PLS建模的经典论文The use of PLS path modeling in international marketing

PartialLeastSquaresPathModelinginInternationalMarketing

x11x12x13x21x22x23x24Outer Model(Here: Formative Mode)??2InnerModel??2??4??1??1??3285

x31x32??31??32x41x42x43??41??42??43Outer Model(Here: Re?ective Mode)Fig.1.ExampleofaPLSPathModel.

standardizedsothatthelocationparameterscanbediscardedinthefollowingequations.Theinnermodelforrelationshipsbetweenlatentvariablescanbewrittenas:

x?Bxtz

(1)

wherexisthevectoroflatentvariables,Bdenotesthematrixofcoef?cientsoftheirrelationships,andzrepresentstheinnermodelresiduals.ThebasicPLSdesignassumesarecursiveinnermodelthatissubjecttopredictorspeci?cation.Thus,theinnermodelconstitutesacausalchainsystem(i.e.withuncorrelatedresidualsandwithoutcorrelationsbetweentheresidualtermofacertainendogenouslatentvariableanditsexplanatorylatentvariables).Predictorspeci?cationreducesEq.(1)to:

exjxT?Bx

(2)

PLSpathmodelingincludestwodifferentkindsofoutermodels:re?ective(ModeA)andformative(ModeB)measurementmodels.Theselectionofacertainoutermodeissubjecttotheoreticalreasoning(Diamantopoulos&Winklhofer,2001).

There?ectivemodehascausalrelationshipsfromthelatentvariabletothemanifestvariablesinitsblock.Thus,eachmanifestvariableinacertainmeasurementmodelisassumedtobegeneratedasalinearfunctionofitslatentvariablesandtheresiduale:

Xx?Lxxt??x

(3)

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whereLrepresentstheloading(pattern)coef?cients.Theouterrelationships

arealsosubjecttopredictorspeci?cation–implyingthattherearenocorrelationsbetweentheouterresidualsandthelatentvariableofthesameblock–thatreducesEq.(3)to:

eXxjxT?Lxx

(4)

Theformativemodeofameasurementmodelhascausalrelationshipsfromthemanifestvariablestothelatentvariable.Forthoseblocks,thelinearrelationshipsaregivenasfollows:

x?PxXxt??x

(5)

Intheformativemode,predictorspeci?cationisalsoineffect,reducingEq.(5)to:

exjXxT?PxXx

(6)

Moreover,itisimportanttoseethattheterms‘‘formative’’and‘‘re?ective,’’aswellastheconnotationwhichisassociatedwiththeclassi?cationof‘‘causal’’and‘‘effect,’’pointatadifferencebetweenthecharacterizationofthelatentvariablemeasurementmodels’mode.Althoughalatentvariablewasoriginallyconsideredanexactlinearcombinationofitsindicatorsforformativeindicatorspeci?cations,acausalindicatorspeci?cation–theoriginaltermfortheseindicators–maybemoregeneralinthatitholdsbothinthecaseofanexactlinearcombination,aswellaswhentheindicatorsdonotcompletelydeterminethelatentvariable(Bollen,1989).Inthischapter,weconsistentlyusetheterms‘‘formative’’and‘‘re?ective’’measurementmodels–inthewaytheyaredescribed,forexample,byJarvisetal.(2003).Itmustbenoted,though,thatwhileweusetheterms‘‘re?ective’’and‘‘formative’’constructstorefertolatentvariablesthataremeasuredwithre?ectiveorformativeindicators,‘‘strictlyspeaking,itisthe(observable)measures(i.e.theindicators)thatarebeingmodeledasre?ectiveorformativeandnotthe(unobservable)constructsassuch’’(Diamantopoulos,2006,p.15).ThefollowingsectionintroducesthebasicPLSalgorithm,whichstartswiththedatamatrixofmanifestvariablesandsuccessivelycomputesthelatentvariablescoresandallunknownrelationships.

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2.2.ThePLSPathModelingAlgorithm

ThePLSalgorithmisessentiallyasequenceofregressionsintermsofweightvectors.Theweightvectorsobtainedatconvergencesatisfy?xedpointequations(seeDijkstra,2009,forageneralanalysisofsuchequationsandensuingconvergenceissues).ThebasicPLSalgorithm,assuggestedbyLohmoller(1989),includesthefollowingthreestages:Stage1:Iterativeestimationoflatentvariablescores,consistingofafour-stepiterativeprocedurethatisrepeateduntilconvergenceisobtained:(1)(2)(3)(4)

outerapproximationofthelatentvariablescores,estimationoftheinnerweights,

innerapproximationofthelatentvariablescores,andestimationoftheouterweights.

Stage2:Estimationofouterweights/loadingandpathcoef?cients.Stage3:Estimationoflocationparameters.

WedrawonTenenhaus,EspositoVinzi,Chatelin,andLauro’s(2005)descriptionofthe?rststageofthePLSpathmodelingalgorithm:Step1:Outerapproximationofthelatentvariablescores.Outerproxiesofthe

^outer,arecalculatedaslinearcombinationsoftheirrespectivelatentvariables,xn

indicators.Theseouterproxiesarestandardized;i.e.theyhaveameanof0andastandarddeviationof1.Theweightsofthelinearcombinationsresultfromstep4ofthepreviousiteration.Whenthealgorithmisinitialized,andnoweightsareavailableyet,anyarbitrarynontriviallinearcombinationofindicatorscanserveasanouterproxyofalatentvariable.

Step2:Estimationoftheinnerweights.Innerweightsarecalculatedforeachlatentvariableinordertore?ecthowstronglytheotherlatentvariablesareconnectedtoit.Therearethreeschemesavailablefordeterminingtheinnerweights.Wold(1982)originallyproposedthecentroidscheme.Later,Lohmoller(1989)developedthefactorweightingandpathweightingschemes.Thecentroidschemeusesthesignofthecorrelationsbetweenalatentvariable–or,moreprecisely,theouterproxy–anditsadjacentlatentvariables;thefactorweightingschemeusesthecorrelations.Thepathweightingschemepaystributetothearroworientationsinthepathmodel.Theweightsofthoselatentvariablesthatexplainthefocallatentvariablearesettotheregressioncoef?cientsstemmingfromaregressionofthefocallatentvariable(regressant)onitslatentrepressorvariables.Theweightsofthoselatentvariables,whichareexplainedbythefocallatent

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variable,aredeterminedinasimilarmannerasinthefactorweighting

scheme.Regardlessoftheweightingscheme,aweightofzeroisassignedtoallnonadjacentlatentvariables.

Step3:Innerapproximationofthelatentvariablescores.Innerproxiesof

^inner,arecalculatedaslinearcombinationsoftheouterthelatentvariables,xn

proxiesoftheirrespectiveadjacentlatentvariables,usingtheafore-determinedinnerweights.

Step4:Estimationoftheouterweights.Theouterweightsarecalculatedeitherasthecovariancesbetweentheinnerproxyofeachlatentvariableanditsindicators(inModeA,re?ective),orastheregressionweightsresultingfromtheordinaryleastsquaresregressionoftheinnerproxyofeachlatentvariableonitsindicators(inModeB,formative).

Thesefourstepsarerepeateduntilthechangeinouterweightsbetweentwoiterationsdropsbelowaprede?nedlimit.Thealgorithmterminatesafterstep1,deliveringlatentvariablescoresforalllatentvariables.Loadingsandinnerregressioncoef?cientsarethencalculatedinastraightforwardway,giventheconstructedindicesandusingEqs.(4)and(5).Inordertodeterminethepathcoef?cients,foreachendogenouslatentvariablea(multiple)linearregressionisconducted.

2.3.MethodologicalCharacteristics

MethodologicalliteratureonPLSpathmodelingorpublicationsoncausalmodelingapplicationsthatutilizethePLSpathmodelingapproachusuallyrefertocertainadvantageousfeaturesofthistechnique(e.g.,Fornell&Bookstein,1982;Joreskog&Wold,1982;Dijkstra,1983;Lohmoller,1989;SchneeweiX,1991;Falk&Miller,1992).ThepopularityofPLSpathmodelingamongscientistsandpractitionersisrootedinfourgenuinecharacteristics:First,insteadofsolelydrawingonthecommonre?ectivemode,thePLSpathmodelingalgorithmallowstheunrestrictedcomputa-tionofcause–effectrelationshipmodelsthatemploybothre?ectiveandformativemeasurementmodels(Diamantopoulos&Winklhofer,2001).Second,PLScanbeusedtoestimatepathmodelswhensamplesizesaresmall(Chin&Newsted,1999).Third,PLSpathmodelscanbeverycomplex(i.e.consistofmanylatentandmanifestvariables)withoutleadingtoestimationproblems(Wold,1985).PLSpathmodelingismethodologicallyadvantageoustoCBSEMwheneverimproperornonconvergentresultsarelikelytooccur(i.e.Heywoodcases;seeKrijnen,Dijkstra,&Gill,1998).