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What is included with this book?
Complete reference for applied statisticians and data analysts that uniquely covers the new statistical methodologies that enable deeper data analysis
An Introduction to Cochran-Mantel-Haenszel Testing and Nonparametric ANOVA provides readers with powerful new statistical methodologies that enable deeper data analysis. The book offers applied statisticians an introduction to the latest topics in nonparametrics. The worked examples with supporting R code provide analysts the tools they need to apply these methods to their own problems.
Co-authored by an internationally recognised expert in the field and an early career researcher with broad skills including data analysis and R programming, the book discusses key topics such as:
Applied statisticians and data analysts, as well as students and professors in data analysis, can use this book to gain a complete understanding of the modern statistical methodologies that are allowing for deeper data analysis.
John Charles William Rayner is an Honorary Professorial Fellow, National Institute for Applied Statistics Research Australia, University of Wollongong, and Conjoint Professor of Statistics, School of Mathematical and Physical Sciences, University of Newcastle, Australia.
Glen Livingston, Jr., is a Lecturer, School of Mathematical and Physical Sciences, University of Newcastle, Australia.
Contents
Preface xiii
1 Introduction 1
1.1 What are the CMH and NP ANOVA tests? . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 The Basic CMH Tests 13
2.1 Genesis: Cochran (1954), and Mantel and Haenszel (1959) . . 13
2.2 The basic CMH tests . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 The Nominal CMH tests . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 The CMH mean scores test . . . . . . . . . . . . . . . . . . . . . 26
2.5 The CMH correlation test . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 The CMH C test defined . . . . . . . . . . . . . . . . 28
2.5.2 An alternative presentation of the CMH C test . . . 30
2.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.4 Derivation of the CMH C test statistic for the RBD
with the same treatment scores in every stratum . . 34
2.5.5 The CMH C test statistic is not, in general, locationscale
invariant. . . . . . . . . . . . . . . . . . . . . . . 38
vii
3 The Completely Randomised Design 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 The design and parametric model . . . . . . . . . . . . . . . . . 42
3.3 The Kruskal-Wallis tests . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Relating the Kruskal-Wallis and ANOVA F tests . . . . . . . . . 47
3.5 The CMH tests for the CRD . . . . . . . . . . . . . . . . . . . . 49
3.6 The KW tests are CMH MS tests . . . . . . . . . . . . . . . . . 52
3.7 Relating the CMH MS and ANOVA F tests . . . . . . . . . . . . 54
3.8 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.9 Wald test statistics in the CRD . . . . . . . . . . . . . . . . . . . 61
3.9.1 The Wald test statistic of general association for
the CMH design . . . . . . . . . . . . . . . . . . . . . 61
3.9.2 The Wald test statistic for the CMH MS design . . 67
3.9.3 The Wald test statistic for the CMH C design . . . 69
4 The Randomised Block Design 71
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 The design and parametric model . . . . . . . . . . . . . . . . . 72
4.3 The Friedman tests . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 The CMH test statistics in the RBD . . . . . . . . . . . . . . . . 77
4.4.1 The CMH OPA test for the RBD . . . . . . . . . . . 78
4.4.2 The CMH GA test statistic for the RBD . . . . . . . 78
4.4.3 The CMH MS test statistic for the RBD . . . . . . . 79
4.4.4 The CMH C test statistic for the RBD . . . . . . . . 84
viii
4.5 The Friedman tests are CMH MS tests . . . . . . . . . . . . . . 86
4.6 Relating the CMH MS and ANOVA F tests . . . . . . . . . . . . 88
4.7 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.8 Wald test statistics in the RBD . . . . . . . . . . . . . . . . . . . 94
5 The Balanced Incomplete Block Design 101
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 The Durbin tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 The relationship between the adjusted Durbin statistic and the
ANOVA F statistic . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5 Orthogonal contrasts for balanced designs with ordered treatments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5.1 Orthogonal contrasts . . . . . . . . . . . . . . . . . . 113
5.5.2 Orthogonal contrasts for nonparametric testing in
balanced designs . . . . . . . . . . . . . . . . . . . . . 114
5.5.3 F orthogonal contrasts . . . . . . . . . . . . . . . . . 119
5.5.4 Simulation study . . . . . . . . . . . . . . . . . . . . . 124
5.6 A CMH MS analogue test statistic for the BIBD . . . . . . . . 124
6 Unconditional Analogues of CMH Tests 129
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Unconditional univariate moment tests . . . . . . . . . . . . . . 132
6.3 Generalised correlations . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 Bivariate generalised correlations . . . . . . . . . . . 137
ix
6.3.2 Trivariate generalised correlations . . . . . . . . . . . 142
6.4 Unconditional bivariate moment tests . . . . . . . . . . . . . . . 147
6.5 Unconditional general association tests . . . . . . . . . . . . . . 152
6.6 Stuart’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7 Higher Moment Extensions To The Ordinal CMH Tests 167
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.2 Extensions to the CMH mean scores test . . . . . . . . . . . . . 168
7.3 Extensions to the CMH correlation test . . . . . . . . . . . . . . 172
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8 Unordered Nonparametric ANOVA 183
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.2 Unordered NP ANOVA for the CMH design . . . . . . . . . . . 187
8.3 Singly ordered three-way tables . . . . . . . . . . . . . . . . . . . 189
8.4 The Kruskal-Wallis and Friedman tests are NP ANOVA tests . 193
8.4.1 The Kruskal-Wallis, ANOVA F, and NP ANOVA F
tests on the ranks are all equivalent . . . . . . . . . 193
8.4.2 The Friedman, ANOVA F, and NP ANOVA F tests
are all equivalent . . . . . . . . . . . . . . . . . . . . . 195
8.5 Are the CMH MS and extensions NP ANOVA tests? . . . . . . 197
8.6 Extension to other designs . . . . . . . . . . . . . . . . . . . . . . 199
8.7 Latin squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.8 Balanced incomplete blocks . . . . . . . . . . . . . . . . . . . . . 204
x
9 The Latin Square Design 207
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.2 The Latin square design and parametric model . . . . . . . . . . 208
9.3 The RL test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
9.4 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.7 Orthogonal trend contrasts for ordered treatments . . . . . . . . 232
9.8 Technical derivation of the RL test . . . . . . . . . . . . . . . . . 238
10 Ordered Nonparametric ANOVA 243
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.2 Ordered NP ANOVA for the CMH design . . . . . . . . . . . . . 247
10.3 Doubly ordered three-way tables . . . . . . . . . . . . . . . . . . 249
10.4 Extension to other designs . . . . . . . . . . . . . . . . . . . . . . 252
10.5 Latin square rank tests . . . . . . . . . . . . . . . . . . . . . . . . 255
10.6 Modelling the moments of the response variable . . . . . . . . . 257
10.7 Lemonade sweetness data . . . . . . . . . . . . . . . . . . . . . . 262
10.8 Breakfast cereal data revisited . . . . . . . . . . . . . . . . . . . 271
11 Conclusion 275
11.1 CMH or NP ANOVA? . . . . . . . . . . . . . . . . . . . . . . . . 275
11.2 Homosexual marriage data revisited for the last time! . . . . . . 277
11.3 Job satisfaction data . . . . . . . . . . . . . . . . . . . . . . . . . 280
xi
11.4 The end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
A Appendix 289
A.1 Kronecker Products and Direct Sums . . . . . . . . . . . . . . . 289
A.2 The Moore-Penrose Generalised Inverse . . . . . . . . . . . . . . 292
xii
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