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1、ConfirmatoryFactorAnalysiswithRJamesH.SteigerPsychology312Spring2013TraditionalExploratoryfactoranalysis(EFA)isoftennotpurelyexploratoryinnature.Thedataanalystbringstotheenterpriseasubstantialamountofintellectualbaggagethataffectstheselectionofvariables,choiceofanumberoffactors
2、,thenamingoffactors,andinsomecasesthewayfactorsarerotatedtosimplestructure.Sotosomeextent,EFAisactuallyconfirmatoryinnature.Confirmatoryfactoranalysis(CFA)providesamoreexplicitframeworkforconfirmingpriornotionsaboutthestructureofadomainofcontent.CFAaddstheabilitytotestconstrain
3、tsontheparametersofthefactormodeltothemethodologyofEFA.Inpractice,peoplefrequentlycombineEFAandCFA,totheextentthattheappropriatestatisticalmodelisnotactuallydeterminable.However,we’llbeginwithanexampleofpurelyconfirmatoryfactoranalysis.1.“Pure”ConfirmatoryFactorAnalysisConsider
4、theAthleticsDataexampleweexaminedinconjunctionwithEFA.Supposethat,priortoanalyzingthedata,wehypothesizedthattherewere3uncorrelatedfactorscalledEndurance,Strength,andHand-EyeCoordination,andthateachfactorhasnon-zeroloadingsononly3variables.Suchahypothesisis,ofcourse,extremelyunl
5、ikelytobetrue,apointwewillreturntolater.Takenliterally,withasuitableorderingofthe9observedvariables,thishypothesisimpliesthatthecommonfactorpatternisoftheforméq00ùê1úêq00úê2úêúêq300úêúê00q4úêúF=ê00qú5êúê00qú6êúê00qúê7úê00qúê8úê00qúêë9úûThereareanumberofequivalentwaysofwritingth
6、isCFAmodel.OnestatesthatyF=+xewhereFhastheformshownabove,andxandearevectorsofrandomvariablessuch22thatE()xe¢=0,E()ee¢=U,andE()xx¢=I.Uisadiagonalmatrixofpositivevalues,andhencemaybewrittenintheform(zeroentriesnotshown):éùqêú10êúqêú11êúêúq12êúêúq132êúU=êúq14êúêúq15êúêúqêú16êúqêú1
7、7êúqêúëû181.1DiagrammingaConfirmatoryFactorModelThismodelmaybewrittenasapathdiagram,asshownonthenextpage.Notethatthevariancesofthecommonfactorsarenotshownexplicitlyinthediagram.Accordingtoourconventions,theyarethereforeassumedtohaveavarianceof1.Notealsothatthecoefficientfromare
8、sidualtoanobservedvariableisnotlabeledinthediagram,whi
