A Unified Maximum Likelihood Approach for Analyzin

:10.1177/00491241062923572007;35;352SociologicalMethodsResearchSik-YumLeeandXin-YuanSongEquationModelsWithMissingNonstandardDataAUnifiedMaximumLikelihoodApproachforAnalyzingStructural:Publishedby:://smr.sagepub.com/cgi/alertsEmailAlerts::::):(thisarticlecites33articleshostedontheCitationsdistribution.©2007SAGEPublications.Allrightsreserved.NotforcommercialuseorunauthorizedatPENNSYLVANIASTATEUNIVonApril14,2008fiedapproachformaximumlikelihoodanalysisofstructuralequationmodelsthatinvolvesubtlemodelformulationsandnonstandarddatastructures.Basedontheideaofdataaugmentation,theydescribeagenericMonteCarloexpectation-maximizationalgorithmforesti-mation.Theyproposepathsamplingforcomputingtheobserveddatalikeli-hoodfunctionsthatusuallyinvolvecomplicatedintegralsandshowhowtoapplythismethodforcomputingtheBayesianinformationcriterionformodelcomparison.Anapplicationoftheproposedunifiedapproachtoatwo-levelnonlinearstructuralequationmodelwithmissingcontinuousandorderedcate-goricaldataispresented.Anillustrativeexamplewitharealdatasetisgiven.Keywords:MCEMalgorithm;Gibbssampler;Bayesianinformationcriterion;pathsamplingtructuralequationmodels(SEMs;seeBollen1989)arewellrecog-nizedasusefulmodelsforexploringandconfirmingrelationshipsamongmanifestandlatentvariables.Bymeansofmorethanadozenuser-friendlysoftware,suchasLISREL(Jo¨reskogandSo¨rbom1996)andEQS6.0(BentlerandWu2002),theyhavebeenextensivelyappliedtobehavioralandsocialsciences.Thestandardmethodsthatareusedintheexistingsoftwareforobtainingthemaximumlikelihood(ML)estimateSociologicalMethods&ResearchVolume35Number3February2007352-381
2007SagePublications10.1177/0049124106292357’Note:ThisresearchisfullysupportedbyagrantfromtheResearchGrantCouncilofTheHongKongSpecialAdministrativeRegion(CUHK4243/03H)andadirectgrant(CUHK2060279).TheauthorsarethankfultotheeditorandanonymousreviewersforhelpfulcommentsandgratefultotheFacultyofEducationandHongKongInstituteofEducationalResearch,TheChineseUniversityofHongKongforprovidingthedataintheexample.S352distribution.©2007SAGEPublications.Allrightsreserved.NotforcommercialuseorunauthorizedatPENNSYLVANIASTATEUNIVonApril14,2008(1996)developedanexpectationandmaximization(EM)–typealgorithm(Dempster,Laird,andRubin1977)toincorporatedatafromfixedcovariatesinanexploratoryfactoranalysis(EFA)model.Foranalyzingthefullinformationitemfactormodel,BockandAitkin(1981)implementedtheEMalgorithminwhichtheE-stepisevaluatedbythefixed-pointGauss-Hermitequadrature.SongandLee(2005a)consideredageneralizationofthismodelanddevelopedaMonteCarloEM(MCEM)algorithminwhichtheE-stepisevaluatedbymeansofobservationsthataresimulatedfromtheappropriateconditionaldistributionsbysomeMarkovchainMonteCarlo(MCMC)methods.ShiandLee(2000)derivedasimilarMCEMalgorithmforfittingaconfirmatoryfactoranalysis(CFA)modelwithmixedcontinuousandorderedcategoricaldata.IntheSEMliterature,modelswithnonlineartermsoflatentvariablesarecallednonlinearmodels—forexample,thenonlinearfactoranalysismodel(ZhuandLee1999)andthenonlinearSEM(LeeandSong2003b),whichisdefinedbyastructuralequationwithnonlineartermsoflatentvariables.AnMCEMalgorithmhasbeendevelopedfortheMLestimationofnonlinearSEM(LeeandZhu,2002;SongandLee2005a).Toaccountforthecorrelatednatureofthehier-archicallystructureddatathatarecollectedfromunitsthatarenestedwithinalargenumberofclusters,multilevelSEMshavebeenproposedandana-lyzed.Raudenbush(1995)formulatedthemodelasamissingdataproblembyconsideringtheobservedunbalancedtwo-leveldataas‘‘incomplete’’andthe‘‘completedata’’asbalancedandappliedtheEMalgorithmtogettheMLsolutionofatwo-levelfactoranalysismodel.Recently,LeeandShi(2001)establishedanMCEMalgorithmforatwo-levelCFAmodelwithhierarchicallycontinuousandorderedcategoricaldata,andLeeandSong(2004,2005)constructedasimilaralgorithmforatwo-levelnonlinearSEMthatalsoaccommodatesfixedcovariateswithhierarchicallycontinuousdata.Finally,asmissingdataarefrequentlyencounteredinpractice,muchattentionhasbeendevotedtoanalyzeSEMswiththis