Latent Variable Models: An Introduction to Factor, Path and Structural Equation Analysis PDF
Preview Latent Variable Models: An Introduction to Factor, Path and Structural Equation Analysis
Latent Variable Models An Introduction to Factor, Path, and Structural Equation Analysis FifthEdition JohnC.Loehlin A.AlexanderBeaujean RO Routledge U TLED Taylor & Francis Group G E LONDON AND NEW YORK Firstpublished2017 byRoutledge 711ThirdAvenue,NewYork,NY10017 andbyRoutledge 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness ©2017Taylor&Francis LibraryofCongressCataloginginPublicationData Names:Loehlin,JohnC.|Beaujean,A.Alexander. Title:Latentvariablemodels:anintroductiontofactor,path,andstructuralequationanalysis. Description:Fifthedition/JohnC.Loehlin,A.AlexanderBeaujean.|NewYork,NY:Routledge,2017.| Includesbibliographicalreferencesandindex. Identifiers:LCCN2016038052|ISBN9781138916067(hb:alk.paper)| ISBN9781138916074(pb:alk.paper)|ISBN9781315643199(epub) Subjects:LCSH:Latentvariables.|Latentstructureanalysis.|Factoranalysis.| Pathanalysis(Statistics)|Structuralequationmodeling. Classification:LCCQA278.6.L642017|DDC519.5/35–dc23 LCrecordavailableathttps://lccn.loc.gov/2016038052 ISBN:978-1-138-91606-7(hbk) ISBN:978-1-138-91607-4(pbk) ISBN:978-1-315-64319-9(ebk) TypesetinHelvetica byJohnC.LoehlinandA.AlexanderBeaujean Visitthecompanionwebsite:routledge.com/cw/loehlin Contents Preface ix Chapter1: PathModelsinFactor,Path,andStructuralEquation Analysis 1 PathDiagrams 2 PathAnalysis 7 FactorModels 16 StructuralEquations 23 OriginalandStandardizedVariables 24 ManifestVersusLatentVariableModels 28 ExtendedExample 28 Notes 31 Exercises 33 Chapter2: FittingPathModels 37 IterativeSolutionofPathEquations 37 MatrixFormulationofPathModels 42 Full-FledgedModel-FittingPrograms 46 FitFunctions 54 2 Hierarchicalffl Tests 63 DescriptiveCriteriaofModelFit 69 ThePowertoRejectanIncorrectModel 72 Identification 76 MissingData 78 CorrelationsVersusCovariancesinModelFitting 82 ExtendedExample 84 Notes 86 Exercises 91 Chapter3: FittingPathandStructuralModelstoDatafromaSingle GrouponaSingleOccasion 95 StructuralandMeasurementModels 95 ConfirmatoryFactorAnalysis 100 SomePsychometricApplicationsofPathandStructuralModels 103 StructuralModels—ControllingExtraneousVariables 111 ModelswithReciprocalInfluencesandCorrelatedErrors 115 NonlinearEffectsAmongLatentVariables 120 ExtendedExample 124 Notes 126 Exercises 129 Chapter4: FittingModelsInvolvingRepeatedMeasures,Multiple Groups,orMeans 133 ModelsofEventsOverTime 133 ModelsComparingDifferentGroups 142 FittingModelstoMeansaswellasCovariances 151 TheVersatilityofMultiple-GroupDesigns 159 ModelswithCategoricalIndicators 160 AConcludingComment 163 ExtendedExample 163 Notes 166 Exercises 169 Chapter5: ExploratoryFactorAnalysis—Basics 171 FactorExtraction 174 EstimatingCommunalities 179 DeterminingtheNumberofFactors 183 Rotation 189 AnExample: Thurstone’sBoxProblem 197 FactorAnalysisUsingPackagedPrograms 201 ExtendedExample 204 Notes 205 Exercises 208 Chapter6: ExploratoryFactorAnalysis—Elaborations 211 Rescalings—AlphaandCanonicalFactors 211 AlternativeStoppingCriteria 214 AlternativeRotationMethods 217 EstimatingFactorScores 220 HierarchicalFactors 225 NonlinearFactorAnalysis 232 ExtendedExample 235 Notes 240 Exercises 241 Chapter7: IssuesintheApplicationofLatentVariableModels 243 ExploratoryModificationofaModel 243 AlternativeModels 248 CanPathDiagramsbeConstructedAutomatically? 252 ModesofLatentVariableAnalysis 255 AdvancedTopicsinLatentVariableModels 260 CriticismsofLatentVariableModeling 263 Notes 268 Exercises 271 Appendices 273 A. SimpleMatrixOperations 273 B. DerivationofMatrixVersionofPathEquations 281 C. LISRELMatricesandExamples 284 D. ModelFitIndices 289 E. TableofChi-squareValues 299 F. NoncentralChi-squareValuesforEstimatingPower 301 G. PowerofaTestofPoorFitandSampleSizesNeededforPower 303 H. CommunicatingAboutLatentVariableModels 305 AnswerstoExercises 309 References 321 Index 367 Preface to the First Edition Thisbookisintendedasanintroductiontoanexcitinggrowthareainsocialscience methodology—theuseofmultiple-latent-variablemodels. Psychologistsandother socialscientistshavelongbeenfamiliarwithonesubvarietyofsuchmodeling,factor analysis—moreproperly,exploratoryfactoranalysis. Inrecentdecades,confirmatory factoranalysis,pathanalysis,andstructuralequationmodelinghavecomeoutof specializednichesandaremakingtheirbidtobecomebasictoolsintheresearch repertoireofthesocialscientist,particularlytheonewhoisforcedtodealwithcomplex real-lifephenomenaintheround: thesociologist,thepoliticalscientist,thesocial, educational,clinical,industrial,personalityordevelopmentalpsychologist,the marketingresearcher,andthelike. Allthesemethodsareatheartone,asIhavetriedtoemphasizeinthechaptersto follow. Ihaveusedearlierversionsofthisbookinteachinggraduatestudentsfrom psychologyandrelateddisciplines,andhavefoundtheparticularapproachused—via pathdiagrams—tobeeffectiveinhelpingnot-too-mathematicalstudentsgrasp underlyingrelationships,asopposedtomerelygoingthroughthemotionsofrunning computerprograms. Insomesectionsofthebookacertainamountofelementary matrixalgebraisemployed;anappendixonthetopicisprovidedforthosewhomay needhelphere. Intheinterestsofaccessibility,Ihavetriedtomaintainarelativelyinformalstyle, andtokeepthemaintextfairlyunclutteredwithreferences. Thenotesattheendof eachchapterareintendedloprovidetheseriousstudentwithapathintothetechnical literature,aswellastodrawhisorherattentiontosomeissuesbeyondthescopeof thebasictreatment. Thebookisnotcloselytiedtoaparticularcomputerprogramorpackage,although thereissomespecialattentionpaidtoLISREL.Iassumethatmostuserswillhave accesstoalatent-variablemodel-fittingprogramontheorderofLISREL,COSAN, EQS,orMILS,andanexploratoryfactoranalysispackagesuchasthoseinSPSS, BMDP,orSAS.Insomeplaces,amatrixmanipulationfacilitysuchasthatinMINITAB orSASwouldbehelpful. Ihaveprovidedsomeintroductorymaterialbuthavenottried totellstudentsalltheyneedtoknowtorunactualprograms—suchinformationisoften local,ephemeral,orboth. Theinstructorshouldexpecttoprovidesomehandoutsand perhapsabitofhands-onassistanceingettingstudentsstarted. Thereadergoingit onhisorherownwillrequireaccesstocurrentmanualsforthecomputerprogramsto beused. Finally,itgivesmegreatpleasuretoacknowledgethehelpandencouragement thatothershaveprovided. Perhapsfirstcreditshouldgotothestudentswhoendured earlyversionsofthemanuscriptandcheerfullypointedoutvariouserrorsand obscurities. ThesebravepioneersincludedMikeBailey,CherylBeauvais,Alan Bergman,BethGeer,SteveGregorich,PriscillaGriffith,JeanHart,PamHenderson, WesHoover,VivianJenkins,TockLim,ScottLiu,JacquelineLovette,FrankMulhern, StevePredmore,NaftaliRaz,andLoriRoggman. Amongothercolleagueswhohave beenkindenoughtoreadandcommentonvariouspartsofthemanuscriptareCarole Holahan,PhilGough,MariaPennock-Roman,PeterBentler,andseveralanonymous reviewers. IamespeciallygratefultoJackMcArdleforextensivecommentsonthe manuscriptasawhole,andtoJackCohenforhispersuasivevoicewiththepublishers. Ofcourse,thesepersonsshouldnotbeblamedforanydefectsthatmayremain. For onething,Ididn’talwaystakeeverybody’sadvice. IamgratefultotheUniversityofChicagoPress,toMultivariateBehavioral Research,andtotheHafnerPublishingCo. forpermissiontoreprintoradapt publishedmaterials,andtothemanypreviousresearchersandwriterscitedinthe book—or,forthatmatter,notcited—whosecontributionshavedefinedthisrapidly developingandexcitingfield. Finally,Ioweaspecialdebttothemembersofmyfamily: JenniferandJames,who workedtheirtermpapersinaroundmysessionsattheMacintosh,andMarj,who providedunfailingsupportthroughout. J.C.L. Notetothesecondedition: Muchofthefirsteditionisstillhere,butacertain amountofnewmaterialhasbeenadded,someexerciseschanged,andonetopic (multidimensionalscaling)droppedtomakeroom. Also,I’vetriedtomakethebook morehelpfultothosewhoareusingprogramsotherthanLISREL.Istillappreciatethe contributionsofthepeopleIthankedbefore. Inaddition,IamgratefultoPeterBentler, RobertCudeck,andJeffTanakafortheirhelpfulcommentsondraftmaterialforthe presentedition,andtotheAmericanMathematicalSocietyforpermissiontoadaptthe tableinAppendixF. Notetothethirdedition: Itisstillthecasethatmoreremainsthanhasbeen changed. What’sgone: IPSOL,BMDP,EzPATH,andafewotheritemssupplantedby themarchofeventsinourfield. What’snew: moreSEMprograms,morefitindices, manynewreferences,connectionstotheInternet,moreonmeans,moreonpower, and,maybeasimportantasanything,emphasisontheRMSEAanditsusein rejectingnullhypothesesofpoorfit. IremaingratefultoallthoseIthankedinthefirstandsecondeditions,andhavea goodmanynamestoadd—peoplewhogavemeadviceorencouragement,sentme reprintsorpreprintsorprograms,spottederrors,answeredqueries. Thesehelpful personsinclude: JimArbuckle,KennethBollen,MichaelBrowne,DavidBurns,Hsin-Yi Chen,MikeCoovert,StanGaines,SteveGregorich,GregHancock,DavidKaplan, TimothyKeith,RobertMacCallum,HerbertMarsh,TorNeilands,FrankNorman,Eddie Oczkowski,EdRigdon,DorisRubio,BillShipley,JimSteiger,BobThorndike,and NielsWaller. AndifI’veleftanybodyout—well,them,too. Notetothefourthedition: Thebasicapproachofthefourtheditionremainsthe sameasthatofpreviouseditions,and,mostly,sodothecontentsofthebook,with somemildreorganization. Chapters3and4arenowdividedslightlydifferently,sothat Chapter3coverssingle-group,single-occasionmodels,andChapter4dealsjustwith modelsinvolvingmultiplegroupsormultipleoccasions. Chapters5and6,exploratory factoranalysis,havealsobeenrearranged,sothatChapter5coversafewbasicfactor extractionandrotationmethods,forthebenefitofinstructorswhopreferabriefer brushwithEFA,andChapter6treatsmoreadvancedmatters. Chapter7hasbecome lessofagrabbagofspecializedtopics,withsomeofthese(e.g.,modelswithmeans, nonlinearmodels,andhigher-orderfactors)beingpromotedtoappropriateearlier chapters,andothers(e.g.,phantomvariables)movingtoanappendix. Thedetailed descriptionofmostgoodness-of-fitindicesisnowinanappendixforreferencerather thanencumberingthemaintext. Afewitems,suchasthecentroidmethodand multivariatepathmodels,havedisappearedfromthebookaltogether,andafewitems havebeenadded,suchassectionsonmissingdata,nonnormality,mediation,factorial invariance,andautomatingtheconstructionofpathdiagrams. Tosavestudentslabor intyping,aCDissuppliedcontainingthevariouscorrelationandcovariancematrices usedintheexercises(detailsaregivenattheendofChapter2). Afewneweasy exerciseshavebeenaddedintheearlychapters,andanumberoftheexisting exerciseshavemovedorchangedinconformitywiththetextshifts. Overall,therehas beenasubstantialexpansionandupdatingofthereferencelistandtheend-of-chapter notes. Icontinuetobegratefultothepeoplementionedpreviously,aswellastoseveral additionalanonymousreferees,andtothefolksatErlbaum: DebraRiegerthasbeen veryhelpfulaseditor,ArtLizzacontinuesasaninvaluableresourceontheproduction side,andofcourseLarryErlbaumbeamsbenevolentlyuponusall. Ifyouhappentonoticeanyerrorsthathaveslippedby,Iwouldbegratefulifyou wouldcallthemtomyattention: [email protected]. Enjoythebook. Notetothefifthedition: Althoughthefiftheditionaddsaco-author,thebook’s approachremainsthesameasthatofpreviouseditions,asdoesmuchofthecontent, althoughwehavereorganizedsomeofitandaddedsomeaswell. Inadditionto expandingthereferencelist,internetsources,andtheend-of-chapternotes,thereare afewotherchangesthatapplythroughoutthebook. First,weaddeddiscussionof packagesfromtheRstatisticalprogramatseveralpointsinthetext. Whilewerealize noteveryoneusesR,wechosetoinclude(andinsomeplacesemphasize)the programbecauseitisfree,opensource,andavailableforalmostallcomputer operatingsystems. Second,weaddedanExtendedExampleattheendofChapters 1–6,usuallyusingR,whereweworkthroughoneofthechapter’sexamplesinmore detailtoaidreadersinconductingsimilaranalyseswiththeirowndata. Third,wehave removedthedataCDand,instead,createdanaccompanyingwebsite (routledge.com/cw/loehlin)thatcontainsbothdataandsyntaxforreplicatingthebook’s examples. Fourth,were-typesettheentirebookandre-createdallthefiguresusing LATEX. Thefiguresarealsoavailableonthebook’swebsite. Changestotheindividualchaptersareasfollows. InChapter3,weaddeda sectiononcompositereliability,andinChapter4weaddedasectionaboutlatent variablemodelswithcategoricaldata. InChapter5weaddedasectiononthe minimumaveragepartialprocedure. InChapter6weexpandedthesectionon hierarchicalfactormodelstodiscussbi-factormodelsinmoredepth,includinga bi-factoranalyticalrotation. InChapter7,weaddedanewsectiononadvancedtopics. Weaddedoneappendix(H),whereweprovidesomesuggestionsforcommunicating aboutlatentvariablemodels. WeremovedtheappendixwithThurstone’sBoxdata (nowontheaccompanyingwebsite),andtheappendixonphantomvariableshas beenreplacedbyaChapter3note. Wecontinuetobegratefultothepeoplementionedpreviously,aswellastoseveral anonymousreferees,andtoalltheindividualsatRoutledgewhohaveaidedinthis book’spublication. Inaddition,AlexwouldliketothankWilliam,Lela,Byron,Susana, Aleisa,Susanna,andespeciallyChristinefortheirperpetualhelpandsupport. Ifyounoticeanyerrorsthathaveslippedbyus,wewouldbegratefulifyouwould bringthemtoourattention: [email protected][email protected]. We hopeyouenjoythebook. Chapter 1: Path Models in Factor, Path, and Structural Equation Analysis Scientistsdealingwithbehavior,especiallythosewhoobserveitoccurringinits naturalsettings,rarelyhavetheluxuryofthesimplebivariateexperiment,inwhicha singleindependentvariableismanipulatedandtheconsequencesobservedfora singledependentvariable. Eventhosescientistswhothinktheydoareoftenmistaken: Thevariablestheydirectlymanipulateandobservearetypicallynottheonesofreal theoreticalinterestbutaremerelysomeconvenientvariablesactingasproxiesor indexesforthem. Afullexperimentalanalysiswouldagainturnouttobemultivariate, withanumberofalternativeexperimentalmanipulationsontheoneside,anda numberofalternativeresponsemeasuresontheother. Overmanyyears,numerousstatisticaltechniqueshavebeendevelopedfor dealingwithsituationsinwhichmultiplevariables,someunobserved,areinvolved. Suchtechniquesofteninvolvelargeamountsofcomputation. Untiltheadventof powerfuldigitalcomputersandassociatedsoftware,theuseofthesemethodstended toberestrictedtothededicatedfew. Butinthelastfewdecadesithasbeenfeasible foranyinterestedbehavioralscientiststotakeamultivariateapproachtotheirdata. Manyhavedoneso. Theexplosivegrowthintheuseofcomputersoftwarepackages suchasSPSS,SAS,andRisoneevidenceofthis. Thecommonfeaturesofthemethodsdiscussedinthisbookarethat(a)multiple variables—threeormore—areinvolved,andthat(b)oneormoreofthesevariablesis unobserved,orlatent. Neitherofthesecriteriaprovidesadecisiveboundary. Bivariate methodsmayoftenberegardedasspecialcasesofmultivariatemethods. Someof themethodswediscusscanbe—andoftenare—appliedinsituationswhereallthe variablesare,infact,observed. Nevertheless,themainfocusofourinterestisonwhat wecall,followingBentler(1980),latentvariableanalysis,atermencompassingsuch specificmethodsasfactoranalysis,pathanalysis,andstructuralequationmodeling (SEM),allofwhichsharethesedefiningfeatures. 1
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