multivariate tests, (b) such effects are more pronounced for small samples than for large samples, (c) increasing skewness tends to result in liberal Type I error rates, (d) increasing kurtosis tends to results in conservative Type I error rates, and (e) the distributional behavior of normal-theory-multivariate tests is affected more by non-normal skewness than by non-normal kurtosis. - ar e c- u va e Analytic results for the distributional properties of nonparametric-multivariate tests are available for a few special cases. Available analytic studies on the asymptotic efficiency of nonparametric-multivariate tests relative to their normal-theory counterparts have shown results similar to those of the univariate case (Puri 5: Sen, 1971, p. 177). For example, the A.R.E. of the multivariate-nonparametric one-sample test of location with normal- scores relative to Hotelling's 12 is equal to 1.00 for a multivariate normal distribution, and sometimes greater than 1.00 for other multivariate distributions. (Puri 6: Sen, 1971, p. 177). Recall that similar results were reported by Marascuilo and McSweeney (1977, p. 87) for the one-sample, normal-scores and t tests in the univariate case (Table 3). 31 Additional results for multivariate tests of location are available for special cases: (a) the A.R.E. of the rank procedure to the normal-theory, two-sample test oflocation is always less than 1.00 in dhe 'bivariate normal case, (b) the normal-scores test is more efficient than the normal-theory test for any mixture of multivariate normal distributions (i.e., a combination of two normal deviates) and heavy-tailed. multivariate distributions, and (c) the normal-scores test is more efficient for a multivariate distribution with marginal densities that have light tails, a result that parallels the univariate case (see Zwick, 1984, p. 9). In general, the nonparametric rank tests for' hypotheses subsumed. under’ the multivariate-general- linear model are asymptotically power-equivalent to the normal-theory likelihood-ratio test for a parent normal distribution (Puri & Sen, 1985, p. 184). The nonparametric rank tests for location are asymptotically as efficient as their normal-theory counterparts for a parent normal distribution and more efficient for a parent non-normal distribution (Zwick, 1984, p. 27). The analytic results indicated that multivariate-nonparametric rank and normal scores tests are more efficient than their normal- theory counterparts for non-normal distributions, and are almost as efficient for the normal distribution. u - et c- u vs C e Surprisingly few simulation studies have been done comparing the Type I error rate and power of multivariate nonparametric tests against their normal-theory counterparts. The available results are presented in some detail since they have important implications for 32 the conduct of the present study. Tiku and Singh (1982) studied the Type I error rate and power of the two-sample Hotelling's 12 and rank tests using samples of size 20 drawn from six bivariate distributions [normal, 3;, two chi-square (v - 2, 4), and two mixed-normal]. In all cases the two outcome variables had a correlation of .5. Their results indicated that the rank test was robust with respect to the Type I error rate for three distributions [normal, chi-square (u - 2), and one mixed-normal], and was conservative for the remaining distributions. The 12 test was robust with respect to Type I error rate for the normal and .g distributions, and conservative for the chi-square and mixed-normal distributions. The rank test proved more powerful than the 12 test for all distributions except the normal. The results of Tiku and Singh suggest that the two-sample rank test should be the procedure of choice for testing the equality of mean vectors for even moderately non-normal distributions. Zwick (1984) studied the empirical Type I error rate and power of the multivariate-nonparametric two-sample rank and normal-scores alternatives to the 12 test under mild non-normality and heterogeneity of variance-covariance conditions. Just as in the univariate case, the Type I error rate was affected mainly by variance-sample size combinations and not by the parent distribution. Power was affected by both the variance-sample size combination and parent distribution, with all tests producing approximately the same power values. In summarizing the results, Zwick recommended that (a) under the conditions of normality and homogeneity of variance the normal-theory test was the 33 best procedure with respect to both Type I error and power, (b) under normality and heterogeneity of variance with equal sample sizes or when the larger group had the larger variance the rank test was the best choice, and (c) for negatively-skewed distributions the normal- scores test appeared to be the best overall choice except when the smaller group had the larger variance. Harwell and Serlin (1985) examined the Type I error rate and power of the Rao E (1951), the nonparametric rank-transform Rao E (Conover & Iman, 1981), and the pure- and mixed-rank tests illustrated by Puri and Sen (1985, p. 312). The simulation conditions included in this study were form of distribution (normal, uniform, double- exponential, exponential, Cauchy), sample size (20, 40, 100), correlation within each set of variables (.3, .7), and correlation among the two sets of variables (Type I error, power). The Cauchy was represented. by a symmetric distribution 'with a kurtosis value of twenty. The Type I error results suggested that the Rao E test was robust with respect to the Type I error for the normal and uniform distributions, became liberal for the Cauchy distribution, and produced. mixed results for the double exponential and. exponential distributions. There was no clear pattern for the liberal Type I error rates with respect to sample size and within-set correlation. As a measure of the Type I error behavior of these tests, the Rao E overall produced 38% liberal Type I error rates, taking into account sampling error, while the rank-transform Rao E produced only 7% liberal Type I error rates. The latter test performed most satisfactorily for 34 extremely non-normal distributions and poorly for the smallest sample size and within-set correlation - .3 conditions. In contrast, the pure- and mixed-rank tests did not produce a single liberal Type I error rate across all simulation conditions. With respect to power, the results indicated that under a normal or uniform distribution the Rao E test was most powerful across all within-set correlation and sample size conditions. In general, the rank-transform Rao E produced the largest power values among the three nonparametric tests, especially for small samples. However, all four tests produced similar power values for the sample size of 100. The power values of all four tests for the double-exponential were comparable for a sample size of 100, and slightly less for a sample size of 40. In general the mixed-rank power values were slightly higher than those of the pure-rank procedure. The power results for the Cauchy and exponential distributions showed that the pure- and mixed-rank tests performed poorly for the sample size of 20 and the .01 level of significance. Once again the power values for all three nonparametric tests were quite similar for larger sample sizes. The three nonparametric tests overall produced power values substantially larger then the reported values of the Rao E test for the sample size of 100. In general, the mixed-rank test produced slightly lower power values than the pure-rank test. Based on the Type I error and power results, the authors recommended that the Rao E test be used for symmetric, light-tailed distributions, the ranketransform Rao F for small samples for any of the non-normal distribution investigated, and the pure- and mixed-rank 35 statistics for larger samples and moderate to distinctly non-normal distributions. The results also confirmed earlier findings that a normal-theory test is affected by moderate-to-large skewness and by a large kurtosis. 1W Studies that examined the effects of the degree of dependency among variables on the distributional behavior of multivariate tests suggest that such dependency affects the power of nonparametric tests. Bhattacharyaa, Johnson, and Neave (1971) examined the power of the two-sample Hotelling's T2 and nonparametric rank-sum tests. The A.R.E.'s of the rank test relative to 12 test for correlation values of .0, .3, .6, and .9 are .955, .947, .924, and .884, respectively. These results suggest that the A.R.E. of the rank test to T2 decreases as the degree of dependency among variables increases. Similar results were found by Puri and Sen (1971, p. 176) for negative correlation values. The results of Harwell and Serlin (1985) showed that the nonparametric pure- and mixed-rank tests produced slightly lower power values for extremely non-normal distributions (e.g., exponential, Cauchy) as the correlation among outcome and predictor variables increased from .3 to .7. These studies suggest that a high degree of dependency among outcome and predictor variables slightly reduces the power of some nonparametric tests. 52mm The present review of the literature leads to the following conclusions with respect to the effects of skewness and kurtosis on Type I error probability and power of normal-theory and nonparametric- 36 multivariate statistical tests: (a) the effects of non-normality on normal-theory-multivariate tests appear to parallel those in the univariate case, (b) for large samples slight departures from normality have negligible effects on the Type I error rate and power of most normal-theory tests, (c) for small samples moderate to large departures from normality affect the Type I errors and power of these tests, (d) increasing skewness results in normal-theory tests that become liberal while increasing kurtosis results in tests that become conservative (except for the Harwell & Serlin 1985 study in which increasing kurtosis results in normal-theory tests that become liberal), (e) normal-theory tests are more sensitive to non-normal skewness than to non-normal kurtosis, (f) nonparametric tests are superior at controlling Type I errors within nominal levels and are asymptotically’ more efficient compared. to their' normal-theory competitors when the underlying distributions are at least moderately non-normal, and (g) a high degree of dependency among variables slightly decreases the power of some nonparametric tests. The review of the literature indicates that most of the studies on distributional properties of multivariate tests were confined to the MANOVA procedure. The present study of the tests for canonical- correlation/multivariate-multiple-regression complements previous work. The present study also extends the results of Harwell and Serlin (1985) by including the number-of-variables factor and some additional parent distributions and within-set correlations. The focus was the interaction of the form of parent distribution and the sample size, the within-set correlation, and the number-of-variables. CHAPTER III METHODOLOGY This chapter presents the methodology employed in the present study. The following topics are discussed (a) multivariate statistical models, (b) test statistics and their assumptions, (c) data generation method, (d) simulation conditions, and (e) presentation of simulation results. V dc This section describes the multivariate-multiple-regression and canonical-correlation models and their relationship to the multivariate general linear model. The term general linear model refers to a family of algebraic models characterized by the linearity of the parameters of the equations specifying the models (Gittins, 1985, p. 19). The multivariate-multiple-regression model is a member of one such family. Let X be an N x p (i-l,2,...,N; j-l,2,...,p) data matrix of N observations on p outcome variables, 2; an N x q (k-l,2,...,q) matrix of regression constants, E a p x q matrix of unknown parameters of the model, and E an N x p matrix of unobserved random errors. The multivariate-multiple-regression model can be written: X-Xfl+E~ (4) pr qu qxp pr 37 38 A canonical-correlation model may be conceived of as a special case of the multivariate-general-linear model (Gittins, 1985, pp. 19-20). Recall that the focus of the present study is testing whether there is a linear relationship among two sets of variables, and that testing the hypothesis of independence among two sets of variables is equivalent to testing the hypothesis of no regression. This linear relationship may be represented and studied using a canonical- correlation model (Gittins, 1985, p. 19). The following paragraph introduces canonical correlation and canonical variables. Let Y1, Y2, ..., Y1) and X1, X2, ..., Xq be two sets of random variables. Define 9 - lel + h2Y2 + ... + hpr (5) as a weighted linear combination of the Y variables, and J i - mlx1 + m2X2 + ... + quq (6) as a weighted linear combination of the Xk variables. Define also I; - (h1’ h2, ..., hp) and m_ - (m1, m2, ..., mq) as the vectors of constants that maximize the correlation between the Yj and Xk variables. The correlation between the canonical variates Y and K is the canonical correlation and h and g are the canonical weights. Given two sets of variables a total of s - minimum (p, q) pairs of linear combinations can be constructed, and hence s canonical correlations can be obtained. The canonical correlations are found as solutions of a determinantal equation and the canonical weights as solutions of an eigen equation. The process of obtaining these 39 quantities is outlined below (Morrison, 1976, pp. 254-257). Let 9 represents a sample canonical correlation, S the sample covariance matrix of the X.j variables, syy the sample covariance matrix of the YJ variables, and 57x the sample covariance matrix of the Y3 and KR variables. By definition p2 is given by 2 (hf m) 2 p - 57" . <7) (h' 573' mm' 5“ it) Since we wish to maximize the correlation between the linear combinations 9 and 2, the problem can be solved by obtaining the values of h and m that maximizes expression (7). To simplify the "maximization" process while assuring the uniqueness of h and m the variance of both linear combinations is set equal to l: b' fiyy h - m' §xx m - 1 . (8) Hence we need only to maximize (h' ny m)2 subject to the constraint in (8). This problem can be solved by introducing Lagrangian multipliers A and 9 as follows: , 2 r _ _ ' - (h Syx m) - A(b Syy h 1) 0(m fixx a 1). (9) The first partial derivatives of expression (9) with respect to h and m are then taken. Setting these equations equal to zero produces a homogenous system of two simultaneous matrix equations, namely - Xfiyy h + (h' Syx m)§yx a - Q. (10) (h' Syx m)§yx h - ofixx m - Q. 40 Premultiplication of the first equation by h' and the second equation by m' produces l-O-(h'%xm)2. (11) Hence each of the Lagrangian multipliers is equal to the squared maximum correlation between Y and K. In order for the equations in (10) to have a nontrivial solution their determinant must vanish. This leads to the determinantal equation I39; fiyx fiii §§x - All - 0. (12) where L is an identity matrix. Morrison (1976, p. 257) shows that the eigenvalues of equation (12) may also be obtained by replacing the covariance matrices in equation (12) with their corresponding sum-of- cross-product (SCP) matrices: IAQ; Ayx A;: A§x - All - 0. (13) where Axx is the SCP matrix of the centered Xk variables, Ayy is the SCP matrix of the centered Yj variables, and Ayx is the SCP matrix of the centered Y3 and KR variables. The maximum 92 is the largest eigenvalue (A) of equation (12) or (13) (Morrison, 1976, p. 256). The canonical correlations are ordered 1 > 91 > 92 > .. > 95 >0. The canonical weights are the solutions to the associated eigen equations (Morrison, 1976, p. 263). Note that each pair of canonical variates is orthogonal to all other pairs and. that the total ‘number' of ‘pairs equals a. 41 WW Five test statistics will be used to test the hypothesis of no linear relationship among the two sets of variables (Yj’ Xk). These tests can be conceived of as a test of the matrix of the regression parameters 5 against zero, or, synonomously, as an omnibus test that all squared population canonical correlations (pi, pg, ..., pi) are simultaneously equal to zero (Gittins, 1985, p. 57). Hence the null hypothesis for the canonical problem can be written as 2 -...-pS-O. (14) Retaining H0 is equivalent to concluding that there is no Yj’ X'k relationship, while rejecting H implies the existence of such a 0 relationship. Each test is performed using the eigen values (squared canonical correlations) obtained from expression (13). No normal-theory and three nonparametric omnibus tests of the hypothesis of expression (14) will be employed. The normal-theory tests are the Bartlett (1938) and Rao E (1951) procedures. The nonparametric tests are the pure- and mixed-‘rank procedures discussed in Puri and Sen (1985, pp. 307-328) and the rank-transform Rao E (Conover & Iman, 1981). Although all five tests provide tests of an omnibus statistical hypothesis it is important to emphasize the differences in the nature of the variables upon which the canonical correlations are computed. For the normal-theory tests the canonical correlations are obtained using the original values of the outcome and predictor variables; for the pure-rank and rank-transform Rao E the canonical correlations are obtained using the ranks of the original 42 values of the outcome and predictor variables; and for the mixed-rank test the canonical correlations are obtained using the original values of the predictors and the ranks of the outcome variables. Note, however, that the hypothesis being tested involves the raw score canonical correlations or regression coefficients. The test statistics, their computations, and assumptions are described below. W Given two sets of random variables, there exist several measures that summarize the strength of the relationship among them. The best known is Wilk's lambda (A), which is defined as (Anderson, 1958, p. 233) s A - H (1 - Ar), (r-l,2,...,s), (15) r-l where the it's are the solutions (eigenvalues) of expression (13). The values of A have a range between. 0 and l, with smaller values indicating a strong relationship between the Xk and Y3 variables and larger 'values indicating a weak. relationship (Marascuilo & Levin, 1983, p. 185). Given that two sets of variables are independent, Wilk's A has been shown to follow Wilk's A distribution (Anderson, 1958, p. 242). Unfortunately, tables of the exact A distribution are needed to perform the test. To obviate the need for these tables, Bartlett (1938) introduced a large-sample chi-square approximation to the exact Wilks's A distribution. Under the truth of the hypothesis of expression (14), Bartlett (1938) showed that Bartlett statistic (BAR) is asymptotically distributed as a central chi-square variable with pq 43 degrees of freedom. The BAR statistic can be computed using the following formula (Marascuilo 6: Levin, 1983, p. 185): 2 BAR - -[(N-1) - (p + q + 1)/2] loseA ~ qu (16) If BAR exceeds the 100(1 - a) percentile of the chi-square distribution with pq degrees of freedom, the hypothesis of expression (14) is rejected (Bartlett, 1938). The Bartlett procedure assumes that the observations are independently and identically distributed random variables (i.i.d.r.v.'s) with a common (multivariate-normal) distribution function (Gittins, 1985, p. 242). B§2_E_§£e£l§£12 A more precise approximation to the exact Wilk's A was developed by Rao (1951) (Marascuilo 6: Levin, 1983, p. 185). In testing the independence of two sets of variables, Rao's procedure yields an exact test when the smaller of the two sets contains two or less variables (Marascuilo & Levin, 1983, p. 187). In contrast, no exact test is possible with the Bartlett test. The Rao E statistic (RAO) can be computed using the following formula (Marascuilo & Levin, 1983, p. 186): (1 - Al/bm1 ~ V1, V2 _ Al/b/u2 Where All ' pq. :12 - 1 + ab - pq/2, a - (N - 1) - (p + q +l)/2, and b - [(p2q2 - 4)/(p2 + q2 -5)]1/2. If RAO exceeds the 100(1 - 0:) percentile of the E distribution the hypothesis of expression ”1' ”2 44 (14) is rejected (Rao, 1951). The assumptions of the Rao F procedure are the same as those of the Bartlett. W The rank-transform. approach (Conover a Iman, 1981) involves transforming the original values of the outcome and predictor variables into their corresponding ranks and then applying the normal-theory Rao E procedure. It is important to emphasize that the theoretical E distribution is used as an approximation to the unknown distribution of the rank-transform Rao E statistic (RIP). The decision rule for the rank-transform Rao E is the same as that of the normal- theory Rao E test. However, unlike the previous two tests the p: of expression (13) are based on the ranks of the XR and the YJ variables. The rank-transfonm procedure assumes that the observations are i.i.d.r;v.'s ‘with a common distribution function (Conover' & Iman, 1981). We The nonparametric pure- and mixed-rank statistics illustrated in Puri and Sen (1985, pp. 307-328), and discussed by Harwell and Serlin (1985), are used when both predictor and outcome variables are random. It should be noted that the model originally presented by Puri and Sen (1969) requires the predictor variables to be known regression constants. However, since this condition rarely obtains in practice interest centers on models in which the predictors are assumed to be random. The pure-rank test is particularly useful when only the ranks of the predictor or outcome variables are available. 45 In the pure-rank model all p outcomes and q predictors are assumed to be i.i.d.r.v.'s. To represent the pure-rank test in a canonical correlation context, let C(XIX) be the conditional distribution function of the i(th) subject's vector of outcomes Xi’ given a vector of predictor values Xi. Recalling the multivariate- multiple-regression model illustrated in expression (4), the conditional distribution function of the 1 given the E can be i i written as: c - “0‘11 - r, a). (18) where GO is some continuous distribution function. Expression (18) implies that the conditional distribution function for each subject is identical, and that this function depends on the observed predictor values. As noted earlier, this implies that inferences are limited to subpopulations having the same configuration of predictor values as those in the sample. Puri and Sen (1985, pp. 307-328) used the form given in expression (18) to write the hypothesis of no relationship among the two sets of variables as Ho: C(xilri) - 60(ri>. (19) Retention of the above hypothesis implies the two sets of variables are independent, while rejection implies that they are related. In order to compute the pure-rank statistic (PUR), the N observations for each of the p outcomes and q predictors must be separately ranked. Let R31 and Rki represent the rank of the i(th) subject on the j(th) 46 outcome and k(th) predictor variables, respectively. Since C(Xilzi) is assumed to be continuous the theoretical probability of tied Rji or Rki values is zero. In practice, as long as the proportion of ties is small, assigning midranks to tied values will have a negligible effect on the test statistic (Lehmann, 1975, p. 18). V Puri and Sen (1985, pp. 307-312) presented a.1arge-samp1e form of the pure-rank statistic based on the SCP matrix 5 of the centered Rj and RR values, with elements N where Rj and RR are the rank means for the j(th) dependent and k(th) predictor variables, respectively. In the construction of the pure-rank test, Puri and Sen show that the E(§) - Q and that the elements of E are asymptotically mmltivariate-normal given that the sets of the Xj and the xk variables are independent. Details of the construction of this statistic appear in Appendix C. The form of the pure-rank statistic as presented by Puri and Sen does not easily permit the use of existing computer software packages. Harwell and Serlin (1985) provide a form of the pure-rank test that allows existing computing packages to be used. Details of the derivation of the alternative form also appear in Appendix C. Assuming the XR and Y3 have been separately ranked, the ranks are submitted to a standard computing package (e.g., SAS, 1982), and the canonical correlations obtained from the output. The form of the PUR statistic presented by Harwell and Serlin (1985) is 47 N PUR - (N-l) 2 6r , (21) i-l where Dr represents the (squared) canonical correlation (eigenvalue) between the ranks of the Xk and the Y5. These are obtained by replacing the SCP matrices based on the original values of the XR and Y3 with the SCP matrices based on their ranks in expression (13). Puri and Sen (1985, p. 312) showed. that 'under the truth. of' the ‘hypothesis of expression (14), the pure-rank statistic is asymptotically distributed as a central chi-square ‘variable *with. pq degrees of freedom. The decision rule for the pure-rank test is to reject the hypothesis of expression (14) if PUR. exceeds the 100(1. - a) percentile of the chi-square distribution with pq degrees of freedom (Puri & Sen, 1985, p. 312). Rejection of the hypothesis of expression (14) implies that the population regression coefficients are not all simultaneously equal to zero, or, synonomously, that the population canonical correlations are not all equal to zero. The pure-rank test assumes that the Xk and Yj observations are i.i.d.r.v.'s whose common distribution function is G(EiIEi). The difference in assumptions between the normal-theory and pure-rank procedures is that 'normality' of" C(Xilzi) is not required for the pure-rank test. Mini-We As presented by Puri and Sen (1985, pp. 307-328), and discussed by Harwell and Serlin (1985), the mixed-rank statistic is computed using the original Xk values and the ranks of the Yj' This test assumes that all outcomes and predictors are i.i.d.r.v.'s and provides a test 48 of the hypothesis that all regression coefficients equal to zero, or, synonomously, all squared canonical correlations simultaneously equal to zero. A procedure similar to that of the pure-rank test was employed by Puri and Sen to obtain a large-sample form of the mixed-rank statistic. In the mixed-rank case, however, the original Xk values are used instead of their ranks. The mixed-rank (MIX) test statistic has exactly the same form as that of the pure-rank test given in expression (21). The decision rule for the mixed-rank test is the same as that of the pure-rank test, and rejection of the hypothesis of expression (14) implies that the two sets of variables are related. The assumptions of the mixed-rank test are the same as those of the pure-rank test. ta e This section outlines the method that was used to generate the multivariate data for the present study. The data generation and analysis were performed on an IBM 3090-180 computer at Michigan State University. The program was coded in FORTRAN V and incorporated a number of subroutines from the International Mathematical and Statistical Libraries (IMSL) (1983). A summary of the IMSL subroutines used in the present study is given in Appendix D. In all cases the data were in standard form (i.e., p - 0, 02 - l). The Basic Uniform Number Generator (GGUBS) subroutine of the International Mathematical and Statistical Libraries (IMSL, 1983) was used to generate random uniform deviates in the range of (0, 1). GGUBS has been extensively tested and has been found to produce deviates with good statistical 49 properties (Learmonth & Lewis, 1973). The resulting uniform deviates were transformed. into normal deviates using the Box-Muller (1958) approach. This procedure transforms a pair of uniform deviates (ul, u2) into a pair of standard normal deviates (21, 22) using the following transformations: z1 - (-2 logeul)1/2cos(2 n u (22) 2). - (-2 logeu2)1/zsin(2 x uz). 22 The resulting variable has (approximately) a mean of 0 and a variance of 1. In generating multivariate data the following structure was assumed to underlie the E values: I - d X + E . (23) W mm mm M where the X (predictor) and ,E (residual) matrices contain. random deviates from a specified distribution. These deviates were generated by specifying population correlations among the Xk variables and among the residuals and using the method described below. The Yj values were then obtained using specified values of the E in expression (23). In generating the multivariate data, a (p+q) x N matrix of standard normal deviates was initially generated using the transfor- mation given in (22). The first p rows of this matrix represented the uncorrelated residuals (E), and the remaining q rows the predictor values for a sample of size N. In all cases the Xk and Yj variables had the same distribution. The two matrices of uncorrelated normal deviates were then separately transformed such that the resulting deviates were 50 correlated with a specified distribution. To generate these correlated deviates the procedure due to ‘Vale and. Maurelli (1983) was used. Details of this procedure, which combines the approaches of Kaiser and Dickman (1962) and Fleishman (1978), are presented in Appendix B. The same procedure was followed separately for the predictor and residual correlation matrices. The Vale and Maurelli procedure begins by using the Kaiser and Dickman (1962) method to generate. a sample: of' multivariate-normal deviates using a matrix decomposition of the desired population correlation matrix, say E . A matrix Z of multivariate-normal deviates can be obtained using the following transformation: a - E z. . (24) (p+q)xN (p+q)x(p+q) (p+q)xN where E is a matrix of principal components (or some other decomposition) of the population correlation matrix E and g is a matrix of uncorrelated standard normal deviates. Multiplication of E and g produces variables with a mean and variance approximately equal to 0 and 1, respectively, and intervariable correlations approximately equal to those in E. To generate non-normal deviates the Vale and Maurelli procedure combines the Fleishman (1978) and Kaiser-Dickman methods. Fleishman (1978) developed a technique for generating a (univariate) non-normal variable, say wi , by finding the first four central moments (mean, variance, skewness, and kurtosis) of the distribution of the variable. The technique uses a polynomial involving the first three powers of a standard normal deviate zi: 51 2 3 wi - a + bzi + czi + dz1 , i - 1, 2, ..., N, (25) where a, b, c, and d are the so-called Fleishman power function constants. These constants were computed using the nonlinear equation- solving routine NEQNF (IMSL, 1983). Fleishman's (1978) procedure has been shown. to 'produce ‘non-normal deviates ‘with. the desired distributional properties (i.e., mean, variance, skewness, kurtosis) (Fleishman, 1978). In generating multivariate non-normal random deviates the processes of decomposition of the population correlation matrix and the Fleishman transformation interact, which leads to non-normal deviates with correlations different from those of the desired population. Vale and Maurelli (1983) developed a method to counteract this effect such that the resulting non-normal deviates would possess (approximately) the desired correlations. Essentially, this method involves the creation of an intermediate correlation matrix, 2*, from the desired correlation matrix E. The 2* matrix is then factored to obtain the E matrix of expression (24), and the matrix of multivariate- normal deviates is transformed. to ‘non-normal. multivariate deviates using expression (25). It should be noted that the data-generation process is not based on the probability density function of any theoretical multivariate distribution (e.g., multivariate-exponential), and hence the method does not actually produce data from such a distribution. Rather, the method produces data that have the same marginal skewness and kurtosis values as those of a theoretical multivariate distribution. However, 52 the Vale and Maurelli (1983) procedure has been shown to produce multivariate data with (asymptotically) the expected marginal (univariate) mean, variance, skewness, kurtosis, and correlations (Vale 6x Maurelli, 1983). The present study used five univariate summary measures (i.e., mean, ‘variance, skewness, kurtosis, correlation) to determine if the simulated data actually possessed the desired distributional properties. In addition, Mardia's (1974) measures of multivariate skewness and kurtosis were computed. S u at o ditio As noted earlier there are several factors which are expected to influence the distributional behavior of the tests. They include differences in the parent distributions as specified by 71 and 12, numbers-of-variables, between-set correlations, within-set correlations, and sample sizes. The simulation factors and their levels are shown in Table 4 and are discussed in detail below. Table 4 Simulation Factors Factor Level skewness and kurtosis [71, 12] [0, 0], [0, -l.12], [0, 3], [0, 20], [.5, 0], [1, .5], [1, 3], [2, 6] number-of-variables (p, q) (2, 2), (3, 3), (4, 4) between-set correlation .0 (Type I error), or >.0 (power) within-set correlation (py, px) (.3, .3), (.3, .7), (.7, .7) sample size (N) 25, 50, 100 53 Data were generated to represent observations from eight selected distributions representing a range of skewness and kurtosis values. The (univariate) skewness and kurtosis values of four known distributions (normal [0,0], uniform [0, -l.12], double-exponential [0, 3], and exponential [2, 6]), and four additional distributions were included. Pairings of skewness and kurtosis values allowed an examination of the effects of this factor over a broad range of non-normal conditions. Specifically, the {[0, -l.12], [.5, 0], [1, .5]} pairings represent three mildly non-normal distributions, {[0, 3], [1, 3]} two moderately non-normal distributions, and [[2, 6], [0, 20]} two extremely non-normal distributions. The three combinations of' the number-of- variables used [(2,2), (3,3), (4,4)] were chosen to examine the effects of increased dimensionality. Recall that the Rao E test is exactly distributed as an E variate under the null hypothesis when the smaller variable set contains two or fewer variables. The (3, 3) and (4, 4) combinations also reflected the increasing non-normality that tends to be associated with increasing numbers of variables. The three sets of values of the correlations within the set of outcome and predictor variables [(.3, .3), (.3, .7), (.7, .7)] represented a range of correlations encountered in practice. These combinations of within-set correlations permitted an examination of the effects of equal and unequal within-set correlation on the distributional behavior of the tests. For example, the power value of nonparametric rank tests appears to decrease slightly as the (absolute) within-set correlation increases for extremely non-normal data. The within-set correlation values allowed the behavior of the 54 tests under these conditions to be examined. Three different sample sizes (25, 50, 100) were also included in the present study. This range permitted an examination of the effect of varying sample size on the tests. Marascuilo and Levin (1983, p. 204) recommended that samples of larger than 10(p+q) should be used in multivariate studies. According to this recommendation, the chosen sample sizes represent small, small-moderate, and moderate-large samples, depending on the numbers of variables used. Finally, a range of between-set correlations (.0 s pxy S .366) were included to examine the Type I error and power properties of the tests. A zero between-set correlation corresponds to the Type I error case while a non-zero between-set correlation corresponds to the power case. The between-set correlation for the power case was obtained analytically using a procedure due to Muller and Peterson (1984) and the tabled power values of the E test due to Pearson and Hartley (1951). This procedure uses an approximation involving the non- centrality parameter of the E distribution. The non-centrality value was found such that a power of .8 would be achieved at an alpha level of .05 for a sample size of 100 and a multivariate-normal distribution. P se ta 10 o i at o su ts The 8x3x2x3x3 fully-crossed design employed in the present study generated a total of 432 simulation conditions. Three thousand replications were carried out for each condition, and the five test statistics (Bartlett, Rao E, rank-transform Rao E, pure- and mixed- rank) were calculated for each replication. The resulting Type I error 55 and power values were tabulated at three levels of significance (.01, .05, .10). The Type I error probability was estimated by the proportion of the number of rejections of the null hypothesis of expression (14) when the null condition was true (i.e. data were sampled from a multivariate population with a between-set correlation equal to zero). The robustness of the Type I error probability of the five tests was determined using a 95% confidence interval of the Type I error probability (i.e., a izlu96J[a(l - a)/3000], where a is the nominal Type I error probability}. The 95% confidence interval of the average Type I error probability was obtained using the standard error of the average empirical Type I error rate (i.e., a i l.96/[a(l - a)/3000n], where n is the number of the Type I errors involved in computing the average). A test was considered robust with respect to the Type I error probability if its empirical Type I error rate fell inside the confidence interval. Otherwise, the test was considered either conservative or liberal. The empirical power value was estimated by the proportion of the number of rejections of the null hypothesis of expression (14) when the null condition was false (i.e., data were sampled from a multivariate population with a nonzero between-set correlation). Evidence that the data generation method was actually producing data with the desired distributional characteristics was obtained by computing five (marginal) summary measures: average mean, variance, skewness, kurtosis, and correlations. In. addition to the marginal measures, multivariate measures of skewness and kurtosis proposed by 56 Mardia (1974) were computed and used to examine the skewness and kurtosis of the data with a parent multivariate-normal distribution. A brief description of these measures is given below with a detailed presentation of the statistics provided in Appendix G. Let g1, Q2, ..., EN be N vectors of random observations on t - (p + q) variables, Q the vector of sample means, and y the matrix of sample covariances: 91 ' l“11 ] fl ' la1 ] y ' lvll v12 V1: 1 (26) “21 u2 v21 v22 ”2: L“11:1 J Lut J L"t1 Vt2 " vtt ] Mardia (1974) proposed the following sample measures of multivariate skewness (11 t) and kurtosis (12 t) for a multivariate distribution with t dimensions: 11 t - N'2 2 2 [(Q1 - fir y'lm. - EH3 <27) ' i-l,j-l J -1 N - -1 — 2 vzt-N 2mg, wry (uj -.u>1 - c.08 g; i ’3 u “ g :3 8 RTF H MIX .4 PUR g.06 3 -6 r RAO >5 ”-4 "‘ 3: 8 5 'C 4 'H 0 D 3.04 PUR ”3 MIX I a L n l L L A Distribution Distribution .10. 1.0 . Q) a 23 Q) :.08, £3 .8_ CD 8 > L1 as 3 ... S m m E, '3 .6. TF .3 UR 1; _: RAO S Q. IX .H E ‘H LL} E. .4. {-5.01 [1.5] [1.3] [2.6] {-5.0} [1,-5] [1.3] [2.6] Distribution Distribution Figure 3. Type I Error and Power Curves by Distribution ( a -.05) 69 values on the Type I error and power values of the four tests. The average Type I error and power values and the total number of conservative and liberal Type I errors are presented in Table 10 and displayed in Figure 3. The Type I error results of Table 10 and Figure 3 indicate that the (a) Type I error rate of the Rao E test increased subtantially only for the [1, 3], exponential, and Cauchy distributions, and (b) Type I error rates of the rank-transform Rao E and the pure- and mixed-rank tests were not affected by distribution. The conservative and liberal Type I error results of Table 11 indicate that the Rao E test tended to produce more liberal Type I errors for the [1, 3], exponential, and Cauchy distributions. The percentage of liberal Type I errors increased from 0% for the normal distribution to 18% for the [1, 3] distribution, 83% for the exponential distribution, and 100% for the Cauchy distribution. In contrast, the number of liberal and conservative Type I errors for the rank-transform. Rao E and. the pure-rank tests did. not seem. to 'be affected by the degree of non-normality. The power results of Table 10 and Figure 3 indicate that the (a) power of the Rao ,E test tended. to increase only for the Cauchy distribution, and (b) power of the rank-transform Rao E, the pure-rank, and the mixed-rank tests tended to increase with increases in the kurtosis value of the parent distributions. Was Three sample sizes were used to examine the effects of varying sample sizes on the Type I error and power values of the four tests. The average Type I error and power values and the total number of 70 conservative and liberal Type I errors are presented in Table 11 and displayed in Figure 4. The Type I error results of Table 11 and Figure 4 indicate that the (a) Type I error rate of the Rao E tended to shrink toward the nominal alpha level as the sample size increased, (b) Type I error rate of the rank-transform Rao E test did not seem to be affected by increases in the sample size, and (c) Type I error rates of the pure- and mixed-rank tests tended to increase subtantially toward the nominal alpha level with increases in the sample size. Table 11 Average Type I Error and Power Values and Numberaof Conservative and Liberal Errors by Sample Size Type I Error Power N .01 .05 .10 ( C L ) .01 .05 .10 25 RAO .0159 .0593 .1088 ( 1 75) .0967 .2395 .3500 RTF .0108 .0515 .1012 ( 0 11) .0938 .2437 .3581 PUR .0026 .0287 .0747 (199 0) .0299 .1611 .2922 MIX .0020 .0264 .0702 (211 0) .0184 .1302 .2566 50 RAO .0150 .0581 .1073 ( 2 50) .2755 .4943 .6182 RTF .0101 .0498 .0986 ( 6 l) .2957 .5253 .6493 PUR .0054 .0390 .0873 (144 0) .2098 .4685 .6159 MIX .0045 .0378 .0862 (159 0) .1881 .4591 .6144 100 RAO .0145 .0564 .1070 ( 2 58) .6694 .8411 .9037 RTF .0103 .0503 .1031 ( 7 17) .7128 .8689 .9231 PUR .0077 .0452 .0970 ( 46 2) .6671 .8549 .9162 MIX .0073 .0442 .0952 ( 61 0) .6892 .8711 .9276 a The tabled values represent the average Type I error and power values across all distributions, within-set correlations, and numbers-of4variables (n-72, [.0491, .0509]). The ‘number' of conservative (C) and liberal (L) Type I errors are the total across three alpha levels (216 cases). 71 The conservative and liberal Type I error results of Table 12 indicate that the (a) number of liberal Type I errors of the Rao E decreased as the sample size increased, (b) number of liberal Type I errors of the rank-transform Rao E was largest for the moderate-large sample size, and (c) number of liberal Type I errors of the pure- and mixed-rank tests decreased subtantially as the sample size increased. The power results of Table 12 and Figure 4 indicate that (a) the power of all four tests increased subtantially with increases in the sample size, and (b) the increment in the power values was higher for the pure- and mixed-rank tests as the sample size increased from small-moderate to moderate-large. .00 3 7.; .03. "9————""“'“""""' RTF q, > —~———-- ——-——-—-—-- .3 . a 3 PER - .5 xxx ; E T .04. 5 ‘9 a. —‘ >~ :3 F .2. v-1 1- fl "" U D. 1: .03. 5 .02. 1 I 25 50 100 25 50 100 Sample Size Sample size Figure 4. Type I Error and Power Curves by Sample Size (a - .05) 72 W - e t Three combinations of the within-set correlation among the Y. variables and among the Xk variables were included to examine the effect of varying correlations on the Type I error and power values of the four tests. The average Type I error and power values and the total number of conservative and liberal Type I errors are presented in Table 12 and displayed in Figure 5. The Type I error results of Table 12 and Figure 5 indicate that the (a) Type I error rate of the Rao E tended to increase slightly as the within-set correlation among the Y variables and among the Xk J variables increased, and (b) Type I error rates of the rank-transform Table 12 Average Type I Error and Power Values and Number of Cogservative and Liberal Errors by Within-Set Correlations Type I Error Power (py, px) .01 .05 .10 ( C L ) .01 .05 .10 (.3 .3) RAO .0144 .0568 .1063 ( 3 56) .3007 .4823 .5862 RTF .0106 .0507 .1019 ( 3 14) .3269 .5079 .6092 FUR .0054 .0378 .0869 (127 l) .2672 .4584 .5738 MIX .0046 .0361 .0838 (139 0) .2737 .4615 .5759 (.3 .7) RAO .0150 .0579 .1080 ( 1 61) .4354 .6072 .6970 RTF .0103 .0506 .1007 ( 7 8) .4537 .6251 .7137 PUR .0053 .0376 .0863 (132 l) .3781 .5698 .6771 MIX .0045 .0359 .0838 (150 0) .3716 .5594 .6666 (.7 .7) RAO .0160 .0591 .1088 ( l 66) .3056 .4854 .5887 RTF .0102 .0504 .1004 ( 3 7) .3217 .5049 .6077 PUR .0050 .0375 .0858 (130 0) .2616 .4563 .5733 MIX .0047 .0365 .0841 (142 0) .2505 .4396 .5560 a The tabled values represent the average Type I error and power values across all distributions, sample sizes, and. numbers-of- variables (n972, [.0491, .0509]). The number of conservative (C) and liberal (L) Type I errors is the total across three alpha levels (216 cases). 73 Rao E and the pure-and mixed-rank tests were not affected by the increment in the within-set correlations. The conservative and liberal Type I error results of Table 12 indicate that the (a) number of liberal Type I errors of the Rao E was slightly higher for the largest within-set correlation, and (b) number of liberal Type I errors of the rank-transform Rao E and the pure- and mixed-rank tests varied slightly with the values of the within-set correlation. The power results of Table 12 and Figure 5 indicate that the (a) increment in the correlation among the XR variables tended to increase the power of all tests, (b) the increment in the correlation among the .06 -7 0 3 H m s g 3 ...05. ' ”‘3” g .6. o —_-—_—-—-_ -------- C. h M h u m 3 o H D. g. H 550’“ §.5_ RTF ... 5 : 4P“ 3:; RAG : ‘ 4—331x 0. 3 s PUR T: Lu MIX --¢ 2:03 '4- Lu 1 ‘ ‘ L (.3,.3) (.3..7) (.7..7) (.3..3) (.3..7) (.7..7) Within-Set Correlation Within-Set Correlation Figure 5. Type I Error and Power Curves by Within-Set Correlation (a - .05) 74 Y3 variables tended to decrease the power of all tests, and (c) change in the power values for all tests was approximately the same. W Three combinations of the number of Y .1 included to examine the effects of varying numbers-of-variables on the and Xk variables were Type I error and power values of the four tests. The average Type I error and power values and the total number of conservative and liberal Type I errors are presented in Table 13 and displayed in Figure 6. Table 13 Average Type I Error and Power Values and Number 2f Conservative Liberal Errors by Number-of-Variables b Type I Error Power V .01 .05 .10 ( C L ) .01 .05 .10 4 RAD .0137 .0556 .1031 ( 2 54) .3452 .5302 .6291 RTF .0107 .0506 .1002 ( 6 15) .3650 .5493 .6476 PUR .0065 .0425 .0934 ( 90 2) .3216 .5222 .6342 MIX .0057 .0414 .0913 (100 0) .3342 .5376 .6491 6 RAO .0155 .0584 .1088 ( 2 65) .3423 .5203 .6204 RTF .0102 .0515 .1025 ( 3 7) .3627 .5431 .6408 PUR .0051 .0381 .0874 (135 0) .2974 .4911 .6057 MIX .0045 .0355 .0842 (153 0) .2938 .4814 .5965 8 RAO .0162 .0599 .1112 ( l 64) .3542 .5245 .6224 RTF .0102 .0496 .1002 ( 4 7) .3746 .5455 .6422 PUR .0041 .0323 .0783 (164 0) .2878 .4711 .5844 MIX .0036 .0316 .0761 (178 0) .2678 .4415 .5529 a The tabled values represent the average Type I error and power values across all distributions, sample sizes, and within-set correlations (n-72, [.0491, .0509]). The number of conservative (C) and liberal (L) Type I errors is the total across three alpha b levels. V - number of variables. 75 The Type I error results of Table 13 and Figure 6 indicate that the (a) Type I error rate of the Rao ,E tended to increase ‘with increases in the number of variables, (b) Type I error rate of the rank-transform Rao E did not seem to be affected by the number of variables, and (c) Type I error rates of the pure- and mixed-rank tests tended to decrease as the number of variables increased. The conservative and liberal Type I error results of Table 13 indicate that (a) the number of liberal Type I errors of the Rao E increased. as the number of 'variables increased, (b) no particular pattern was found for the number of liberal Type I errors of the rank- F, (c) the number of conservative Type I errors of the transform Rao .06. RAO 7. 2 / To > E3105. 3.6. h RTF '; w i» H La 0 g ¥=RTF :1 o rr=RAO 9.01., 9‘05. H "‘ (U 3 g PUR 3 :: @- PUR 2 MIX” m.03. IX “.4. 1 a L n Ji 4 6 8 4 6 8 Number of Variables Number of Variables Figure 6. Type I Error and Power Curves by Number-of-Variables (a - .05) 76 pure-and mixed-rank tests increased substantially as the number of variables increased. The power results of Table 13 and Figure 6 indicate that the power of the Rao E and rank-transform Rao E tests did not vary with increases in the number of variables, and that the power of the pure— and mixed-rank tests tended to decrease as the number of variables increased. tera o t o u a Co d The next three sections present the Type I error and ‘power results categorized by distribution and sample size, distribution and within-set correlation, and distribution and number-of-variables. Complete results appear in Tables E3, E4, and E5 of Appendix E. e ze The average Type I error and power values and the total number of conservative and liberal Type I errors categorized by distribution and sample size for the .05 alpha level are presented in Table 14. The average Type I error and power values for the normal (thin-tailed), double-exponential (moderately non-normal/moderate-tailed), and exponential and Cauchy distributions (extremely non-normal/heavy- tailed) are displayed in Figures 7 and 8, respectively. The Type I error results of Table 14 indicate that the (a) Type I error rate of the Rao E varied only slightly for the normal and mildly and moderately non-normal distributions and decreased subtantially for the extremeLy non-normal distributions as the sample size increased, (b) Type I error rate of the rank-transform Rao E did 77 not vary much with sample size for all distributions, and (c) Type I error rates of the pure-and mixed-rank tests increased substantially toward the nominal alpha level as the sample size increased for all distributions . The power results of Table 14 indicate that as the sample size increased the (a) power values of all four tests increased subtantially for all distributions, (b) power values of the rank-transform Rao E and the pure- and mixed-rank tests increased at higher rates for all non-normal distributions other than the uniform and the [.5, 0] distributions, (c) power of the Rao E was largest for the normal, uniform, and the [.5, 0] distributions across all sample sizes, (d) power value of the rank-transform Rao E was largest for the [1, .5], double-exponential, [1, 3] , exponential, and Cauchy distributions and small and small-moderate sample sizes, and (e) the mixed-rank test produced the highest power value for the double-exponential, [1, 3], exponential, and Cauchy distributions and moderately-large sample size. 78 Table 14. Average Type I Error and Power Values and Number of Conservative and Liberal Errors by Distribution and Sample Size (a - .05)a Type I Error Power [11, 12] N-25(C L) N-50(C L) N-100(C L) N-25 N-50 N-100 [0. 0] RAO .0474(0 0) .0539(0 0) .0511(0 0) .2090 .4773 .8509 RTF .0499(0 1) .0500(0 0) .0500(2 0) .2047 .4548 .8202 PUR .0274(9 0) .0394(6 0) .0440(3 0) .1325 .3961 .8039 MIX .0256(9 0) .0423(6 0) .0446(2 0) .1253 .3995 .8159 [0, -l.12] RAD .0504(0 l) .0532(0 0) .0490(0 0) .1951 .4674 .8420 RTF .0528(0 l) .0515(0 0) .0483(0 0) .1879 .4291 .7924 PUR .0302(7 0) .0404(5 0) .043l(3 0) .1219 .3740 .7734 MIX .0299(7 0) .0409(4 0) .0439(3 0) .1237 .3787 .7764 [.5, .0] RAO .0521(0 0) .0487(0 0) .0512(0 0) .2041 .4754 .8435 RTF .0517(0 0) .0484(0 0) .0499(0 0) .2023 .4563 .8191 PUR .0287(9 0) .0385(5 0) .0458(1 0) .1299 .4026 .8012 MIX .0277(9 0) .0366(6 0) .0459(1 0) .1182 .3980 .8106 [1, .5] RAD .0514(1 1) .0503(l 0) .0483(0 0) .2245 .4802 .8401 RTF .0507(0 0) .0501(0 0) .0493(0 0) .2417 .5315 .8885 PUR .0282(9 0) .0399(7 0) .0450(3 0) .1598 .4751 .8739 MIX .0272(9 0) .0385(6 0) .0437(2 0) .1164 .4352 .8849 [0. 3] RAO .0535(0 0) .0516(0 0) .0519(0 0) .2279 .5037 .8459 RTF .0494(0 0) .0521(0 0) .0488(0 0) .2319 .5184 .8700 PUR .0279(9 0) .04l4(3 0) .0439(4 0) .1515 .4606 .8573 MIX .0259(9 0) .0376(6 0) .0440(1 0) .1339 .4735 .8826 [1. 3] RAO .0566(0 3) .0503(0 0) .0557(0 4) .2340 .4993 .8415 RTF .0529(0 l) .0472(l 0) .0520(0 2) .2372 .5139 .8665 PUR .029l(9 0) .0357(9 0) .0463(3 0) .1551 .4564 .8511 MIX .0264(9 0) .0355(9 0) .0458(3 0) .1253 .4474 .8768 [2. 6] RAO .0665(0 9) .0687(0 9) .0603(0 5) .2799 .5005 .8230 RTF .0515(0 1) .0516(0 0) .0533(0 l) .2951 .6144 .9366 PUR .0287(8 0) .0399(8 0) .0479(0 0) .1980 .5553 .9277 MIX .0250(9 0) .036l(9 0) .0440(2 0) .1074 .4707 .9427 [0, 20] RAO .096l(0 9) .0884(0 9) .0840(0 9) .3417 .5509 .8420 RTF .0535(0 1) .0471(l 0) .0510(0 2) .3469 .6839 .9577 PUR .0296(7 0) .0366(9 0) .0459(2 0) .2397 .6277 .9510 MIX .0235(9 0) .0351(9 0) .04l9(4 0) .1918 .6698 .9784 a The tabled values represent the average and the number of conservative (C) and liberal (L) Type I error rates and the average power values across all within-set correlations and numbers-of- variables (n~9, [.0474, .0526]). 79 Normal D-Exponential .03. .08. 3 3 23 I; >.06. >.06. "" L. 3 e {:3 _ X0 9.. RAG RIF *9 RT? 1'04 P PLR :- 04 ,. x >~ F H '7‘ H 1‘: 2 :23. k u 2.02 . 3,02 _ 5 e m m l L l L 25 50 100 25 50 100 Sample Size Sample Size Exponential Cauchy .08. RAG .08. .06_."fl”fl.~“‘\‘-‘-~\gRAG W RTF PUR m 3 33’ m > :3 “ u 8 e h u “. m.06, H "‘ MIX H 3- a :7. >‘ RTF 9‘ PUR 3 p1 MIX m U u 0 fi fi “ u ”a. -4 If§.02. 3. In .02. 1 r L 1 l— 25 50 100 25 50 100 Sample Size Sample Size Figure 7. Type I Error Curves by Distribution and Sample Size (a - .05) Empirical Power Value Empirical Power Value Normal RAO RTF . MIX PUR RAG - RTF PUR MIX 25 50 100 Sample Size MIX Exponential RTF PUR 25 50 Sample Size 100 80 Empirical Power Value Empirical Power Value D-Exponential 50 Sample Size Cauchy 25 50 100 Sample Size Figure 8. Power Curves by Distribution and Sample Size (a - .05) 81 o W - e The average Type I error and power values and the total number of conservative and liberal Type I errors categorized by distribution and within-set correlation for the .05 alpha level are presented in Table 15. The average Type I error and power values for the normal, doub le - exponential , exponential , and Cauchy dis tributions are displayed in Figures 9 and 10, respectively. The Type I error results of Table 16 indicate that (a) the Type I error rates of the rank-transform Rao E and the pure- and mixed-rank tests were not affected by the within-set-correlation values for all distributions, and (b) except for the Cauchy in which the Type I error rate increased with increases in the within-set correlation, the Type I error rate of the Rao E was minimally affected by the within-set- correlation values. The power results of Table 15 indicate that the power of all four tests increased as the within-set correlation among the predictor variables increased from .3 to .7, and decreased as the within-set correlation among the outcome variables increased from .3 to .7 for all distributions. 82 Table 15 Average Type I Error and Power Values and Number of Liberal Errors by Distribution and Within-Set Correlation (a - .05)a Type I Error Power [11, 12] (.3 .3)(C L)(.3 .7)(C L)(.7 .7)(C L) (.3 .3)(.3 .7)(.7 .7) [0. 0] RAD .0508(0 0) .0508(0 0) .0508(0 0) .4687 .5999 .4686 RTF .0499(1 0) .0504(0 l) .0494(1 0) .4472 .5814 .4511 PUR .0369(6 0) .0369(6 0) .0369(6 0) .4003 .5383 .4039 MIX .0370(6 0) .0370(6 0) .0384(5 0) .4070 .5288 .4050 [0, -l.12] RAD .0514(0 0) .0502(0 0) .0510(0 l) .4583 .5897 .4565 RTF .0513(0 0) .0507(0 0) .0506(0 l) .4257 .5560 .4296 PUR .0385(5 0) .0373(6 0) .0379(4 0) .3817 .5038 .3838 MIX .0387(4 0) .0377(6 0) .0382(4 0) .3837 .5084 .3866 [.5, .0] RAD .0505(0 0) .0505(0 0) .0510(0 0) .4653 .5919 .4658 RTF .0496(0 0) .0511(0 0) .0493(0 0) .4461 .5789 .4526 PUR .0378(6 0) .0382(4 0) .0370(5 0) .3997 .5251 .4090 MIX .0369(6 0) .0367(5 0) .0366(5 0) .4027 .5236 .4006 [1, .5] RAD .0488(2 0) .0495(0 0) .0516(0 l) .4710 .6013 .4725 RTF .0502(0 0) .0494(0 0) .0505(0 0) .5118 .6389 .5111 PUR .0383(5 0) .0374(7 0) .0374(7 0) .4636 .5822 .4631 MIX .0355(5 0) .0370(6 0) .0369(6 0) .4504 .5573 .4288 [0. 3] RAD .0520(0 0) .0517(0 0) .0533(0 0) .4831 .6087 .4857 RTF .0506(0 0) .0496(0 0) .0501(0 0) .5004 .6208 .4992 PUR .0374(5 0) .038l(6 0) .0377(5 0) .4521 .5652 .4520 MIX .0359(5 0) .0359(6 0) .0357(5 0) .4688 .5697 .4515 [1. 3] RAD .0543(0 2) .0542(0 2) .054l(0 3) .4821 .6084 .4842 RTF .0511(0 2) .0507(0 0) .0503(1 l) .4976 .6203 .4997 PUR .0373(7 0) .0367(8 0) .0370(6 0) .4482 .5646 .4498 MIX .0357(7 0) .0355(7 0) .0365(7 0) .4559 .5571 .4365 [2. 6] RAD .0623(0 7) .0666(0 8) .0666(0 8) .4922 .6123 .4989 RTF .0514(0 0) .0527(0 0) .0523(0 2) .5868 .6853 .5740 PUR .0385(6 0) .0389(5 0) .039l(5 0) .5318 .6272 .5220 MIX .0354(7 0) .0350(7 0) .0348(6 0) .5019 .5708 .4482 [0, 20] RAD .084l(0 9) .0898(0 9) .0946(0 9) .5381 .6453 .5513 RTF .0512(0 2) .0502(l l) .0503(0 0) .6479 .7190 .6216 PUR .0378(7 0) .037l(5 0) .0372(6 0) .5894 .6617 .5672 MIX .0334(8 0) .0321(8 0) .0351(6 0) .6214 .6591 .5594 a The tabled values represent the average and the number of conservative (C) and liberal (L) Type I error rates and the average power values across all sample sizes and numbers-of- variables (np9, [.0474, .0526]). Empirical Type 1 Error Value Empirical Type I Error Value .08 .06 .04 .02 (. .08 Normal F r -“- --- - Cg ....... - RA E-------§ ———————— 1 RT? . ‘ fidulx ‘::* *' PUR L n L 3,.3) (.3,.7) (.7..7) Within-Set Correlation Exponential / f: RAD .06. ------ ~ g RTF '04 "cw 4* 4mm .02 l- l L l (.3,.3) (.3,.7) (.7..7) Within-Set Correlation 83 .08. Empirical Type I Error Value Empirical Type I Error Value '08? O 0‘ I o b O N j D-Exponential RAD :5: A- ——oRTF 1' ----d-‘-.U---- ------ . “tr“ ~—:PUR :: 3 4MIX L L (.3..3) (.3,.7) (.7..7) Within-Set Correlation .Ior Cauchy /RAO . 06.. a: ---- fin ........ fl ' 7 RTF .04. H r 4 PUR H , #4 MIX .02 . 1 L l (.3..3) (.3,.7) (.7..7) Within-Set Correlation Figure 9. Type I Error Curves by Distribution and Within- Set Correlation (a -.05) Empirical Power Vaiue Empirical Power Value 0‘ o b Normal [— (.3,.3) (.3,.7) (.7:.7) Within-Set Correlation Exponential ',/’//”’/.\\\\\\\\\.RTF PUR RAD MIX I L (.3,.3) (.3,.7) (.7..7) Within-Set Correlation 84 Empirical Power Value Empirical Power Value D-Exponential L 6. RTF RAD iii 4 L .2. (.3,.3) (.3,.7) (.7..7) Within-Set Correlation Cauchy .8. L J (.3..3) (.3,.7) (.7..7) Within-Set Correlation Figure 10. Power Curves by Distribution and Within-Set Correlation (a -.05) 85 e - -V The average Type I error and power values and the total number of conservative and liberal Type I errors categorized by distribution and number-of—variables for the .05 alpha level are presented in Table 16. The average Type I error and power values for the normal, double- exponential, exponential, and Cauchy distributions are displayed in Figures 11 and 12, respectively. The Type I error results of Table 16 indicate that (a) only for the exponential and Cauchy distributions did the Type I error rate of the Rao E increased as the number of variables increased, (b) the Type I error rate of the rank-transform Rao E was not affected by the number-of-variables factor for all distributions, and (c) the Type I error rates of the pure-and mixed-rank tests decreased as the number of variables increased for all distributions. The power results of Table 16 indicate that the (a) power values of the Rao E and rank-transform Rao E tests were not affected by the number-of-variables factor for all distributions, and (b) power values of the pure- and mixed-rank tests decreased with increases in the number of variables for all distributions. Table 16 86 Average Type I Error and Power Values and Number of Conservative and Liberal Errors by Distribution and Number-of-Variables (a - .05)a Type I Error Power [11, 12] V-4(C L) V-6(C L) V—8(C L) V-4 V-6 V-8 [0. 0] . RAD .0494(0 0) .0528(0 0) .0502(0 0) .5273 .5031 .5067 RTF .0468(2 0) .0526(0 1) .0504(0 0) .4984 .4850 .4963 PUR .0386(6 0) .0389(6 0) .0332(6 0) .4719 .4340 .4267 MIX .0409(5 0) .0386(6 0) .0329(6 0) .4829 .4354 .4224 [0, -1.l2] RAD .053l(0 l) .0512(0 0) .0483(0 0) .5149 .4960 .4937 RTF .0518(0 l) .0519(0 0) .0489(0 0) .4784 .4646 .4683 PUR .0439(2 0) .0390(4 0) .0308(9 0) .4526 .4170 .3998 MIX .0444(1 0) .0392(4 0) .03ll(9 0) .4578 .4186 .4023 [.5, .0] RAD .0515(0 0) .0504(0 0) .0501(0 0) .5141 .5045 .5044 RTF .0500(0 0) .0526(0 0) .0474(0 0) .4906 .4908 .4963 PUR .0426(3 0) .0393(S 0) .0310(7 0) .4640 .4402 .4295 MIX .0422(3 0) .0363(6 0) .0317(7 0) .4746 .4397 .4126 [1, .5] RAD .0503(1 0) .0485(1 0) .0512(0 l) .5223 .5067 .5159 RTF .0494(0 0) .0520(0 0) .0487(0 0) .5633 .5487 .5497 PUR .0422(5 0) .0387(6 0) .0323(8 0) .5363 .4970 .4755 MIX .0426(3 0) .0357(6 0) .03ll(8 0) .5386 .4717 .4262 [0. 3] RAD .0480(0 0) .0553(0 0) .0539(0 0) .5335 .5233 .5207 RTF .0487(0 0) .0540(0 0) .0476(0 0) .5393 .5394 .5417 PUR .0410(5 0) .0402(3 0) .0320(8 0) .5130 .4879 .4685 MIX .0400(3 0) .0360(7 0) .03l6(6 0) .5373 .4929 .4599 [1. 3] RAD .054l(0 3) .053l(0 3) .0553(0 l) .5260 .5237 .5250 RTF .0510(l 2) .0504(0 0) .0507(0 l) .5355 .5365 .5456 PUR .0426(6 0) .0362(8 0) .0323(7 0) .5083 .4859 .4685 MIX .0419(6 0) .0323(9 0) .0334(6 0) .5244 .4820 .4430 [2. 6] RAD .0590(0 6) .0662(0 9) .0702(0 8) .5320 .5330 .5385 RTF .0526(0 l) .0509(0 0) .0529(0 1) .6247 .6169 .6044 PUR .0441(4 0) .0378(6 0) .0346(6 0) .5954 .5604 .5251 MIX .0394(7 0) .0344(6 0) .03l4(7 0) .5970 .5025 .4213 [0. 20] RAD .0793(0 9) .0894(0 9) .0999(0 9) .5714 .5723 .5909 RTF .0543(0 3) .0474(l 0) .0500(0 0) .6639 .6629 .6617 PUR .0453(4 0) .0344(7 0) .0323(7 0) .6630 .6067 .5756 MIX .0397(6 0) .03l4(8 0) .0294(8 0) .6881 .6081 .5439 a The tabled values represent the average and the number of conservative (C) and liberal (L) Type I error rates and the average power values across all sample sizes and within-set correlations (II-9. [.0474, .0526]). .08 Empirical Type I Error Value .06 .10 Empirical Type I Error Value .08 .06 .04 .02 Normal 1 l 8 Number 60f Variables . Exponential RAO L 6 Number of Variables 87 Empirical Type I Error Value .03 . .04. .02- Empirical Type I Error Value D-Exponential L j n 6 8 Number of Variables .00 p .08 err/zzzzr:i:::::/””’.RAG .06. RTF '04” \ t\~\~\\\‘\._—__—-_—::PUR MIX .02» 4 6 8 Number of Variables Figure 11. Type I Error Curves by Distribution and Number- of-Variables (a -.05) 88 0‘ Empirical Power Value 5 o N 0‘ J.\ Empirical Power Value 0 N Number of Variables Normal D-Exponential b 08? n 306- E .\ RAG > i 3&8 k 4‘ :gRTF g PUR MIX \_=3PUR ° . MIX “.4. H m 0 6H H w-i o. E _ ”.2. 4 6 8 4 6 8 Number of Variables Number of Variables Exponential Cauchy ' 08' ——3RTF *— + 44m 2; 6 D :0 ..-——v $88 RAD >’ P ’5 - 2'32 where E is the expected matrix product of 11 and 22: l 0 1 0 0 pz 2 0 3P2 2 l 0 2,02 2 + l 0 l 2 0 3p 0 6p + 9p _ 2122 2122 2122 . Returning to the vector and matrix product of expression (40) the correlation pw W is given by the scalar expression 1 2 pw W - p (b2 + 6bd + 9d2 + 2c2pz z + 6d2p: z ). (41) 1 2 2122 l 2 l 2 The values of pz obtained by solving the polynomial of expression 2 l 2 * (41) provides the intermediate correlation matrix E . naltizariate_§keznesa_and_surtcaia The sample measures of Mardia (1974) multivariate skewness and kurtosis values are given in expressions (27) and (28) of Chapter III. The following algorithms were used to compute these values: _2 N N 3 71,: ' N 1E1 j§1 313 ' (“2) _1 N 2 72,: ' N 1§1 311 ' t(t + 2)’ (43) 108 where gij is the (ij)th element of the matrix 1 (In - i] [u - I]..[u - El) 2‘ {[u - I] [u - u]..[u - El). (44) 1 2 N Nxt txt 1 2 N th where Q1, Q2, ..., EN and-E are defined in expression (26) of Chapter III. Iaata_2f.flaltixariata.fl2:malitI Mardia (1970) introduced two tests of multivariate normality based on the sampling distributions of the multivariate skewness and kurtosis statistics. The null hypotheses for the tests are that both the population skewness and kurtosis values are equal to zero. The test statistics are: 2 A - (N71,t)/5 “ Xt(t+l)(t+2)/5 B - 12,t//[8t(t+2)/N1 ~ N<0. 1). (46) (45) where t is the number of variables. The sample measures of the multivariate skewness and kurtosis and the corresponding test-statistic values are given in Table Bl. Table Bl Tests For Multivariate Normality (Mardia, 1974) ta Measure Nb Value Statistic dfc cvd dce 4 skewness 300 .3919 19.595 20 31.41 NS kurtosis 300 -.3094 -.387 -l.96 NS 6 skewness 300 1.0867 54.335 56 74.45 NS kurtosis 300 -.6064 -.536 -l.96 NS 8 skewness 300 2.3257 116.285 120 124.34 NS kurtosis 300 -l.03l8 -.706 -l.96 NS a t - number of variables, b N - sam le size, c df -degrees of freedom, cv - critical value, dc - decision, NS - not significant. 109 WW1»: The power value of the Rao E test was computed analytically using a method due to Muller and Peterson (1983) and the tabled power values of the E test due to Pearson and Hartley (1951). Given the matrices of regression coefficients (fl) and the within-set correlations among the dependent variables (311) and among the predictor variables (322), the power value of the Rao E test can be determined using the following procedures: (1) (2) (3) Let E be the matrix of intercorrelations among the dependent and predictor variables 8 B 11 12 B - [ ] (47) R21 322 312, the matrix of between-set correlations of the dependent and predictor variables, is given by (Timm, 1975, p. 309): Wilks' lambda (A) can be obtained using the ratio of the determinants of E, R _11, and 322 (Anderson, 1958, p. 233) A - Isl/(Inullezzh. (49) The nonrcentrality parameter for the Rao F can be obtained using the formulas given by Muller and Peterson (1983) and Pearson and Hartley (1951): A, - <1 - Al/b)/(A1/b/v2). (50) d - (AA/v1 + 1)]'/2 (51) 110 where b, and v are given by expressions (17) of Chapter III. ”1' 2 (4) The power value is obtained using the power charts of the E test (Pearson & Hartley, 1951) and. the non-centrality' parameter of expression (51). The power values of the Rao E test are given in Table B2. Table B2 a Theoretical and Empirical Power Values of the Rao E Test Within-set correlation Measure (.3, .3) (.3, .7) (.7, .7) fi .180 .180 .180 312 . .234 .306 .306 A .870 .831 .870 AA 13.798 18.680 13.798 d 1.660 1.933 1.660 Theoretical power .840 .940 .840 Empirical power .815 .910 .815 a Tabled power values were based on N-100 and t - 4. The empirical power values were obtained using 3,000 replications for a normal distribution. As totic Relat ve Ef enc The efficiencies of two test statistics can be compared using their power values for various alternative hypotheses and sample sizes. However, a single measure of their relative effeciency can be obtained using the so-called asymptotic relative efficiency (A.R.E.). The computation of the A.R.E. of the pure-rank test to the normal-theory likelihood-ratio test for regression (Puri & Sen, 1985, pp. 316-317) is outlined here. Let {Z' - (1', z') - (Y1, Y2,..., Y , X1, X2,..., Xq)} be a set P of random variables with a normal density function (d.f.) F(;), F1(3) 111 be the marginal d.f. for the X variables, and Go(y - £0 - fl'x) be the conditional d.f. of I given X - 3. Let f(z), f1(x), and go(y) be the probability distribution functions (p.d.f.'s) corresponding to the F, F1, and C respectively. Then f(;) can be written in terms of the 0’ conditional and marginal p.d.f's as follows: f(z) - sou: - £0 - fl'x)f1(x) (52) Under Ho: 5 - Q, expression (52) becomes f(z) - 20(2 - 20>£1<2) (53) The likelihood function can then be written as n L(Zl. 1290--o an) - n1 1— n f(zi) -iglgo(xi -flo - fl xi)£1(zi)° (54) Denoting the maximum likelihood estimator of E by a, the likelihood- ratio statistic is given by n g (X > An - n 0 1 . (55> 1-1 30(11 - 2X) Under Ho (i.e., fl - Q) -2 log(An) is asymptotically distributed as the central chi-square distribution with pq degrees of freedom. Under H1, the statistic has a noncentral chi-square distribution with pq degrees of freedom and a noncentrality parameter, say A The A.R.E. of the A' pure-rank test (L) with respect to An is A e(L, A) - __L_ , (56) A A 112 where AL is the noncentrality parameter for the chi-square distribution for the pure-rank test under H It has been shown that for a parent 1. multivariate normal population the pure-rank test and the normal-theory likelihood-ratio test are asymptotically power equivalent (Puri & Sen, 1969). For other forms of continuous distributions the A.R.E. of the pure-rank test to the likelihood ratio-test is bounded below by 0.864. APPENDIX C Ihg Eggg-Rang Statistig: Egg; and §en Fog; The construction of the pure-rank statistic (Puri & Sen, 1985, pp. 307-312) is outlined in this appendix. Let [11, Xi] - [Y11, YZi’ ., Ypi’ X11, X21... xqil’ i -. 1, 2, .., N, be a vector of random observations for the i(th) subject on Y1, Y2,.., Yp dependent and X1, X2..., Xq predictor variables having an identical (p + q) -variate continuous distribution function. Let Rji and Rk1 represent the rank of the i(th) subject on the j(th) dependent and k(th) predictor variables, respectively. The original and the rank values of the Yj and Xk can be represented as matrices E and E, respectively: :11 :12 §1N :11 :12 :IN 21 21 2N 21 22 " 2N Y Y .. Y R R .. R H _ pl p2 pN B _ p1 p2. pN (57) :11 £12 filN gp+l,1 §p+1,2 §p+l,N 21 22 '° 2N p+2,l p+2,2 " p+2,N X X .. X R ql q2 qN p+q,l Rp+q,2 .. Rp+q,N Since the N vectors of observations are independent each row of E represents a permutation of of integers l, 2,..., N (assuming no ties 'by virtue of continuity of the distribution function of the Yj and Xk), ‘With a total of N! permutations. Since 3 contains (p + q) rows, under the truth of the hypothesis of independence of the YJ. and Xk, the total number of possible realizations of E is (N!)(p+q). ll3 114 Following Chatterjee and Sen (1964), two rank matrices are said to be permutationally equivalent if it is possible to obtain the second matrix by permutations of the columns of the first matrix. Suppose the columns of R.are rearranged in such a way that the first row has the elements in the natural order 1, 2, ..., N, and denote the corresponding matrix by 3*. E is said to be permutationally equivalent to 3* if it is possible to obtain 3* by permutations of the columns of E. Since there are N columns in E, for a given 3* there will be a total of N! possible realizations of E 'which. are permutationally equivalent to 3*. In general, the probability distribution of 3 over (N!)(p+q) possible realizations of 3 depends on the joint distribution of the Yj and Xk. However, given a particular 3* the conditional distribution of 3 over the N! permutations of the columns of 3* is uniform under the truth of the hypothesis of independence of the Y and Xk. An exact test J of independence of Y and Xk may be computed using the distribution of J 3. However, the arithmetic is excessive and. a large-sample approximation is of principal interest. Puri and Sen (1985, pp. 307-312) presented a large-sample test based on the sum-of-cross-products vector 5 of the centered R1 and Rk and the covariance matrices E of the Rj and Q of the Rk with elements sjk -igl(Rji ’ Rj)(Rki ' R19’ (58) -1 - - - " . ' . .'- 3": o 59 mjj. N 121011, Rszj.i R1.) 3.3 1.2 p < > -1 - — ' c... -N Elma-Rammed ...-1,2,..,., <60) 115 where R3, R1,, Rk, and Ek' are the rank means for the j(th) and j'(th) dependent and k(th) and k'(th) predictor variables, respectively. Puri and Sen showed that for a large N the expected values of E and E'S are: 3(5) - Q. (61) E(fi'fi) - N(fl 0 Q). (62) where (M O Q) represents the Kronecker product of u and Q (see Anderson, 1958, p. 347). The large-sample pure-rank statistic (L) is given by L - N'lts (M e 9'1 3'1. (63) Puri and Sen showed that the L statistic is distributed as a chi-square variable with pq degrees of freedom when the YJ and Xk variables are independent. The L statistic is genuinely distribution- free for p - q - l, but only (conditionally) permutationally distribution-free for p > 1 or q > 1. e - nk Test: a e 1 and Se 0 Harwell and Serlin (1985) derived a simpler form of the Puri and Sen L statistic using canonical correlations among the Rj and Rk values. Defining a - E 9-1, where a is a matrix of sample regression coefficients based on ranks, Harwell and Serlin showed that the L statistic has the definitional form: N N N N L - 2 2 2 2 dJk'j'k' 3-1 j'-1 k-l k'-1 sjk sj,k, , (64) 116 N N N N ., , L - 2 2 2 2 mJJ ckk S S , , , 3-1 j'-1 k—l k'-1 3k 3 k where djk’j'k' represent the elements of the (3.1 0 9:1), and mjj'and ckk' represent the elements of M-1 and 9'1, respectively. In this definitional form the elements of 5 appear as part of a quadruple sum across products of the elements of H and 9, thus making the computation of expression (64) extremely difficult. In the derivation of the simpler form of the L statistic, Harwell and Serlin showed that the summation of expression (64) may be written as the matrix product: m4>h(flalfim mm (N-l) Ir (3 9'1 9'1 3' 2 2'1) (N-l) Tr < 3 9'1 3' 1(1) 1" I (N-l) (sum of eigenvalues of resulting matrix). The eigenvalues (squared canonical correlations) of the resulting matrix are the eigenvalues among the sets of the R and Rk values, J at (r - l,2,..,s). The L statistic can then be written in the form 3 L - (N-l) 2 9 , r - l, 2,..., s. (66) r r-l The L statistic, as shown by Puri and Sen, is asymptotically distributed as a central chi-square variable with pq degrees of freedom when the Y and Xk variables are indepenedent. J This APPENDIX D COMPUTER PROGRAMMING appendix describes the subroutines of the Statistical Package for the Social Sciences (SPSSX. 2.2) and. the International Mathematical and Statistical Libraries (1983) used in. the present study. The subroutine FACTOR of the SPSSX was used to obtain the principal component weights of the intermediate correlation matrices. The following subroutines of the IMSL were used for data generation and computation of test statistics: NEQNF GGUBS VMULFP LINVlF NMRANK EIGRF LINV3F To solve a set of simultaneous nonlinear equations for Fleishman power function constants To To To To To To To To generate uniform random deviates multiply matrices multiply the transpose of a matrix A by a matrix B multiply a matrix A by the transpose of a matrix B compute the inverse of a matrix rank the generated deviates compute eigenvalues of a matrix compute determinant of a matrix The complete listing of the computer program, which was coded in FORTRAN V, is given below. 117 CC-------------------------—- 118 PROGRAM NONPAR CC TYPE I ERROR RUN (NV-4,NP-2,NQ-2,CY-0.3,CX-0.3,NS-25,NL-3000). CC DISTRIBUTION: NORMAL AND UNIFORM. cc----------------------------B'-------------------------------------=- 0‘ fibfl‘hhmhbhhh hbfl‘hfl‘fih INTEGER NL,NS,TS,NP,NQ,NV,AL,NT,WA,BC PARAMETER(NL-3000,NV-4,NS-25,TS-100,NP-2,NQ-2,AL-3,NT-5,WA-8, BC-l) INTEGER DGT,EJOB,DJOB,DIS,TST,REJB(AL),REJR(AL),REJT(AL), REJP(AL),REJM(AL),LOOP,IER,TN,IR(NS) REAL RB(NQ,NP),PE(NP,NP),PE1(NP,NP),PE2(NP,NP),PX(NQ,NQ), PXl(NQ,NQ),PX2(NQ,NQ),PH,EPS,D1,AA,BB,CC,DD,SUMD(NV), SUMDl (NV) , SUMD2 (NV) . SUMD3 (NV) , SUMD4 (NV) ,AVD (NV) , AVD1(NV),AVD2(NV),AVD3(NV),AVD4(NV),SUMMS,SUMMK,SUMS, SUMK,UD(TS),ND(TS),UNE(NP,NS),UNX(NQ,NS),MNE(NP,NS), MNX(NQ,NS),MNNE(NP,NS),MNNX(NQ,NS),MBX(NP,NS),CY,CX, MNNY(NP,NS),DATD(NV,NS),VAR(NV),DEVD(NV,NS),AAA(NV,NV), COVN(NV,NV),COVI(NV,NV),WK(WA),DCV(NS,NV),DCD(NS,NS). MULS,MULK,TMEAN.TVARN,TSKEW,TKURT,ACOVN(NV,NV),VARN(NV) SKEW(NV),KURT(NV),OMEAN,OVARN,OUSKW,OUKUR,OMSKW,OMKUR, CORR(NV,NV),CORL(NV,NV),BCR(BC),D2,DCOR,RCP(NV,NV), SUMCC(NV,NV) REAL AYY(NP,NP),AYX(NP,NQ).AXX(NQ,NQ),DTA(NS),R(NS),DTR(NS), S,T,DATR(NV,NS),AVR,DEVR(NV,NS),RRR(NV,NV),RYY(NP,NP), RYX(NP.NQ).RXX(NQ.NQ),DVYR(NP.NS)oDVXD(NQ,NS)3 MYX(NP.NQ).AYYI(NP.NP),AXXI(NQ,NQ),RYYI(NP.NP), RXXI(NQ,NQ),MATDl(NP,NQ).MATD2(NQ,NP),MATD(NP,NP),A1, MATRl(NP,NQ),MATR2(NQ,NP),MATR(NP,NP),MATMl(NP,NQ),Bl, MATMZ(NQ,NP),MATM(NP,NP),PED,PER,SER,SEM,B2,V1,V2,V3, BAR,RAO,RTF,PRN.MRN,C51,CSZ,CS3,CF1,CF2,CF3 COMPLEX ED(NP),ER(NP),Z(NP,NP) DOUBLE PRECISION DSEED DSEED -66901.D0 cc -------------------------------------------------------------------- CC (1) cc -------------------------------------------------------------------- CC CC CC CC TO SPECIFY SIMULATION CONDITIONS. (1A) TO SPECIFY REGRESSION COEFFICIENT MATRIX (RB), PRINCIPAL COMPONENT WEIGHTS FOR ERRORS (PE) AND PREDICTORS (PX), WITHIN- SET CORRELATION (CY,CX), PHI(PH), TIE VALUE (EPS) FOR RANKING, PARAMETER VALUES FOR THE INVERSE (DGT), EIGENVALUE (IZ,IJOB). AND DETERMINAT (D1,DJOB) PROCEDURES. OPEN(20,FILE-’TESTS') DATA RB / .0. .0. .0. .0 / DATA PEl/ .80623, .80623, -.59161, .59161/ DATA PXl/ .80623, .80623, -.59161, .59161/ DATA PEZ/ .81475, .81475, -.57981, .57981/ DATA PXZ/ .81475. .81475, -.57981, .57981/ CY-O.3 CX-0.3 TN-NS'NL PH-3.1428571 BPS-0.000001 DGT-O EJOB-O DJOB-4 Dl-l.0 (13) TO OBTAIN A1, 81, 82, V1, V2 FOR COMPUTING TEST STATISTICS AND TO SPECIFY CRITICAL VALUES: F .01, .05, .10 (CF1,CF2,CF3) AND CHI-SQUARE .01, .05, .10 (CSl.CSZ.CS3). Al-(NS-l.)-(NP+NQ+1.)/2. 119 Bl-SQRT((NP‘NP'NQ‘NQ-4.)/(NP‘NP+NQ‘NQ-5.)) BZ-l./Bl Vl-NP‘NQ v2-A1*Bl-V1/2.+1. v3-v2/v1 IF (V1 .EQ. 4)THEN CSl-13.277 CSZ- 9.488 CS3- 7.779 IF (NS .EQ. 25)THEN CF1- 3.800 CFZ- 2.590 CF3- 2.080 ENDIF IF (NS .EQ. 50)THEN CFl- 3.526 CP2- 2.466 CF3- 2.008 ENDIF IF(NS .EQ. 100) THEN CF1- 3.416 CFZ- 2.422 CF3- 1.972 ENDIF ENDIF IF (V1 .EQ. 9)THEN CSl-21.666 C32-16.919 CS3-14.684 IF (NS .EQ. 25)THEN CFI- 2.820 CFZ- 2.090 CF3- 1.770 ENDIF IF (NS .EQ. SO)THEN CFl- 2.579 CPZ- 1.976 CP3- 1.689 ENDIF IF(NS .EO. 100)THEN CF1- 2.491 CFZ- 1.924 CF3- 1.657 ENDIF ENDIF IF (V1 .EQ. 16)THEN CSl-32.000 CSZ-26.296 CS3-23.542 IF (NS .EQ. 25)THEN CFI- 2.2359 CFZ- 1.841 CF3- 1.604 ENDIF IF (NS .EQ. 50)THEN CFI- 2.145 CFZ- 1.727 CF3- 1.528 ENDIF IF(NS .EQ. 100)THEN CC NW A 56 CC CC CC CC 58 120 CFl- 2.065 CFZ- 1.682 CF3' 1.494 ENDIF ENDIF (18) TO SECIFY FLEISHMAN'S CONSTANTS FOR NORMAL AND UNIFORM DIS. DIS‘l WRITE(20,1) roman/M --------- 1. NORMAL (o, 0) ------------------------ r) AA-0.0 BB-l.0 CC-0.0 DD-0.0 DO 2 I-1,NP DO 3 01-1le PE(I,J)-PEI(I,J) CONTINUE CONTINUE DO 4 I-1,NQ DO 5 J-loNQ PX(I,J)-PX1(I,J) CONTINUE CONTINUE GOTO 55 WRITE(20,7) FORMAT(' --------- 2. UNIFORM (O, -1.12) ------------------ ') AA-0.0 BB-l.34891701 CC‘0.0 DD--.13265955 DO 8 I-1,NP DO 9 J'1,NP PE(I,J)-PEZ(I,J) CONTINUE CONTINUE DO 10 I-leQ DO 11 J-leQ PX(I,J)-PX2(I,J) CONTINUE CONTINUE (1C) TO SET THE NUMBER OF REJECTIONS TO ZERO: BARTLETT (REJB), RAO F (REJR), RANK-TRANSFORM (REJT), PURE-RANK (REJP), AND MIXED-RANK (REJM). REJB(I)'0 REJR(I)‘0 REJT(I)'0 REJP(I)‘0 REJM(I)'0 CONTINUE (1C) TO SET SUNS TO ZERO FOR MULTIVARIATE SKEWNESS (SUMMS) AND KURTOSIS (SUMMK). RAW-SCORE CROSS PRODUCTS (SUMCC), RAW SCORES (SUMDI), RAW~SCORE SQUARES (SUMDZ), RAW-SCORE CUBES (SUMD3), AND RAW-SCORE DUADS (SUMD4). SUMMS-0.0 SUMMK-0.0 DO 57 I-1,NV DO 58 J'1,NV SUMCC(I,J)'0.0 COLTINUE 64 65 70 69 CC CC 82 84 CC 96 121 SUMDl(I)-0.0 SUMDZ(I)-0.0 SUMD3(I)-0.0 SUMD4(I)-0.0 CONTINUE (2A) TO SET SUM OF LOOP RAW SCORES TO ZERO (SUMD)p GENERATE UNIVARIATE RANDOM UNIFORM DEVIATES (UD), TRANSFORM UD INTO RANDOM NORMAL DEVIATES (ND), AND FORM A MATRIX OF UNIVARIATE RANDOM NORMAL ERRORS (UNE) AND PREDICTORS (UNX). LOOP-0 LOOP-LOOP+1 IF(LOOP .GT. NL)GOTO 250 D0 64 I-1,NV SUMD(I)'0.0 CONTINUE SUNS-0.0 SUMK-0.0 CALL GGUBS(DSEED,TS,UD) DO 65 I-1,TS,2 ND(I) -SQRT(-2.*LOG(UD(I)))‘COS(2.*PH‘UD(I+I)) ND(I+1)‘SQRT(~2.*LOG(UD(I)))‘SIN(2.*PH*UD(I+1)) CONTINUE DO 69 I-I'NP K'(I-1)*NS+J UNE(I,J)‘ND(K) CONTINUE CONTINUE DO 71 I-1,NQ DO 72 J-1,NS K'(NP+I-1)*NS+J UNX(I.J)-ND(K) CONTINUE CONTINUE (28) TO OBTAIN MULTIVARIATE NORMAL ERRORS (MNE) AND PREDICTORS (MNX) BY MULTIPLYING PRINCIPAL COMPONENT WEIGHTS (PE,PX) WITH NORMAL ERRORS AND PREDICTORS (UNE,UNX). CALL VMULFF (PE,UNE,NP,NP,NS,NP,NP,MNE,NP,IER) CALL VMULEE (PX,UNX,NQ,NQ,NS,NQ,NQ,MNX,NQ,IER) (2C) TO OBTAIN MULTIVARIATE NON-NORMAL DATA (MNNE, MNNX) BY MULTIPLYING EACH SCORE WITH FLEISHMAN CONSTANTS. DO 81 I-1,NP MNNE(I,J)'AA+BB'MNE(I,J)+CC'(MNE(I,J)"2)+DD*(MNE(I,J)'*3) CONTINUE CONTINUE DO 84 J'1¢NS MNNX(IoJ)'AA+BB'MNX(I.J)*CC'(MNX(I,J)"2)‘3"'(MNX(I,J)'*3) CONTINUE CONTINUE (2D) TO OBTAIN DEPENDENT DEVIATES (MNNY): Y - B’X + E. CALL VMULFM (RB,MNNX,NQ,NP,NS,NQ,NQ,MBX,NP,IER) DO 85 I-1,NP MNNY(I,J) - MBX(I,J) + MNNUIJ) CONTINUE 85 CC 88 87 96 95 121 120 122 CONTINUE (2E)TO FORM A COMBINED DEPENDENT-PREDICTOR DATA MATRIX (DATD). DO 87 I-1,NP DO 88 J-1,NS DATD(I,J)- MNNY(I,J) CONTINUE CONTINUE DO 89 I-1,NQ DO 90 J-1,NS K-NP+I DATD(K,J)-MNNX(I,J) CONTINUE CONTINUE (3A) TO OBTAIN SUM OF RAW SCORES (SUMD,SUMDl), SUM OF RAW-SCORE SQUARES (SUMDZ), SUM OF RAW-SCORE CUBES (SUMD3), SUM OF RAW-SCORE QUADS (SUMD4), MEAN (AVD), AND DEVIATION FROM THE MEAN (DEVD). DO 95 I-1,NV DO 96 J-leS SUMD(I) ‘SUMD(I) +DATD(I:J) SUMDl(I)-SUMD1(I)+DATD(I,J) SUMDZ (I)"SUMD2 (I) +DATD(I,J) **2 SUMD3(I)-SUMD3(I)+DATD(I,J)**3 SUMD4(I)-SUMD4(I)+DATD(I,J)*‘4 CONTINUE AVD(I) 'SUMD(I)/NS CONTINUE DO 98 I'1,NV DEVD(I:J)'DATD(I,J)'AVD(I) CONTINUE CONTINUE (38) TO FIND SUM OF RAW-SCORE CROSS PRODUCT MTRIX (RCP), SUM OF CROSS PRODUCT MATRX (AAA), SUM OF RAW-SCORE CROSS PRODUCTS (SUMCCI AND COVARIANCE (COVN). CALL VMULPP(DATD,DATD,NV,NS,NV,NV,NV,RCP,NV,IER) CALL VMULFP(DEVD,DEVD,NV,NS,NV,NV,NV,AAA,NV,IER) DO 108 I-1,NV DO 109 J-1,NV SUMCC(I.J)'SUMCC(I,J)+RCP(I.J) COVN(I,J)'AAA(I.J)/(NS-1.) CONTINUE CONTINUE (3C) TO OBTAIN MULTIVARIATE SKEWNESS (MULS) AND KURTOSIS (MULK) INVERSE OF COVARIANCE MATRIX (COVI) AND PRODUCT OF THE MATRICES DEVD"COVI'DEVD (DCD). CALL LINVIF(COVN,NV,NV,COVI,DGT,WK,IER) CALL VMULFM(DEVD,COVI,NV,NS,NV,NV,NV,DCV,NS,IER) CALL VMULFE(DCV,DEVD,NS,NV,NS,NS,NV,DCD,NS,IER) DO 120 I'1,NS SUMS'SUMS+DCD(IoJ)"3 CONTINUE SUMK‘SUMK+DCD(I,I)"2 CONTINUE MULS'SUMS/(NS"2) MULK‘SUMK/NS‘NV'(NV‘2.) SUMMS‘SUMMS+MULS 123 SUMMK-SUMMK+MULK cc ---------------------------------------------------------------------- CC (4) TO OBTAIN SUM OF CROSS PRODUCTS MATRIX AND SUBMATRICES cc ---------------------------------------------------------------------- '123 127 133 132 141 142 140 146 145 156 155 161 160 166 165 CC (AVR). (4C) DO 127 I-1,NP DO 128 J-1.NP AYY(I.J)'AAA(I.J) CONTINUE CONTINUE DO 132 I-1,NP DO 133 J-1,NQ AYX(I,J)-AAA(I,NP+J) CONTINUE CONTINUE DO 136 I-1.NQ DO 137 J-1,NQ AXX(I,J)'AAA(NP+I,NP+J) CONTINUE CONTINUE DO 140 I-1,NV DO 141 J-1,NS DTA(J)-DATD(I,J) CONTINUE CALL NMRANK(DTA,NS,EPS,IR,R,DTR,S,T) DO 142 J-IINS DATR(I.J)-DTR(J) CONTINUE CONTINUE AVR-(Ns+l.)/2. DO 145 I-1.NV DO 146 J-1,NS DEVR(I,J)-DATR(I,J) - AVR CONTINUE CONTINUE CALL VMULFP(DEVR,DEVR,NV,NS,NV,NV,NV,RRR,NV,IER) DO 155 I-1,NP DO 156 J-1.NP RYY(I,J)-RRR(I,J) CONTINUE CONTINUE DO 160 I-1,NP DO 161 J-1.NQ RYX(I,J)-RRR(I,NP+J) CONTINUE CONTINUE DO 165 I-1.NQ RXX(I.J)'RRR(NP+I,NP+J) CONTINUE CONTINUE DO 170 I-1,NP DO 171 J'1.NS DVYR(I.J)-DEVR(I.J) CONTINUE MIXED DATA: TO OBTAIN SCP FOR RANKED Y AND ORIGINAL X (4A) ORIGINAL DATA: TO OBTAIN SCP SUBMATRIX FOR Y AND K VARIABLES (AYY,AXX) FROM AAA. (48) RANKED DATA: TO RANK ORIGINAL DATA AND OBTAIN MEAN RANK SCP MATRIX (RRR) AND SCP SUBMATRICES FOR Y,YX,X VARIABLES (RYY,RYX,RXX). (MYX). 124 170 CONTINUE DO 175 I-1,NQ DO 176 J-1.NS K-NP+I DVXD(I,J)-DEVD(K,J) 176 CONTINUE 175 CONTINUE CALL VMULFP(DVYR,DVXD,NP,NS,NQ,NP,NQ,MYX,NP,IER) CC --------------------------------------------------------------------- CC (5) TO OBTAIN WILKS' LAMBDA AND SUM OF EIGENVALUES CC --------------------------------------------------------------------- CC (SA) TO OBTAIN THE INVERSES OF AYY,AXX,RYY,RXX. CALL LINVIF(AYY,NP,NP,AYYI,DGT,WK,IER) CALL LINV1F(AXX,NQ,NQ,AXXI,DGT,WK,IER) CALL LINVIF(RYY,NP,NP,RYYI,DGT,WK,IER) CALL LINVlF(RXX,NQ,NQ,RXXI,DGT,WK,IER) CC (SB) TO OBTAIN PRODUCT OF MATRICES USING ORIGINAL DATA(MATD), CC RANKED DATA(MATR), AND MIXED DATA(MATM). CALL VMULEF(AYYI,AYX.NP,NP,NQ,NP,NP,MATD1,NP,IER) CALL VMULFP(AXXI,AYX,NQ,NQ,NP,NQ,NP,MATD2,NQ,IER) CALL VMULFF(MATDl,MATDZ,NP,NQ,NP,NP,NQ,MATD,NP,IER) CALL VMULFF(RYYI,RYX,NP,NP,NQ,NP,NP,MATR1,NP,IER) CALL VMULFP(RXXI,RYX,NQ,NQ,NP,NQ,NP,MATR2,NQ,IER) CALL VMULFF(MATR1,MATR2,NP,NQ,NP,NP,NQ,MATR,NP,IER) CALL VMULFF(RYYI,MYX,NP,NP,NQ,NP,NP,MATM1,NP,IER) CALL VMULEP(AXXI,MYX,NQ,NQ,NP,NQ,NP,MATM2,NQ,IER) CALL VMULFF(MATM1,MATM2,NP,NQ,NP,NP,NQ,MATM,NP,IER) CC (5C) To OBTAIN EIGENVALUES OF MATD (ED) AND MATR (ER), THE PRODUC' CC OF (l-EIGENVALUE) (PED,PER) AND SUM OF THE EIGENVALUES (SER, SEM) CALL EIGRF(MATD,NP,NP,EJOB,ED,Z,NP,WK,IER) CALL EIGRF(MATR,NP,NP,EJOB,ER,Z,NP,WK,IER) RED-1.0 PER-1.0 SER-0.0 SEM-0.o DO 230 I-1,NP PED-PED'(1.0-ED(I)) PER-PER'(1.0-ER(I)) SER-SER+MATR(I,I) SEM-SEM+MATM(I,I) 230 CONTINUE CC --------------------------------------------------------------------- CC (6) To COMPUTE TEST STATISTICS AND To COUNT REJECTIONS. CC --------------------------------------------------------------------- CC (6A) To CCMPUTE BARTLETT (BAR), RAO F (RAD), RANK- CC TRANSFORM (RTF), PURE-RANK (PRN), MIXED-RANK (MRN). BAR - -A1'LOG(P£D) RAo - ((1.-PED"BZ)/(PED*'82))‘V3 RTF - ((1.-PER**BZ)/(PER*'BZ))~v3 PRN - (NS-1.)*SER MRN - (NS-1.)‘SEM CC (6C) NUMBER OF REJECTIONS FOR ALPHA - .01 IF(BAR .63. C51) REJB(l)-REJB(1)+1 I?(RAO .GE. CFl) REJR(1)-REJR(1)+1 IF(RTF .GE. CFl) REJT(1)-REJT(1)+1 IF(PRN .53. C51) REJP(1)-REJP(1)+1 IF(MRN .62. C51) REJM(1)-REJM(1)+1 CC (SD) NUMBER Of REJECTIONS FOR ALPHA - .05 IP(BAR .62. C52) REJB(2)-REJB(2)+1 IE(RAO .GE. CFZ) REJR(2)-REJR(2)+1 CC 251 CC CC 261 260 263 262 270 125 IF(RTF .GE. CF2) REJT(2)-REJT(2)+1 IF(PRN .GE. C32) REJP(2)-REJP(2)+1 IF(MRN .GE. CS2) REJM(2)-REJM(2)+1 (6E) NUMBER OF REJECTIONS FOR ALPHA - .10 IF(BAR .GE. CS3) REJB(3)'REJB(3)+1 IF(RAO .GE. CF3) REJR(3)-REJR(3)+1 IF(RTF .GE. CF3) REJT(3)-REJT(3)+1 IF(PRN .GE. CS3) REJP(3)-REJP(3)+1 IF(MRN .GE. CS3) REJM(3)-REJM(3)+1 GOTO 62 (7A) TO OBTAIN AVERAGE AND OVERALL MEAN (AVDl, OMEAN), VARIANCE (AVARN, OVARN), UNIVARIATE SKEWNESS (SKEW, OSKEW) AND KURTOSIS (KURT,OSKEW), MULTIVARIATE SKEWNESS (OMSKW) AND KURTOSIS (OMKUR), CORRELATION MATRIX (CORR) AND DETERMINANT OF CORR (DCOR). TMEAN-0.0 TVARN-0.0 TSKEW-0.0 TKURT-0.0 DO 251 I-1,NV AVD1(I)'SUMD1(I)/TN AVD2(I)'SUMD2(I)/TN AVD3(I)'SUMDB(I)/TN AVD4(I)-SUMD4(I)/TN VARN(I)-AVDZ(I)-AVD1(I)'*2 SKEW(I)-(AVD3(I)-3.*AVD1(I)*AVD2(I)+2.*AVDl(I)"3)/VARN(I)**1.5 KURT(I)'((AVD4(I)-4.*AVD1(I)*AVD3(I)+6.*(AVD1(I)**2)*AVD2(I)- 3.'AVD1(I)**4)/(VARN(I)**2)) - 3.0 TMEAN-TMEAN+AVDI(I) TVARN-TVARN+VARN(I) TSKEW-TSKEW+SKEW(I) TKURT-TKURT+KURT(I) CONTINUE OMEAN-TMEAN/NV OVARN-TVARN/NV OUSKW-TSKEW/NV OUKUR-TKURT/NV OMSKW-SUMMS/NL OMKUR-SUMMK/NL (78) TO OBTAIN CORRELATION MATRIX (CORR) AND DETERMINANT OF CORRELATION MATRIX (DCOR). DO 260 I-1.NV DO 261 J-1,NV ACOVN(I.J)‘(SUMCC(I.J)/TN)'(AVD1(I)'AVD1(J)) CONTINUE CONTINUE DO 262 I-1,NV DO 263 J-1.NV CCRR(I.J)' ACOVN(I.J)/SQRT(ACOVN(I,I)'ACOVN(J,J)) CORL(I,J)'CORR(I,J) CONTINUE CONTINUE CALL LINV3F(CORL.BCR.DJOB,NV,NV,D1,DZ,WK,IER) DCOR-Dl'2.**DZ WRITE(20,270)NV.NS.NL.CY,CX FORMAT(/,'NV -’,I1,' NS -',I3,’ NL -',I4,’ CY -',F3.1, ’ CX -’,F3.1) WRITE(20,272)AVD1(1),AVDI(2),AVDl(3),AVDl(4),CMEAN 272 274 276 278 281 280 282 284 286 288 290 126 FORMAT(/.'MEAN’,3X,5F11.6) wRITE(20,274)VARN(1),VARN(2),VARN(3),VARN(4),OVARN FORMAT(/,'VARN',3X,SF11.6) WRITE(20,276)SKEW(1),SKEW(2),SKEW(3),SKEW(4),OUSKW FORMAT(/,’USKEW',2X,5F11.6) WRITE(20,278)KURT(1).KURT(2),KURT(3),KURT(4),OUKUR FORMAT(/,’UKURT',2X.5F11.6) Do 280 I-1,Nv WRITE(20,281) CORR(I,1),CORR(I,2),CORR(I,3),CORR(I,4) FORMAT(/.'CORR’,3X,4F11.6) CONTINUE WRITE(20,282)OMSKW,OMKUR,DCOR FORMAT(/,'M5KEW -',F11.6,' MKURT -',F11.6.' DETCOR - ’,F8.6) WRITE(20,284) FORMAT(/,’ALFA(REJ) BART RAOF RAOR PURR MIXR') WRITE(20,286) REJB(1),REJR(I),REJT(1),REJP(1),REJM(1) FORMAT(/,’0.01( 30)’,5(3X,I4)) WRITE(20,288) REJB(2).REJR(2),REJT(2),REJP(2),REJM(2) FORMAT(/,'0.05(150)’.5(3X,I4)) WRITE(20,290) RBJB(3),REJR(3),REJT(3),REJP(3),REJM(3) FORMAT(/,'0.10(300)',5(3X,I4)./) DIs-DIs+1 IF (DIS .EQ. 2) GOTO 6 STOP END APPENDIX E TABLES Table E1. Fleishman Constants Used for Data Generatione 11 12 a b c d .00 0.00 0.00 1.00 .00 0.00 .00 -1.12 0.00 1.348917 .00 -0.132660 .50 0.00 -0.092624 1.039946 .092624 -0.016461 .00 0.50 -0.258525 1.114655 .258525 -0.066013 .00 3.00 0.00 0.782356 .00 0.067905 .00 3.00 -0.128397 0.832216 .128397 0.048032 .00 6.00 -0.313749 0.826324 .313749 0.022707 .00 20.00 0.00 0.338712 .00 0.184461 - skewness; 72 - kurtosis; a, 127 b, c, d - Fleishman constants. 128 Table 32. Average Mean, Variance, Skewness, Kurtosis, and Within-Set Correlations of the Generated Dataa V 11 12 u 02 11 72 p( 3) p( 7) 4 0.00 0.00 .002 .993 .006 .031 .301 .701 0.00 -1.12 -.001 1.006 .000 -1.149 .299 .694 0.50 0.00 .001 .995 .498 -.008 .299 .699 1.00 0.50 .001 .998 1.006 .522 .303 .701 0.00 3.00 .003 1.001 .020 2.991 .304 .702 1.00 3.00 .001 .992 .980 2.930 .298 .699 2.00 6.00 .000 .997 2.015 6.196 .300 .700 0.00 20.00 .003 1.000 .045 20.551 .297 .698 6 0.00 0.00 .003 .992 -.006 .026 .299 .700 0.00 -1.12 .001 1.005 -.003 -1.163 .297 .694 0.50 0.00 .002 .995 .502 .025 .300 .700 1.00 0.50 .002 .996 1.001 .512 .299 .701 0.00 3.00 .001 .999 .020 3.264 .301 .701 1.00 3.00 .004 1.001 1.027 3.156 .299 .700 2.00 6.00 -.001 .991 2.005 6.159 .299 .700 0.00 20.00 .002 1.005 -.040 20.031 .301 .700 8 0.00 0.00 .002 .996 .005 .014 .302 .701 0.00 -1.12 .000 1.001 .000 -1.162 .299 .694 0.50 0.00 .002 .994 .500 .013 .299 .699 1.00 0.50 .001 .996 1.002 .518 .303 .699 0.00 3.00 .001 .992 -.016 3.061 .298 .699 1.00 3.00 .001 .995 .994 2.932 .300 .700 2.00 6.00 -.001 .991 1.995 6.016 .298 .699 0.00 20.00 -.002 .991 -.027 19.985 .300 .701 4 0.00 0.00 .003 .983 .007 .084 .302 .701 0.00 -1.12 .000 .995 .001 -1.124 .300 .694 0.50 0.00 .002 .981 .506 .089 .298 .699 1.00 0.50 .002 .986 1.008 .593 .304 .700 0.00 3.00 .001 .984 .001 3.228 .302 .701 1.00 3.00 .000 .984 1.012 3.190 .301 .700 2.00 6.00 .001 .986 2.034 6.219 .298 .699 0.00 20.00 .000 .985 .000 20.922 .302 .702 6 0.00 0.00 .001 .984 -.003 .090 .301 .701 0.00 -1.12 -.001 .993 .003 -1.113 .297 .693 0.50 0.00 .000 .984 .508 .080 .299 .699 1.00 0.50 .001 .984 1.012 .606 .300 .701 0.00 3.00 .002 .983 .011 3.207 .300 .700 1.00 3.00 .000 .986 1.012 3.270 .302 .701 2.00 6.00 -.001 .982 2.045 6.382 .300 .700 0.00 20.00 -.001 .986 -.056 20.037 .300 .700 129 Table E2 (continued) N V 11 12 n 02 11 12 p(.3) p( 7) 50 8 0.00 0.00 .003 .984 .008 .085 .300 .700 0.00 -1.12 .002 .992 -.004 -1.103 .298 .694 0.50 0.00 .002 .982 .505 .076 .300 .700 1.00 0.50 .001 .984 1.001 .604 .301 .699 0.00 3.00 -.002 .983 -.009 3.176 .298 .698 1.00 3.00 .001 .985 1.010 3.212 .300 .700 2.00 6.00 .000 .980 2.024 6.185 .298 .699 0.00 20.00 .003 .979 .019 19.879 .299 .699 100 4 0.00 0.00 .002 .978 .006 .021 .301 .700 0.00 -1.12 .001 .986 -.003 -1.077 .298 .694 0.50 0.00 .000 .976 .502 .124 .299 .699 1.00 0.50 .001 .979 1.015 .640 .300 .700 0.00 3.00 .000 .979 .009 3.104 .298 .699 1.00 3.00 .001 .980 1.018 3.250 .301 .700 2.00 6.00 -.002 .972 2.040 6.192 .298 .699 0.00 20.00 .001 .970 .011 19.614 .299 .699 6 0.00 0.00 .001 .978 -.003 .120 .301 .700 0.00 -1.12 .001 .987 -.004 -1.063 .298 .694 0.50 0.00 -.001 .977 .502 .120 .301 .701 1.00 0.50 .001 .978 1.014 .640 .300 .701 0.00 3.00 .000 .977 .020 3.181 .300 .700 1.00 3.00 .001 .978 1.021 3.234 .300 .699 2.00 6.00 .001 .977 2.047 6.299 .300 .700 0.00 20.00 .001 .981 -.047 20.647 .300 .701 8 0.00 0.00 .001 .977 .003 .123 .300 .700 0.00 -1.12 -.001 .986 -.001 -1.087 .297 .693 0.50 0.00 .002 .977 .500 .121 .300 .700 1.00 0.50 -.001 .977 1.016 .643 .300 .700 0.00 3.00 -.001 .978 .003 3.136 .300 .700 1.00 3.00 .001 .980 1.013 3.176 .301 .701 2.00 6.00 -.001 .974 2.055 6.354 .299 .699 0.00 20.00 .000 .987 -.010 21.295 .300 .700 The tabled values represent the average mean, variance, skewness, kurtosis, and within-set correlation values based on 9,000 replications. 130 Table E3. Average of the Type I Error and gower Values by Distribution and Sample Size Type I Error Power a - .01 a - .10 a - .01 a - .10 N 25 50 100 25 50 100 25 50 100 25 50 100 [0. 0] RAO 010 011 009 097 101 102 065 247 675 323 613 913 RTF 010 010 009 099 098 106 069 227 630 313 587 894 PUR 002 005 007 073 088 100 021 153 580 254 551 885 MIX 002 005 008 068 087 099 016 153 594 243 556 893 [0, -1.12] RAD 011 011 008 101 103 102 063 232 660 311 603 912 RTF 011 011 009 101 102 102 063 210 592 297 563 874 PUR 002 006 007 074 091 095 019 141 539 239 528 864 MIX 002 005 006 074 092 096 020 142 541 243 532 865 [.5, 0] RAD 012 011 011 101 099 102 071 249 661 318 605 907 RTF 011 010 011 104 095 105 070 234 626 312 583 890 PUR 003 005 009 075 084 099 021 159 574 251 550 881 MIX 002 005 008 073 085 096 017 149 583 233 551 889 [1, .5] RAD 012 010 010 100 099 100 086 259 661 334 605 905 RTF 010 010 009 101 099 103 090 296 736 361 657 936 PUR 002 005 006 074 088 097 028 209 684 292 624 930 MIX 002 004 007 071 085 095 015 169 692 234 593 940 [0. 3] RAD 014 011 012 102 101 101 083 273 677 344 631 905 RTF 011 011 012 099 098 099 087 287 711 351 645 924 PUR 003 006 009 074 087 093 028 202 663 283 610 917 MIX 002 005 008 070 084 091 019 195 706 270 629 935 [1. 3] RAO 013 010 013 105 100 108 089 279 669 346 623 903 RTF 011 009 010 102 098 103 088 282 702 348 643 923 PUR 003 004 008 076 086 096 030 198 657 283 610 917 MIX 002 004 007 072 087 096 019 179 686 249 604 933 [2. 6] RAD 021 020 018 116 118 107 138 309 658 379 610 887 RTF 010 011 011 101 101 105 124 374 826 414 729 966 PUR 003 007 008 074 090 099 040 272 789 342 697 962 MIX 002 004 008 070 085 097 013 182 801 221 640 975 [0. 20] RAO 036 037 035 149 136 133 181 358 695 446 657 899 RTF 012 010 012 104 097 103 160 456 880 471 787 978 PUR 003 006 009 076 086 097 054 345 851 392 758 975 MIX 002 004 007 065 083 093 029 336 912 359 810 991 a Tabled values represent the average Type I error and power values across all within-set correlations and numbers-of-variables cases). (9 131 Table E4. Average of the Type I Error and Power Values by Distribution and Within-Set Correlationa Type I Error Power a - .01 a - .10 a - .01 a - .10 b (py,px) 1 2 3 1 2 3 1 2 3 1 2 3 [0. [0. 0] RAD 010 010 010 100 100 100 283 419 283 577 694 577 RTF 011 010 010 102 100 101 261 399 266 556 676 563 PUR 005 004 005 089 085 087 211 329 214 520 640 530 MIX 005 005 005 085 085 084 215 334 214 526 639 527 -1.12] RAD 010 010 010 102 103 102 274 410 271 569 686 570 RTF 010 010 010 102 100 102 244 376 244 534 657 543 PUR 005 005 005 087 086 087 195 309 195 501 620 509 MIX 005 005 004 086 086 089 197 310 197 506 624 510 {-5. 0] [1. [0. [1. [2. [0. RAD 011 011 011 101 100 101 231 420 230 573 686 571 RTF 011 011 011 104 100 100 263 401 267 554 671 560 PUR 006 006 005 037 036 035 209 333 211 520 636 526 MIX 006 005 005 035 035 034 211 333 206 524 631 519 .5] RAO 011 010 011 099 102 099 233 423 290 575 692 576 RTF 010 010 009 101 102 100 329 466 327 615 727 612 PUR 005 004 005 036 037 035 267 333 265 573 691 577 MIX 004 004 005 032 035 034 260 372 244 559 663 545 3] RAO 012 012 012 100 102 103 297 434 301 590 699 592 RTF 011 011 011 100 099 097 313 451 316 603 711 605 PUR 006 006 006 036 035 034 257 377 253 567 674 569 MIX 005 005 005 032 031 032 273 332 259 535 681 570 3] RAO 011 012 012 104 104 106 299 434 304 535 693 533 RTF 010 010 010 102 100 100 314 443 311 602 710 603 PUR 005 005 005 037 035 035 257 373 255 563 673 569 MIX 004 004 004 035 035 034 265 373 245 569 665 552 6] RAO 013 020 022 110 115 116 322 452 331 537 696 594 RTF 011 011 011 103 104 101 410 522 393 676 765 668 PUR 006 006 005 037 039 037 339 439 323 640 723 633 MIX 005 004 005 034 035 033 323 393 270 609 670 557 20] RAO 032 035 040 135 140 144 362 437 335 634 726 642 RTF 011 011 011 102 101 101 473 568 450 734 793 709 PUR 006 006 005 037 086 086 401 476 372 696 756 674 MIX 004 004 004 031 073 032 436 471 370 730 761 669 a b Tabled values represent the average Type I error and power values across all sample sizes and numbers-of-variables (9 cases). Within-set correlation 1 - (.3, .3), 2 - (.3, .7), 3 - (.7..7). Table E5. Average of the Type I Error and Power Values by 132 Distribution and Number-of-Variablesa Type I Error Power a - .01 a - .10 a - .01 - V 4 6 8 4 6 8 4 6 8 4 6 8 [0. 0] RAO 009 011 010 098 101 101 336 318 332 628 608 612 RTF 010 010 010 100 105 098 307 299 321 603 591 600 PUR 005 005 004 093 090 078 267 241 245 590 555 545 MIX 006 005 004 089 089 076 274 246 244 597 555 539 [0, -1.12] RAD 011 009 009 103 104 100 322 313 320 624 602 599 RTF 011 010 010 100 104 101 287 284 293 587 572 574 PUR 007 005 004 094 089 078 250 228 221 573 538 519 MIX 006 004 003 095 090 077 252 231 221 577 542 522 [.5, 0] RAD 011 011 012 099 102 101 324 326 331 617 607 606 RTF 011 011 010 099 106 100 301 306 323 596 593 596 PUR 007 006 004 092 089 077 261 250 243 582 558 542 MIX 006 005 004 089 088 077 265 249 235 592 555 527 [1. .5] RAO 010 010 011 100 099 101 336 328 342 621 609 614 RTF 010 010 009 100 103 100 377 366 378 657 648 648 PUR 006 004 003 092 088 078 333 298 290 644 612 589 MIX 005 004 004 092 085 074 337 285 255 649 584 534 [0. 3] RAO 011 012 013 094 108 102 348 341 343 632 626 623 RTF 011 011 011 098 104 094 357 357 371 640 639 640 PUR 007 006 005 092 090 073 315 294 282 626 603 581 MIX 006 005 004 087 084 074 336 303 281 647 611 576 [1. 3] RAO 012 012 012 104 103 106 341 343 352 625 625 621 RTF 009 009 011 099 101 102 350 352 370 638 634 642 PUR 006 004 004 092 086 080 308 291 286 624 601 585 MIX 006 004 003 094 081 079 324 290 269 639 594 552 [2. 5] ' RAO 018 020 021 102 115 124 361 365 379 624 626 627 RTF 011 011 011 103 101 104 447 438 440 710 705 694 PUR 006 006 005 095 086 082 397 361 343 697 670 634 MIX 005 005 004 092 081 079 403 325 267 702 614 521 [0. 20] RAD 029 038 041 125 139 155 395 405 434 663 661 678 RTF 012 011 011 104 096 105 495 499 502 750 744 742 PUR 008 005 004 098 081 081 441 416 392 737 709 680 MIX 006 004 002 093 077 072 484 422 371 790 718 652 3 across all sample sizes and within-set correlations (9 cases). Tabled values represent the average Type I error and power values 133 Table E6. Empirical Type I Error Rates 2nd Power Distribution [0, 0] Values For Type I Error Power (p , px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (a - .01) (.3, .3) 25 4 007 007 012 003- 003- 067 067 064 029- 021- 6 012 011 010 002- 001- 049 048 053 013- 013- 8 011 011 012 002- 001- 045 043 049 009- 006- 50 4 011 011 012 007 007 238 239 212 “166 169 6 011 010 010 004- 003- 187 184 164 108- 111- 8 011 011 010 004- 004- 176 175 156 080- 083- 100 4 009 009 008 006- 006- 606 606 552 525- 543- 6 011 011 012 010 011 582 577 537 479 490 8 008 008 010 005- 005- 610 610 565 487- 500- (.3, 7) 25 4 007 007 010 003- 003- 095 094 090 040- 035- 6 012 011 009 001- 001- 090 087 093 025- 020- 8 011 011 011 002- 001- 089 084 091 017- 010- 50 4 011 011 013 007- 007- 340 340 304 252- 253- 6 011 010 010 004- 003- 324 323 310 209- 209- 8 011 011 009 003- 004- 362 359 350 200- 194- 100 4 009 009 006- 006- 006- 765 765 710- 683- 709- 6 011 011 010 007 011 830 828 776 733 759 8 008 008 009 006- 005- 893 893 869 805- 817- (.7, .7) 25 4 007 007 009 003- 002- 067 067 065 029- 023- 6 012 011 010 001- 002- 049 048 055 015- 013- 8 011 011 010 002- 001- 045 043 060 011- 006- 50 4 011 011 012 009 009 238 239 210 162 168 6 011 010 008 004- 002- 187 184 171 110- 111- 8 011 011 009 002- 004- 176 175 170 091- 082- 100 4 009 009 006- 005- 006- 606 606 553- 521- 541- 6 011 011 010 008 011 582 577 532 481 489 8 008 008 011 007 006- 610 610 575 503- 495- (a - .05) (.3, .3) 25 4 043 043 043 029- 031- 211 211 202 152- 160- 6 052 050 054 028- 027- 180 179 179 110- 103- 8 053 050 051 021- 017- 170 162 155 078- 072- 50 4 057 057 053 046 049 466 466 436 410 421 6 053 053 050 040- 040- 400 399 374 321- 324- 8 052 051 048 032- 036- 386 381 369 282- 287- 100 4 048 048 041 039- 042- 815 814 768 761- 785- 6 055 055 055 050 048 792 791 758 738 752 8 050 050 054 046 043 815 814 784 750 761 (.3, .7) 25 4 043 043 043 031- 031- 270 270 248 203- 205- 6 052 050 058+ 033- 027- 254 252 251+ 164- 151- 8 053 050 052 022- 017- 261 254 251 130- 111- 50 4 057 057 054 048 049 583 584 546 518 531 6 053 053 049 038- 040- 595 595 563 496- 504- 8 052 051 044 030- 036- 626 623 608 507- 501- 100 4 048 048 043 039- 042- 910 910 888 882- 890- 6 055 055 054 046 048 942 942 922 910 920 8 050 050 055 045 043 968 967 956 945 947 134 Table E6 (continued) Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAD RTF PUR MIX (.7, .7) 25 4 043 043 044 027- 029- 211 211 200 156- 154- 6 052 050 054 032- 032- 180 179 186 113- 099- 8 053 050 049 023- 020- 170 162 171 087- 075- 50 4 057 057 057 048 052 466 466 431 405 418 6 053 053 047 037- 040- 400 399 383 324- 320- 8 052 051 046 034- 038- 386 381 383 302- 290- 100 4 048 048 042- 039- 043 815 814 767 759- 783- 6 055 055 052 045 046 792 791 749 730 747 8 050 050 054 045 047 815 814 790 758 760 (a - .10) ( 3, .3) 25 4 090 090 095 081- 074- 321 321 306 283- 278- 6 100 098 110 084- 076- 286 283 280 220- 210- 8 106 103 094 060- 059- 271 266 255 171- 168- 50 4 104 104 104 100 098 594 594 559 549 561 6 099 099 102 090 088- 536 536 513 473 483- 8 101 100 091 077- 080- 540 536 509 448- 455- 100 4 099 099 101 098 095 888 888 860 856 870 6 105 106 111+ 107 106 875 875 848+ 837 852 8 101 102 109 099 092 890 891 871 846 858 (.3, .7) 25 4 090 090 094 078- 074- 397 397 372 346- 343- 6 100 098 105 080- 076- 391 388 363 296- 287- 8 106 103 095 060- 059- 382 378 379 262- 239- 50 4 104 104 104 098 098 699 698 667 656 667 6 099 099 101 088- 088- 723 723 698 663- 665- 8 101 100 089- 072- 080- 761 759 736- 678- 674- 100 4 099 099 101 096 095 953 953 934 933 939 6 105 106 105 100 106 970 970 957 954 961 8 101 102 105 096 092 983 983 976 972 975 (.7, .7) 25 4 090 090 093 081- 074- 321 321 303 280- 281- 6 100 098 105 076- 069- 286 283 287 234- 211- 8 106 103 099 060- 056- 271 266 269 194- 169- 50 4 104 104 107 103 097 594 594 561 548 562 6 099 099 095 083- 083- 536 536 517 479- 483- 8 101 100 091 077- 073- 540 536 526 467- 452- 100 4 099 099 101 097 099 888 888 864 862 874 6 105 106 108 104 107 875 875 853 839 847 8 101 102 110 100 096 890 891 882 865 862 a Tabled values represent the proportion of rejections across 3000 replications at a - .01, .05, and .10, where N - sample size, NV - no. of variables, BAR - Bartlett, RAO - Rao F, RTF - rank- transform Rao F, PUR - pure-rank, MIX - mixed-rank, "+" indicates a liberal Type I error rate, and a "-" indicates a conservative Type I error rate. Table E7. Empirical Type I Error Rates And Power Values For 135 Distribution [0, -1.12]8 Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (0 - .01) (.3, .3) 25 4 013 013 012 004- 004- 062 062 060 025- 029- 6 011 010 012 001- 003- 054 051 051 013- 015- 8 011 010 010 001- 001- 049 046 041 007- 010- 50 4 012 012 011 008 007 219 219 192 157 154 6 010 010 010 006- 006- 172 171 157 098- 101- 8 012 012 012 006- 004- 168 167 151 084- 085- 100 4 009 009 009 007 007 586 586 517 481 488 6 009 008 008 007 005- 582 580 510 457 460- 8 008 008 009 006- 004- 580 579 514 432- 430- (.3, .7) 25 4 014+ 014+ 013 005- 005- 089 089 087 038- 041- 6 010 010 009 003- 002- 084 083 084 023- 024- 8 010 010 009 001- 001- 079 076 076 015- 012- 50 4 010 010 011 008 007 316 317 279 228 224 6 009 009 011 006- 006- 321 319 289 185- 191- 8 010 010 011 004- 004- 352 346 317 183- 182- 100 4 008 008 010 008 007 754 754 675 650 651 6 009 009 008 006- 005- 829 828 755 706- 710- 8 007 007 010 007 006- 883 882 823 752 752- (.7, .7) 25 4 014+ 014+ 014+ 005- 003- 063+ 063+ 065+ 030- 030- 6 008 008 011 001- 002- 051 049 052 017- 016- 8 009 008 009 001- 000- 047 045 047 005- 008- 50 4 012 012 012 009 007 222 222 195 155 157 6 011 011 011 006- 005- 168 167 154 099- 100- 8 013 013 009 003- 002- 161 159 153 080- 085- 100 4 009 009 011 008 008 583 584 517 488 491 6 011 010 009 007 006- 574 571 500 455 458- 8 006- 006- 009 005- 006- 577- 577- 513 430- 427- (a - .05) ( 3, .3) 25 4 054 054 055 042- 043 200 200 184 146- 146 6 048 047 050 029- 027- 170 166 162 098- 097- 8 055 050 054 020- 024- 157 151 149 072- 071- 50 4 055 055 051 043 045 454 454 410 383 390 6 051 051 053 044 043 394 394 364 312 310 8 056 054 050 037- 034- 377 374 345 274- 279- 100 4 049 048 O49 047 045 804 803 749 741 744 6 057 057 050 046 049 790 789 736 715 719 8 047 046 049 039- 039- 795 794 732 696 698 (.3, .7) 25 4 054 055 057 043 042- 250 251 239 191 196- 6 046 045 052 031- 026- 245 242 225 147- 147- 8 047 045 052 018- 020- 242 235 225 109- 115- 50 4 053 054 048 040- 044 571 571 515 485- 498 6 050 050 051 039- 039- 585 585 531 468- 475- 8 057 056 051 036- 035- 615 610 572 471- 474- 100 4 048 048 045 043 046 909 909 859 854 858 6 053 053 051 043 047 939 938 904 890 891 8 046 045 049 042- 040- 968 968 935 919 921 136 Table E7 (continued) Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (.7, .7) 25 4 060+ 060+ 061+ 047 044 195+ 195+ 194+ 156 155 6 050 050 051 028- 027- 172 170 172 105- 105- 8 050 047 043 014- 016- 152 147 156 073- 081- 50 4 053 054 052 044 044 446 447 406 376 386 6 055 055 057 044 047 394 394 358 310 312 8 051 049 050 036- 036- 381 377 362 288- 285- 100 4 049 049 047 046 046 805 804 750 742 747 6 054 053 052 047 048 788 786 729 709 710 8 043 043 043 035- 036- 789 788 739 695- 699- (a - .10) (.3, .3) 25 4 104 104 103 089- 085- 318 319 294 267- 275- 6 098 096 100 071- 073- 280 279 261 210- 207- 8 103 101 104 065- 062- 258 250 241 162- 174- 50 4 104 104 100 097 098 585 585 536 522 531 6 099 099 102 090 088- 535 536 493 460 469- 8 104 100 100 083- 083- 513 511 476 424- 429- 100 4 100 101 100 096 096 884 884 839 836 838 6 110 111+ 110 103 104 871 872+ 832 819 821 8 097 098 101 089- 089- 883 884 836 809- 811- (.3, .7) 25 4 105 105 098 090 085- 383 384 359 331 330- 6 100 098 100 073- 073- 368 364 348 277- 288- 8 103 099 101 059- 058- 365 356 342 239- 247- 50 4 105 104 099 095 097 694 694 647 636 641 6 105 105 103 092 094 717 717 669 632 639 8 108 107 104 088- 085- 754 752 707 639- 646- 100 4 097 097 097 095 096 953 953 925 923 922 6 112+ 113+ 105 099 101 972+ 972+ 948 942 947 8 094 095 096 087- 087- 986 986 970 963- 961- (.7, .7) 25 4 104 104 102 091 096 320 321 298 271 276 6 101 099 104 074- 075- 271 268 273 218- 222- 8 104 100 096 056- 056- 264 258 253 173- 173- 50 4 107 107 103 098 105 586 586 542 532 536 6 103 103 107 096 095, 531 531 496 460 461 8 103 101 102 084- 086- 514 511 503 447- 439- 100 4 097 097 097 095 098 888 888 844 839 842 6 110 111+ 108 099 104 881 881+ 831 821 822 8 094 095 101 090 087- 883 885 842 819 818- a Tabled values represent the proportion of rejections across 3000 replications at a - .01, .05, and .10, where N - sample size, NV - no. of variables, BAR - Bartlett, RAO - Rao F, RTF - rank- transfomm Rao F, PUR - pure-rank, MIX - mixed-rank, "+" indicates a liberal Type I error rate, and a ”-" indicates a conservative Type I error rate. Table E8. Empirical Type I Error Rates And Power Values For 137 Distribution [.5, 0]8 Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (0 - .01) (.3, .3) 25 4 011 011 012 005- 003- 071 071 065 029- 026- 6 012 012 013 003- 003- 059 058 053 013- 012- 8 015+ 014+ 012 001- 001- 050+ 048+ 052 005- 005- 50 4 011 011 012 006- 007 222 222 201 155- 159 6 012 011 012 007 007 205 204 190 124 116 8 010 010 007 003- 005- 175 173 165 087- 078- 100 4 011 011 011 010 009 576 577 546 514 524 6 013 012 012 009 009 574 572 534 481 494 8 011 010 010 007 008 600 599 557 476 483 (.3, .7) 25 4 010 010 013 005- 003- 100 100 089 046- 036- 6 013 012 009 003- 002- 097 093 092 028- 023- 8 014+ 014+ 010 003- 003- 094+ 090+ 102 014- 008- 50 4 011 011 011 009 006- 330 330 297 234 243- 6 010 010 011 005- 006- 353 353 325 238- 230- 8 010 009 006- 002- 004- 360 359 354- 209- 188- 100 4 011 011 011 009 009 742 742 697 671 687 6 011 011 012 009 008 820 818 782 743 762 8 012 012 011 007 007 893 893 871 813 816 (.7, .7) 25 4 010 010 011 003- 002- 071 071 070 028- 027- 6 011 011 012 002- 002- 062 060 055 014- 010- 8 015+ 014+ 011 001- 001- 050+ 047+ 055 007- 003- 50 4 011 011 011 007 006- 220 221 204 161 152- 6 011 011 009 005- 004- 202 202 188 124- 104- 8 010 010 008 003- 004- 176 173 182 096- 073- 100 4 012 012 011 009 008 576 577 542 513 530 6 010 010 012 009 007 570 569 537 481 493 8 012 012 011 008 008 601 600 567 478 459 (a - .05) ( 3, .3) 25 4 051 051 051 037- 037- 199 199 198 160- 152- 6 050 048 053 028- 024- 183 181 171 100- 101- 8 060+ 057 046 020- 022- 163+ 156 157 076- 063- 50 4 050 051 053 047 045 451 452 421 392 409 6 050 050 050 041- 038- 420 420 401 345- 345- 8 047 045 045 028- 031- 383 379 363 291- 280- 100 4 052 052 048 046 047 800 799 755 746 769 6 055 055 055 051 046 786 785 758 732 753 8 048 047 046 042- 042- 817 816 791 754- 753- (.3, .7) 25 4 052 052 055 041- 036- 254 254 245 198- 201- 6 050 049 054 025- 025- 259 255 254 157- 138- 8 059+ 055 054 022- 021- 255+ 247 249 129- 098- 50 4 051 052 050 043 044 570 571 543 512 526 6 049 049 054 043 037- 586 586 571 511 512- 8 046 045 043 031- 030- 622 617 597 502- 485- 100 4 052 052 047 045 048 897 897 872 864 883 6 053 052 056 052 045 934 934 919 909 921 8 050 049 047 043 045 960 965 959 944 947 138 Table E8 (continued) Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAD RTF PUR MIX (.7, .7) 25 4 051 051 051 037- 035- 205 205 193 151- 152- 6 053 052 048 028- 029- 180 178 185 118- 100- 8 059+ 055 053 020- 020- 166+ 162 168 081- 060- 50 4 051 051 049 044 043 450 452 430 407 413 6 049 049 049 039- 033- 422 422 398 347- 340- 8 049 047 043 031- 028- 385 380 382 316- 272- 100 4 053 053 047 044 045 797 797 758 747 767 6 051 050 054 048 049 782 781 761 742 747 8 051 050 049 043 046 819 817 799 772 754 (a - .10) (.3, .3) 25 4 099 099 105 089- 081- 310 310 295 272- 271- 6 098 097 106 076- 075- 290 287 278 216- 205- 8 111+ 106 105 061- 062- 267+ 262 253 177- 156- 50 4 097 097 097 091 090 584 584 554 544 561 6 105 105 104 089- 091 548 548 526 494- 500 8 102 099 090 074- 078- 527 525 496 442- 445- 100 4 102 102 101 098 094 874 874 850 846 864 6 105 106 111+ 105 100 867 868 852+ 839 849 8 097 098 112+ 099 093 894 895 881+ 854 862 (.3, .7) 25 4 099 099 097 088- 082- 382 383 360 331- 327- 6 099 097 107 078- 074- 387 384 376 309- 286- 8 111+ 107 104 061- 063- 378+ 372 372 261- 223- 50 4 097 097 090 085- 091 689 689 662 652- 669 6 101 101 101 088- 091 714 714 692 656- 659 8 099 097 091 075- 076- 740 737 721 660- 649- 100 4 102 102 101 099 091 947 948 928 926 938 6 107 107 110 105 101 964 964 954 950 953 8 098 098 098 091 095 981 981 977 974 974 (.7, .7) 25- 4 098 098 107 092 086- 309 309 303 278 275- 6 100 099 099 071- 072- 289 287 294 234- 209- 8 113+ 108 104 064- 058- 268+ 265 272 185- 145- 50 4 095 095 092 089- 090 581 580 560 545- 561 6 104 104 099 086- 086- 546 546 522 491- 487- 8 099 097 090 075- 074- 523 520 514 464- 430- 100 4 104 104 096 094 095 873 874 848 845 861 6 104 104 113+ 105 103 867 868 844+ 832 845 8 099 100 102 094 093 892 893 880 861 855 a Tabled values represent the proportion of rejections across 3000 replications at a - .01, .05, and .10, where N - sample size, NV - no. of variables, BAR - Bartlett, RAO - Rao F, RTF - rank- transfomm Rao F, PUR - pure-rank, MIX - mixed-rank, "+” indicates a liberal Type I error rate, and a "-" indicates a conservative Type I error rate. 139 Table E9. Empirical Type I Error Rates And Power Values For Distribution [1, .5] Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (a - .01) (.3, .3) 25 4 011 011 009 003- 002- 079 079 081 032- 021- 6 012 012 013 002- 002- 074 073 066 016- 008- 8 015+ 015+ 010 001- 001- 064+ 062+ 060 009- 004- 50 4 012 012 011 007 007 245 245 284 232 218 6 010 010 010 005- 004- 191 191 220 148- 122- 8 007 007 008 003- 001- 186 183 212 118- 073- 100 4 009 009 007 006- 008 595 595 683 650- 693 6 009 009 011 008 007 555 554 662 602 611 8 013 013 010 007 005- 608 608 690 598 589- (.3, .7) 25 4 010 010 015+ 004- 004- 105 105 126+ 065- 038- 6 014+ 013 007 001- 002- 116+ 115 122 032- 018- 8 011 009 010 001- 001- 123 120 121 018- 006- 50 4 008 008 009 008 004- 339 340 382 324 317- 6 010 009 011 004- 004- 359 359 403 287- 244- 8 010 010 010 002- 004- 389 383 436 280- 178- 100 4 011 011 010 007 009 753 753 816 795 837 6 009 008 008 005- 005- 807 806 868 825- 851- 8 009 009 007 004- 005- 867 866 917 870- 864- (.7, .7) 25 4 010 010 010 002- 003- 078 078 090 043- 031- 6 013 013 010 002- 001- 075 074 076 021- 009- 8 015+ 014+ 009 002- 001- 070+ 066+ 065 013- 003- 50 4 010 010 012 008 007 234 234 260 212 199 6 009 009 007 003- 003- 201 201 232 157- 107- 8 013 012 008 005- 005- 193 191 233 126- 061- 100 4 009 009 007 006- 006- 589 589 666 639- 675- 6 009 009 009 006- 006- 577 575 648 594- 591- 8 008 008 009 007 011 600 599 671 581 517 (a - .05) (.3, .3) 25 4 042- 042- 045 032- 030- 211- 211- 235 191- 169- 6 057 056 055 029- 031- 205 203 206 129- 099- 8 052 050 048 022- 018- 187 182 182 083- 050- 50 4 056 057 053 048 045 462 463 513 493 502 6 042- 042- 048 040- 032- 389- 389- 454 396- 369- 8 047 045 051 032- 031- 397 394 443 364- 287- 100 4 047 047 047 046 044 803 803 859 851 886 6 047 047 050 048 044 781 779 843 824 851 8 055 054 054 048 044 815 814 871 842 842 (.3, .7) 25 4 051 051 055 042- 037- 272 272 310 252- 209- 6 048 047 048 027- 029- 286 283 297 202- 144- 8 055 051 046 019- 017- 285 279 305 152- 087- 50 4 051 052 049 041- 046 564 565 628 599- 620 6 052 052 056 045 034- 590 590 655 597 566- 8 052 049 049 036- 037- 645 640 682 584- 502- 100 4 053 053 046 046 050 901 901 937 933 953 6 046 046 050 042- 043 926 926 960 954- 968 8 045 045 045 039- 038- ’957 957 976 966- 966- 140 Table E9 (continued) Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (.7, .7) 25 4 054 054 057 039- 038- 216 216 233 191- 158- 6 052 051 051 024- 024- 195 193 213 140- 082- 8 064+ 061+ 051 020- 020- 188 187 194 098- 050- 50 4 051 051 047 042- 046 469 470 502 471- 474 6 051 051 053 041- 038- 414 414 467 406- 342- 8 054 054 046 033- 037- 400 397 440 366- 256- 100 4 047 047 046 044 046 800 799 853 846 877 6 045 045 057 052 046 783 783 842 826 825 8 053 052 048 041- 038- 801 799 856 824- 796- (a - .10) (.3, .3) 25 4 091 091 097 084- 078- 330 330 355 327- 303- 6 103 102 109 079- 078- 299 296 319 257- 209- 8 101 098 102 063- 053- 295 288 298 197- 131- 50 4 103 103 103 096 092 574 574 629 618 639 6 086- 086- 093 081- 078- 520- 520- 596 554- 523- 8 099 098 097 081- 075- 537 531 586 524- 458- 100 4 106 106 100 098 096 883 883 918 916 936 6 099 100 103 098 099 866 867 908 901 921 8 107 107 104 097 092 883 883 925 910 910 (.3, .7) 25 4 099 100 102 089- 087- 385 385 430 403- 372- 6 101 099 096 071- 068- 405 403 426 347- 287- 8 107 104 093 055- 056- 408 401 441 314- 196- 50 4 099 099 099 094 093 680 679 739 730 752 6 105 105 105 093 091 713 714 773 748 724 8 107 106 104 085- 083- 752 750 801 747- 680- 100 4 105 105 106 103 103 950 950 964 963 978 6 098 099 103 099 099 966 966 982 979 985 8 098 098 105 097 081- 984 984 987 985 991- (.7, .7) 25 4 096 097 104 091 086- 317 318 342 314 291- 6 103 103 099 074- 068- 299 296 327 261- 190- 8 112+ 108 103 062- 063- 290+ 285 313 212- 129- 50 4 095 095 094 087- 094 591 591 624 614- 630 6 106 106 107 096 086- 549 549 593 565 505- 8 097 095 090 075- 076- 536 533 568 513- 422- 100 4 098 099 093 090 097 874 875 916 914 938 6 092 092 111+ 104 099 867 868 905+ 898 910 8 096 097 097 090 088- 868 869 916 902 887- Tabled values represent the proportion of rejections across 3000 replications at a - .01, .05, and .10, where N - sample size, NV - no. of variables, BAR - Bartlett, RAO - Rao F, RTF - rank- . transform Rao F, PUR - pure-rank, MIX - mixed-rank, "+" indicates a liberal Type I error rate, and a ”-" indicates a conservative Type I error rate. Table E10.Empirica1 Type I Error Rates And Power Values For 141 Distribution [0, 31a Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (a - .01) (.3, .3) 25 4 012 012 013 005- 003- 081 081 081 038- 027- 6 015+ 015+ 009 002- 001- 066+ 064+ 069 018- 012- 8 015+ 014+ 012 003- 003- 060+ 058+ 061 012- 011- 50 4 011 011 012 008 007 251 252 258 206 224 6 011 011 010 006- 003- 218 216 227 148- 159- 8 012 012 010 004- 003- 194 190 226 123- 114- 100 4 009 009 010 008 009 614 614 647 619 688 6 013 013 011 008 009 601 598 635 582 651 8 013 013 009 008 007 604 603 655 570 618 (.3, .7) 25 4 011 011 011 006- 004- 113 113 111 052- 041- 6 013 013 012 003- 001- 116 111 113 036- 018- 8 018+ 017+ 010 002- 003- 112+ 105+ 119 020- 014- 50 4 012 012 010 007 007 357 357 350 295 319 6 011 010 011 004- 003- 369 368 390 283- 272- 8 012 012 011 004- 003- 395 393 427 262- 231- 100 4 009 009 009 007 008 759 760 790 766 817 6 013 012 014+ 012 009 825 823 854+ 826 857 8 014+ 014+ 012 009 006 875+ 875+ 900 850 872 (.7, .7) 25 4 013 013 011 005- 003- 083 082 078 039- 026- 6 013 013 012 002- 001- 068 066 077 024- 011- 8 016+ 015+ 010 002- 002- 066+ 062+ 072 010- 007- 50 4 011 011 011 008 007 256 258 247 204 212 6 010 010 010 006- 006- 227 225 226 162- 136- 8 012 012 012 006- 004- 201 199 234 132- 093- 100 4 009 009 009 007 009 614 615 648 616 668 6 015+ 015+ 013 009 009 601+ 600+ 623 570 608 8 013 013 010 009 008 602 602 643 563 571 (a - .05) (.3, .3) 25 4 048 048 054 036- 032- 223 224 215 172- 172- 6 057 057 055 027- 027- 200 197 202 130- 113- 8 059+ 057 048 020- 021- 185+ 179 187 090- 086- 50 4 051 051 056 049 044 479 480 484 459 504 6 053 053 054 045 040- 441 441 456 393 414- 8 051 050 046 033- 028- 417 412 443 363- 365- 100 4 043 043 045 043 043 817 817 838 829 870 6 054 054 053 046 046 800 793 836 818 856 8 057 056 045 038- 043 806 805 842 815- 839 (.3, .7) 25 4 047 048 049 036- 032- 276 277 274 221- 222- 6 055 055 050 028- 025- 285 284 286 185- 160- 8 059+ 056 047 023- 021- 284+ 274 300 157- 112- 50 4 051 052 052 044 044 588 589 597 570 613 6 052 052 057 045 038- 614 614 638 577 598- 8 051 048 048 037- 030- 652 649 660 562- 557- 100 4 044 044 043 041- 046 898 898 913 910- 933 6 057 056 056 050 042- 935 934 948 942 960- 8 056 056 045 040- 044 959 959 970 962- 971 Table E10 (continued) 142 Type I Error Power (py, px) N V BAR RAO RIF PUR MIX BAR RAO RTF PUR MIX (.7, .7) 25 4 048 049 044 033- 028- 227 227 216 172- 168- 6 058+ 057 049 027- 023- 208+ 205 208 137- 105- 8 059+ 056 048 022- 025- 190+ 184 199 099- 067- 50 4 052 053 052 047 046 479 480 480 454 491 6 056 056 056 043 037- 444 444 457 398 389- 8 050 050 048 030- 030- 427 424 452 369- 331- 100 4 045 045 044 040- 044 811 811 836 829- 862 6 057 057 057 052 046 798 797 824 811 841 8 057 057 052 046 043 802 801 821 800 812 (a - .10) (.3, .3) 25 4 091 091 100 088- 076- 333 334 330 304- 312- 6 113+ 112+ 100 075- 070- 320+ 317+ 308 246- 249- 8 107 104 101 061- 066- 291 285 304 204- 192- 50 4 099 099 099 096 092 610 609 614 604 648 6 105 105 105 090 086- 576 576 591 556 575- 8 095 093 096 079- 076- 562 557 580 514- 531- 100 4 090 091 096 093 091 879 879 899 896 919 6 102 102 111+ 105 099 875 875 899+ 893 918 8 105 106 088- 082- 082- 878 880 905- 889- 916- (.3, .7) 25 4 089- 090 105 094 074- 403- 404 400 371 381- 6 113+ 113+ 097 072- 070- 412+ 409+ 420 341- 330- 8 108 103 093 056- 061- 409 402 426 304- 254- 50 4 102 101 096 091 092 705 705 718 706 745 6 107 107 103 092 085- 729 730 751 721 750- 8 101 100 094 079- 073- 759 756 768 713- 727- 100 4 089- 089- 098 095 095 936- 936- 953 951 968 6 103 104 112+ 107 097 966 966 976+ 972 981 8 107 109 089- 082- 083- 978 978 986- 983- 990- (.7, .7) 25 4 092 092 100 090 077- 334 335 332 305 307- 6 114+ 114+ 098 072- 069- 314+ 311+ 320 257- 232- 8 104 102 095 061- 064- 305 298 315 216- 176- 50 4 101 101 092 087- 093 604 604 612 601- 632 6 108 108 105 094 088- 579 579 592 555 555- 8 099 098 092 073- 073- 569 567 576 523- 501- 100 4 094 095 094 093 092 880 880 899 896 914 6 105 106 106 098 091 871 871 897 886 911 8 107 108 095 084- 086- 879 879. 901 882- 897- a Tabled values represent the proportion of rejections across 3000 replications of variables, NV -'no. at a - .01, a liberal Type I error rate, Type I error rate. .05, and .10, BAR - Bartlett , transform Rao F, PUR - pure-rank, MIX - mixed-rank, and a. where N - sample RAO - Rao F, RTF - rank- "+" indicates size, indicates a conservative Table E11.Empirica1 Type I Error Rates And Power Values For 143 Distribution [1, 31a Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (a - .01) (.3, .3) 25 4 011 011 007 003- 003- 089 089 085 038- 030- 6 015+ 014+ 011 003- 003- 073+ 071+ 066 023- 012- 8 012 012 015+ 002- 000- 064 060 064+ 010- 006- 50 4 010 010 008 006- 006- 242 242 251 200- 210- 6 008 008 008 004- 001- 227 226 229 154- 144- 8 011 010 010 004- 003- 214 212 219 127- 099- 100 4 014+ 015+ 012 009 008 596+ 597+ 632 607 662 6 012 011 008 008 007 583 581 619 571 615 8 012 012 010 007 006- 613 613 656 582 606 (.3, .7) 25 4 011 011 008 004- 004- 122 122 110 058- 045- 6 016+ 015+ 010 002- 003- 118+ 113+ 113 039- 020- 8 013 012 013 002- 000- 123 116 129 026- 008- 50 4 011 011 007 006- 005- 333 334 348 286- 303- 6 008 008 009 003- 002- 388 387 385 279- 257- 8 013 013 009 004- 003- 397 393 419 254- 205- 100 4 014+ 014+ 013 010 008 746+ 747+ 772 751 799 6 013 013 009 008 006- 821 819 852 811 849 8 013 013 009 006- 006- 872 872 901 855 871 (.7, .7) 25 4 011 011 009 004- 004- 091 091 080 037- 029- 6 015+ 015+ 010 003- 003- 075+ 073+ 076 025- 011- 8 013 013 012 001- 000- 068 064 073 014- 005- 50 4 010 010 009 005- 005- 248 249 244 197- 189- 6 009 009 007 002- 002- 240 239 221 156- 124- 8 012 011 009 003- 004- 228 226 221 130- 076- 100 4 013 013 011 010 009 598 598 628 600 647 6 013 013 008 007 006- 584 582 610 562 578- 8 014+ 013 011 008 007 615+ 615 647 575 544 (a - .05) (.3, .3) 25 4 056 056 050 036- 032- 224 225 218 177- 167- 6 061+ 060+ 051 030- 028- 199+ 198+ 200 124- 111- 8 058+ 055 060+ 024- 019- 194+ 190 195+ 100- 075- 50 4 048 048 047 038- 039- 464 464 476 453- 475- 6 048 048 048 034- 031- 445 445 449 396- 398- 8 056 055 ‘047 034- 034- 422 417 439 347- 329- 100 4 061+ 060+ 058+ 055 055 803+ 803+ 819+ 810 860 6 051 051 047 040- 037- 791 789 833 814- 847- 8 056 055 052 045 046 810 808 848 814 841 (.3, .7) 25 4 054 054 050 036- 033- 283 284 278 226- 207- . 6 062+ 060+ 054 029- 028- 293+ 290+ 284 186- 150- 8 057 055 055 024- 020- 294 289 307 160- 101- 50 4 049 049 046 038- 040- 579 580 590 560- 601- 6 049 049 052 039- 032- 614 614 627 570- 575- 8 057 054 047 030- 034- 640 635 667 574- 520- 100 4 060+ 060+ 057 054 053 894+ 894+ 913 909 936 6 052 051 047 040- 037- 934 933 949 943- 958- 8 056 056 049 040- 044 958 957 967 953 966 144 Table E11 (continued) Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (.7, .7) 25 4 054 054 050 034- 034- 228 228 224 177- 160- 6 061+ 059+ 057 030- 027- 210+ 208+ 215 138- 095- 8 059+ 056 050 018- 018- 203+ 196 214 107- 062- 50 4 047 047 042- 035- 037- 461 463 481- 451- 460- 6 048 048 051 041- 034- 451 451 453 397- 378- 8 054 054 047 032- 038- 428 423 442 361- 291- 100 4 059+ 059+ 060+ 057 054 794+ 793+ 820+ 812 854 6 051 051 047 043 039- 787 785 819 804 827- 8 059+ 058+ 050 043 048 810 810 830 801 802 (a - .10) ( 3, .3) 25 4 107 107 096 083- 085- 326 327 330 306- 296- 6 108 107 108 083- 075- 314 311 304 247- 226- 8 102 100 106 067- 060- 301 296 299 206- 170- 50 4 096 096 096 091 092 589 589 607 595 626 6 099 099 102 091 086- 575 575 587 553 559- 8 104 103 100 079- 081- 548 545 584 522- 497- 100 4 108 108 108 104 104 877 877 895 892 925 6 102 103 094 089- 085- 872 873 900 892- 912- 8 109 111+ 105 096 098 873 875+ 912 896 908 (.3, .7) 25 4 108 108 093 082- 084- 403 404 398 367- 368- 6 105 104 110 081- 071- 418 414 412 335- 301- 8 105 101 102 065- 055- 414 405 425 307- 220- 50 4 095 095 093 086- 094 692 692 722 711- 734 6 099 099 101 090 086- 726 726 743 710 719- 8 106 104 097 077- 085- 758 757 777 728- 698- 100 4 109 109 108 105 110 940 941 956 954 971 6 102 103 094 088- 085- 962 962 973 971- 979- 8 109 109 103 094 094 977 978 982 979 989 (.7, .7) 25 4 106 107 093 083- 085- 328 328 329 304- 293- 6 107 107 103 076- 073- 320 319 313 260- 213- 8 110 107 104 062- 058- 314 307 320 219- 152- 50 4 099 099 091 085- 086- 593 593 610 599- 619- 6 099 100 101 089- 081- 572 573 581 552- 537- 8 108 107 096 083- 088- 557 554 579 520- 448- 100 4 111+ 111+ 110 106 105 874+ 874+ 894 892 919 6 105 106 096 090 086- 872 872 897 887 901- 8 113+ 113+ 106 095 096 874+ 874+ 902 888 889 Tabled values represent the proportion of rejections across 3000 replications at a - .01, .05, and .10, where N - sample size, NV - no. of variables, BAR - Bartlett, RAO - Rao F, RTF - rank- transform Rao F, PUR - pure-rank, MIX - mixed-rank, "+" indicates a liberal Type I error rate, and a "-" indicates a conservative Type I error rate. Table E12.Empirica1 Type I Error Rates And Power Values For 145 Distribution [2, 61a Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (a - .01) (.3, .3) 25 4 021+ 021+ 014+ 004- 003- 127+ 127+ 122+ 053- 028- 6 021+ 019+ 008 003- 001- 117+ 115+ 104 028- 008- 8 017+ 015+ 008 001- 001- 104+ 101+ 083 013- 001- 50 4 016+ 016+ 009 007 005- 275+ 275+ 365 308 302- 6 018+ 018+ 011 006- 004- 250+ 248+ 311 214- 149- 8 021+ 020+ 014+ 007 003- 260+ 259+ 307+ 171 069- 100 4 015+ 015+ 010 008 008 594+ 594+ 787 767 858 6 017+ 017+ 014+ 011 010 582+ 580+ 798+ 755 811 8 017+ 017+ 015+ 009 007 601+ 600+ 811+ 746 721 (.3, .7) 25 4 020+ 020+ 012 004- 002- 164+ 164+ 167 083- 039- 6 020+ 020+ 009 003- 001- 173+ 171+ 166 050- 010- 8 024+ 023+ 010 001- 001- 175+ 169+ 151 030- 003- 50 4 017+ 017+ 011 007 004- 361+ 361+ 461 402 388- 6 022+ 022+ 011 006- 005- 399+ 398+ 485 361- 236- 8 024+ 023+ 013 006- 003- 429+ 428+ 506 330- 121- 100 4 016+ 016+ 010 008 006- 728+ 728+ 885 867 933- 6 019+ 019+ 011 009 010 801+ 799+ 923 901 930 8 019+ 019+ 011 008 006- 849+ 848+ 953 925 919- (.7, .7) 25 4 021+ 021+ 011 004- 003- 131+ 131+ 122 056- 020- 6 025+ 024+ 011 002- 002- 135+ 132+ 105 030- 004- 8 026+ 025+ 008 001- 001- 133+ 130+ 095 015- 001- 50 4 017+ 017+ 011 007 006- 277+ 277+ 348 291 246- 6 023+ 022+ 010 006- 005- 266+ 264+ 297 204- 092- 8 027+ 026+ 011 006- 003- 273+ 271+ 289 164- 031- 100 4 016+ 016+ 010 008 006- 590+ 591+ 769 743 815- 6 022+ 022+ 011 008 008 579+ 575+ 748 703 688 8 021+ 021+ 011 006- 009 604+ 604+ 762 697 535 (a - .05) (.3, .3) 25 4 067+ 067+ 056 042- 039- 259+ 260+ 302 243- 197- 6 060+ 058+ 047 023- 020- 254+ 252+ 263 163- 088- 8 068+ 065+ 043 018- 020- 242+ 237+ 232 113- 037- 50 4 058+ 059+ 048 040- 036- 468+ 468+ 597 569- 605- 6 065+ 065+ 055 042- 038- 441+ 441+ 574 509- 446- 8 078+ 077+ 057 039- 035- 436+ 433+ 540 448- 309- 100 4 050 049 053 051 045 781 779 919 915 964 6 066+ 065+ 052 048 045 775+ 775+ 923 914 949 8 057 056 052 044 041- 786 784 930 912 921 (.3, .7) 25 4 066+ 067+ 057 039- 037- 315+ 316+ 362 299- 236- 6 071+ 070+ 053 027- 019- 334+ 331+ 356 247- 118- 8 071+ 069+ 049 020- 018- 339+ 331+ 349 184- 041- 50 4 060+ 061+ 051 043 038- 565+ 567+ 696 670- 716- 6 067+ 067+ 051 040- 040- 606+ 606+ 732 676- 616- 8 083+ 080+ 054 037- 031- 626+ 623+ 736 644- 446- 100 4 053 053 051 049 042- 881 881 964 963 985- 6 069+ 069+ 051 046 046 912+ 912+ 980 976 990 8 066+ 065+ 057 048 043 944+ 944+ 990 986 990 Table E12 (continued) 146 Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (.7, .7) 25 4 064+ 064+ 058+ 043 037- 266+ 267+ 290+ 242- 162- 6 069+ 069+ 048 028- 019- 263+ 260+ 266 174- 064- 8 074+ 070+ 052 017- 016- 270+ 266+ 235 117- 023- 50 4 061+ 061+ 048 041- 036- 465+ 466+ 583 556- 558- 6 065+ 065+ 048 039- 037- 453+ 453+ 555 495- 345- 8 086+ 084+ 054 038- 034- 452+ 448+ 515 431- 195- 100 4 051 051 051 049 044 784 784 908 902 949 6 068+ 067+ 053 047 045 769+ 767+ 902 891 906 8 068+ 068+ 059+ 050 045 780+ 780+ 911+ 891 830 (a - .10) (.3, .3) 25 4 115+ 115+ 111+ 096 090 353+ 354+ 410+ 382 348 6 110 110 092 069- 064- 343 345 376 308- 204- 8 117+ 114+ 092 050- 059- 338+ 334+ 339 237- 113- 50 4 100 100 097 091 090 580 580 710 696 755 6 108 108 109 097 085- 559 559 695 664 642- 8 133+ 132+ 110 091 081- 548+ 545+ 673 610 489- 100 4 094 094 101 099 098 861 862 956 954 981 6 110 110 106 097 093 853 853 957 953 980 8 108 108 105 095 093 854 854 968 957 969 (.3, .7) 25 4 113+ 113+ 112+ 099 091 413+ 414+ 483+ 456 405 6 115+ 114+ 098 069- 064- 445+ 443+ 489 408- 256- 8 121+ 119+ 105 061- 056- 449+ 444+ 479 345- 123- 50 4 101 101 098 093 089- 675 674 797 787 836- 6 117+ 117+ 105 092 090 706+ 706+ 831 804 777 8 142+ 141+ 101 089- 079- 729+ 728+ 834 783- 645- 100 4 092 092 099 096 100 928 928 982 981 994 6 114+ 114+ 105 099 094 949+ 950+ 993 991 998 8 118+ 119+ 112+ 104 099 972+ 972+ 997+ 995 998 (.7, .7) 25 4 114+ 114+ 109 098 085- 363+ 365+ 406 377 305- 6 126+ 124+ 095 066- 066- 361+ 359+ 382 315- 159- 8 123+ 120+ 095 062- 056- 360+ 356+ 361 249- 076- 50 4 102 102 091 087- 083- 581 580 699 688- 722- 6 116+ 116+ 098 084- 083- 567+ 568+ 679 648- 549- 8 146+ 144+ 102 085- 084- 557+ 554+ 647 591- 348- 100 4 090 090 104 100 102 860 860 950 947 975 6 117+ 118+ 103 099 090 848+ 848+ 942 938 957 8 115+ 116+ 111+ 103 103 852+ 853+ 950+ 941 926 a Tabled values represent the proportion of rejections across 3000 replications of variables, NV - no. at a - .01, a liberal Type I error rate, Type I error rate. .05, and .10, BAR - Bartlett , transform Rao F, PUR - pure-rank, MIX - mixed-rank, and a: where "+II N - sample RAO - Rao F, RTF - rank- size, indicates indicates a conservative 147 Empirical Type I Error Rates And Power Values For Table E13. Distribution [0, 20]a Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (a - .01) (.3, .3) 25 4 026+ 026+ 013 004- 003- 158+ 158+ 158 083- 060- 6 039+ 039+ 011 003- 001- 144+ 141+ 137 035- 022- 8 034+ 032+ 013 001- 000- 144+ 142+ 115 019- 010- 50 4 027+ 027+ 012 007 005- 305+ 305+ 426 364 470- 6 036+ 036+ 008 006- 002- 295+ 294+ 422 314- 338- 8 037+ 037+ 010 004- 003- 303+ 302+ 421 266- 231- 100 4 026+ 026+ 014+ 012 009 628+ 629+ 865+ 848 938 6 029+ 029+ 010 008 009 628+ 626+ 872 842 937 8 037+ 037+ 011 009 004- 664+ 663+ 886 842 918 (.3, .7) 25 4 030+ 030+ 012 005- 004- 203+ 203+ 195 103- 069- 6 040+ 040+ 012 003- 001- 221+ 217+ 204 064- 022- 8 039+ 037+ 008 001- 000- 231+ 227+ 207 036- 011- 50 4 027+ 028+ 010 007 004- 399+ 400+ 516 451 541- 6 037+ 036+ 008 006- 003- 430+ 429+ 565 442- 422- 8 043+ 043+ 011 005- 003- 496+ 493+ 588 407- 282- 100 4 029+ 029+ 013 011 010 746+ 747+ 916 906 961 6 035+ 035+ 012 009 007 808+ 806+ 954 938 970 8 037+ 036+ 009 006- 004- 856+ 856+ 964 941 965 (.7 .7) 25 4 032+ 032+ 012 005- 003- 174+ 174+ 145 074- 043- 6 044+ 044+ 012 002- 001- 185+ 182+ 139 046- 016- 8 047+ 045+ 010 001- 001- 189+ 186+ 136 024- 009- 50 4 030+ 030+ 009 006- 006- 313+ 313+ 404 336- 378- 6 040+ 040+ 008 004- 004- 323+ 322+ 381 281- 226- 8 055+ 054+ 011 005- 003- 363+ 361+ 377 241- 137- 100 4 030+ 030+ 013 011 010 625+ 626+ 826 809 898 6 041+ 041+ 012 007 007 630+ 627+ 818 781 844 8 048+ 048+ 010 004- 004- 673+ 673+ 821 754- 776- (a - .05) (.3, .3) 25 4 082+ 082+ 064+ 045 033- 321+ 321+ 353+ 295 323- 6 090+ 088+ 055 030- 020- 298+ 296+ 318 210- 178- 8 104+ 100+ 051 021- 016- 296+ 289+ 298 139- 096- 50 4 073+ 074+ 047 040- 041- 514+ 515+ 658 636- 762- 6 083+ 083+ 044 034- 035- 489+ 489+ 666 611- 682- 8 096+ 095+ 051 036- 031- 501+ 496+ 667 567- 601- 100 4 076+ 076+ 058+ 056 047 807+ 807+ 947+ 942 982 6 079+ 078+ 046 040- 037- 809+ 807+ 961 953 988 8 082+ 081+ 045 038- 040- 824+ 823+ 963 952- 981- (.3, .7) 25 4 090+ 090+ 055 043 034- 385+ 385+ 412 353 358- 6 095+ 093+ 052 029- 020- 386+ 383+ 400 285- 202- 8 111+ 108+ 049 016- 015- 408+ 401+ 399 219- 103- 50 4 077+ 077+ 048 039- 038- 600+ 601+ 740 716- 827- 6 092+ 092+ 041- 030- 028- 630+ 630+ 775- 726- 777- 8 098+ 097+ 051 037- 034- 683+ 681+ 794 713- 685- 100 4 074+ 074+ 058+ 053 046 877+ 877+ 973+ 971 991 6 085+ 084+ 047 043 038- 915+ 913+ 986 984 994 8 094+ 093+ 051 044 035- 937+ 936+ 992 989 995- 148 Table E13 (continued) Type I Error Power (py, px) N V BAR RAO RTF PUR MIX BAR RAO RTF PUR MIX (.7, .7) 25 4 092+ 092+ 054 037- 034- 332+ 333+ 335 282- 264- 6 103+ 102+ 050 027- 023- 331+ 329+ 311 210- 129- 8 115+ 111+ 051 020- 015- 344+ 337+ 296 165- 074- 50 4 075+ 076+ 048 041- 037- 508+ 509+ 626 602- 714- 6 092+ 092+ 044 034- 034- 505+ 505+ 614 558- 559- 8 111+ 110+ 052 036- 038- 537+ 533+ 615 521- 421- 100 4 074+ 073+ 057 054 045 795+ 795+ 932 928 '972 6 093+ 093+ 048 044 047 799+ 798+ 934 924 963 8 103+ 102+ 049 043 043 824+ 822+ 932 916 940 (a - .10) (.3, .3) 25 4 128+ 128+ 107 097 083- 430+ 430+ 482 457 513- 6 144+ 142+ 106 076- 069- 343+ 345+ 376 308- 204- 8 161+ 158+ 104 060- 049- 417+ 409+ 422 285- 240- 50 4 117+ 117+ 088- 085- 090 621+ 621+ 772- 763- 869 6 134+ 134+ 092 080- 077- 598+ 598+ 766 740- 826- 8 151+ 149+ 105 091 078- 611+ 608+ 781 724 782- 100 4 123+ 123+ 119+ 115+ 104 875+ 875+ 972+ 971+ 994 6 127+ 127+ 093 088- 086- 872+ 873+ 981 979- 995- 8 132+ 132+ 104 091 091 887+ 888+ 981 975 994 (.3, .7) 25 4 135+ 136+ 103 090 080- 484+ 484+ 538 509 562- 6 146+ 144+ 104 077- 060- 484+ 481+ 530 448- 401- 8 177+ 173+ 103 058- 045- 518+ 512+ 526 379- 251- 50 4 119+ 119+ 096 092 092 704+ 704+ 836 828 902 6 137+ 137+ 089- 078- 075- 735+ 735+ 860- 835- 892- 8 154+ 152+ 106 088- 080- 773+ 772+ 873 828- 845- 100 4 123+ 124+ 119+ 117+ 105 926+ 926+ 986+ 986+ 996 6 127+ 128+ 089- 083- 086- 955+ 955+ 993- 992- 998- 8 146+ 146+ 101 090 083- 962+ 962+ 996 996 998- (.7, .7) 25 4 134+ 134+ 099 090 083- 435+ 436+ 458 430 454- 6 155+ 154+ 106 076- 065- 422+ 419+ 425 357- 279- 8 174+ 171+ 106 060- 050- 448+ 441+ 410 298- 172- 50 4 115+ 115+ 095 088- 093 621+ 621+ 742 729- 830 6 143+ 144+ 092 082- 078- 616+ 616+ 728 699- 726- 8 162+ 161+ 110 090 086- 636+ 635+ 725 677 613- 100 4 129+ 129+ 109 106 104 868+ 868+ 961 959 986 6 138+ 138+ 091 087- 093 867+ 867+ 969 964- 983 8 149+ 150+ 103 096 087- 878+ 878+ 962 954 976- a Tabled values represent the proportion of rejections across 3000 replications at a - .01, .05, and .10, where N - sample size, NV - no. of variables, BAR - Bartlett, RAO - Rao F, RTF - rank- transform Rao F, PUR - pure-rank, MIX - mixed-rank, "+" indicates a liberal Type I error rate, and a "-" indicates a conservative Type I error rate. 149 Table E14. Frequency Distributions of Simulated Data Interval [0, 0] [0,-1.12] [.5, 0] [1, .5] [0, 3] [1, 3] [2, 6] [0,20] <-8.0 0 0 0 0 0 0 0 1 -8.0 -7.0 0 0 0 0 0 0 0 3 -7.0 -6.0 0 0 0 0 1 0 0 5 -6.0 -5.0 0 1 0 0 3 0 0 7 -5.0 -4.0 0 0 0 0 8 0 0 24 -4.0 -3.0 9 1 0 0 45 4 0 56 -3.0 -2.7 15 0 0 0 29 7 0 28 -2.7 -2.5 32 0 0 0 30 5 0 24 -2.5 -2.3 38 0 0 0 38 15 0 33 -2.3 -2.1 65 0 0 0 51 35 0 37 -2.1 -1.9 108 0 23 0 87 44 0 52 -1.9 -1.7 149 0 153 0 95 77 0 56 -1.7 -1.5 202 720 258 0 158 136 0 77 -1.5 -1.3 317 575 371 0 186 236 0 119 -1.3 -1.1 384 572 531 1197 316 375 0 119 -1.1 -0.9 508 538 644 1287 377 549 809 214 -0.9 -0.7 596 544 694 896 584 798 1940 303 -0.7 -0.5 630 585 718 770 681 861 1256 447 -0.5 -0.3 782 550 772 674 844 984 1002 820 -0.3 -0.1 727 586 751 681 914 964 873 1469 -0.1 0.1 801 580 783 563 1038 890 724 2151 0.1 0.3 751 591 686 549 935 795 587 1455 0.3 0.5 740 577 629 494 837 630 503 797 0.5 0.7 648 572 564 431 689 561 416 431 0.7 0.9 608 540 514 412 529 429 328 313 0.9 1.1 465 575 415 355 385 328 254 210 1.1 1.3 387 528 343 289 300 273 234 150 1.3 1.5 316 562 280 277 215 219 185 102 1.5 1.7 215 802 230 228 143 169 160 86 1.7 1.9 174 0 164 198 124 116 128 77 1.9 2.1 130 0 142 160 91 98 87 49 2.1 2.3 66 0 108 134 68 87 95 55 2.3 2.5 56 0 72 121 44 71 74 34 2.5 2.7 42 0 48 99 33 51 53 25 2.7 3.0 20 0 59 93 38 52 85 33 3.0 4.0 17 0 41 92 62 102 118 76 4.0 5.0 2 0 7 0 14 22 59 32 5.0 6.0 0 0 0 0 2 10 16 11 6.0 7.0 0 0 0 0 3 3 7 8 7.0 8.0 0 0 0 0 1 2 3 4 >8.0 0 0 0 0 2 2 4 7 Tabled values represent the frequency distributions of the simulated data using 10,000 deviates. BIBLIOGRAPHY THE BIBLIOGRAPHY Anderson, T. W. (1958). t v i Statist cal Mg. New York: John Wiley & Sons. Arnold, H. J. (1964). Permutation support for multivariate techniques. Biggesrika. 21. 65°70. Bartlett, M. S. (1938). regression. ' - - 14, 33-40. Further aspects of the o q. -no 0 theory of multiple ,o,,o-q . __ , Belli, G. M. (1983). Robustness and power of multivariate tests for trends in repeated measures data under variance-covariance heterogeneity. Unpublished doctoral dissertation, Michigan State University. Bhattacharrya, G. R., Johnson, R. A., 8: Neave, H. R. (1971). A comparative power study of the bivariate rank sum test and t-test. W. 1.3.0.1. 191-198- Box, G. E. P., 8: Cox, D. R. (1964). An analysis of transformations. l2arna1_2fiBexa1_§£a£i§£isal_§esie£1. 826. 211-246. Box, G. E. P., & Muller, M. E. (1958). A note on the generation of random normal deviates. AW. 22. 610-613. Boyer Jr., J. E., Palachek, A. D., 8: Schucany, W. R. (1983). An empirical study of related correlation coefficients. W Edssasisnal_§£atistiss. 8. 75-86. Chase, G. R., 6: Bulgren, W. 02 (1971). A Monte Carlo investigation of the robustness of the I . Jo a o ic Stat stical Assgsiasisn. 56. 499-502. Chatterjee, S. R., & Sen, P. K. (1964). Nonparametric tests for the bivariate two-sample location problem. Ca utt St istical Assesiarien_fisllesin. 13. 18-58. Conover» ‘W. J. (1980). Eras2isa1_Hennsramerrie_§tatistiss (2nd ed ). New York: John Wiley 8: Sons. Conover, W. J., 8: Iman, R. L. (1976). On some alternative procedures using ranks for the analysis of experimental designs. Q2mmunisatign§_in_§ta£istiss. Series A. 5. 1348-1368. 150 151 (1980a). Small sample efficiency of Fisher's randomization test when applied to experimental designs. Unpublished manuscript presented at the annual meeting of the American Statistical Association, Houston, August 1980. (1980b). The rank transformation as a method of discrimination with some examples. Qemmenieeeiene_in_§eeeieeiee. Series A. 2. 465-487. (1981). Rank transformation as a bridge between parametric and nonparametric statistics. Ihe_Ame;19en_§§e§1e§1eieh, 35, 124-129. (1982). Analysis of covariance using the rank transformation. W! 1&1 715-724- Davis, A. U. (1980). On the effects of moderate nonnormality on Wilks's likelihood ratio criterion. Biemegrihe, 61, 419-427. (1982a). 0n the distribution of Hotelling's one-sample 12 under moderate nonnormality. We; 1.9.9.. 207-216. (1982b). 0n the effects of moderate nonnormality on Roy's largest root test. W. 12. 896-900. Fawcett, R. P., & Salter, K. C. (1987). Distributional studies and the computer: An analysis of Durbin's rank test. The Amegicah Seesiesieien. £1. 31-83- Fair-Walsh, B. J., & Toothaker, L. E. (1974). An empirical comparison of the ANOVA-E test, normal scores test, and Kruskal-Wallis test under violation of assumptions. du t o a d s o cal Meeeereeeee. 33. 789-799. Fleishman, A. (1978). A method for simulating nonnormal distributions. Exehemeerike. 5;, 521-532. Gaito, J. (1970). Non-parametric methods in psychological research. In Reagihge in §§8E1§§12§ fog ehe Beheviegel §ciehee§, E. F. Heermann 8: L. A. Braskamp (eds.), 38-49. Englewood Cliffs, N.J.: Prentice-Hall, Inc. Class, C. V., Peckham, P. D., & Sanders, J. R. (1972). Consequences of failure to meet assumptions underlying the fixed-effects analysis of variance and covariance. Review ef fleecagiehel Reeeegeh, 3;, 237-288. Gibbons. J. D. (1971). HeD2eremeerie_§eeeieeieel_lnfereeee. New York: McGraw-Hill Book Company. Gittins, R. (1985). C no na ' ev w W ca io in Eeelegy. New York: Springer-Verlag. 152 Harris. R. J. (1975). W. New York: Academic Press . Hartley, H. 0. (1976). The impact of computers on statistics. In c , D. B. Owen (ed.), 421-442. New York: Marcel Dekker, Inc. Harwell, M. R., 6: Serlin, R. C. (1985). A nonparametric test statistic for the general linear model. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago. Hogg. R. V., 8: Randles, R. H. (1975). Adaptive distribution-free regression methods and their applications. W, 11, 399-407. Hollander, M., 6: Wolfe, D. A. (1973). e a1 11m. New York: John Wiley 8. Sons. Hopkins, J. W., 8: gay, P. P. F. (1963). Some empirical distributions of bivariate I and homoscedasticity criterion M under unequal variance and leptokurtosis. WW Wen. 18. 1048-1053. Iman, R. L. (1974a). Use of a e-statistic as an approximation to the exact distribution of the Wilcoxon signed ranks test statistic. WW. 3.. 795-806. (1974b). A power study of a rank transform for the two-way classification model when interaction may be present. The Canadian W. 21.21. 227-239. (1976). An approximation to the exact distribution of the Wilcoxon-Mann-Whitney rank sum test statistic. gomunieagiohe in W. 2. 537-598. Iman, R. L. and Conover, W. J. (1976). A comparison of several rank tests for the two-way layout. Technical Report SAND76-0631, Sandia Laboratories, Albuquerque, New Maxico. (1978). Approximations of the critical region for Spearman's rho with or without ties present. W, Series B, 1, 269-282. (1979). The use of rank transform in regression. W, 21, 499-509. (1980a). A comparison of distribution free procedures for the analysis of complete blocks. Unpublished manuscript presented at the annual meeting of the American Institute of Decision Sciences, Las Vegas, November, 1980. 153 (1980b). Multiple comparisons procedures based on the rank transformation. Unpublished manuscript presented at the annual meeting of the American Statistical Association, Houston, August, 1980. International Mathematical and Statistical Libraries. (1983). IMSL W. Houston. Texas. Ito, R., a Schull, w. J. (1964). On the robustness of the 12 test in multivariate analysis of variance when variance-covariance matrices are not equal. EIQEQSIIKB. 51, 71-82. Ito, P. K. (1980). Robustness of ANOVA and MANOVA test procedures. In flehghegk_ef_§§e;1e;1ee, P. R. Krishnaiah (ed.), Vol. 1, 199-236. New York: North Holland Publishing Company. Johnson. N. L.. 6: Kotz. S. (1970). WW 1. New York: John Wiley & Sons. _(1970). W- New York: John Wiley & Sons. - Kaiser, H. P., 8: Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from an. arbitrary population correlation matrix. Beyehemeezihe, 21, 179-182. Kaskey, G., Kolman, B., Krishnaiah, P. R., 8: Steinberg, L. (1980). Transformations to normality. In W, P. R. Krishnaiah (ed.), Vol. 1, 321-341. New York: North-Holland Publishing Company. Kendall. M. 6.. & Stuart. A. (1969). WWW (Vol. I). New York: Hafner Publishing Company. _(1969). Wee (Vol. II). New York: Hafner Publishing Company. Knapp, T. R. (1978). Canonical correlation analysis: A general parametric significance testing system. £52£h21281££l_§2112£13. 82, 410-416. Kottkamp, R., Mulhern, J. A., & Hoy, W. K. (1985). Secondary School Climate: A Revision of the 0CDQ. Rutgers University. Learmonth, G. P., & Lewis, P. A. W. (1973). Statistical tests of some widely used and recently' proposed. uniform random number generators. Naval Postgraduate School, Montery, California. Lehmann, E. L. (1975). o a et ' a 0 se on Rehke. San Francisco, CA: Holden-Day. 154 Lester, P. E. (1983). Development of An Instrument to Measure Teacher Job Satisfaction. Unpublished doctoral dissertation, New York University, New York. Marascuilo, L. A. , 6: Levin, J. R. (1983). u t va t n oc e e ' Re a e ' u . Monterey, CA: Brooks/Cole. Marascuilo, L. A., 6: McSweeney, M. (1977). W DietIiheeienn;Eree_Methede_£er_£he_§eeiel_§eieneee. Monterey. CA: Brooks/Cole. Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. W, 51, 519-530. (1971). The effect of nonnormality on some multivariate tests and robustness to nonnormality in the linear model. 112mm, 28, 105-121. (1974). Applications of some measures of multivariate skewness and kurtosis to testing normality and to robustness studies. $33M. Series, B, 16, 115-128. Morrison. D. F. (1976). Be1ti2erieee__§te£ieeieel__ue2heee (2nd ed.) New York: McGraw-Hill Book Company. Muirhead, R. J., 8: Waternaux C. M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. W, 61, 31-43. Muller, K. E., 8: Peterson, B. L. (1984). Practical methods for computing power in testing the multivariate general linear hypothesis. Qem2eeeeienel_Seeeistiee_ene_neee_eee1xeie. 2. 143-158. Noether, G. E. (1984). Nonparametrics: The early years - impressions and recollections. Ihe_Amerieen_§£eeieeieien. 33. 173-178. Olson, C. L. (1974). Comparative robustness of six tests in multivariate analysis of variance. c o a u , 83, 579-586. (1976). On choosing a test statistic in multivariate analysis of variance. Eexehelezieel_nelleein. 33. 579-586. Pearson, E. S., & Hartley, H. 0. (1951). Charts of the power function for analysis of variance tests derived from the non-central 1‘: distribution. W, §_8_, 112-130. Penfield, D. A., 8: Xoffler, S. L. (1985). A power study of selected nonparametric K-sample tests. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago. 155 Puri, M. L., & Sen, P. K. (1969). A class of rank order tests for a general linear hypothesis. Annals—oueehemetieeLiteeifiiee. $9.. 1325-1343. _(197l). WW3. New York: John Wiley 6 Sons. _(1985). W. New York: John Wiley & Sons. Rao, C. R. (1951). An asymptotic expansion of the distribution of Wilk's criterion. W W: 13. 177-180- Rogosa, D. (1980). Comparing nonparallel regression lines. W. 33. 307-321. SAS Institute. Inc. (1982). MW. Cary. NC: SAS Institute, Inc. Scheffe', H. (1959). Ihe_Ahely§1e_efi_ye;1ehee. New York: John Wiley & Sons. Srisukho, D. (1974). Monte Carlo Study of the Power of fl-test Compared to E-test When Population Distributions are Different in Form. Unpublished doctoral dissertation, University of California, Berkeley. Takeuchi, R., Yanai, H., & Mukherjee, B. N. (1982). oun on of hel§12e11e£e_ehely§1e. New Delhi, India: Wiley Eastern Limited. Tiku, M. L., 8: Singh, M. (1982). Robust statistics for testing mean vectors of multivariate distributions. 0 u a io s in W. 11. 985-1001. Timm, N. H. (1975). v w' 0 WW. Monterey, Cal.: Brooks/Cole Publishing Company. Tracy, D. 8., & Conley, W. C. (1982). Exact sampling distributions of the coefficient of kurtosis using the computer. Joughal of MW. 13. 262-265. Vale,C. D., & Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions. Beyehemeegihe, 48(3), 465-471. Williams, E. J. (1959). The comparison of regression variables. Jeurhal Wen. Series B. 2.1. 396-399. Zwick, R. J. (1984). A Monte Carlo Investigation of Non-parametric One-Way Multivariate Analysis of Variance in the No-Group Casef Unpublished doctoral dissertation, University of California, Berkeley. HICHIGRN STATE UN IV. LIBRRRIES WI! HUI] IIHI lllllNIIIHHIIWIIWI 312 300600466