SPSS混合线性模型课件

MixedAnalysisofVarianceModelswithSPSSRobertA.Yaffee,Ph.D.Statistics,SocialScience,andMappingGroupInformationTechnologyServices/AcademicComputingServicesOfficelocation:75ThirdAvenue,LevelC-3Phone:212-998-34021MixedAnalysisofVarianceModOutlineClassificationofEffectsRandomEffectsTwo-WayRandomLayoutSolutionsandestimatesGenerallinearmodelFixedEffectsModelsTheone-waylayoutMixedModeltheoryPropererrortermsTwo-waylayoutFull-factorialmodelContrastswithinteractiontermsGraphingInteractions2OutlineClassificationofEffecOutline-Cont’dRepeatedMeasuresANOVAAdvantagesofMixedModelsoverGLM.3Outline-Cont’dRepeatedMeasureDefinitionofMixedModels bytheircomponenteffects MixedModelscontainbothfixedandrandomeffectsFixedEffects:factorsforwhichtheonlylevelsunderconsiderationarecontainedinthecodingofthoseeffectsRandomEffects:Factorsforwhichthelevelscontainedinthecodingofthosefactorsarearandomsampleofthetotalnumberoflevelsinthepopulationforthatfactor.4DefinitionofMixedModels byExamplesofFixedandRandomEffectsFixedeffect:Sexwherebothmaleandfemalegendersareincludedinthefactor,sex.Agegroup:MinorandAdultarebothincludedinthefactorofagegroupRandomeffect:Subject:thesampleisarandomsampleofthetargetpopulation5ExamplesofFixedandRandomEClassificationofeffectsTherearemaineffects:LinearExplanatoryFactorsThereareinteractioneffects:Jointeffectsoverandabovethecomponentmaineffects.6ClassificationofeffectsThere77ClassificationofEffects-cont’dHierarchicaldesignshavenestedeffects.Nestedeffectsarethosewithsubjectswithingroups.AnexamplewouldbepatientsnestedwithindoctorsanddoctorsnestedwithinhospitalsThiscouldbeexpressedbypatients(doctors)doctors(hospitals)8ClassificationofEffects-cont99BetweenandWithin-SubjecteffectsSucheffectsmaysometimesbefixedorrandom.Theirclassificationdependsontheexperimentaldesign

Between-subjectseffects

arethosewhoareinonegrouporanotherbutnotinboth.

Experimentalgroupisafixedeffectbecausethemanagerisconsideringonlythosegroupsinhisexperiment.Onegroupistheexperimentalgroupandtheotheristhecontrolgroup.Therefore,thisgrouping

factorisa

between-subjecteffect.

Within-subjecteffects

areexperiencedbysubjectsrepeatedlyovertime.Trialisarandomeffectwhenthereareseveraltrialsintherepeatedmeasuresdesign;allsubjectsexperienceallofthetrials.Trialisthereforeawithin-subjecteffect.

Operatormaybeafixedorrandomeffect,dependinguponwhetheroneisgeneralizingbeyondthesample

Ifoperatorisarandomeffect,thenthemachine*operatorinteractionisarandomeffect.

Therearecontrasts:Thesecontrastthevaluesofonelevelwiththoseofotherlevelsofthesameeffect.

10BetweenandWithin-SubjecteffBetweenSubjecteffectsGender:Oneiseithermaleorfemale,butnotboth.Group:Oneiseitherinthecontrol,experimental,orthecomparisongroupbutnotmorethanone.11BetweenSubjecteffectsGender:Within-SubjectsEffectsThesearerepeatedeffects.Observation1,2,and3mightbethepre,post,andfollow-upobservationsoneachperson.Eachpersonexperiencesalloftheselevelsorcategories.Thesearefoundinrepeatedmeasuresanalysisofvariance.12Within-SubjectsEffectsTheseaRepeatedObservationsareWithin-Subjectseffects Trial1Trial2Trial3GroupGroupisabetweensubjectseffect,whereasTrialisawithinsubjectseffect.13RepeatedObservationsareWithTheGeneral

LinearModelThemaineffectsgenerallinearmodelcanbeparameterizedas14TheGeneral

LinearModelThemAfactorialmodelIfaninteractiontermwereincluded,theformulawouldbeTheinteractionorcrossedeffectisthejointeffect,overandabovetheindividualmaineffects.Therefore,themaineffectsmustbeinthemodelfortheinteractiontobeproperlyspecified.15AfactorialmodelIfaninteracHigher-OrderInteractionsIf3-wayinteractionsareinthemodel,thenthemaineffectsandalllowerorderinteractionsmustbeinthemodelforthe3-wayinteractiontobeproperlyspecified.Forexample,a3-wayinteractionmodelwouldbe:16Higher-OrderInteractionsIf3-TheGeneralLinearModelInmatrixterminology,thegenerallinearmodelmaybeexpressedas17TheGeneralLinearModelInmatAssumptionsOfthegenerallinearmodel18AssumptionsOfthegenerallineGeneralLinearModelAssumptions-cont’d 1.ResidualNormality.2.Homogeneityoferrorvariance3.FunctionalformofModel: LinearityofModel4.NoMulticollinearity5.Independenceofobservations6.Noautocorrelationoferrors7.NoinfluentialoutliersWehavetotestforthesetobesurethatthemodelisvalid.Wewilldiscusstherobustnessofthemodelinfaceofviolationsoftheseassumptions.Wewilldiscussrecourseswhentheseassumptionsareviolated.19GeneralLinearModelAssumptioExplanationoftheseassumptionsFunctionalformofModel:LinearityofModel:Thesemodelsonlyanalyzethelinearrelationship.IndependenceofobservationsRepresentativenessofsampleResidualNormality:Sothealpharegionsofthesignificancetestsareproperlydefined.Homogeneityoferrorvariance:Sotheconfidencelimitsmaybeeasilyfound.NoMulticollinearity:Preventsefficientestimationoftheparameters.Noautocorrelationoferrors:AutocorrelationinflatestheR2,Fandttests.Noinfluentialoutliers:Theybiastheparameterestimation.20ExplanationoftheseassumptioDiagnostictestsfortheseassumptionsFunctionalformofModel:LinearityofModel:PairplotIndependenceofobservations:RunstestRepresentativenessofsample:InquireaboutsampledesignResidualNormality:SKorSWtestHomogeneityoferrorvarianceGraphofZresid*ZpredNoMulticollinearity:CorrofXNoautocorrelationoferrors:ACFNoinfluentialoutliers:LeverageandCook’sD.21DiagnostictestsfortheseassTestingforoutliersFrequenciesanalysisofstdrescksd.Lookforstandardizedresidualsgreaterthan3.5orlessthan–3.5AndlookforCook’sD.22TestingforoutliersFrequencieStudentizedResidualsBelsleyetal(1980)recommendtheuseofstudentizedResidualstodeterminewhetherthereisanoutlier.23StudentizedResidualsBelsleyeInfluenceofOutliersLeverageismeasuredbythediagonalcomponentsofthehatmatrix.ThehatmatrixcomesfromtheformulafortheregressionofY.24InfluenceofOutliersLeverageLeverageandtheHatmatrixThehatmatrixtransformsYintothepredictedscores.Thediagonalsofthehatmatrixindicatewhichvalueswillbeoutliersornot.Thediagonalsarethereforemeasuresofleverage.Leverageisboundedbytwolimits:1/nand1.Theclosertheleverageistounity,themoreleveragethevaluehas.Thetraceofthehatmatrix=thenumberofvariablesinthemodel.Whentheleverage>2p/nthenthereishighleverageaccordingtoBelsleyetal.(1980)citedinLong,J.F.ModernMethodsofDataAnalysis(p.262).Forsmallersamples,VellmanandWelsch(1981)suggestedthat3p/nisthecriterion.25LeverageandtheHatmatrixTheCook’sDAnothermeasureofinfluence.Thisisapopularone.Theformulaforitis:CookandWeisberg(1982)suggestedthatvaluesofDthatexceeded50%oftheFdistribution(df=p,n-p)arelarge.26Cook’sDAnothermeasureofinfCook’sDinSPSSFindingtheinfluentialoutliersSelectthoseobservationsforwhichcksd>(4*p)/nBelsleysuggests4/(n-p-1)asacutoffIfcksd>(4*p)/(n-p-1);27Cook’sDinSPSSFindingtheinWhattodowithoutliers1.Checkcodingtospottypos2.Correcttypos3.Ifobservationaloutlieriscorrect,examinethedffitsoptiontoseetheinfluenceonthefittingstatistics.4.Thiswillshowthestandardizedinfluenceoftheobservationonthefit.Iftheinfluenceoftheoutlierisbad,thenconsiderremovalorreplacementofitwithimputation.28Whattodowithoutliers1.ChDecompositionoftheSumsofSquaresMeandeviationsarecomputedwhenmeansaresubtractedfromindividualscores.Thisisdoneforthetotal,thegroupmean,andtheerrorterms.MeandeviationsaresquaredandthesearecalledsumsofsquaresVariancesarecomputedbydividingtheSumsofSquaresbytheirdegreesoffreedom.ThetotalVariance=ModelVariance+ errorvariance29DecompositionoftheSumsofSFormulaforDecompositionofSumsofSquares

SStotal=SSerror+SSmodel30FormulaforDecompositionofSVarianceDecompositionDividingeachofthesumsofsquaresbytheirrespectivedegreesoffreedomyieldsthevariances.Totalvariance=errorvariance+modelvariance.31VarianceDecompositionDividingProportionofVarianceExplainedR2=proportionofvarianceexplained.SStotal=SSmodel+SSerrrorDivideallsidesbySStotalSSmodel/SStotal=1-SSError/SStotalR2=1-SSError/SStotal32ProportionofVarianceExplainTheOmnibusFtestTheomnibusFtestisatestthatallofthemeansofthelevelsofthemaineffectsandaswellasanyinteractionsspecifiedarenotsignificantlydifferentfromoneanother.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)33TheOmnibusFtestTheomnibusTestingdifferentLevelsofaFactoragainstoneanotherContrastaretestsofthemeanofonelevelofafactoragainstotherlevels.34TestingdifferentLevelsofaContrasts-cont’dAcontraststatementcomputesTheestimatedV-isthegeneralizedinverseofthecoefficientmatrixofthemixedmodel.TheLvectoristhek’bvector.Thenumeratordfistherank(L)andthedenominatordfistakenfromthefixedeffectstableunlessotherwisespecified.35Contrasts-cont’dAcontraststaConstructionoftheFtestsindifferentmodelsTheFtestisaratiooftwovariances(MeanSquares).ItisconstructedbydividingtheMSoftheeffecttobetestedbyaMSofthedenominatorterm.Thedivisionshouldleaveonlytheeffecttobetestedleftoverasaremainder.AFixedEffectsmodelFtestfora=MSa/MSerror.ARandomEffectsmodelFtestfora=MSa/MSabAMixedEffectsmodelFtestforb=MSa/MSabAMixedEffectsmodelFtestforab=MSab/MSerror36ConstructionoftheFtestsinDataformatThedataformatforaGLMisthatofwidedata.37DataformatThedataformatforDataFormatforMixedModelsisLong38DataFormatforMixedModelsiConversionofWidetoLongDataFormatClickonDataintheheaderbarThenclickonRestructureinthepop-downmenu39ConversionofWidetoLongDatArestructurewizardappearsSelectrestructureselectedvariablesintocasesandclickonNext40ArestructurewizardappearsSeAVariablestoCases:NumberofVariableGroupsdialogboxappears.Weselectoneandclickonnext.41AVariablestoCases:NumberoWeselecttherepeatedvariablesandmovethemtothetargetvariablebox42WeselecttherepeatedvariablAftermovingtherepeatedvariablesintothetargetvariablebox,wemovethefixedvariablesintotheFixedvariablebox,andselectavariableforcaseid—inthiscase,subject.

ThenweclickonNext43AftermovingtherepeatedvariAcreateindexvariablesdialogboxappears.Weleavethenumberofindexvariablestobecreatedatoneandclickonnextatthebottomofthebox44AcreateindexvariablesdialoWhenthefollowingboxappearswejusttypeintimeandselectNext.45WhenthefollowingboxappearsWhentheoptionsdialogboxappears,weselecttheoptionfordroppingvariablesnotselected.

WethenclickonFinish.46WhentheoptionsdialogboxapWethusobtainourdatainlongformat47WethusobtainourdatainlonTheMixedModelTheMixedModeluseslongdataformat.Itincludesfixedandrandomeffects.Itcanbeusedtomodelmerelyfixedorrandomeffects,byzeroingouttheotherparametervector.TheFtestsforthefixed,random,andmixedmodelsdiffer.BecausetheMixedModelhastheparametervectorforbothoftheseandcanestimatetheerrorcovariancematrixforeach,itcanprovidethecorrectstandarderrorsforeitherthefixedorrandomeffects.48TheMixedModelTheMixedModeTheMixedModel49TheMixedModel49MixedModelTheory-cont’dLittleetal.(p.139)notethatuandeareuncorrelatedrandomvariableswith0meansandcovariances,GandR,respectively.V-isageneralizedinverse.BecauseVisusuallysingularandnoninvertibleAVA=V-isanaugmentedmatrixthatisinvertible.ItcanlaterbetransformedbacktoV.TheGandRmatricesmustbepositivedefinite.IntheMixedprocedure,thecovariancetypeoftherandom(generalized)effectsdefinesthestructureofGandarepeatedcovariancetypedefinesstructureofR.50MixedModelTheory-cont’dLittlMixedModelAssumptionsAlinearrelationshipbetweendependentandindependentvariables51MixedModelAssumptionsAlineaRandomEffectsCovarianceStructureThisdefinesthestructureoftheGmatrix,therandomeffects,inthemixedmodel.PossiblestructurespermittedbycurrentversionofSPSS:ScaledIdentityCompoundSymmetryAR(1)Huynh-Feldt52RandomEffectsCovarianceStruStructuresofRepeatedeffects(Rmatrix)-cont’d53StructuresofRepeatedeffectsStructuresofRepeatedEffects(Rmatrix)54StructuresofRepeatedEffectsStructuresofRepeatedeffects(Rmatrix)–con’td55StructuresofRepeatedeffectsRmatrix,definesthecorrelationamongrepeatedrandomeffectsOnecanspecifythenatureofthecorrelationamongtherepeatedrandomeffects.56Rmatrix,definesthecorrelatGLMMixedModelTheGeneralLinearModelisaspecialcaseoftheMixedModelwithZ=0(whichmeansthatZudisappearsfromthemodel)and57GLMMixedModelTheGeneraMixedAnalysisofaFixedEffectsmodelSPSSteststhesefixedeffectsjustasitdoeswiththeGLMProcedurewithtypeIIIsumsofsquares.Weanalyzethebreakingpressureofbondsmadefromthreemetals.Weassumethatwedonotgeneralizebeyondoursampleandthatoureffectsareallfixed.TestsofFixedEffectsisperformedwiththehelpoftheLmatrixbyconstructingthefollowingFtest:Numeratordf=rank(L)Denominatordf=RESID(n-rank(X)df=Satherth58MixedAnalysisofaFixedEffeEstimation:NewtonScoring59Estimation:NewtonScoring59Estimation:MinimizationoftheobjectivefunctionsUsingNewtonScoring,thefollowingfunctionsareminimized60Estimation:MinimizationofthSignificanceofParameters61SignificanceofParameters61TestonecovariancestructureagainsttheotherwiththeICTheruleofthumbissmallerisbetter-2LLAICAkaikeAICCHurvichandTsayBICBayesianInfoCriterionBozdogan’sCAIC62TestonecovariancestructureMeasuresofLackoffit:TheinformationCriteria-2LLiscalledthedeviance.Itisameasureofsumofsquarederrors.AIC=-2LL+2p(p=#parms)BIC=SchwartzBayesianInfocriterion=2LL+plog(n)AICC=HurvichandTsay’ssmallsamplecorrectiononAIC:-2LL+2p(n/(n-p-1))CAIC=-2LL+p(log(n)+1)63MeasuresofLackoffit:TheiProceduresforFittingtheMixedModelOnecanusetheLRtestorthelesseroftheinformationcriteria.Thesmallertheinformationcriterion,thebetterthemodelhappenstobe.Wetrytogofromalargertoasmallerinformationcriterionwhenwefitthemodel.64ProceduresforFittingtheMixLRtestTotestwhetheronemodelissignificantlybetterthantheother.TotestrandomeffectforstatisticalsignificanceTotestcovariancestructureimprovementTotestboth.DistributedasaWithdf=p2–p1wherepi=#parmsinmodeli65LRtestTotestwhetheronemodApplyingtheLRtestWeobtainthe-2LLfromtheunrestrictedmodel.Weobtainthe-2LLfromtherestrictedmodel.Wesubtractthelatterfromthelargerformer.Thatisachi-squarewithdf=thedifferenceinthenumberofparameters.Wecanlookthisupanddeterminewhetherornotitisstatisticallysignificant.66ApplyingtheLRtestWeobtainAdvantagesoftheMixedModelItcanallowrandomeffectstobeproperlyspecifiedandcomputed,unliketheGLM.Itcanallowcorrelationoferrors,unliketheGLM.Itthereforehasmoreflexibilityinmodelingtheerrorcovariancestructure.Itcanallowtheerrortermstoexhibitnonconstantvariability,unliketheGLM,allowingmoreflexibilityinmodelingthedependentvariable.Itcanhandlemissingdata,whereastherepeatedmeasuresGLMcannot.67AdvantagesoftheMixedModelIProgrammingARepeatedMeasuresANOVAwithPROCMixedSelecttheMixedLinearOptioninAnalysis68ProgrammingARepeatedMeasureMovesubjectIDintothesubjectsboxandtherepeatedvariableintotherepeatedbox.Clickoncontinue69MovesubjectIDintothesubjeWespecifysubjectsandrepeatedeffectswiththenextdialogboxWesettherepeatedcovariancetypeto“Diagonal”&clickoncontinue70WespecifysubjectsandrepeatDefiningtheFixedEffectsWhenthenextdialogboxappears,weinsertthedependentResponsevariableandthefixedeffectsofanxietyandtensionClickoncontinue71DefiningtheFixedEffectsWhenWeselecttheFixedeffectstobetested72WeselecttheFixedeffectstoMovethemintothemodelbox,selectingmaineffects,andtypeIIIsumofsquaresClickoncontinue73Movethemintothemodelbox,WhentheLinearMixedModelsdialogboxappears,selectrandom74WhentheLinearMixedModelsdUnderrandomeffects,selectscaledidentityascovariancetypeandmovesubjectsoverintocombinationsClickoncontinue75Underrandomeffects,selectsSelectStatisticsandcheckofthefollowinginthedialogboxthatappearsThenclickcontinue76SelectStatisticsandcheckofWhentheLinearMixedModelsboxappears,clickok77WhentheLinearMixedModelsbYouwillgetyourtests78Youwillgetyourtests78EstimatesofFixedeffectsandcovarianceparameters79EstimatesofFixedeffectsandRmatrix80Rmatrix80

RerunthemodelwithdifferentnestedcovariancestructuresandcomparetheinformationcriteriaThelowertheinformationcriterion,thebetterfitthenestedmodelhas.Caveat:Ifthemodelsarenotnested,theycannotbecomparedwiththeinformationcriteria.81

RerunthemodelwithdifferenGLMvs.MixedGLMhasmeanslsmeanssstype1,2,3,4estimatesusingOLSorWLSonehastoprogramthecorrectFtestsforrandomeffects.lossescaseswithmissingvalues.Mixedhaslsmeanssstypes1and3estimatesusingmaximumlikelihood,generalmethodsofmoments,orrestrictedmaximumlikelihoodMLMIVQUE0REMLgivescorrectstderrorsandconfidenceintervalsforrandomeffectsAutomaticallyprovidescorrectstandarderrorsforanalysis.Canhandlemissingvalues82GLMvs.MixedGLMhas82MixedAnalysisofVarianceModelswithSPSSRobertA.Yaffee,Ph.D.Statistics,SocialScience,andMappingGroupInformationTechnologyServices/AcademicComputingServicesOfficelocation:75ThirdAvenue,LevelC-3Phone:212-998-340283MixedAnalysisofVarianceModOutlineClassificationofEffectsRandomEffectsTwo-WayRandomLayoutSolutionsandestimatesGenerallinearmodelFixedEffectsModelsTheone-waylayoutMixedModeltheoryPropererrortermsTwo-waylayoutFull-factorialmodelContrastswithinteractiontermsGraphingInteractions84OutlineClassificationofEffecOutline-Cont’dRepeatedMeasuresANOVAAdvantagesofMixedModelsoverGLM.85Outline-Cont’dRepeatedMeasureDefinitionofMixedModels bytheircomponenteffects MixedModelscontainbothfixedandrandomeffectsFixedEffects:factorsforwhichtheonlylevelsunderconsiderationarecontainedinthecodingofthoseeffectsRandomEffects:Factorsforwhichthelevelscontainedinthecodingofthosefactorsarearandomsampleofthetotalnumberoflevelsinthepopulationforthatfactor.86DefinitionofMixedModels byExamplesofFixedandRandomEffectsFixedeffect:Sexwherebothmaleandfemalegendersareincludedinthefactor,sex.Agegroup:MinorandAdultarebothincludedinthefactorofagegroupRandomeffect:Subject:thesampleisarandomsampleofthetargetpopulation87ExamplesofFixedandRandomEClassificationofeffectsTherearemaineffects:LinearExplanatoryFactorsThereareinteractioneffects:Jointeffectsoverandabovethecomponentmaineffects.88ClassificationofeffectsThere897ClassificationofEffects-cont’dHierarchicaldesignshavenestedeffects.Nestedeffectsarethosewithsubjectswithingroups.AnexamplewouldbepatientsnestedwithindoctorsanddoctorsnestedwithinhospitalsThiscouldbeexpressedbypatients(doctors)doctors(hospitals)90ClassificationofEffects-cont919BetweenandWithin-SubjecteffectsSucheffectsmaysometimesbefixedorrandom.Theirclassificationdependsontheexperimentaldesign

Between-subjectseffects

arethosewhoareinonegrouporanotherbutnotinboth.

Experimentalgroupisafixedeffectbecausethemanagerisconsideringonlythosegroupsinhisexperiment.Onegroupistheexperimentalgroupandtheotheristhecontrolgroup.Therefore,thisgrouping

factorisa

between-subjecteffect.

Within-subjecteffects

areexperiencedbysubjectsrepeatedlyovertime.Trialisarandomeffectwhenthereareseveraltrialsintherepeatedmeasuresdesign;allsubjectsexperienceallofthetrials.Trialisthereforeawithin-subjecteffect.

Operatormaybeafixedorrandomeffect,dependinguponwhetheroneisgeneralizingbeyondthesample

Ifoperatorisarandomeffect,thenthemachine*operatorinteractionisarandomeffect.

Therearecontrasts:Thesecontrastthevaluesofonelevelwiththoseofotherlevelsofthesameeffect.

92BetweenandWithin-SubjecteffBetweenSubjecteffectsGender:Oneiseithermaleorfemale,butnotboth.Group:Oneiseitherinthecontrol,experimental,orthecomparisongroupbutnotmorethanone.93BetweenSubjecteffectsGender:Within-SubjectsEffectsThesearerepeatedeffects.Observation1,2,and3mightbethepre,post,andfollow-upobservationsoneachperson.Eachpersonexperiencesalloftheselevelsorcategories.Thesearefoundinrepeatedmeasuresanalysisofvariance.94Within-SubjectsEffectsTheseaRepeatedObservationsareWithin-Subjectseffects Trial1Trial2Trial3GroupGroupisabetweensubjectseffect,whereasTrialisawithinsubjectseffect.95RepeatedObservationsareWithTheGeneral

LinearModelThemaineffectsgenerallinearmodelcanbeparameterizedas96TheGeneral

LinearModelThemAfactorialmodelIfaninteractiontermwereincluded,theformulawouldbeTheinteractionorcrossedeffectisthejointeffect,overandabovetheindividualmaineffects.Therefore,themaineffectsmustbeinthemodelfortheinteractiontobeproperlyspecified.97AfactorialmodelIfaninteracHigher-OrderInteractionsIf3-wayinteractionsareinthemodel,thenthemaineffectsandalllowerorderinteractionsmustbeinthemodelforthe3-wayinteractiontobeproperlyspecified.Forexample,a3-wayinteractionmodelwouldbe:98Higher-OrderInteractionsIf3-TheGeneralLinearModelInmatrixterminology,thegenerallinearmodelmaybeexpressedas99TheGeneralLinearModelInmatAssumptionsOfthegenerallinearmodel100AssumptionsOfthegenerallineGeneralLinearModelAssumptions-cont’d 1.ResidualNormality.2.Homogeneityoferrorvariance3.FunctionalformofModel: LinearityofModel4.NoMulticollinearity5.Independenceofobservations6.Noautocorrelationoferrors7.NoinfluentialoutliersWehavetotestforthesetobesurethatthemodelisvalid.Wewilldiscusstherobustnessofthemodelinfaceofviolationsoftheseassumptions.Wewilldiscussrecourseswhentheseassumptionsareviolated.101GeneralLinearModelAssumptioExplanationoftheseassumptionsFunctionalformofModel:LinearityofModel:Thesemodelsonlyanalyzethelinearrelationship.IndependenceofobservationsRepresentativenessofsampleResidualNormality:Sothealpharegionsofthesignificancetestsareproperlydefined.Homogeneityoferrorvariance:Sotheconfidencelimitsmaybeeasilyfound.NoMulticollinearity:Preventsefficientestimationoftheparameters.Noautocorrelationoferrors:AutocorrelationinflatestheR2,Fandttests.Noinfluentialoutliers:Theybiastheparameterestimation.102ExplanationoftheseassumptioDiagnostictestsfortheseassumptionsFunctionalformofModel:LinearityofModel:PairplotIndependenceofobservations:RunstestRepresentativenessofsample:InquireaboutsampledesignResidualNormality:SKorSWtestHomogeneityoferrorvarianceGraphofZresid*ZpredNoMulticollinearity:CorrofXNoautocorrelationoferrors:ACFNoinfluentialoutliers:LeverageandCook’sD.103DiagnostictestsfortheseassTestingforoutliersFrequenciesanalysisofstdrescksd.Lookforstandardizedresidualsgreaterthan3.5orlessthan–3.5AndlookforCook’sD.104TestingforoutliersFrequencieStudentizedResidualsBelsleyetal(1980)recommendtheuseofstudentizedResidualstodeterminewhetherthereisanoutlier.105StudentizedResidualsBelsleyeInfluenceofOutliersLeverageismeasuredbythediagonalcomponentsofthehatmatrix.ThehatmatrixcomesfromtheformulafortheregressionofY.106InfluenceofOutliersLeverageLeverageandtheHatmatrixThehatmatrixtransformsYintothepredictedscores.Thediagonalsofthehatmatrixindicatewhichvalueswillbeoutliersornot.Thediagonalsarethereforemeasuresofleverage.Leverageisboundedbytwolimits:1/nand1.Theclosertheleverageistounity,themoreleveragethevaluehas.Thetraceofthehatmatrix=thenumberofvariablesinthemodel.Whentheleverage>2p/nthenthereishighleverageaccordingtoBelsleyetal.(1980)citedinLong,J.F.ModernMethodsofDataAnalysis(p.262).F