Comparative Power Of The Anova, Randomization Anova, And Kruskal-Wallis Test

96 Pages·2015·0.38 MB·English

Wayne State University
Wayne State University Dissertations
1-1-2013
Comparative Power Of The Anova,
Randomization Anova, And Kruskal-Wallis Test
Jamie Gleason
Wayne State University,
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Gleason, Jamie, "Comparative Power Of The Anova, Randomization Anova, And Kruskal-Wallis Test" (2013).Wayne State University
Dissertations.Paper 658.
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COMPARATIVE POWER OF THE ANOVA, APPROXIMATE RANDOMIZATION
ANOVA, AND KRUSKAL-WALLIS TEST

by

JAMIE H. GLEASON
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
2013
MAJOR: EVALUATION AND RESEARCH

Approved by:
___________________________________
Advisor Date

___________________________________

___________________________________

___________________________________
DEDICATION
I dedicate this to my wonderful wife, Amanda, who somehow agreed it would be a
great idea to move 2,400 miles across the country so her fiancé could begin this
journey, and then proceeded to marry him. Not to be outdone, though, are the
Buggels, Caoimhe and Tobaira, who provided me with the daily passion and
reminder to finish what I started a lifetime ago. I will forever associate their love of
Mary Poppins with my desire for peace and quiet. In the end, the bulk chapter 4 and
5 were written while humming to myself about a spoonful of sugar. Now that I have
completed my project, there's time to go fly a kite.

ii
ACKNOWLEDGMENTS
First and foremost, I need to acknowledge my mother, Judi, for supporting me
throughout my educational career and always finding a way to help me to do the
things I needed, or sometimes just wanted, to do. Also to my stepfather, Ed. I truly
appreciate how difficult it was to meet me as a teenager, but you have always filled
the role you stepped into all of those years ago.
Academically, I would like to acknowledge my committee, and thank them for
making the process as painless as possible. Dr. Sawilowsky always assured me that
the main goal of enduring this process is to get my degree, something that he
empowered me to do through the entire dissertation process. Dr. Fahoome almost
single-handedly taught me everything that I learned throughout the program. Every
semester for nearly three years we were reunited… hockey season or no hockey
season. Dr. Bridge and, the most recent addition, Dr. Castronova, for being willing to
be active participants in my dissertation endeavor, and for the latter, after the ball had
already begun rolling.
Finally, I could have never done this without the support of my family. Thank
you. Now it’s my turn to give back to the Buggeldome.

iii
TABLE OF CONTENTS
Dedication_____________________________________________________ ii
Acknowledgments_______________________________________________ iii
List of Tables___________________________________________________ vi
List of Figures__________________________________________________ viii
Chapter 1 Introduction___________________________________________ 1
Problem_________________________________________________ 2
Purpose of the Study_______________________________________ 4
Assumptions and Limitations_________________________________ 5
Definition of Terms_________________________________________ 5
Chapter 2 Literature Review_______________________________________ 8
Hypothesis Testing_________________________________________ 8
Parametric Tests__________________________________________ 9
Permutation and Randomization Tests_________________________ 11
Nonparametric Tests_______________________________________ 14
Underlying Assumptions____________________________________ 16
Chapter 3 Methodology__________________________________________ 17
Design__________________________________________________ 17
Distributions______________________________________________ 17
Sample Sizes and Effect Sizes_______________________________ 21
Fortran Programming_______________________________________ 23
Presentation of Results_____________________________________ 24

iv
Chapter 4 Results_______________________________________________ 25
Type I Error______________________________________________ 25
Comparative Power Analysis________________________________ 27
Normal Distribution________________________________________ 28
Uniform Distribution________________________________________ 38
Chi-Square (df=2) Distribution________________________________ 48
Chapter 5 Discussion____________________________________________ 70
Type I Error______________________________________________ 70
Comparative Statistical Power_______________________________ 71
Implications______________________________________________ 74
Conclusion______________________________________________ 75
References____________________________________________________ 77
Abstract______________________________________________________ 83
Autobiographical Statement_______________________________________ 85
v
LIST OF TABLES

Table 1: Sample Size and Treatment Conditions……………….. 22

Table 2: Rejections Under Null Condition for Normal
Distribution (Type I Error)………………………………… 25

Table 3: Rejections Under Null Condition for Uniform
Distribution (Type I Error)………………………………… 26

Table 4: Rejections Under Null Condition for Chi-Square
(df=2) Distribution (Type I Error)………………………… 26

Table 5: Rejections of the Null Under Treatment Condition for
n =n =n (tr)=10……………………………………………. 58
1 2 3

Table 6: Rejections of the Null Under Treatment Condition for
n =n (tr)=n (tr)=10………………………………………… 59
1 2 3

Table 7: Rejections of the Null Under Treatment Condition for
n =n =n (tr)=30……………………………………………. 60
1 2 3

Table 8: Rejections of the Null Under Treatment Condition for
n =n (tr)=n (tr)=30………………………………………… 61
1 2 3

Table 9: Rejections of the Null Under Treatment Condition for
n =n =n =n =n (tr)=10……………………………………. 62
1 2 3 4 5

Table 10: Rejections of the Null Under Treatment Condition for
n =n =n =n (tr)=n (tr)=10………………………………… 63
1 2 3 4 5

Table 11: Rejections of the Null Under Treatment Condition for
n =n =n (tr)=n (tr)=n (tr)=10…………………………….. 64
1 2 3 4 5

Table 12: Rejections of the Null Under Treatment Condition for
n =n (tr)=n (tr)=n (tr)=n (tr)=10…………………………. 65
1 2 3 4 5

Table 13: Rejections of the Null Under Treatment Condition for
n =n =n =n =n (tr)=30……………………………………. 66
1 2 3 4 5

Table 14: Rejections of the Null Under Treatment Condition for
n1=n2=n3=n4(tr)=n5(tr)=30……………………………… 67

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Table 15: Rejections of the Null Under Treatment Condition for
n =n =n (tr)=n (tr)=n (tr)=30…………………………….. 68
1 2 3 4 5

Table 16: Rejections of the Null Under Treatment Condition for
n =n (tr)=n (tr)=n (tr)=n (tr)=30…………………………. 69
1 2 3 4 5

vii
LIST OF FIGURES

Figure 1: Gaussian (Normal) Distribution, Sawilowsky &
Fahoome (2003)………………………………………….. 18

Figure 2: Uniform Distribution, Sawilowsky & Fahoome (2003)… 19

Figure 3: Chi-Square Distribution, Sawilowsky &
Fahoome (2003)………………………………………….. 20

Figure 4: Shift vs. Power in the Normal Distribution for
Sample Condition n =n =n (tr)=10……………………… 28
1 2 3

Figure 5: Shift vs. Power in the Normal Distribution for Sample
Condition n =n (tr)=n (tr)=10……………………………. 29
1 2 3

Figure 6: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n (tr)=30………………………….……. 30
1 2 3

Figure 7: Shift vs. Power in the Normal Distribution for Sample
Condition n =n (tr)=n (tr)=30.…………………………… 31
1 2 3

Figure 8: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n =n =n (tr)=10………….……………. 32
1 2 3 4 5

Figure 9: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n =n (tr)=n (tr)=10……………………. 33
1 2 3 4 5

Figure 10: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n (tr)=n (tr)=n (tr)=1………………….. 33
1 2 3 4 5

Figure 11: Shift vs. Power in the Normal Distribution for Sample
Condition n =n (tr)=n (tr)=n (tr)=n (tr)=10…………….. 34
1 2 3 4 5

Figure 12: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n =n =n (tr)=30……………………….. 35
1 2 3 4 5

Figure 13: Shift vs. Power in the Normal Distribution for Sample
Condition n =n (tr)=n (tr)=n (tr)=n (tr)=30……………... 36
1 2 3 4 5

Figure 14: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n =n (tr)=n (tr)=30……………………. 37
1 2 3 4 5

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Figure 15: Shift vs. Power in the Normal Distribution for Sample
Condition n =n =n (tr)=n (tr)=n (tr)=30………………… 37
1 2 3 4 5

Figure 16: Shift vs. Power in the Uniform Distribution for
Sample Condition n =n =n (tr)=10……………………… 38
1 2 3

Figure 17: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n (tr)=n (tr)=10………………..………….. 39
1 2 3

Figure 18: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n (tr)=30………………………….……. 40
1 2 3

Figure 19: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n (tr)=n (tr)=30.………………..…………. 41
1 2 3

Figure 20: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n =n =n (tr)=10…………..…………… 42
1 2 3 4 5

Figure 21: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n =n (tr)=n (tr)=10……………………. 43
1 2 3 4 5

Figure 22: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n (tr)=n (tr)=n (tr)=10………………… 43
1 2 3 4 5

Figure 23: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n (tr)=n (tr)=n (tr)=n (tr)=10…………..… 44
1 2 3 4 5

Figure 24: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n =n =n (tr)=30……………………….. 45
1 2 3 4 5

Figure 25: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n (tr)=n (tr)=n (tr)=n (tr)=30……………… 46
1 2 3 4 5

Figure 26: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n =n (tr)=n (tr)=30……………………. 47
1 2 3 4 5

Figure 27: Shift vs. Power in the Uniform Distribution for Sample
Condition n =n =n (tr)=n (tr)=n (tr)=30………………… 47
1 2 3 4 5

Figure 28: Shift vs. Power in the Chi-Square (df=2) Distribution
for Sample Condition n =n =n (tr)=10………………….. 48
1 2 3

Figure 29: Shift vs. Power in the Chi-Square (df=2) Distribution
for Sample Condition n =n (tr)=n (tr)=10………………. 49
1 2 3

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