An Introduction to Cochran-Mantel-Haenszel Testing and Nonparametric ANOVA (eBook)

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.

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

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

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

4.5 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

5.6 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

6.4 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

8.5 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

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

11.4 The end 286

A Appendix 289

A.1 Kronecker Products and Direct Sums 289

A.2 The Moore-Penrose Generalised Inverse 292