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Comparing the Statistical Tests for Homogeneity of Variances. Zhiqiang Mu East Tennessee State University

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Comparing the Statistical Tests for Homogeneity of Variances. Zhiqiang Mu East Tennessee State University East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations Student Works 8-2006 Comparing the Statistical Tests for Homogeneity of Variances. Zhiqiang Mu East Tennessee State University Follow this and additional works at: https://dc.etsu.edu/etd Part of the Statistical Theory Commons Recommended Citation Mu, Zhiqiang, "Comparing the Statistical Tests for Homogeneity of Variances." (2006). Electronic Theses and Dissertations. Paper 2212. https://dc.etsu.edu/etd/2212 This Thesis - Open Access is brought to you for free and open access by the Student Works at Digital Commons @ East Tennessee State University. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons @ East Tennessee State University. For more information, please contact [email protected]. Comparing the Statistical Tests for Homogeneity of Variances _______________________________ A thesis presented to the faculty of the Department of Mathematics East Tennessee State University In partial fulfillment of the requirements for the degree Master of Science in Mathematical Sciences _______________________________ by Zhiqiang Mu August, 2006 _______________________________ Edith Seier, PhD., Chair Robert Price, PhD. Yali Liu, PhD. Key words: Equality of Variances, Robust, ANOVA, Resampling, Bootstrap Test, Permutation Test ABSTRACT Comparing the Statistical Tests for Homogeneity of Variances by Zhiqiang Mu Testing the homogeneity of variances is an important problem in many applications since statistical methods of frequent use, such as ANOVA, assume equal variances for two or more groups of data. However, testing the equality of variances is a difficult problem due to the fact that many of the tests are not robust against non-normality. It is known that the kurtosis of the distribution of the source data can affect the performance of the tests for variance. We review the classical tests and their latest, more robust modifications, some other tests that have recently appeared in the literature, and use bootstrap and permutation techniques to test for equal variances. We compare the performance of these tests under different types of distributions, sample sizes and true variance ratios of the populations. Monte-Carlo methods are used in this study to calculate empirical powers and type I errors under different settings. 2 DEDICATION This thesis is dedicated to all of those people who have supported me throughout my educational journey at East Tennessee State University. 3 ACKNOWLEDGEMENTS The writer of this dissertation strongly feels indebted to those who provided assistance and encouragement with this work. My first appreciation goes to Dr. Seier, my committee chairperson: Your invaluable suggestions that helped to improve my performance, your deep sense of caring for your students, and your willingness to be available at any time will be long remembered. I would also like to express my deep appreciation to my committee members: Dr. Price and Dr. Liu. Thank you all for your words of insight and wisdom throughout the development of this project. Thank you all for your sincere and generous cooperation. Lastly, I express my gratitude and appreciation to my wife, Yu, who encouraged and helped me to finish this work despite all the mishaps. I must also thank my dearest daughter, Michelle, who has the sweetest smile in the world and who made a difference in my life, which encouraged me to move farther. 4 CONTENTS ABSTRACT........................................................................................................................ 2 DEDICATION.................................................................................................................... 3 ACKNOWLEDGEMENTS................................................................................................ 4 LIST OF FIGURES ............................................................................................................ 7 1 INTRODUCTION ...................................................................................................... 8 2 TESTS FOR EQUAL VARIABILITY....................................................................... 9 2.1 F Test ............................................................................................................. 9 2.2 F-Test Improved Version – Shoemaker's Adjustment................................ 10 2.3 Levene Test.................................................................................................. 12 2.4 Bartlett's Test ............................................................................................... 13 2.5 Bhat’s Version of Bartlett's Test.................................................................. 14 2.6 Count Five Test............................................................................................ 15 2.7 Introduction to Randomization Test ............................................................ 16 2.8 The Bootstrap Test....................................................................................... 17 2.9 The Permutation Test................................................................................... 20 3 EXPERIMENTAL CONDITIONS/DESIGN........................................................... 21 3.1 Introduction to Distributions........................................................................ 21 3.2 Systems of Distributions.............................................................................. 22 3.3 Transformed Distributions........................................................................... 24 3.4 Laplace Distribution..................................................................................... 26 3.5 Extreme Value Type I Distribution/Gumbel................................................ 26 3.6 Chi-Square Distribution............................................................................... 29 3.7 Weibull Distribution .................................................................................... 30 3.8 Tukey-Lambda Distribution......................................................................... 31 3.9 Logistic Distribution .................................................................................... 32 3.10 Student's T-Distribution............................................................................... 33 3.11 Half-Normal Distribution............................................................................. 34 3.12 Log Normal Distribution.............................................................................. 35 3.13 The Experimental Design/Simulations ........................................................ 36 5 3.14 The Computations........................................................................................ 37 4 SIMULATION RESULTS ....................................................................................... 39 5 CONCLUSIONS ...................................................................................................... 50 REFERENCES ................................................................................................................. 51 VITA................................................................................................................................. 53 6 LIST OF FIGURES 1 Sample pdfs of SB Distributions ................................................................................... 25 2 Sample pdfs of SU Distributions .................................................................................. 25 3 pdf and cdf of Laplace Distribution............................................................................... 26 4 pdf of Extreme Value Type 1 Minimum Distribution ................................................... 27 5 pdf of Extreme Value Type 1 Maximum Distribution................................................... 28 6 pdfs and cdfs of χ 2 Distribution................................................................................... 29 7 pdfs of Weibull Distribution .......................................................................................... 30 8 pdfs of Tukey-Lambda Distribution .............................................................................. 31 9 pdf and cdf of Logistic Distribution............................................................................... 32 10 pdf and cdf of Student's T-Distribution ....................................................................... 33 11 pdf and cdf of Half-Normal Distribution..................................................................... 34 12 pdf and cdf of Log Normal Distribution...................................................................... 35 13 Simulation Result χ 2 Distribution.............................................................................. 43 14 Simulation Result SU Distribution ............................................................................... 43 15 Simulation Result Extreme Distribution...................................................................... 44 16 Simulation Result Half Normal Distribution............................................................... 44 17 Simulation Result Lambda Distribution ...................................................................... 45 18 Simulation Result Laplace Distribution....................................................................... 45 19 Simulation Result Location Contaminated Distribution.............................................
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