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Trimsize:170mmx244mm Galwey ffirs.tex V2-07/25/2014 11:34A.M. Pageii Trimsize:170mmx244mm Galwey ffirs.tex V2-07/25/2014 11:34A.M. Pagei Introduction to Mixed Modelling Trimsize:170mmx244mm Galwey ffirs.tex V2-07/25/2014 11:34A.M. Pageii Trimsize:170mmx244mm Galwey ffirs.tex V2-07/25/2014 11:34A.M. Pageiii Introduction to Mixed Modelling Beyond Regression and Analysis of Variance Second Edition N. W. Galwey Statistical Consulting Group, GlaxoSmithKline, UK Trimsize:170mmx244mm Galwey ffirs.tex V2-07/25/2014 11:34A.M. Pageiv Thiseditionfirstpublished2014 ©2014JohnWiley&Sons,Ltd Registeredoffice JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,United Kingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapply forpermissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. Therightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewith theCopyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor otherwise,exceptaspermittedbytheUKCopyright,DesignsandPatentsAct1988,withouttheprior permissionofthepublisher. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprint maynotbeavailableinelectronicbooks. Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.All brandnamesandproductnamesusedinthisbookaretradenames,servicemarks,trademarksor registeredtrademarksoftheirrespectiveowners.Thepublisherisnotassociatedwithanyproductor vendormentionedinthisbook. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbestefforts inpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Itissoldontheunderstandingthatthepublisheris notengagedinrenderingprofessionalservicesandneitherthepublishernortheauthorshallbeliable fordamagesarisingherefrom.Ifprofessionaladviceorotherexpertassistanceisrequired,theservices ofacompetentprofessionalshouldbesought. LibraryofCongressCataloging-in-PublicationData Galwey,Nicholas Introductiontomixedmodelling:beyondregressionandanalysisofvariance/N.W.Galwey. – Secondedition. pagescm Includesbibliographicalreferencesandindex. ISBN978-1-119-94549-9(cloth) 1. Multilevelmodels(Statistics) 2. Experimentaldesign. 3. Regressionanalysis. 4. Analysisof variance. I.Title. QA276.G332014 519.5–dc23 2014021670 AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN:978-1-119-94549-9 Setin10/11.5ptTimesLTstd-RomanbyLaserwordsPrivateLimited,Chennai,India. 1 2014 Trimsize:170mmx244mm Galwey ftoc.tex V1-07/25/2014 6:00P.M. Pagev Contents Preface xi 1 Theneedformorethanonerandom-effecttermwhenfittingaregressionline 1 1.1 AdatasetwithseveralobservationsofvariableYateachvalueof variableX 1 1.2 Simpleregressionanalysis:UseofthesoftwareGenStattoperform theanalysis 2 1.3 Regressionanalysisonthegroupmeans 9 1.4 Aregressionmodelwithatermforthegroups 10 1.5 ConstructionoftheappropriateFtestforthesignificanceofthe explanatoryvariablewhengroupsarepresent 13 1.6 Thedecisiontospecifyamodeltermasrandom:Amixedmodel 14 1.7 Comparisonofthetestsinamixedmodelwithatestoflackoffit 16 1.8 TheuseofREsidualMaximumLikelihood(REML)tofitthemixed model 17 1.9 Equivalenceofthedifferentanalyseswhenthenumberofobservations pergroupisconstant 21 1.10 Testingtheassumptionsoftheanalyses:Inspectionoftheresidualvalues 26 1.11 UseofthesoftwareRtoperformtheanalyses 28 1.12 UseofthesoftwareSAStoperformtheanalyses 33 1.13 FittingamixedmodelusingGenStat’sGraphicalUserInterface(GUI) 40 1.14 Summary 46 1.15 Exercises 47 References 51 2 Theneedformorethanonerandom-effectterminadesignedexperiment 52 2.1 Thesplitplotdesign:Adesignwithmorethanonerandom-effectterm 52 2.2 Theanalysisofvarianceofthesplitplotdesign:Arandom-effecttermfor themainplots 54 2.3 Consequencesoffailuretorecognizethemainplotswhenanalysingthe splitplotdesign 62 2.4 Theuseofmixedmodellingtoanalysethesplitplotdesign 64 2.5 AmoreconservativealternativetotheFandWaldstatistics 66 2.6 Justificationforregardingblockeffectsasrandom 67 Trimsize:170mmx244mm Galwey ftoc.tex V1-07/25/2014 6:00P.M. Pagevi vi Contents 2.7 Testingtheassumptionsoftheanalyses:Inspectionoftheresidualvalues 68 2.8 UseofRtoperformtheanalyses 71 2.9 UseofSAStoperformtheanalyses 77 2.10 Summary 81 2.11 Exercises 82 References 86 3 Estimationofthevariancesofrandom-effectterms 87 3.1 Theneedtoestimatevariancecomponents 87 3.2 Ahierarchicalrandom-effectsmodelforathree-stageassayprocess 87 3.3 Therelationshipbetweenvariancecomponentsandstratummeansquares 91 3.4 Estimationofthevariancecomponentsinthehierarchicalrandom-effects model 93 3.5 Designofanoptimumstrategyforfuturesampling 95 3.6 UseofRtoanalysethehierarchicalthree-stageassayprocess 98 3.7 UseofSAStoanalysethehierarchicalthree-stageassayprocess 100 3.8 Geneticvariation:Acropfieldtrialwithanunbalanceddesign 102 3.9 Productionofabalancedexperimentaldesignby‘padding’with missingvalues 106 3.10 Specificationofatreatmenttermasarandom-effectterm:Theuseof mixed-modelanalysistoanalyseanunbalanceddataset 110 3.11 Comparisonofavariancecomponentestimatewithitsstandarderror 112 3.12 Analternativesignificancetestforvariancecomponents 113 3.13 Comparisonamongsignificancetestsforvariancecomponents 116 3.14 Inspectionoftheresidualvalues 117 3.15 Heritability:Thepredictionofgeneticadvanceunderselection 117 3.16 UseofRtoanalysetheunbalancedfieldtrial 122 3.17 UseofSAStoanalysetheunbalancedfieldtrial 125 3.18 Estimationofvariancecomponentsintheregressionanalysison groupeddata 128 3.19 Estimationofvariancecomponentsforblockeffectsinthesplit-plot experimentaldesign 130 3.20 Summary 132 3.21 Exercises 133 References 136 4 Intervalestimatesforfixed-effecttermsinmixedmodels 137 4.1 Theconceptofanintervalestimate 137 4.2 Standarderrorsforregressioncoefficientsinamixed-modelanalysis 138 4.3 Standarderrorsfordifferencesbetweentreatmentmeansinthesplit-plot design 142 4.4 Asignificancetestforthedifferencebetweentreatmentmeans 144 4.5 Theleastsignificantdifference(LSD)betweentreatmentmeans 147 4.6 Standarderrorsfortreatmentmeansindesignedexperiments: Adifferenceinapproachbetweenanalysisofvarianceand mixed-modelanalysis 151 4.7 UseofRtoobtainSEsofmeansinadesignedexperiment 157 Trimsize:170mmx244mm Galwey ftoc.tex V1-07/25/2014 6:00P.M. Pagevii vii Contents 4.8 UseofSAStoobtainSEsofmeansinadesignedexperiment 159 4.9 Summary 161 4.10 Exercises 163 References 164 5 Estimationofrandomeffectsinmixedmodels:BestLinearUnbiased Predictors(BLUPs) 165 5.1 Thedifferencebetweentheestimatesoffixedandrandomeffects 165 5.2 Themethodforestimationofrandomeffects:Thebestlinearunbiased predictor(BLUP)or‘shrunkestimate’ 168 5.3 TherelationshipbetweentheshrinkageofBLUPsandregressiontowards themean 170 5.4 UseofRfortheestimationoffixedandrandomeffects 176 5.5 UseofSASfortheestimationofrandomeffects 178 5.6 TheBayesianinterpretationofBLUPs:Justificationofarandom-effect termwithoutinvokinganunderlyinginfinitepopulation 182 5.7 Summary 187 5.8 Exercises 188 References 191 6 Moreadvancedmixedmodelsformoreelaboratedatasets 192 6.1 Featuresofthemodelsintroducedsofar:Areview 192 6.2 Furthercombinationsofmodelfeatures 192 6.3 Thechoiceofmodeltermstobespecifiedasrandom 195 6.4 Disagreementconcerningtheappropriatesignificancetestwhenfixed- andrandom-effecttermsinteract:‘Thegreatmixed-modelmuddle’ 197 6.5 Argumentsforspecifyingblockeffectsasrandom 204 6.6 Examplesofthechoiceoffixed-andrandom-effectspecificationofterms 209 6.7 Summary 213 6.8 Exercises 215 References 216 7 Threecasestudies 217 7.1 Furtherdevelopmentofmixedmodellingconceptsthroughtheanalysis ofspecificdatasets 217 7.2 Afixed-effectsmodelwithseveralvariatesandfactors 218 7.3 UseofRtofitthefixed-effectsmodelwithseveralvariatesandfactors 233 7.4 UseofSAStofitthefixed-effectsmodelwithseveralvariatesandfactors 237 7.5 Arandomcoefficientregressionmodel 242 7.6 UseofRtofittherandomcoefficientsmodel 246 7.7 UseofSAStofittherandomcoefficientsmodel 247 7.8 Arandom-effectsmodelwithseveralfactors 249 7.9 UseofRtofittherandom-effectsmodelwithseveralfactors 266 7.10 UseofSAStofittherandom-effectsmodelwithseveralfactors 274 7.11 Summary 282 7.12 Exercises 282 References 294 Trimsize:170mmx244mm Galwey ftoc.tex V1-07/25/2014 6:00P.M. Pageviii viii Contents 8 Meta-analysisandthemultipletestingproblem 295 8.1 Meta-analysis:Combinedanalysisofasetofstudies 295 8.2 Fixed-effectmeta-analysiswithestimationonlyofthemaineffectof treatment 296 8.3 Random-effectsmeta-analysiswithestimationofstudy×treatment interactioneffects 301 8.4 Arandom-effectinteractionbetweentwofixed-effectterms 303 8.5 Meta-analysisofindividual-subjectdatausingR 307 8.6 Meta-analysisofindividual-subjectdatausingSAS 312 8.7 Meta-analysiswhenonlysummarydataareavailable 318 8.8 Themultipletestingproblem:ShrinkageofBLUPsasadefenceagainst theWinner’sCurse 326 8.9 FittingofmultiplemodelsusingR 338 8.10 FittingofmultiplemodelsusingSAS 340 8.11 Summary 342 8.12 Exercises 343 References 348 9 Theuseofmixedmodelsfortheanalysisofunbalancedexperimentaldesigns 350 9.1 Abalancedincompleteblockdesign 350 9.2 Imbalanceduetoamissingblock:Mixed-modelanalysisofthe incompleteblockdesign 354 9.3 UseofRtoanalysetheincompleteblockdesign 358 9.4 UseofSAStoanalysetheincompleteblockdesign 360 9.5 Relaxationoftherequirementforbalance:Alphadesigns 362 9.6 Approximatebalanceintwodirections:Thealphalphadesign 368 9.7 UseofRtoanalysethealphalphadesign 373 9.8 UseofSAStoanalysethealphalphadesign 374 9.9 Summary 376 9.10 Exercises 377 References 378 10 Beyondmixedmodelling 379 10.1 Reviewoftheusesofmixedmodels 379 10.2 Thegeneralizedlinearmixedmodel(GLMM):Fittingalogistic (sigmoidal)curvetoproportionsofobservations 380 10.3 UseofRtofitthelogisticcurve 388 10.4 UseofSAStofitthelogisticcurve 390 10.5 FittingaGLMMtoacontingencytable:Trouble-shootingwhenthe mixedmodellingprocessfails 392 10.6 Thehierarchicalgeneralizedlinearmodel(HGLM) 403 10.7 UseofRtofitaGLMMandaHGLMtoacontingencytable 410 10.8 UseofSAStofitaGLMMtoacontingencytable 415 10.9 Theroleofthecovariancematrixinthespecificationofamixedmodel 418 10.10 Amoregeneralpatterninthecovariancematrix:Analysisofpedigrees andgeneticdata 421