SPSS混合线性模型.ppt

1,MixedAnalysisofVarianceModelswithSPSS,RobertA.Yaffee,Ph.D.Statistics,SocialScience,andMappingGroupInformationTechnologyServices/AcademicComputingServicesOfficelocation:75ThirdAvenue,LevelC-3Phone:212-998-3402,2,Outline,ClassificationofEffectsRandomEffectsTwo-WayRandomLayoutSolutionsandestimatesGenerallinearmodelFixedEffectsModelsTheone-waylayoutMixedModeltheoryPropererrortermsTwo-waylayoutFull-factorialmodelContrastswithinteractiontermsGraphingInteractions,3,Outline-Contd,RepeatedMeasuresANOVAAdvantagesofMixedModelsoverGLM.,4,DefinitionofMixedModelsbytheircomponenteffects,MixedModelscontainbothfixedandrandomeffectsFixedEffects:factorsforwhichtheonlylevelsunderconsiderationarecontainedinthecodingofthoseeffectsRandomEffects:Factorsforwhichthelevelscontainedinthecodingofthosefactorsarearandomsampleofthetotalnumberoflevelsinthepopulationforthatfactor.,5,ExamplesofFixedandRandomEffects,Fixedeffect:Sexwherebothmaleandfemalegendersareincludedinthefactor,sex.Agegroup:MinorandAdultarebothincludedinthefactorofagegroupRandomeffect:Subject:thesampleisarandomsampleofthetargetpopulation,6,Classificationofeffects,Therearemaineffects:LinearExplanatoryFactorsThereareinteractioneffects:Jointeffectsoverandabovethecomponentmaineffects.,7,8,ClassificationofEffects-contd,Hierarchicaldesignshavenestedeffects.Nestedeffectsarethosewithsubjectswithingroups.AnexamplewouldbepatientsnestedwithindoctorsanddoctorsnestedwithinhospitalsThiscouldbeexpressedbypatients(doctors)doctors(hospitals),9,10,BetweenandWithin-Subjecteffects,Sucheffectsmaysometimesbefixedorrandom.TheirclassificationdependsontheexperimentaldesignBetween-subjectseffectsarethosewhoareinonegrouporanotherbutnotinboth.Experimentalgroupisafixedeffectbecausethemanagerisconsideringonlythosegroupsinhisexperiment.Onegroupistheexperimentalgroupandtheotheristhecontrolgroup.Therefore,thisgroupingfactorisabetween-subjecteffect.Within-subjecteffectsareexperiencedbysubjectsrepeatedlyovertime.Trialisarandomeffectwhenthereareseveraltrialsintherepeatedmeasuresdesign;allsubjectsexperienceallofthetrials.Trialisthereforeawithin-subjecteffect.Operatormaybeafixedorrandomeffect,dependinguponwhetheroneisgeneralizingbeyondthesampleIfoperatorisarandomeffect,thenthemachine*operatorinteractionisarandomeffect.Therearecontrasts:Thesecontrastthevaluesofonelevelwiththoseofotherlevelsofthesameeffect.,11,BetweenSubjecteffects,Gender:Oneiseithermaleorfemale,butnotboth.Group:Oneiseitherinthecontrol,experimental,orthecomparisongroupbutnotmorethanone.,12,Within-SubjectsEffects,Thesearerepeatedeffects.Observation1,2,and3mightbethepre,post,andfollow-upobservationsoneachperson.Eachpersonexperiencesalloftheselevelsorcategories.Thesearefoundinrepeatedmeasuresanalysisofvariance.,13,RepeatedObservationsareWithin-Subjectseffects,Trial1Trial2Trial3,Group,Groupisabetweensubjectseffect,whereasTrialisawithinsubjectseffect.,14,TheGeneralLinearModel,Themaineffectsgenerallinearmodelcanbeparameterizedas,15,Afactorialmodel,Ifaninteractiontermwereincluded,theformulawouldbe,Theinteractionorcrossedeffectisthejointeffect,overandabovetheindividualmaineffects.Therefore,themaineffectsmustbeinthemodelfortheinteractiontobeproperlyspecified.,16,Higher-OrderInteractions,If3-wayinteractionsareinthemodel,thenthemaineffectsandalllowerorderinteractionsmustbeinthemodelforthe3-wayinteractiontobeproperlyspecified.Forexample,a3-wayinteractionmodelwouldbe:,17,TheGeneralLinearModel,Inmatrixterminology,thegenerallinearmodelmaybeexpressedas,18,Assumptions,Ofthegenerallinearmodel,19,GeneralLinearModelAssumptions-contd,1.ResidualNormality.2.Homogeneityoferrorvariance3.FunctionalformofModel:LinearityofModel4.NoMulticollinearity5.Independenceofobservations6.Noautocorrelationoferrors7.Noinfluentialoutliers,Wehavetotestforthesetobesurethatthemodelisvalid.Wewilldiscusstherobustnessofthemodelinfaceofviolationsoftheseassumptions.Wewilldiscussrecourseswhentheseassumptionsareviolated.,20,Explanationoftheseassumptions,FunctionalformofModel:LinearityofModel:Thesemodelsonlyanalyzethelinearrelationship.IndependenceofobservationsRepresentativenessofsampleResidualNormality:Sothealpharegionsofthesignificancetestsareproperlydefined.Homogeneityoferrorvariance:Sotheconfidencelimitsmaybeeasilyfound.NoMulticollinearity:Preventsefficientestimationoftheparameters.Noautocorrelationoferrors:AutocorrelationinflatestheR2,Fandttests.Noinfluentialoutliers:Theybiastheparameterestimation.,21,Diagnostictestsfortheseassumptions,FunctionalformofModel:LinearityofModel:PairplotIndependenceofobservations:RunstestRepresentativenessofsample:InquireaboutsampledesignResidualNormality:SKorSWtestHomogeneityoferrorvarianceGraphofZresid*ZpredNoMulticollinearity:CorrofXNoautocorrelationoferrors:ACFNoinfluentialoutliers:LeverageandCooksD.,22,Testingforoutliers,Frequenciesanalysisofstdrescksd.Lookforstandardizedresidualsgreaterthan3.5orlessthan3.5AndlookforCooksD.,23,StudentizedResiduals,Belsleyetal(1980)recommendtheuseofstudentizedResidualstodeterminewhetherthereisanoutlier.,24,InfluenceofOutliers,Leverageismeasuredbythediagonalcomponentsofthehatmatrix.ThehatmatrixcomesfromtheformulafortheregressionofY.,25,LeverageandtheHatmatrix,ThehatmatrixtransformsYintothepredictedscores.Thediagonalsofthehatmatrixindicatewhichvalueswillbeoutliersornot.Thediagonalsarethereforemeasuresofleverage.Leverageisboundedbytwolimits:1/nand1.Theclosertheleverageistounity,themoreleveragethevaluehas.Thetraceofthehatmatrix=thenumberofvariablesinthemodel.Whentheleverage2p/nthenthereishighleverageaccordingtoBelsleyetal.(1980)citedinLong,J.F.ModernMethodsofDataAnalysis(p.262).Forsmallersamples,VellmanandWelsch(1981)suggestedthat3p/nisthecriterion.,26,CooksD,Anothermeasureofinfluence.Thisisapopularone.Theformulaforitis:,CookandWeisberg(1982)suggestedthatvaluesofDthatexceeded50%oftheFdistribution(df=p,n-p)arelarge.,27,CooksDinSPSS,FindingtheinfluentialoutliersSelectthoseobservationsforwhichcksd(4*p)/nBelsleysuggests4/(n-p-1)asacutoffIfcksd(4*p)/(n-p-1);,28,Whattodowithoutliers,1.Checkcodingtospottypos2.Correcttypos3.Ifobservationaloutlieriscorrect,examinethedffitsoptiontoseetheinfluenceonthefittingstatistics.4.Thiswillshowthestandardizedinfluenceoftheobservationonthefit.Iftheinfluenceoftheoutlierisbad,thenconsiderremovalorreplacementofitwithimputation.,29,DecompositionoftheSumsofSquares,Meandeviationsarecomputedwhenmeansaresubtractedfromindividualscores.Thisisdoneforthetotal,thegroupmean,andtheerrorterms.MeandeviationsaresquaredandthesearecalledsumsofsquaresVariancesarecomputedbydividingtheSumsofSquaresbytheirdegreesoffreedom.ThetotalVariance=ModelVariance+errorvariance,30,FormulaforDecompositionofSumsofSquares,SStotal=SSerror+SSmodel,31,VarianceDecomposition,Dividingeachofthesumsofsquaresbytheirrespectivedegreesoffreedomyieldsthevariances.Totalvariance=errorvariance+modelvariance.,32,ProportionofVarianceExplained,R2=proportionofvarianceexplained.SStotal=SSmodel+SSerrrorDivideallsidesbySStotalSSmodel/SStotal=1-SSError/SStotalR2=1-SSError/SStotal,33,TheOmnibusFtest,TheomnibusFtestisatestthatallofthemeansofthelevelsofthemaineffectsandaswellasanyinteractionsspecifiedarenotsignificantlydifferentfromoneanother.,Supposethemodelisaonewayanovaonbreakingpressureofbondsofdifferentmetals.Supposetherearethreemetals:nickel,iron,andCopper.H0:Mean(Nickel)=mean(Iron)=mean(Copper)Ha:Mean(Nickel)neMean(Iron)orMean(Nickel)neMean(Copper)orMean(Iron)neMean(Copper),34,TestingdifferentLevelsofaFactoragainstoneanother,Contrastaretestsofthemeanofonelevelofafactoragainstotherlevels.,35,Contrasts-contd,Acontraststatementcomputes,TheestimatedV-isthegeneralizedinverseofthecoefficientmatrixofthemixedmodel.TheLvectoristhekbvector.Thenumeratordfistherank(L)andthedenominatordfistakenfromthefixedeffectstableunlessotherwisespecified.,36,ConstructionoftheFtestsindifferentmodels,TheFtestisaratiooftwovariances(MeanSquares).ItisconstructedbydividingtheMSoftheeffecttobetestedbyaMSofthedenominatorterm.Thedivisionshouldleaveonlytheeffecttobetestedleftoverasaremainder.,AFixedEffectsmodelFtestfora=MSa/MSerror.ARandomEffectsmodelFtestfora=MSa/MSabAMixedEffectsmodelFtestforb=MSa/MSabAMixedEffectsmodelFtestforab=MSab/MSerror,37,Dataformat,ThedataformatforaGLMisthatofwidedata.,38,DataFormatforMixedModelsisLong,39,ConversionofWidetoLongDataFormat,ClickonDataintheheaderbarThenclickonRestructureinthepop-downmenu,40,Arestructurewizardappears,SelectrestructureselectedvariablesintocasesandclickonNext,41,AVariablestoCases:NumberofVariableGroupsdialogboxappears.Weselectoneandclickonnext.,42,Weselecttherepeatedvariablesandmovethemtothetargetvariablebox,43,Aftermovingtherepeatedvariablesintothetargetvariablebox,wemovethefixedvariablesintotheFixedvariablebox,andselectavariableforcaseidinthiscase,subject.ThenweclickonNext,44,Acreateindexvariablesdialogboxappears.Weleavethenumberofindexvariablestobecreatedatoneandclickonnextatthebottomofthebox,45,WhenthefollowingboxappearswejusttypeintimeandselectNext.,46,Whentheoptionsdialogboxappears,weselecttheoptionfordroppingvariablesnotselected.WethenclickonFinish.,47,Wethusobtainourdatainlongformat,48,TheMixedModel,TheMixedModeluseslongdataformat.Itincludesfixedandrandomeffects.Itcanbeusedtomodelmerelyfixedorrandomeffects,byzeroingouttheotherparametervector.TheFtestsforthefixed,random,andmixedmodelsdiffer.BecausetheMixedModelhastheparametervectorforbothoftheseandcanestimatetheerrorcovariancematrixforeach,itcanprovidethecorrectstandarderrorsforeitherthefixedorrandomeffects.,49,TheMixedModel,50,MixedModelTheory-contd,Littleetal.(p.139)notethatuandeareuncorrelatedrandomvariableswith0meansandcovariances,GandR,respectively.,V-isageneralizedinverse.BecauseVisusuallysingularandnoninvertibleAVA=V-isanaugmentedmatrixthatisinvertible.ItcanlaterbetransformedbacktoV.TheGandRmatricesmustbepositivedefinite.IntheMixedprocedure,thecovariancetypeoftherandom(generalized)effectsdefinesthestructureofGandarepeatedcovariancetypedefinesstructureofR.,51,MixedModelAssumptions,Alinearrelationshipbetweendependentandindependentvariables,52,RandomEffectsCovarianceStructure,ThisdefinesthestructureoftheGmatrix,therandomeffects,inthemixedmodel.PossiblestructurespermittedbycurrentversionofSPSS:ScaledIdentityCompoundSymmetryAR(1)Huynh-Feldt,53,StructuresofRepeatedeffects(Rmatrix)-contd,54,StructuresofRepeatedEffects(Rmatrix),55,StructuresofRepeatedeffects(Rmatrix)contd,56,Rmatrix,definesthecorrelationamongrepeatedrandomeffects,Onecanspecifythenatureofthecorrelationamongtherepeatedrandomeffects.,57,GLMMixedModel,TheGeneralLinearModelisaspecialcaseoftheMixedModelwithZ=0(whichmeansthatZudisappearsfromthemodel)and,58,MixedAnalysisofaFixedEffectsmodel,SPSSteststhesefixedeffectsjustasitdoeswiththeGLMProcedurewithtypeIIIsumsofsquares.Weanalyzethebreakingpressureofbondsmadefromthreemetals.Weassumethatwedonotgeneralizebeyondoursampleandthatoureffectsareallfixed.,TestsofFixedEffectsisperformedwiththehelpoftheLmatrixbyconstructingthefollowingFtest:,Numeratordf=rank(L)Denominatordf=RESID(n-rank(X)df=Satherth,59,Estimation:NewtonScoring,60,Estimation:Minimizationoftheobjectivefunctions,UsingNewtonScoring,thefollowingfunctionsareminimized,61,SignificanceofParameters,62,TestonecovariancestructureagainsttheotherwiththeIC,Theruleofthumbissmallerisbetter-2LLAICAkaikeAICCHurvichandTsayBICBayesianInfoCriterionBozdogansCAIC,63,MeasuresofLackoffit:TheinformationCriteria,-2LLiscalledthedeviance.Itisameasureofsumofsquarederrors.AIC=-2LL+2p(p=#parms)BIC=SchwartzBayesianInfocriterion=2LL+plog(n)AICC=HurvichandTsayssmallsamplecorrectiononAIC:-2LL+2p(n/(n-p-1)CAIC=-2LL+p(log(n)+1),64,ProceduresforFittingtheMixedModel,OnecanusetheLRtestorthelesseroftheinformationcriteria.Thesmallertheinformationcriterion,thebetterthemodelhappenstobe.Wetrytogofromalargertoasmallerinformationcriterionwhenwefitthemodel.,65,LRtest,Totestwhetheronemodelissignificantlybetterthantheother.TotestrandomeffectforstatisticalsignificanceTotestcovariancestructureimprovementTotestboth.DistributedasaWithdf=p2p1wherepi=#parmsinmodeli,66,ApplyingtheLRtest,Weobtainthe-2LLfromtheunrestrictedmodel.Weobtainthe-2LLfromtherestrictedmodel.Wesubtractthelatterfromthelargerformer.Thatisachi-squarewithdf=thedifferenceinthenumberofparameters.Wecanlookthisupanddeterminewhetherornotitisstatisticallysignificant.,67,AdvantagesoftheMixedModel,Itcanallowrandomeffectstobeproperlyspecifiedandcomputed,unliketheGLM.Itcanallowcorrelationoferrors,unliketheGLM.Itthereforehasmoreflexibilityinmodelingtheerrorcovariancestructure.Itcanallowtheerrortermstoexhibitnonconstantvariability,unliketheGLM,allowingmoreflexibilityinmodelingthedependentvariable.Itcanhandlemissingdata,whereastherepeatedmeasuresGLMcannot.,68,ProgrammingARepeatedMeasuresANOVAwithPROCMixed,SelecttheMixedLinearOptioninAnalysis,69,MovesubjectIDintothesubjectsboxandtherepeatedvariableintotherepeatedbox.,Cli