An Introduction to Cochran-Mantel-Haenszel Testing and Nonparametric ANOVA

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:

• NP ANOVA methodology
• Cochran-Mantel-Haenszel (CMH) methodology and design
• Latin squares and balanced incomplete block designs
• Parametric ANOVA F tests for continuous data
• Nonparametric rank tests (the Kruskal-Wallis and Friedman tests)
• CMH MS tests for the nonparametric analysis of categorical response data

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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