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 xiii1 Introduction 11.1 What are the CMH and NP ANOVA tests? 11.2 Outline 31.3 51.4 Examples 62 The Basic CMH Tests 132.1 Genesis: Cochran (1954), and Mantel and Haenszel (1959) 132.2 The basic CMH tests 182.3 The Nominal CMH tests 222.4 The CMH mean scores test 262.5 The CMH correlation test 283 The Completely Randomised Design 413.1 Introduction 413.2 The design and parametric model 423.3 The Kruskal-Wallis tests 433.4 Relating the Kruskal-Wallis and ANOVA F tests 473.5 The CMH tests for the CRD 493.6 The KW tests are CMH MS tests 523.7 Relating the CMH MS and ANOVA F tests 543.8 Simulation study 583.9 Wald test statistics in the CRD 614 The Randomised Block Design 714.1 Introduction 714.2 The design and parametric model 724.3 The Friedman tests 744.4 The CMH test statistics in the RBD 774.5 The Friedman tests are CMH MS tests 864.6 Relating the CMH MS and ANOVA F tests 884.7 Simulation study 914.8 Wald test statistics in the RBD 945 The Balanced Incomplete Block Design 1015.1 Introduction 1015.2 The Durbin tests 1015.3 The relationship between the adjusted Durbin statistic and the ANOVA F statistic 1035.4 Simulation study 1105.5 Orthogonal contrasts for balanced designs with ordered treatments 1135.6 A CMH MS analogue test statistic for the BIBD 1246 Unconditional Analogues of CMH Tests 1296.1 Introduction 1296.2 Unconditional univariate moment tests 1326.3 Generalised correlations 1376.4 Unconditional bivariate moment tests 1476.5 Unconditional general association tests 1526.6 Stuart’s Test 1637 Higher Moment Extensions To The Ordinal CMH Tests 1677.1 Introduction 1677.2 Extensions to the CMH mean scores test 1687.3 Extensions to the CMH correlation test 1727.4 Examples 1768 Unordered Nonparametric ANOVA 1838.1 Introduction 1838.2 Unordered NP ANOVA for the CMH design 1878.3 Singly ordered three-way tables 1898.4 The Kruskal-Wallis and Friedman tests are NP ANOVA tests 1938.5 Are the CMH MS and extensions NP ANOVA tests? 1978.6 Extension to other designs 1998.7 Latin squares 2028.8 Balanced incomplete blocks 2049 The Latin Square Design 2079.1 Introduction 2079.2 The Latin square design and parametric model 2089.3 The RL test 2109.4 Alignment 2129.5 Simulation study 2169.6 Examples 2259.7 Orthogonal trend contrasts for ordered treatments 2329.8 Technical derivation of the RL test 23810 Ordered Nonparametric ANOVA 24310.1 Introduction 24310.2 Ordered NP ANOVA for the CMH design 24710.3 Doubly ordered three-way tables 24910.4 Extension to other designs 25210.5 Latin square rank tests 25510.6 Modelling the moments of the response variable 25710.7 Lemonade sweetness data 26210.8 Breakfast cereal data revisited 27111 Conclusion 27511.1 CMH or NP ANOVA? 27511.2 Homosexual marriage data revisited for the last time! 27711.3 Job satisfaction data 28011.4 The end 286A Appendix 289A.1 Kronecker Products and Direct Sums 289A.2 The Moore-Penrose Generalised Inverse 292

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 xiii1 Introduction 11.1 What are the CMH and NP ANOVA tests? 11.2 Outline 31.3 51.4 Examples 62 The Basic CMH Tests 132.1 Genesis: Cochran (1954), and Mantel and Haenszel (1959) 132.2 The basic CMH tests 182.3 The Nominal CMH tests 222.4 The CMH mean scores test 262.5 The CMH correlation test 283 The Completely Randomised Design 413.1 Introduction 413.2 The design and parametric model 423.3 The Kruskal-Wallis tests 433.4 Relating the Kruskal-Wallis and ANOVA F tests 473.5 The CMH tests for the CRD 493.6 The KW tests are CMH MS tests 523.7 Relating the CMH MS and ANOVA F tests 543.8 Simulation study 583.9 Wald test statistics in the CRD 614 The Randomised Block Design 714.1 Introduction 714.2 The design and parametric model 724.3 The Friedman tests 744.4 The CMH test statistics in the RBD 774.5 The Friedman tests are CMH MS tests 864.6 Relating the CMH MS and ANOVA F tests 884.7 Simulation study 914.8 Wald test statistics in the RBD 945 The Balanced Incomplete Block Design 1015.1 Introduction 1015.2 The Durbin tests 1015.3 The relationship between the adjusted Durbin statistic and the ANOVA F statistic 1035.4 Simulation study 1105.5 Orthogonal contrasts for balanced designs with ordered treatments 1135.6 A CMH MS analogue test statistic for the BIBD 1246 Unconditional Analogues of CMH Tests 1296.1 Introduction 1296.2 Unconditional univariate moment tests 1326.3 Generalised correlations 1376.4 Unconditional bivariate moment tests 1476.5 Unconditional general association tests 1526.6 Stuart’s Test 1637 Higher Moment Extensions To The Ordinal CMH Tests 1677.1 Introduction 1677.2 Extensions to the CMH mean scores test 1687.3 Extensions to the CMH correlation test 1727.4 Examples 1768 Unordered Nonparametric ANOVA 1838.1 Introduction 1838.2 Unordered NP ANOVA for the CMH design 1878.3 Singly ordered three-way tables 1898.4 The Kruskal-Wallis and Friedman tests are NP ANOVA tests 1938.5 Are the CMH MS and extensions NP ANOVA tests? 1978.6 Extension to other designs 1998.7 Latin squares 2028.8 Balanced incomplete blocks 2049 The Latin Square Design 2079.1 Introduction 2079.2 The Latin square design and parametric model 2089.3 The RL test 2109.4 Alignment 2129.5 Simulation study 2169.6 Examples 2259.7 Orthogonal trend contrasts for ordered treatments 2329.8 Technical derivation of the RL test 23810 Ordered Nonparametric ANOVA 24310.1 Introduction 24310.2 Ordered NP ANOVA for the CMH design 24710.3 Doubly ordered three-way tables 24910.4 Extension to other designs 25210.5 Latin square rank tests 25510.6 Modelling the moments of the response variable 25710.7 Lemonade sweetness data 26210.8 Breakfast cereal data revisited 27111 Conclusion 27511.1 CMH or NP ANOVA? 27511.2 Homosexual marriage data revisited for the last time! 27711.3 Job satisfaction data 28011.4 The end 286A Appendix 289A.1 Kronecker Products and Direct Sums 289A.2 The Moore-Penrose Generalised Inverse 292

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 xiii1 Introduction 11.1 What are the CMH and NP ANOVA tests? 11.2 Outline 31.3 51.4 Examples 62 The Basic CMH Tests 132.1 Genesis: Cochran (1954), and Mantel and Haenszel (1959) 132.2 The basic CMH tests 182.3 The Nominal CMH tests 222.4 The CMH mean scores test 262.5 The CMH correlation test 283 The Completely Randomised Design 413.1 Introduction 413.2 The design and parametric model 423.3 The Kruskal-Wallis tests 433.4 Relating the Kruskal-Wallis and ANOVA F tests 473.5 The CMH tests for the CRD 493.6 The KW tests are CMH MS tests 523.7 Relating the CMH MS and ANOVA F tests 543.8 Simulation study 583.9 Wald test statistics in the CRD 614 The Randomised Block Design 714.1 Introduction 714.2 The design and parametric model 724.3 The Friedman tests 744.4 The CMH test statistics in the RBD 774.5 The Friedman tests are CMH MS tests 864.6 Relating the CMH MS and ANOVA F tests 884.7 Simulation study 914.8 Wald test statistics in the RBD 945 The Balanced Incomplete Block Design 1015.1 Introduction 1015.2 The Durbin tests 1015.3 The relationship between the adjusted Durbin statistic and the ANOVA F statistic 1035.4 Simulation study 1105.5 Orthogonal contrasts for balanced designs with ordered treatments 1135.6 A CMH MS analogue test statistic for the BIBD 1246 Unconditional Analogues of CMH Tests 1296.1 Introduction 1296.2 Unconditional univariate moment tests 1326.3 Generalised correlations 1376.4 Unconditional bivariate moment tests 1476.5 Unconditional general association tests 1526.6 Stuart’s Test 1637 Higher Moment Extensions To The Ordinal CMH Tests 1677.1 Introduction 1677.2 Extensions to the CMH mean scores test 1687.3 Extensions to the CMH correlation test 1727.4 Examples 1768 Unordered Nonparametric ANOVA 1838.1 Introduction 1838.2 Unordered NP ANOVA for the CMH design 1878.3 Singly ordered three-way tables 1898.4 The Kruskal-Wallis and Friedman tests are NP ANOVA tests 1938.5 Are the CMH MS and extensions NP ANOVA tests? 1978.6 Extension to other designs 1998.7 Latin squares 2028.8 Balanced incomplete blocks 2049 The Latin Square Design 2079.1 Introduction 2079.2 The Latin square design and parametric model 2089.3 The RL test 2109.4 Alignment 2129.5 Simulation study 2169.6 Examples 2259.7 Orthogonal trend contrasts for ordered treatments 2329.8 Technical derivation of the RL test 23810 Ordered Nonparametric ANOVA 24310.1 Introduction 24310.2 Ordered NP ANOVA for the CMH design 24710.3 Doubly ordered three-way tables 24910.4 Extension to other designs 25210.5 Latin square rank tests 25510.6 Modelling the moments of the response variable 25710.7 Lemonade sweetness data 26210.8 Breakfast cereal data revisited 27111 Conclusion 27511.1 CMH or NP ANOVA? 27511.2 Homosexual marriage data revisited for the last time! 27711.3 Job satisfaction data 28011.4 The end 286A Appendix 289A.1 Kronecker Products and Direct Sums 289A.2 The Moore-Penrose Generalised Inverse 292