ROBUSTNESS AND COMPARATIVE STATISTICAL POWER OF THE REPEATED MEASURES ANOVA AND FRIEDMAN TEST WITH REAL DATA By OPEOLUWA BOLU FADEYI DISSERTATION Submitted to the Graduate School Of Wayne State University, Detroit, Michigan In partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 2021 MAJOR: EVALUATION AND RESEARCH Approved by: ____________________________________ Advisor Date ____________________________________ ____________________________________ ____________________________________ i DEDICATION To my husband, children, and parents. ii ACKNOWLEDGEMENTS I would like to express my profound gratitude to my dissertation advisor Dr. Shlomo Sawilowsky for his insightful guidance. It is difficult for me to comprehend how he made out time to review my dissertation progressively and thoroughly. His demise was shocking and devastating as my study was concluding. However, he has written his name in gold in my heart. I sincerely appreciate Dr. Barry Markman for all his helpful comments and also for stepping out of retirement to chair my final defense. The course that I took with Dr. Monte Piliawsky impacted the quality of my research significantly. On several instances, Dr. Aguwa reviewed my work and contributed immensely to the richness of my research. I am very grateful to these professors for servin g on my dissertation committee. Also, I will like to express my love and thankfulness to my husband, Johnson, who spent sleepless nights with me proofreading my work. Our children, Wisdom, Delight, and Goodness, that endured a busy mom are much appreciated. I sincerely acknowledge several people that have been a part of my life during the doctoral study who are too numerous to be included. Above all, I want to express my unquantifiable gratitude to God. His grace sustained and enabled me to reach the zenith of the academic ladder. iii TABLE OF CONTENTS DEDICATION II ACKNOWLEDGEMENTS III LIST OF TABLES V LIST OF FIGURES VII CHAPTER ONE: OVERVIEW OF THE PARAMETRIC TESTS 1 CHAPTER TWO: THEORETICAL FOUNDATIONS AND LITERATURE REVIEW 11 CHAPTER THREE: METHODOLOGY 46 CHAPTER FOUR : RESULTS AND DISCUSSION 60 CHAPTER FIVE CONCLUSIONS AND IMPLICATIONS 90 APPENDIX A 98 APPENDIX B 107 APPENDIX C 109 ABSTRACT 157 AUTOBIOGRAPHICAL STATEMENT 159 iv LIST OF TABLES Table 1. Hypothesis Table. ......................................................................................... 28 Table 2. Type I Error Rates when α=0.05 with G-G correction. ................................. 60 Table 3. Type I Error Rates for k=3,and G-G corrections. ......................................... 61 Table 4. Type I Error Rates for k=5,and G-G corrections .......................................... 62 Table 5. Type I Error Rates for k=7,and G-G corrections .......................................... 62 Table 6. Partial Eta squared for RM & Kendall’s W for Fr test ................................... 62 Table 7. The Sphericity Assumption Results. ............................................................. 64 Table 8. The power rates for n=5, k=3. ....................................................................... 83 Table 9. The power rates for n=8, k=3. ....................................................................... 83 Table 10. The power rates for n=12, k=3. .................................................................. 84 Table 14. The power rates for n=5, k=5...................................................................... 84 Table 15. The power rates for n=8, k=5...................................................................... 85 Table 16. The power rates for n=12, k=5 ................................................................... 85 Table 17. The power rates for n=18, k=5 ................................................................... 85 Table 18. The power rates for n=25, k=5 ................................................................... 85 Table 19. The power rates for n=33, k=5. .................................................................. 86 Table 20. The power rates for n=5, k=7...................................................................... 86 Table 21. The power rates for n=8, k=7...................................................................... 86 Table 22. The power rates for n=12, k=7. .................................................................. 86 Table 23. The power rates for n=18, k=7. .................................................................. 86 Table 24. The power rates for n=25, k=7. .................................................................. 87 v Table 25. The power rates for n=33, k=7. .................................................................. 87 Table 26. Power differences for all Samples, when K=3. .......................................... 87 Table 27. Power differences for all Samples, when K=5. .......................................... 88 Table 28 Power differences for all Samples, when K=7. ........................................... 89 vi LIST OF FIGURES Figure 1. Partition of Errors for One-factor Repeated Measures ANOVA. ................ 16 Figure 2.Gaussian (Normal) Distribution. ................................................................... 51 Figure 3.Chi-Square Distribution. ................................................................................ 52 Figure 4.Uniform Distribution. ..................................................................................... 53 Figure 5. Multivariate Normal Distribution for Sample Size of 5, k=7. ....................... 55 Figure 6. Multivariate Normal Distribution for Sample Size of 8, k=7. ....................... 55 Figure 7. Multivariate Normal Distribution for Sample Size of 12, k=7. ..................... 56 Figure 8. Multivariate Normal Distribution for Sample Size of 18, k=7. ..................... 56 Figure 9. Multivariate Normal Distribution for Sample Size of 25, k=7. ..................... 57 Figure 10. Multivariate Normal Distribution for Sample Size of 33, k=7. ................... 57 Figure 11. Comparative Power rate for the RM & Fr for n=5,k=3. ............................. 66 Figure 12. Comparative Power rate for the RM & Fr for n=5,k=5. ............................. 67 Figure 13. Comparative Power rate for the RM & Fr for n=5,k=7. ............................. 68 Figure 14. Comparative Power rate for the RM & Fr for n=8,k=3. ............................. 69 Figure 15. Comparative Power rate for the RM & Fr for n=8,k=5. ............................. 70 Figure 16. Comparative Power rate for the RM & Fr for n=8,k=7. ............................. 71 Figure 17. Comparative Power rate for the RM & Fr for n=12,k=3. ........................... 72 Figure 18. Comparative Power rate for the RM & Fr for n=12,k=5. ........................... 73 Figure 19. Comparative Power rate for the RM & Fr for n=12,k=7. ........................... 74 Figure 20. Comparative Power rate for the RM & Fr for n=18,k=3. ........................... 75 Figure 21. Comparative Power rate for the RM & Fr for n=18,k=5. ........................... 76 vii Figure 22. Comparative Power rate for the RM & Fr for n=18,k=7. ........................... 77 Figure 23. Comparative Power rate for the RM & Fr for n=25,k=3. ........................... 78 Figure 24. Comparative Power rate for the RM & Fr for n=25,k=5. ........................... 79 Figure 25. Comparative Power rate for the RM & Fr for n=25,k=7. ........................... 80 Figure 26. Comparative Power rate for the RM & Fr for n=33,k=3. ........................... 81 Figure 27. Comparative Power rate for the RM & Fr for n=33,k=5. ........................... 82 Figure 28. Comparative Power rate for the RM & Fr for n=33,k=7. ........................... 83 viii © COPYRIGHT BY OPEOLUWA BOLU FADEYI 2021 All Rights Reserved ix 1 CHAPTER ONE INTRODUCTION OVERVIEW OF THE PARAMETRIC TESTS Parametric tests are those which based the necessary assumptions on the parameters of the underlying population distribution from which the samples are drawn. It is generally believed that parametric tests are robust to the violation of some of the assumptions, this means that the tests have the power to control the probability of rejecting the false null hypothesis. For example, ANOVA can be used to analyze ordinal scale data such as Likert scales, even without any consequences (Leys & Schumann, 2010; Nanna & Sawilowsky, 1998; Zimmerman & Zumbo, 1993). Another peculiar characteristic of a parametric test is that it is uniformly most powerful unbiased (UMPU). “This means that when all underlying assumptions are met based on the inference from the samples, no other test has greater ability to detect a true difference for a given sample” (Bridge & Sawilowsky, 1999, p. 229). For example, the t-test is uniformly most powerful unbiased when the assumptions of independence, homoscedasticity, and normality are met (Bradley, 1968b; Kelley & Sawilowsky, 1997). However, a “light shift” in the shapes of the distribution of the variables when the number of samples in each treatment group gets close to 30 or more still generates robust results (Glass, Peckham, & Sanders, 1972; Leys & Schumann, 2010; Lix, Keselman, & Keselman, 1996; Lumley, Diehr, Emerson, & Chen, 2002). Studies have been carried out to examine the violation of the assumption of homogeneity of variances which may have a severe impact on the type I error rate of F-tests, and It has been established that F-test will yield statistically significant results when the group sample sizes are equal and size of the groups greater