spss-mixed-model

1MixedAnalysisofVarianceModelswithSPSSRobertA.Yaffee,Ph.D.Statistics,SocialScience,andMappingGroupInformationTechnologyServices/AcademicComputingServicesOfficelocation:75ThirdAvenue,LevelC-3Phone:[phone]Outline1.ClassificationofEffects2.RandomEffects1.Two-WayRandomLayout2.Solutionsandestimates3.Generallinearmodel1.FixedEffectsModels1.Theone-waylayout4.MixedModeltheory1.Propererrorterms5.Two-waylayout6.Full-factorialmodel1.Contrastswithinteractionterms2.GraphingInteractions3Outline-Cont’d•RepeatedMeasuresANOVA•AdvantagesofMixedModelsoverGLM.4DefinitionofMixedModelsbytheircomponenteffects1.MixedModelscontainbothfixedandrandomeffects2.FixedEffects:factorsforwhichtheonlylevelsunderconsiderationarecontainedinthecodingofthoseeffects3.RandomEffects:Factorsforwhichthelevelscontainedinthecodingofthosefactorsarearandomsampleofthetotalnumberoflevelsinthepopulationforthatfactor.5ExamplesofFixedandRandomEffects1.Fixedeffect:2.Sexwherebothmaleandfemalegendersareincludedinthefactor,sex.3.Agegroup:MinorandAdultarebothincludedinthefactorofagegroup4.Randomeffect:1.Subject:thesampleisarandomsampleofthetargetpopulation6Classificationofeffects1.Therearemaineffects:LinearExplanatoryFactors2.Thereareinteractioneffects:Jointeffectsoverandabovethecomponentmaineffects.7InteractionsareCrossedEffectsVariableYVariableXLevel1Level2Level3Level4Level1Level2Level3X11X12X13X14X21X22X23X24X31X32X33X34AllofthecellsarefilledEachlevelofXiscrossedwitheachlevelofY8ClassificationofEffects-cont’dHierarchicaldesignshavenestedeffects.Nestedeffectsarethosewithsubjectswithingroups.AnexamplewouldbepatientsnestedwithindoctorsanddoctorsnestedwithinhospitalsThiscouldbeexpressedbypatients(doctors)doctors(hospitals)9Hospital1Hospital2Doctor1Doctor2Doctor3Doctor4Doctor5Pat1Pat2Pat3Pat4Pat5Pat6Pat7Pat8NestingofpatientswithinDoctorsandDoctorswithinHospitals10BetweenandWithin-Subjecteffects•Sucheffectsmaysometimesbefixedorrandom.TheirclassificationdependsontheexperimentaldesignBetween-subjectseffectsarethosewhoareinonegrouporanotherbutnotinboth.Experimentalgroupisafixedeffectbecausethemanagerisconsideringonlythosegroupsinhisexperiment.Onegroupistheexperimentalgroupandtheotheristhecontrolgroup.Therefore,thisgroupingfactorisabetween-subjecteffect.Within-subjecteffectsareexperiencedbysubjectsrepeatedlyovertime.Trialisarandomeffectwhenthereareseveraltrialsintherepeatedmeasuresdesign;allsubjectsexperienceallofthetrials.Trialisthereforeawithin-subjecteffect.Operatormaybeafixedorrandomeffect,dependinguponwhetheroneisgeneralizingbeyondthesampleIfoperatorisarandomeffect,thenthemachine*operatorinteractionisarandomeffect.Therearecontrasts:Thesecontrastthevaluesofonelevelwiththoseofotherlevelsofthesameeffect.11BetweenSubjecteffects•Gender:Oneiseithermaleorfemale,butnotboth.•Group:Oneiseitherinthecontrol,experimental,orthecomparisongroupbutnotmorethanone.12Within-SubjectsEffects•Thesearerepeatedeffects.•Observation1,2,and3mightbethepre,post,andfollow-upobservationsoneachperson.•Eachpersonexperiencesalloftheselevelsorcategories.•Thesearefoundinrepeatedmeasuresanalysisofvariance.13RepeatedObservationsareWithin-SubjectseffectsExperimentalGroupPre-testControlGroupPre-testExperimentalGroupPost-testControlGroupPost-testExperimentalGroupFollow-upControlGroupFollow-upRepeatedMeasuresDesignTrial1Trial2Trial3GroupGroupisabetweensubjectseffect,whereasTrialisawithinsubjectseffect.14TheGeneralLinearModel1.Themaineffectsgenerallinearmodelcanbeparameterizedas()()()exp~(,)ijijijijiijjijYbwhereYobservationforithgrandmeananunknownfixedparmeffectofithvalueofabeffectofjthvalueofbberimentalerrorN2015AfactorialmodelIfaninteractiontermwereincluded,theformulawouldbeijiiijijyeTheinteractionorcrossedeffectisthejointeffect,overandabovetheindividualmaineffects.Therefore,themaineffectsmustbeinthemodelfortheinteractiontobeproperlyspecified.()()ijijijyy16Higher-OrderInteractionsIf3-wayinteractionsareinthemodel,thenthemaineffectsandalllowerorderinteractionsmustbeinthemodelforthe3-wayinteractiontobeproperlyspecified.Forexample,a3-wayinteractionmodelwouldbe:ijkijkijikjkijkijkyabcabacbcabce17TheGeneralLinearModel•Inmatrixterminology,thegenerallinearmodelmaybeexpressedasYXwhereYtheobserveddatavectorXthedesignmatrixthevectorofunknownfixedeffectparametersthevectoroferrors18AssumptionsOfthegenerallinearmodel()var()var()()EIYIEYX22019GeneralLinearModelAssumptions-cont’d1.ResidualNormality.2.Homogeneityoferrorvariance3.FunctionalformofModel:LinearityofModel4.NoMulticollinearity5.Independenceofobservations6.Noautocorrelationoferrors7.NoinfluentialoutliersWehavetotestforthesetobesurethatthemodelisvalid.Wewilldiscusstherobustnessofthemodelinfaceofviolationsoftheseassumptions.Wewilldiscussrecourseswhentheseassumptionsareviolated.20Explanationoftheseassumptions1.FunctionalformofModel:LinearityofModel:Thesemodelsonlyanalyzethelinearrelationship.2.Independenceofobservations3.Representativenessofsample4.ResidualNormality:Sothealpharegi