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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).
