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(cid:2) AnIntroductiontoCochran–Mantel–HaenszelTestingand NonparametricANOVA (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) An Introduction to Cochran–Mantel–Haenszel Testing and Nonparametric ANOVA J.C.W. Rayner and G. C. Livingston Jr. UniversityofNewcastle Australia (cid:2) (cid:2) (cid:2) (cid:2) Thiseditionfirstpublished2023 ©2023JohnWiley&SonsLtd Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrieval system,ortransmitted,inanyformorbyanymeans,electronic,mechanical, photocopying,recordingorotherwise,exceptaspermittedbylaw.Adviceonhowto obtainpermissiontoreusematerialfromthistitleisavailableathttp://www.wiley.com/ go/permissions. TherightofJ.C.W.RaynerandG.C.LivingstonJr.tobeidentifiedastheauthorsofthis workhasbeenassertedinaccordancewithlaw. 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LibraryofCongressCataloging-in-PublicationDataisAppliedfor: HardbackISBN:9781119831983 CoverDesign:Wiley CoverImages:©metamorworks/Shutterstock,KatieDobies/GettyImages Setin9.5/12.5ptSTIXTwoTextbyStraive,Chennai,India (cid:2) (cid:2) John:ForJohnBest,myfriendandcolleagueforoverhalfacentury Glen:Formyboys,HuxleyandDuke (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) vii Contents Preface xiii 1 Introduction 1 1.1 WhatAretheCMHandNPANOVATests? 1 1.2 Outline 2 1.3 4 1.4 Examples 5 1.4.1 StrawberryData 5 (cid:2) (cid:2) 1.4.2 HomosexualMarriageData 6 Bibliography 8 2 TheBasicCMHTests 9 2.1 Genesis:Cochran(1954),andMantelandHaenszel(1959) 9 2.2 TheBasicCMHTests 12 2.2.1 HomosexualMarriageData 14 2.2.2 JamsData 14 2.3 TheNominalCMHTests 15 2.4 TheCMHMeanScoresTest 18 2.5 TheCMHCorrelationTest 19 2.5.1 TheCMHCTestDefined 19 2.5.2 AnAlternativePresentationoftheCMHCTest 21 2.5.3 Examples 22 2.5.3.1 HomosexualMarriageData 22 2.5.3.2 WhiskeyData 23 2.5.4 DerivationoftheCMHCTestStatisticfortheRBDwiththe SameTreatmentScoresinEveryStratum 24 2.5.4.1 JamsDataRevisited 25 2.5.5 TheCMHCTestStatisticIsNot,inGeneral,Location-Scale Invariant 27 Bibliography 28 (cid:2) (cid:2) viii Contents 3 TheCompletelyRandomisedDesign 29 3.1 Introduction 29 3.2 TheDesignandParametricModel 30 3.3 TheKruskal–WallisTests 30 3.3.1 Mid-RankLemma 32 3.3.2 WhiskeyDataRevisited 33 3.4 RelatingtheKruskal–WallisandANOVAFTests 33 3.5 TheCMHTestsfortheCRD 35 3.5.1 WhiskeyDataRevisited 36 3.6 TheKWTestsAreCMHMSTests 37 3.7 RelatingtheCMHMSandANOVAFTests 38 3.7.1 TheOne-WayANOVAFTest 38 3.7.2 W inTermsoftheANOVAF 38 MS 3.7.3 WhiskeyDataRevisited 39 3.7.4 HomosexualMarriageDataRevisited 39 3.7.5 CornData 39 3.8 SimulationStudy 41 3.9 WaldTestStatisticsintheCRD 43 3.9.1 TheWaldTestStatisticofGeneralAssociationfortheCMH Design 43 (cid:2) (cid:2) 3.9.2 TheWaldTestStatisticfortheCMHMSDesign 47 3.9.3 TheWaldTestStatisticfortheCMHCDesign 49 Bibliography 49 4 TheRandomisedBlockDesign 51 4.1 Introduction 51 4.2 TheDesignandParametricModel 52 4.3 TheFriedmanTests 53 4.3.1 JamsData 55 4.4 TheCMHTestStatisticsintheRBD 56 4.4.1 TheCMHOPATestfortheRBD 56 4.4.2 TheCMHGATestStatisticfortheRBD 56 4.4.3 TheCMHMSTestStatisticfortheRBD 57 4.4.3.1 FoodProductsData 58 4.4.4 TheCMHCTestStatisticfortheRBD 60 4.4.4.1 HumanResourceData 60 4.5 TheFriedmanTestsareCMHMSTests 62 4.6 RelatingtheCMHMSandANOVAFTests 63 4.6.1 JamsDataRevisited 65 4.7 SimulationStudy 65 4.8 WaldTestStatisticsintheRBD 68 Bibliography 72 (cid:2)