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SEM
Basics:
ASupplementtoMultivariateDataAnalysis
MultivariateDataAnalysisPearsonPrenticeHallPublishing
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TableofContents
LearningObjectives.....................................................................................................................................................3Preview.........................................................................................................................................................................3FundamentalsofStructuralEquationModeling......................................................................................................4
EstimatingRelationshipsUsingPathAnalysis..........................................................................................................4IdentifyingPaths.................................................................................................................................................5
EstimatingtheRelationship..............................................................................................................................6
UnderstandingDirectandIndirectEffects...............................................................................................................7IdentificationofCausalversusNonCausaleffects...........................................................................................7
DecomposingEffectsintoCausalversusNoncausal........................................................................................8
CalculatingIndirectEffects................................................................................................................................9
ImpactofModelRespecification......................................................................................................................11
OtherAbsoluteFitIndices.......................................................................................................................................11SpecificationIssuesinSEMPrograms......................................................................................................................12
TheMultivariateRelationshipsinSEM.....................................................................................................................12TheMainStructuralEquation...........................................................................................................................12
UsingConstructstoExplainMeasuredVariables:TheMeasurementModel................................................13
CompleteStructuralandMeasurementModelEquations.............................................................................14
SpecifyingAModelinLISRELNotation...................................................................................................................18SpecificationofaCFAModelwithLISREL.......................................................................................................18
ChangingTheCFASetupinLISRELtoaStructuralModelTest......................................................................19
HBAT:TheCFAModel...............................................................................................................................................21HBAT:TheStructuralModel...................................................................................................................................23HowtoFixFactorLoadingstoaSpecificValueinLISREL.......................................................................................25MeasuredVariableandConstructInterceptTerms................................................................................................27PathModelSpecificationwithAMOS.....................................................................................................................27ResultsUsingDifferentSEMPrograms...................................................................................................................28
AdditionalSEMAnalyses...........................................................................................................................................28TestingforDifferencesinConstructMeans............................................................................................................29ItemParcelinginCFAandSEM................................................................................................................................29WhenIsParcelingAppropriate?......................................................................................................................30
HowShouldItemsBeCombinedintoParcels?................................................................................................31
MeasurementBias..................................................................................................................................................31ContinuousVariableInteractions...........................................................................................................................33
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SEMBasics:
ASupplement
to
MultivariateDataAnalysis
LEARNINGOBJECTIVES
Inthecourseofcompletingthissupplement,youwillbeintroducedtothefollowing:
Thebasicsofestimatingpathcoefficientsbasedonthespecifiedpathmodel. Determinationofthedirectandindirecteffectsimpliedinapathmodel,plusdetermination
whethertheycanbecharacterizedascausalornoncausal.
Someadditionalabsolutefitindicesusedincertainsituations.
Specificationofthepathmodelasaseriesofequationsforboththestructuralmodeland
themeasurementmodel.
Use of LISREL notation to represent these equations and the relationships in the path
model.
Testingformeandifferencesbetweenlatentconstructsindifferentgroups.
Itemparcelingtoreducethenumberofitemsperconstruct.
Assessmentofmeasurementbiasbyintroductionofanadditionallatentconstruct.
Estimationofmoderatingeffectsforcontinuousmultiitemconstructs.
PREVIEW
Thissupplement to the textMultivariateDataAnalysisprovidesadditionalcoverageofsomebasicconceptsthatarethefoundationsforstructuralequationmodeling(SEM). Whilethereis
considerablecoverageofthe technique inthetext,theauthors feltthatreadersmaybenefit
from further reviewof certain topicsnot covered in the text,but issuesaddressedbymany
researchers. Moreover,thereisamorecomprehensivediscussionofthenotationusedinSEM,
particularly those associated with LISREL. There will be some overlap with material in the
chapterssoastofullyintegratetheconcepts.
The supplement is not intended to be a comprehensive primer on all of the SEM
topicsnotcovered inthetest,butonlythoseselected issuesthatmaybeencountered inthe
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courseofbehavioral research. Weencourage readers to complement this supplementwith
othertreatmentsandtextsontheseconceptsasneeded.
The supplement focuseson threebroad areas related to SEM. The first area covers
someofthefundamentalconceptsrelatedthebasicsofpathmodelsestimationofthepath
estimates and determining and interpreting direct and indirect effects. The second area isspecificationoftheSEMmodelinmoreformalterms. Theprimarilyinvolvesdiscussionofwhat
is termed LISRELnotation. This involves the notation used in the LISREL software program
whichhasbecomeacommonmethodofdescribingtherelationshipsinboththestructuraland
measurementmodels. Severalexamples, including theHBATCFAandstructuralmodels,are
used to illustrate how those models can be expressed in this notation. Included in the
discussionarealsosometechniquestoaccomplishspecializedtasksinLISREL,aswellasabrief
introduction toAMOS,anotherpopularSEMsoftwarepackage.Finally,somemoreadvanced
topicsarediscussed toprovide theuseran introduction intosomeofthemorecomplex,but
oftenused,techniquesavailableinSEManalyses.
FUNDAMENTALSOFSTRUCTURALEQUATIONMODELING
The use of SEM is predicated on a strong theoreticalmodel bywhich latent constructs are
defined(measurementmodel)andtheseconstructsarerelatedtoeachotherthroughaseries
ofdependencerelationships(structuralmodel). Theemphasisonstrongtheoreticalsupportfor
anyproposedmodelunderliestheconfirmatorynatureofmostSEMapplications.
Butmanytimesoverlooked isexactlyhowtheproposedstructuralmodel istranslatedinto structural relationships and how their estimation is interrelated. Path analysis is the
processwhereinthestructuralrelationshipsareexpressedasdirectandindirecteffectsinorder
to facilitate estimation. The importance of understanding this process is not so that the
research can understand the estimation process, but instead to understand how model
specification (and respecification) impacts theentire setof structural relationships. Wewill
first illustrate theprocessofusingpathanalysis forestimating relationships inSEManalyses.
Thenwewilldiscusstherolethatmodelspecificationhasindefiningdirectandindirecteffects
andclassificationofeffectsascausalversusspurious. Wewillseehowthisdesignationimpacts
theestimationofstructuralmodel.
ESTIMATINGRELATIONSHIPSUSINGPATHANALYSIS
Whatwasthepurposeofdevelopingthepathdiagram?Pathdiagramsarethebasisforpath
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analysis, the procedure for empirical estimation of the strength of each relationship (path)
depicted in thepathdiagram.Pathanalysiscalculates thestrengthof the relationshipsusing
only a correlation or covariance matrix as input. We will describe the basic process in the
following section, using a simple example to illustrate how the estimates are actually
computed.
IdentifyingPaths
The first step is to identify all relationships that connect any two constructs. Path analysis
enablesustodecomposethesimple(bivariate)correlationbetweenanytwovariablesintothe
sum of the compound paths connecting these points. The number and types of compound
paths between any two variables are strictly a function of the model proposed by the
researcher.
Acompoundpathisapathalongthearrowsofapathdiagramthatfollowthreerules:
1. Aftergoingforwardonanarrow,thepathcannotgobackwardagain;butthepathcango
backwardasmanytimesasnecessarybeforegoingforward.
2. Thepathcannotgothroughthesamevariablemorethanonce.
3. Thepathcanincludeonlyonecurvedarrow(correlatedvariablepair).
Whenapplying these rules,eachpathorarrow representsapath. Ifonlyonearrow links
two constructs (path analysis can also be conducted with variables), then the relationship
betweenthosetwoisequaltotheparameterestimatebetweenthosetwoconstructs.Fornow,
this relationship can be called adirect relationship. If there aremultiple arrows linkingone
constructtoanotherasinXYZ,thentheeffectofXonZseemquitecomplicatedbutan
examplemakesiteasytofollow:
Thepathmodelbelowportraysasimplemodelwithtwoexogenousconstructs(X1andX2)
causallyrelatedtotheendogenousconstruct(Y1).ThecorrelationalpathAisX1correlatedwith
X2,pathBistheeffectofX1predictingY1,andpathCshowstheeffectofX2predictingY1.
ThevalueforY1canbestatedsimplywitharegressionlikeequation:
B
C
X1
Y1
X2
A
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Wecannow identify thedirectand indirectpaths inourmodel.Forease in referring to the
paths,thecausalpathsarelabeledA,B,andC.
DirectPaths IndirectPathsA=X1toX2
B=X1toY1 AC=X1toY1
C=X2toY1 AB=X2toY1
EstimatingtheRelationship
Withthedirectandindirectpathsnowdefined,wecanrepresentthecorrelationbetweeneach
constructasthesumofthedirectandindirectpaths.Thethreeuniquecorrelationsamongthe
constructscanbeshowntobecomposedofdirectandindirectpathsasfollows:
First,thecorrelationofX1andX2issimplyequaltoA.ThecorrelationofX1andY1(CorrX1,Y1)can
berepresentedastwopaths:BandAC.ThesymbolBrepresentsthedirectpathfromX1toY1,andtheotherpath(acompoundpath)followsthecurvedarrowfromX1toX2andthentoY1.
Likewise,thecorrelationofX2andY1canbeshowntobecomposedoftwocausalpaths:Cand
AB.
Once all the correlations are defined in terms of paths, the values of the observed
correlations canbe substituted and theequations solved for each separatepath. Thepaths
then represent either the causal relationships between constructs (similar to a regression
coefficient)orcorrelationalestimates.
Assuming that the correlationsamong the three constructsareas follows:CorrX1 X2=
.50,CorrX1 Y1= .60andCorrX2 Y1= .70,we can solve the equations for each correlation (seebelow)andestimate the causal relationships representedby the coefficientsb1andb2. We
know thatAequals .50, sowecan substitute thisvalue into theotherequations.By solving
thesetwoequations,wegetvaluesofB(b1)= .33andC(b2)= .53.Thisapproachenablespath
analysis to solve for any causal relationship based only on the correlations among the
constructsandthespecifiedcausalmodel.
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SolvingfortheStructuralCoefficients
.50=A
.60=B+AC
.70=C+AB
SubstitutingA=.50
.60=B+.50C
.70=C+.50B
SolvingforBandC
B=.33
C=.53
Asyoucansee from thissimpleexample, ifwechange thepathmodel insomeway, the
causal relationshipswill change aswell. Sucha changeprovides thebasis formodifying the
modeltoachievebetterfit,iftheoreticallyjustified.
With these simple rules, the larger model can now be modeled simultaneously, using
correlationsorcovariancesastheinputdata.Weshouldnotethatwhenusedinalargermodel,we can solve for any number of interrelated equations. Thus, dependent variables in one
relationshipcaneasilybe independentvariables inanotherrelationship.Nomatterhow large
thepathdiagramgetsorhowmanyrelationshipsareincluded,pathanalysisprovidesawayto
analyzethesetofrelationships.
UNDERSTANDINGDIRECTANDINDIRECTEFFECTS
Whilepathanalysisplaysakeyroleinestimatingtheeffectsrepresentedinastructuralmodel,
it also provides additional insight into not only the direct effects of one construct versus
another,butallofthemyriadsetofindirecteffectsbetweenanytwoconstructs. Whiledirect
effects can always be considered causal if a dependence relationship is specified, indirect
effects require furtherexamination todetermine if theyarecausal (directlyattributable toa
dependence relationship) or noncausal (meaning that they represent relationship between
constructs,butitcannotbeattributedtoaspecificcausalprocess).
IdentificationofCausalversusNonCausaleffects
The prior section discussed the process of identifying all of the direct and indirect effects
betweenanytwoconstructsbyaseriesofrulesforcompoundpaths. Herewewilldiscusshow
tocategorizethemintocausalversusnoncausalandthenillustratetheiruseinunderstanding
theimplicationsofmodelspecification.
An importantquestion is:Why isthedistinction important? Theparameterestimates
aremadewithoutanydistinctionasdescribedabove. But theestimatedparameters in the
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C1 C3.Thenextrelationship isC1withC3. Herewecanseetwoeffects: thedirect
effect(P3,1)andtheindirecteffect(P3,2xP2,1). Sincethedirectionofthepathsneverreversesin
the indirecteffect, itcanbecategorizedascausal. Sothedirectand indirecteffectsareboth
causaleffects.
C2
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Thisrelationshipintroducesthefirstnoncausaleffectswehaveseen. ThereisthedirecteffectofB3,2,butthere isalsothenoncausaleffect(duetocommoncause)seen in
B3,1xB2,1. Hereweseetheresultoftwocausaleffectscreatinganoncausaleffectsincethey
bothoriginatefromacommonconstruct(C1C2andC1C3).
C1 C4. In this relationshipwewill see thepotential fornumerous indirect causal
effectsinadditiontodirecteffects. Inadditiontothedirecteffect(B4,1),weseethreeother
indirecteffectsthatarealsocausal:B4,2xB2,1;B4,3xB3,1;andB4,3xB3,2xB2,1.
C3 C4.Thisfinalrelationship wewillexaminehasonlyonecausaleffects(B4,2),but
four different noncausal effects, all a result of C1 or C2 acting as common causes. The two
noncausal effects associatedwith C1 are B4,1 x B3,1 and B4,1 x B2,1 x B3,2. The two othernoncausaleffectsareassociatedwithC2(B4,2xB3,2 andB4,2xB2,1x B3,1).
TheremainingrelationshipisC2C4. Seeifyoucanidentifythecausalandnoncausal
effects. Hint: There are all three types of effects direct and indirect causal effects and
noncausaleffectsaswell.
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Effects
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C3C4 P4,3 None P4,1xP3,1
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P4,2xP2,1xP3,1
CalculatingIndirectEffects
In theprevious sectionwediscussed the identificationand categorizationofbothdirectand
indirecteffectsforanypairofconstructs. Thenextstepistocalculatetheamountoftheeffect
basedonthepathestimatesofthemodel. Assumethispathmodelwithestimatesasfollows:
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ImpactofModelRespecification
Theimpactofmodelrespecificationonboththeparameterestimatesandthecausal/noncausal
effects can be seen in our example aswell. Look back at the C3 C4 relationship. What
happensifweeliminatetheC1C4relationship? DoesitimpacttheC3
C4relationshipinanyway? Ifwe lookbackat the indirecteffects,we can see that twoof the fournoncausal
effectswouldbeeliminated (B4,1 xB3,1andB4,1 x B2,1 x B3,2). Howwould this impact the
model? IftheseeffectsweresubstantialbuteliminatedwhentheC1C4pathwaseliminated,
then most likely the C3 C4 relationship would be underestimated, resulting in a larger
residualforthiscovarianceandoverallpoorermodelfit. Plus,anumberofothereffectsthat
usedthispathwouldbeeliminatedaswell. This illustrateshowtheremovaloradditionofa
path in thestructuralmodelcan impactnotonly thatdirect relationship (e.g.,C1C4),but
manyotherrelationshipsaswell.
OTHERABSOLUTEFITINDICES
Most SEM programs today provide the userwith many different fit indices. In the textwe
focusedmorecloselyonthosethataremostwidelyused.Inthissection,webrieflytouchona
fewotherabsolutefitindicesthataresometimesreported:
Theexpectedcrossvalidation index (ECVI) isanapproximationof thegoodnessoffit the
estimatedmodelwouldachieve inanothersampleofthesamesize.Basedonthesample
covariancematrix,ittakesintoaccounttheactualsamplesizeandthedifferencethatcould
beexpected inanothersample.TheECVIalsotakesintoaccountthenumberofestimated
parametersforagivenmodel.Itismostusefulincomparingtheperformanceofonemodel
toanother.
The actual crossvalidation index (CVI) canbe formedbyusing the computed covariance
matrixderivedfromamodelinonesampletopredicttheobservedcovariancematrixtaken
from a validation sample. Given a sufficiently large sample (i.e., N > 500 for most
applications), the researcher can create a validation sample by splitting the original
observationsrandomlyintotwogroups.
GammaHat also attempts to correct forboth the sample size andmodel complexityby
includingeachinitscomputation.TypicalGammaHatvaluesrangebetween.9and1.0.Its
primaryadvantageisthatithasaknowndistribution[10].
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SPECIFICATIONISSUESINSEMPROGRAMS
InthissectionweprovideanoverviewofspecificationissuesinSEMfortwosoftwarepackages.
Wewill firstdiscuss thenotationused inLISREL,apopularSEMprogram. Thisnotationhas
becomeastandardlanguageofSEMinreferringtobothmeasurementandstructuralmodel
relationships.WewillthenexaminehowtheformulationofthepathmodelistranslatedintoprogramcommandswhileconformingtotheLISRELnotation. Thefirstexamplewillbeasimple
path model to illustrate the basic issues involved. The discussion then shifts to the HBAT
analysisfromthetextbookforboththeCFAandstructuralmodels.Wethenreviewtheseissues
brieflyforAMOSaswell. Inthefinalsectionseveralmorecomplexissuesinmodelspecification
arediscussed.
THEMULTIVARIATERELATIONSHIPSINSEM
Aswediscussed inthetext,SEMmodelsaredefinedbytwosubmodelsthemeasurement
model and the structural model. Each submodel can be expressed is a set of multivariate
equations.Itisntcalledstructuralequationsmodelingfornothing!Eventhoughitispossible
to learnhowtorunaSEMmodelwithoutafullandcompleteunderstandingof itsequations,
knowingthebasicequationscanbehelpfulinunderstandingthedistinctionbetweenmeasured
variablesandconstructsandbetweenexogenousandendogenousconstructs.Moreover, the
equations introduce thenotationused in LISREL,whichwewilldiscuss inmoredetail in the
followingsection. Finally,theequationsalsohelpshowhowSEMissimilartoothertechniques.
TheMainStructuralEquation
Inregression,ourgoalwastobuildamodelthatpredictedasingledependentvariable.Here,
we are trying to predict and explain a set of endogenous constructs. Therefore, we need
equationsthatexplainendogenousconstructs()inadditiontothoseexplainingthemeasured
items (individual x and y variables used as indicators). Not surprisingly, we find that these
equationsaresimilartothemultipleregressionequationthatexplainsthedependentvariable
(y) with multiple independent variables (i.e.,x1 andx2). This fundamental equation for the
structuralmodelisasfollows(refertotheabbreviationguideintheAppendixofthisdocument
foranyneededhelpwithpronunciationsordefinitions):
The representstheendogenousconstructsinamodel.Wewillhaveaseparateequation
for each endogenous construct. The appears on both sides of the equation because
endogenousconstructscanbedependentononeanother(i.e.,oneendogenousconstructcan
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beapredictorofanotherendogenousconstruct).TheBrepresentstheparametercoefficients
thatlinkendogenousconstructswithotherendogenousconstructs.TheBisamatrixconsisting
of as many rows and columns as there are endogenous constructs. So is there are two
endogenousconstructs inthemodel,Bwouldbea22matrix,with2rowsand2columns.
The individual elements of B are designated by a . The is the corresponding matrix of
parametercoefficientslinkingtheexogenousconstructs()withtheendogenousconstructs().It also is a matrix that has as many rows as there are exogenous constructs and as many
columnsasthereareendogenousconstructs.Iftherearethreeexogenousconstructsandtwo
endogenousconstructs,therewouldbea32 matrix.Itsindividualelementsaredesignated
by as shown in the figure. Finally, represents theerror in thepredictionof . It canbe
thoughtofastheresidualorconverseoftheR2conceptfromregression(i.e.,1R
2).
Another way to think of the structural equation is as a multiple regression equation
predicting(aconstruct)insteadofy,withtheothervaluesandthevaluesaspredictors.
The B (1,1,) and (1,1,) provide structural parameter estimates. In the regression
equation, the predictor values were represented by x and the standardized parameter
estimates by the regression coefficients. In both cases, the parameter estimate depicts thelinearrelationshipbetweenapredictorandanoutcome.Thus,clearsimilaritiesexistbetween
SEMandregressionanalysis.
UsingConstructstoExplainMeasuredVariables:TheMeasurementModel
Oncevaluesfor areknown,wecanalsopredicttheyvariablesusinganequationoftheform:
,
Here,eachmeasuredvariableyispredictedbyitsloadingsontheendogenousconstructs.
Typicallyameasuredvariableonlyhasaloadingononeconstruct,butthatcanvaryincertain
situations.Predicted values foreachxalso canbe computed in the samemannerusing the
followingequation:
,
Thepredictedvaluesforeachobservedvariable,(whetherapredictedxory)canbeusedto
computecovarianceestimatesthatcouldbecomparedtotheactualobservedcovarianceterms
inassessingmodelfit.Inotherwords,wecanusetheparameterestimatestomodeltheactual
observedvariables.Theestimatedcovariancematrixobtainedbycomputingcovariationamongpredictedvalues for themeasured items is k.Recall that thedifferencebetween theactual
covariancematrixforobserved items(S)andtheestimatedcovariancematrix isan important
partofanalyzingthevalidityofanySEMmodel.
Rarely is it necessary inmost applications to actually list predicted values based on the
valuesoftheothervariablesorconstructs.Although it isusefultounderstandhowpredicted
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valuescanbeobtainedbecause ithelpsdemonstratethewaySEMworks,the focus insocial
sciencetypicallyisonexplainingrelationships.
CompleteStructuralandMeasurementModelEquations
As noted earlier, LISREL notation has become in some sense the language of SEM. A
researcher, therefore, must have a basic understanding of the notation no matter what
softwareprogram isbeingused.Theexamplebelow illustrates thecompleteequations fora
model consisting of three exogenous constructs, two endogenous constructs and four
indicatorseachforthesetsofendogenousandexogenousconstructs.
StructuralModelEquations
Endogenous
Construct
Exogenous
Construct
Endogenous
Construct
Error
1 = 111+122+133 + 111+123 + 12 = 211+222+233 + 212+222 + 2
MeasurementModelEquations
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= x111 +
x122 +
x133 + 1
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x211 +
x222 +
x233 + 2
X3 = x
311 +
x
322 +
x
333 + 3
X4=
x411 +
x422 +
x433 + 4
EndogenousIndicator EndogenousConstructs Error1 y11 1 + y12 2 12 y21 1 + y22 2 23 y31 1 + y32 2 34 y41 1 + y42 2 4
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StructuralEquationCorrelationsAmongConstructs
AmongExogenousConstructs(Phi) Among EndogenousConstructs(Psi)
1 2 3 1 2
1 1 2 21 2 21
3 31 32
CorrelationsAmongIndicators
AmongExogenousIndicators(Thetadelta )
AmongEndogenousIndicators(Thetaepsilon )
X1 X2 X3 X4 1 2 3 4
X1 1
X2 21 2 21
X3 31 32 3 31 32
X4 41 42 43 4 41 42 43
Thepathmodelnotonly representsthestructuralrelationshipsbetweenconstructs,butalso
provides a means of depicting the direct and indirect effects implied in the structural
relationships. Aworkingknowledgeofthedirectandindirecteffectsofanypathmodelgives
theresearchernotonlythebasisforunderstandingthefoundationsofmodelestimation,but
also insight into the total effects of one construct upon another. Moreover, the indirecteffects can be further subdivided into casual and noncausal/spurious to provide greater
specificity intothetypesofeffects involved. Finally,anunderstandingofthe indirecteffects
allows forgreaterunderstandingof the implicationsofmodel respecification,either through
additionordeletionofadirectrelationship.
Thefollowingtableprovidesanoverviewofthenotationusedformatrices,constructs
and indicators commonly used in SEM. SEM terminology often is abbreviated with a
combinationofGreekcharactersandromancharacterstohelpdistinguishdifferentpartsofa
SEMmodel. Itisfollowedbyaguidetoaidinthepronunciationandunderstandingofcommon
SEMabbreviations.
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Matrices,Construct/IndicatorsandModelEquationNotationoftheLISRELModel
LISRELModel
Element
Description Notation
Matrix Element
Matrices
StructuralModelBeta Relationshipsofendogenoustoendogenous
constructs
nn
Gamma Relationshipsofexogenoustoendogenous
constructs
nm
Phi Correlationamongexogenousconstructs mm
Psi Correlationofstructuralequationsor
endogenousconstructs
n
MeasurementModel
LambdaX Correspondence(loadings)ofexogenous
indicators
x xpm
LambdaY Correspondence(loadings)ofendogenous
indicators
y yqn
Thetadelta Matrixofpredictionerrorforexogenous
constructindicators
pp
Thetaepsilon Matrixofpredictionerrorforendogenous
constructindicators
qq
Construct/Indicators
Construct
Exogenous Exogenousconstruct
Endogenous Endogenousconstruct Indicator
Exogenous Exogenousindicator X
Endogenous Endogenousindicator Y
StructuralandMeasurementModelEquations
StructuralModel Relationshipsbetweenexogenousand
endogenousconstructs = + +
MeasurementModel
Exogenous Specificationofindicatorsforexogenous
constructs X=x +
Endogenous Specificationofindicatorsforendogenousconstructs Y=y +
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PronunciationGuidetoSEMNotation
SymbolPronunciation Meaning
xi(KSIorKZI) An exogenous construct associated with measured X
variables
eta(eightta) An endogenous construct associated with measured Y
variables
Xlambdax A path representing the factor loading between an
exogenouslatentconstructandameasuredxvariable
Ylambday A path representing the factor loading between an
endogenouslatentconstructandameasuredyvariable
capitallambda Thesetof loadingestimatesrepresented inamatrixwhere
rows representmeasuredvariablesand columns represent
latentconstructs
phi(fi) Anarcedtwoheadedarrowdenotingthecovariationoftwo
exogenous()constructs
capitalphi Away of referring to the covarianceor correlationmatrix
betweenasetofexogenous()constructs
gamma Apathrepresentingacausalrelationshipfromanexogenous
construct()toanendogenousconstruct()
capitalgamma Awayof referring totheentiresetof relationships fora
givenmodel
beta(bayta) A path representing a causal relationship from one
endogenous()constructtoanotherconstruct
capitalbeta Awayof referringto theentiresetof relationships fora
givenmodel
delta The error term associatedwith an estimated,measuredx
variable
theta(theyta)
delta
Theresidualvariancesandcovariancesassociatedwiththex
estimates;theerrorvarianceitemsarethediagonal
epsilon The error term associatedwith an estimated,measured y
variable
thetaepsilon Theresidualvariancesandcovariancesassociatedwiththey
estimates;theerrorvarianceitemsarethediagonal
zeta(zayta) Thecovariationbetweenconstructerrors
tau(likenow) Theintercepttermsforameasuredvariable
kappa Theintercepttermsforalatentconstruct
2
chi(ki)squared Thelikelihoodratio
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SPECIFYINGAMODELINLISRELNOTATION
ForLISRELandAMOS, theusercaneitheruse thedropdownmenus togenerate the syntax
thatmatchesthemeasurementmodel,drawthemeasurementmodelusingapathdiagram,or
write the appropriate program commands into a syntax window. If either of the first two
alternativesisdonecorrectly,theprogramsgeneratetheprogramsyntaxautomatically.Wewill
discuss this thirdapproach forLISRELsincethisbest illustrateshowtouseLISRELnotation in
specifyingthemodel.
SpecificationofaCFAModelwithLISREL
SpecificationisquitedifferentusingCFAcomparedtoEFA.Thecommandsbelowillustratehow
thesimpleCFAmodelshownbelow iscommunicatedusingLISRELprogramstatements.Note
thathereweonlyprovide the commandsneeded todefine themodel.The complete setof
programcommandsaregiveninourHBATexampleinalatersection.Also,linenumbershavebeenaddedtothecommandsforreference,buttheyarenotneededasinputtoLISREL.
Inourexamplewehave fourconstructs,eachwith four indicators. See theCFApath
modelbelow:
TheLISRELcommandsforthisCFAareasfollows:
01 MO NX=16 NK=4 PH=SY,FR
02 VA 1.0 LX 1 1 LX 5 2 LX 9 3 LX 13 403 FR LX 2 1 LX 3 1 LX 4 1 LX 6 2 LX 7 2 LX 8 204 FR LX 10 3 LX 11 3 LX 12 3 LX 14 4 LX 15 4 LX 16 4
WebeginwiththeModelcommand(MO) indicatingthenumbersofmeasuredand latent
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variablesanddescriptionsof the keymatricesofparameters. InaCFAmodelweonlyhave
exogenousconstructsandthusofxvariables. NXstandsfornumberofxvariables,inthiscase
16.NKstandsforthenumberofexogenous()constructs,inthiscase4.PHindicatesthatthe
matrixof covariancesbetween the4 constructs ( )willbe symmetric (SY)and free (FR). In
otherwords,theconstructvariances(thediagonalof )andthecovariancebetweeneachpair
ofconstructswillbeestimated.Line2isavaluestatement(VA)whereweassignavaluetoafixedparameter.Inthiscase,
eachof theparameters listedon this line is fixedto1.0toset thescale fortheconstructs.
Oneitemisfixedto1.0oneachconstruct.LX1,1representstheparameterforthefirstloading
onthefirstconstruct(x1,1).TheLstandsforlambda,theXisanxvariableand11standforthe
measured variable number and construct number, respectively. Thus, LX2,1 stands for the
parameter representing the factor loadingof the secondmeasured variable (x2)on the first
latentconstruct(1),or x2,1.
Lines3and4designatethefreeloadingestimates(FR).The12loadingsreferredtoonthese
lines will be estimated and shown as factor results in the output (in x). Thus, this model
estimatesatotalof16loadings,oneforeachindicator(actually12areestimatedandfourfixedtoavalueof1.0)asshowninthepathdiagram. ThiscomparestoEFA,wheretherewouldbea
totalof64loadings(oneforeachindicatoroneachconstruct).
ChangingTheCFASetupinLISRELtoaStructuralModelTest
As discussed in the text, the CFA model forms the foundation from which the structural
model is formulated. In making the conversion from a CFA to a structural model, the
research must make two fundamental decisions: distinguish between exogenous and
endogenousconstructsandspecifythestructuralrelationshipsbetweenconstructs. Notethat
inmostinstances,themeasurementmodelwillbespecifiedandanalyzedintheCFAstage.
Toillustratetheprocess,weutilizetheCFAexamplediscussedinthesectionabove.As
canbeseenfromthepathmodelbelow,twooftheconstructsaredefinedasendogenouswith
relationshipstothetworemainingexogenousconstructs.
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ShownbelowarethemodifiedLISRELsyntaxchangescorrespondingtothestructuralmodel.
Asbefore,linenumbers(notrequiredintheactualLISRELsyntax)areincludedforreference
andonlythecommandsrelatingtothemodelspecificationareshown.
MO NY=8 NE=2 NX=8 NK=2 PH=SY,FR PS=DI,FR GA=FU,FI BE=FU,FIVA 1.0 LX 1 1 LX 5 2 LY 1 1 LY 5 2FR LX 2 1 LX 3 1 LX 4 1 LX 6 2 LX 7 2 LX 8 2
FR LY 2 1 LY 3 1 LY 4 1 LY 6 2 LY 7 2 LY 8 2FR GA 1 1 GA 1 2FR BE 2 1
Thestructuralmodelcommandshaveseveralchanges:
1. TheMOstatementnowprovidesvaluesfor:
a. Thenumberofindicatorsofendogenousconstructs(NY=8)
b. Thenumberofendogenousconstructs(NE=2)
c. Thenewnumberofindicatorsofexogenousconstructs(NX=8)
d. Thenewnumberofexogenousconstructs(NK=2)
2. The MO statement now provides the parameter matrices for the structural parameter
estimates:
a. GAstands for therelationshipsbetweenexogenousandendogenousconstructs ( ,or
gamma).Itisspecifiedasfull(FU)andfixed(FI).Theconventionistospecifyindividual
freeelementsbelow.
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Line01issimplyatitlestatement.Theusercanenteranythingonthislinethathelps
identifytheanalysis.Line02isadatastatement.ItmustbeginwithDAandtellstheSEM
programthat28variablesareincludedinthedatasetof399observations.Althoughthedata
setoriginallycontained400observations,oneresponsepointwasdeletedforbeingoutof
rangeandanotherwassimplymissing.Usingpairwisedeletionandthepreviousruleofthumb,
thenumberofobservationswassetattheminimumnumberofobservationsforanycovariancecomputation.Inthiscase,atleast399observationsareinvolvedinanysinglecovariance
computation.Thisnumbercanbeverifiedbyexaminingthestatisticaloutputforthecovariance
computations.Iflistwisedeletionhadbeenused,thenNOwouldbesetat398sincebothcases
withamissingresponsewouldbedeletedfromanycomputations.MA=CMdenotesthatthe
inputmatrixisacovariancematrix.Line03indicatesthatacovariancematrix(CM)isstoredina
file(FI)namedHBAT.COV.Line04isalabelsstatementandmustbeginwithLA.Thelabelsare
listedbeginningonthelinebelow.Lines05and06showthelabelsforthe28variables.Users
canchooseanylabelstherespectiveprogramwillallow.Inthiscase,HBATlabeledthevariables
withinitialsfromtheconstructnameslikeJS1,JS2,...,SI4.TheycouldhaveusedX1X28orV1
V28oranyothersimilarabbreviation.Onelabelisnecessaryforeachvariableinthedataset.
Line07isaselectstatementandmustbedenotedwithSE.Itindicatesthatthevariables
listedonthenextline(s)aretheonestobeusedintheanalysis.A/indicatestheendofthe
selectedvariableslist.Theorderisparticularlyimportant.Whateverislistedfirstwillbecome
thefirstobservedvariable.Forexample,thefirstmeasuredvariableintheCFAprogram,
designatedasx1(thesmallxwithsubscripthererepresentingthefirstobservedvariable
selectedandcorrespondstotheloadingestimate x1,1),willberepresentedbytheinputted
variablelabeledJS1.SI4,the21stvariableontheSEline,willbecomethe21stmeasured
variableorx21,andtheloadingestimatesassociatedwiththisvariablewillbefoundinthe21st
rowofthefactorloadingmatrix(x21,5of xinthiscase).
Onlyinrarecircumstanceswillthevariablesbestoredintheoriginaldatafileintheexact
orderthatwouldmatchtheconfigurationcorrespondingtothetheorybeingtested.Also,theuserseldomincludesallvariablesintheCFAbecausemostdatawillalsocontainsome
classificationvariablesoridentifyingvariablesaswellaspotentialvariablesthatweremeasured
butnotincludedintheCFA.Theselectprocess,whetherthroughastatementoradropdown
menu,isthewaythevariablesinvolvedintheCFAareselected.
Line09isamodelstatementandmustbeginwithMO.Modelstatementsindicatethe
respectivenumbersofmeasuredandlatentvariablesandcanincludedescriptionsofthekey
matricesofparameters.Theabbreviationsshownherearerelativelyeasytofollow.NXstands
fornumberofxvariables,inthiscase21.NKstandsforthenumberofconstructs,inthiscase
5.PHindicatesthatthematrixofcovariancesbetweenthe5constructs( )willbesymmetric
(SY)andfree(FR).Inotherwords,theconstructvariances(thediagonalof )andthecovariancebetweeneachpairofconstructswillbeestimated.TDisthematrixoferror
variancesandcovariances.Itissetasdiagonal(DI)andfree(FR),soonlytheerrorvariancesare
estimated.AnyparametermatrixnotlistedintheMOlineissetattheprogramdefaultvalue.
Thereadercanconsulttheprogramdocumentationforotherpossibleabbreviationsand
defaults.
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Line10isavaluestatement(VA).Valuestatementsassignavaluetoafixedparameter.In
thiscase,eachoftheparameterslistedonthislineisfixedto1.0.Thisstatementsetsthescale
fortheconstructssothatoneitemisfixedto1.0oneachconstruct.LX1,1representsthe
parameterforthefirstloadingonthefirstconstruct(x1,1).TheLstandsforlambda,theXisan
xvariableand11standforthemeasuredvariablenumberandconstructnumber,respectively.
Thus,LX2,1standsfortheparameterrepresentingthefactorloadingofthesecondmeasuredvariable(x2)onthefirstlatentconstruct(1),or x2,1.Factorloadingsinareflectivefactormodel
canbeexpressedequallyascausalpaths.Usingthisterminology,LX21,5standsforthepath
fromconstruct5tox21(x21,5).
Lines11and12startwithFRanddesignatethefreeloadingestimates.The16loadings
referredtoontheselineswillbeestimatedandshownasfactorresultsintheoutput(in x).
Withthefiveestimatesfixedat1online10and16loadingsestimated,84elementsremainin
thefactorpatternfixedatzero(21variables5constructs=105potentialloadings;10516
5=84).RecallthatEFAwouldproduceanestimateforall105loadings.Thepatternoffreeand
fixedloadingsmatchesthetheoreticalstructureproposedinthemeasurementmodel.
Consistentwiththecongenericmodelproposed,onlyoneloadingestimateisfreeforeachmeasuredindicatorvariable.Inotherwords,eachmeasuredindicatorvariableloadsononly
oneconstruct.
Line13isanotherlabelline.Itiswherethelabelsforthelatentconstructscanbelisted.LK
standsforlabelsforksi().Theactuallabelsappearonthenextlineorlinesifnecessary.Inthis
case,thelabelsmatchtheconstructabbreviationsprovided(JS,OC,SI,EP,andAC).Line15,
withtheabbreviationPD,requeststhatapathdiagrambedrawnbytheprogramdepictingthe
specifiedmodelandpathestimates.TheOUline(16)isrequiredandiswhereanyoneof
numerousoptionscanberequested.Forexample,theSCisrequestingthatcompletely
standardizedestimatesbeincludedintheoutput.RSrequeststhatallmodelresidualsresulting
fromestimatingthemodelbeshown,includingboththestandardizedandnonstandardized
residuals.ND=2meansthatresultswillbeshowntotwosignificantdigits.
AttimesaresearchermaywishtoplaceadditionalconstraintsonaCFAmodel.For
instance,itissometimesusefultosettwoormoreparametersasequal.Itwouldproducea
solutionthatrequiresthevaluesfortheseparametersbethesame.Iftauequivalenceis
assumedforinstance,thisconstraintisneeded.WithLISREL,thistaskcanbedoneusingtheEQ
commandline.Similarly,researcherssometimeswishtosetaspecificparametertoaspecific
valuebyusingtheVAcommandline.Additionalinformationaboutconstraintscanbefoundin
thedocumentationfortheSEMprogramofchoice.
HBAT:THESTRUCTURALMODEL
TheHBATCFAcanbetransformedintotheHBATstructuralmodelaswasdoneearlierinthe
example. The LISREL commands for the structural model are shown below, followed by a
discussionofmaking thechanges from theCFA to the structuralmodel.Again, linenumbers
havebeenaddedtothefarlefttoaidindescribingthesyntax.
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01 TI HBAT EMPLOYEE RETENTION MODEL02 DA NI=28 NO=399 NG=1 MA=CM03 CM FI=HBAT.COV04 LA05 ID JS1 OC1 OC2 EP1 OC3 OC4 EP2 EP3 AC1 EP4 JS2 JS3 AC2 SI1 JS4
SI2 JS5 AC3 SI3 AC4 SI4
06 C1 C2 C3 AGE EXP JP07 SE08 JS1 JS2 JS3 JS4 JS5 OC1 OC2 OC3 OC4 SI1 SI2 SI3 SI4 EP1 EP2
EP3 EP4 AC1 AC2 AC3 AC4/09 MO NY=13 NE=3 NX=8 NK=2 PH=SY,FR PS=DI,FR BE=FU,FI GA=FU,FI
TD=DI,FR TE=DI,FR10 VA 1.00 LX 1 1 LX 5 2 LY 1 1 LY 6 2 LY 10 311 FR LX 2 1 LX 3 1 LX 4 1 LX 6 2 LX 7 2 LX 8 212 FR LY 2 1 LY 3 1 LY 4 1 LY 5 1 LY 7 2 LY 8 2 LY 9 2 LY 11 3 LY
12 3 LY 13 313 FR GA 1 1 GA 2 1 GA 1 2 GA 2 214 FR BE 2 1 BE 3 1 BE 3 215 LK
16 EP AC17 LE18 JS OC SI19 PD20 OU RS SC MI EF ND=2
The firstchange from theCFA setup isnoted in line09.Themodel statementmustnow
specifyanumberofvariablesandconstructsforbothexogenousandendogenousconstructs.
Thus,theMOlinespecifiesNY=13(5itemsforJS,4itemsforOC,4itemsforSI).Eventhough
these are the same items as represented by these constructs in the CFA model, they now
becomeyvariablesbecausetheyareassociatedwithanendogenousconstruct.Their loading
parametersarenowchangedtobeconsistentwiththisto y(LY).Next,theMOlinespecifiesNE
= 3, indicating three endogenous constructs. This process is repeated for the exogenousconstructs(NX=8andNK=2).PHandTDremainthesame.
Severalnewmatricesarespecified.BE=FU,FImeansthatB,whichwill listallparameters
linkingendogenousconstructswithoneanother (), isset to fulland fixed. Itmeanswewill
freetheelementscorrespondingtothefollowinghypotheses.GArepresenting ,whichwilllist
allparameters linkingexogenousconstructswithendogenousconstructs (), istreated in the
sameway.Becausewenowhaveendogenousconstructs,theerrorvariancetermsassociated
withthe13yvariablesarenowshownin ,whichisabbreviatedwithTE=DI,FR,meaningitis
adiagonalmatrixandthediagonalelementswillbeestimated.
Line10setsthescaleforfactorsjustasintheCFAmodelwiththeexceptionthatthreeofthesetvaluesare foryvariables(yvalues:LY1,1;LY6,2;LY10,3).Lines11and12specifythe
freevaluesforthemeasureditemsjustasintheCFA.Wearefollowingtheruleofthumbthat
thefreefactorloadingparametersshouldbeestimatedratherthanfixedeventhoughwehave
someideaoftheirvaluebasedontheCFAresults.Lines13and14specifythepatternoffree
structuralparameters.Line13specifiesthefreeelementsof .ThesecorrespondwithH1H4
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inFigure126(1,1islistedasGA1,1).Similarly,line14specifiesthefreeelementsofB.Lines15
and16liststhelabelsfortheconstructs(LK).Lines17and18dothesamefortheconstructs
(LE).Line19containsaPDthattellstheprogramtogenerateapathdiagramfromthe input.
Line20istheoutputlineandisthesameasintheCFAexampleexceptfortheadditionofEF,
whichwillprovideaseparatelistingofalldirectandindirecteffects.
Iftheuserisusingagraphicalinterface(e.g.,AMOSorLISREL),theuserwillneedtomake
thecorrespondingchangestothepathdiagram.Thesechangeswouldincludemakingsurethe
constructsareproperlydesignatedasexogenousorendogenousand thatobservedvariables
eachhaveacorrespondingerrorvariance term.Theneachof thecurved twoheadedarrows
thatdesignatedcovariancebetweenconstructs inCFAwillhave tobe replacedwithasingle
headedarrow to representhypothesized relationships.Arrowsbetweenconstructs forwhich
no relationship is hypothesized are unnecessary. Therefore, the twoheaded paths between
these constructs in the CFA can be deleted. Once these changes are made, the user can
reestimate themodeland the results shouldnow reflect the structuralmodel results. If the
programsyntaxhasbeenchangedasindicated,theprogramwillproducetheappropriatepath
diagramautomatically.
AvisualdiagramcorrespondingtotheSEMcanbeobtainedbyselectingStructuralModel
fromtheviewoptionsandrequestingthatthecompletelystandardizedestimatesbedisplayed
bytheSEMprogram.InLISREL,forexample,thevaluesonthepathdiagramcanberequested
sothateithertheestimatesareshownonthediagram,thetvaluesforeachestimate,orother
keyestimatesareshownincludingthemodificationindices.
HOWTOFIXFACTORLOADINGSTOASPECIFICVALUEINLISREL
IfaresearcherwishedtofixthefactorloadingsofaSEMmodeltothevaluesidentifiedinthe
CFA,proceduressuchasthosedescribedherecanbeused.Tospecifythevaluesshowninthe
pathmodelbelow,theresearcherwouldtakethefollowingstepsifusingtheLISRELsoftware.
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Thefollowingloadingestimateswouldbefixedandtheirvaluessetasfollows:
FI LX 1 1 LX 2 1 LX 3 1 LX 4 1 LX 5 2 LX 6 2 LX 7 2 LX 8 2
FI LY 1 1 LY 2 1 LY 3 1 LY 4 1 LY 5 2 LY 6 2 LY 7 2 LY 8 2VA .80 LX 1 1VA .70 LX 2 1VA .80 LX 3 1VA .75 LX 4 1VA .90 LX 5 2VA .80 LX 6 2VA .75 LX 7 2VA .70 LX 8 2VA .70 LY 1 1VA .90 LY 2 1VA .75 LY 3 1VA .75 LY 4 1VA .85 LY 5 2
VA .80 LY 6 2VA .80 LY 7 2VA .70 LY 8 2
TheerrorvariancetermsalsocanbefixedtotheirCFAestimatesasshownhere:
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FI TD 1 1 TD 2 2 TD 3 3 TD 4 4 TD 5 5 TD 6 6 TD 7 7 TD 8 8
FI TE 1 1 TE 2 2 TE 3 3 TE 4 4 TE 5 5 TE 6 6 TE 7 7 TE 8 8VA .36 TD 1 1VA .51 TD 2 2VA .36 TD 3 3VA .44 TD 4 4
VA .19 TD 5 5VA .36 TD 6 6VA .44 TD 7 7VA .36 TD 8 8VA .51 TE 1 1VA .81 TE 2 2VA .44 TE 3 3VA .44 TE 4 4VA .28 TE 5 5VA .36 TE 6 6VA .36 TE 7 7VA .51 TE 8 8
Theresearchercouldthenproceedtospecifythefreeelementsofthestructuraltheory.
MEASUREDVARIABLEANDCONSTRUCTINTERCEPTTERMS
Itoftenbecomesnecessarytousethemeasuredvariableandlatentvariablemeansindrawing
conclusionsaboutsimilaritiesanddifferencesbetweengroups.Untilnow,noSEMequationhas
shownameanvalue.Now,however,themeansmaybeconsidered.
Onewaythatwecouldthinkaboutthemeanvalueofanymeasuredvariableistothinkofit
asthesumofitszerointerceptterm,plusthefactorloading,timestheaveragevalueofthe
latentconstruct.Inequationform,itwouldlooklikethefollowingexpressedintermsofx1:
The1representsthemeanvalueforthefirstlatentconstruct1,the X1representsthe
meanofthemeasuredvariablex1,andthe X1isthezerointerceptforx1.Moregenerally,
representsthemeanforanylatentconstruct.Mathematically,itisalsothezerointerceptterm
whensolvingfor.Eventhoughthemathematicsinthiscalculationmaybedifficulttofollow,it
isimportanttoknowthatunlessspecificinstructionsareprovidedtotheSEMprogram,itwill
notconsidernorestimateconstructmeansofanytype.
Thisequationcanberearrangedtosolveforeither X1or .Ifanyhypothesesconcern
differencesbetweenconstructmeans,thosedifferencescanbefoundinthevaluesfor .
PATHMODELSPECIFICATIONWITHAMOS
ProgramstatementscanalsobewrittenforAMOSthatwouldformthemodelinthesameway
astheLISRELstatement.However,theassumptionwithAMOSisthattheuserwillworkwitha
pathdiagram.Inessence,thepathdiagramprovidestheframeworkfromwhichtobuildthe
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model.However,theusermustassignvariablestoeachrectangle,whichrepresentsa
measuredvariable,andassignconstructnamestoeachoval.Likewise,theusermustspecific
namesforeachmeasuredvariableerrorterm.Then,theappropriatearrowsmustbedrawnto
formthemodel.Theusermustbecarefulthatvariablesareassignedcorrectly.Dropdown
windowscanbeusedtoaddconstraintstothemodelandtoperformadvancedapplications
likemultiplegroupanalysis. WhileitispossibletospecifytheSEMmodelthroughcommands,AMOSisdesignedtobeusedthroughthegraphicalinterfaceandthisistherecommended
routeformostusers.
RESULTSUSINGDIFFERENTSEMPROGRAMS
AlthoughtheinputfordifferentSEMprogramsvaries,theresultsshouldbeessentiallythe
same.Thealgorithmsmayvaryslightly,butamodelthatdisplaysgoodfitusingoneSEM
programalsoshoulddisplaygoodfitinanother.Eachhasitsownidiosyncrasiesthatmay
preventthesamemodelspecificationfrombeingestimated.Forinstance,somemakeitmore
orlessdifficulttouseeachofthemissingvariableoptionsjustmentioned.Eachapproachcan
beeasilyspecifiedwithLISREL,butAMOSusesEMalone.Listwisedeletion,forexample,canbe
performedwithAMOSbyscreeningobservationswithmissingdatapriortobeginningthe
AMOSroutine(e.g.,withSPSS).
Theoverallmodelfitstatistics,includingthe2andallfitindices,shouldnotvaryinany
consequentialwaybetweentheprograms.Similarly,theparameterestimatesshouldalsonot
varyinanyconsequentialway.Differencescanbeexpectedintwoareas.
Oneareawheredifferencesinthenumericalestimatesmayvaryisintheresiduals.In
particular,somedifferencesmaybefoundbetweenAMOSandtheotherprograms.Without
gettingintothedetails,AMOSusesadifferentmethodforscalingtheerrortermsofmeasured
variablesthandotheotherprograms.Thisformathastodowithsettingthescalefortheerror
terms,muchaswesetthescaleforthelatentconstructsinaSEMmodel.Thismethodmay
causerelativelysmalldifferencesinthevaluesforresidualsandstandardizedresiduals
computedwithAMOS.However,thedifferencesdonotaffecttherulesofthumbgiveninthe
text.
Anotherareawherenumericalestimatesmayvaryisinthemodificationindices.Again,
AMOStakesadifferentcomputationalapproachthandosomeoftheotherSEMprograms.The
differenceliesinwhetherthechangeinfitisisolatedinoneorseveralparameters.Onceagain,
althoughtheusercomparingresultsbetweenAMOSandotherprogramsmayfindsome
differencesinMI,thedifferencesshouldnotbesolargeastoaffecttheconclusionsinmost
situations.So,onceagain,therulesofthumbfortheMIholdusinganySEMprogram.
ADDITIONALSEMANALYSES
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TESTINGFORDIFFERENCESINCONSTRUCTMEANS
Afinaltypeofmultigroupcomparisonisthetestfordifferencesinconstructmeans.Ifatleast
partialscalarinvarianceispresent,wecanoperationalizeavalueforthemeansofthelatent
constructs.Inthisway,wwtelltheSEMprogramthatweareinterestedinanalyzingmeans.An
earlierdiscussionshowedtheequationtorepresentlatentconstructmeans.Inonewayor
anotherthough,theSEMprogrammustbetoldweareinterestedinthemeansofthelatent
constructs.
SEMprogramscomparemeansonlyinarelativesense.Inotherwords,theycantellyou
whetherthemeanishigherorlowerrelativetoanothergroup.Onereasonforthislimitation
hastodowithidentificationgiventhattheintercepttermsarenowbeingestimated.Aresultis
thatthevectoroflatentconstructmeans(containedinthekappamatrix)hastobefixedtozero
inonegrouptoidentifythemodel.Werefertothisgroupasgroup1.Itcanbefreelyestimated
intheothergroup(s)andtheresultingvaluescanbeinterpretedashowmuchhigherorlower
thelatentconstructmeansareinthisgrouprelativetogroup1.
Assumewehaveatwogroupmodelwiththreeconstructsineachgroup. TheSEM
outputwillnowincludeestimatesforthevectoringroup2(i.e.,thecomparisonofgroup2
relativetogroup1).Typically,thisoutputwouldincludeanestimatedvalue,astandarderror,
andatvalueassociatedwitheachvalue.Forinstance,itmaylooklikethis:
KAPPA()
Construct1Construct2 Construct3
2.60.09 3.50
(0.45) (0.60) (1.55)
5.780.10 2.25
Thesevaluessuggestthatthemeanfortheconstruct1is2.6greateringroup2thaningroup1.
Thisdifferenceissignificantasevidencedbythetvalueof5.78(p
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scales may contain more than 100 items to capture only two or three basic personality
dimensions. Thus, evenwith a few constructs one could end upwith farmore than 100
measureditems.SEMapplicationsaredifficulttomanagewithsomanymeasuredvariables.
Using itemparcelingasingle latentconstructwith40measured items(x1x40)could
berepresentedbyeightparcels,eachconsistingof5ofthe40measureditems.Aparcelisamathematical combination summarizing multiple variables into one. In the extreme, all
measureditemsforaconstructcanbecombinedintooneaverageorsumofthosevariables.
InChapter3,wediscussedhow tocreatea summatedconstruct in this fashion.The term
compositeindicatorisgenerallyusedtorefertoparcelingresultinginonlyoneparcelfromallthemeasureditemsforaconstruct.
Numerous issues are associated with item parceling. These issues include the
appropriatenessofparceling,which itemsshouldbecombined intoaparcel,andwhat the
effectsofparcelingareonevaluatingmodels.Parcelinghasthepotentialtoimprovemodel
fitsimplybecause itreducesthecomplexityofthemodel,andmodelswithfewervariables
have thepotential forbetter fit.Better fitalone,however, isnota sufficient rationale for
combiningmultiple items intoonebecause theprimarygoal is creatingamodel thatbest
represents the actual data. Further, item parcels can often mask problems with item
measuresandsuggestabetterfitthanactuallyexistsinreality.Parcelingalsocanhideother
latent constructs thatexist in thedata.So,a covariancematrix thatactually contains five
latentconstructsmaybeadequatelybutfalselyrepresentedbythreelatentconstructsusing
parceling.
WhenIsParcelingAppropriate?
Itemparcelingshouldonlybeconsideredwhenaconstructhasalargenumberofmeasured
variable indicators.For instance,applications involving fewer than15 itemsdonotcall for
parceling.Similarly,parcelingisnotusedwithformativemodelsbecauseitisimportantthat
allcausesofaformativefactorbeincluded.Parcelingisappropriatewhenalltheitemsfora
construct areunidimensional. That is, evenwitha largenumberofmeasured items, they
shouldallloadhighlyononlyoneconstructanditshoulddisplayhighreliability(.9orbetter).
Most importantly, parceling is appropriate when information is not lost by using parcels
instead of individual items. Thus, some simple checks prior to parceling would involve
runningaCFAontheindividualfactortocheckforunidimensionalityandtoseewhetherthe
construct reflectedbyall individual items relates tootherconstructs inthesamewayasa
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constructreflectedbyasmallernumberofparcels.
HowShould
Items
Be
Combined
into
Parcels?
Traditionally,littlethoughtwasgiventohowitemsshouldbecombined.However,the
combinationstrategycanaffectthelikelihoodthataCFAisactuallysupportingafalse
measurementtheory.Althoughmanyintricaciesareassociatedwiththecombination
strategies,twosimpleconsiderationsleadtothebestperformancewhenaresearchermustuse
itemparcels.Oneconsiderationisempiricalandtheotheristheoretical.Giventhatthe
individualitemssuggestunidimensionality,thebestparcelsareformedbyitemsthatdisplay
approximatelythesamecovariance,whichshouldleadthemtohaveapproximatelythesame
factorloadingestimates.Further,theparcelsshouldcontaingroupsofitemswiththemost
conceptualsimilarity.Thatis,itemswiththeclosestcontentvalidity.Thus,parcelswithitems
showingapproximatelythesameamountofcovarianceandthatshareaconceptualbasiswill
tendtoperformwellandrepresentthedatamostaccurately.
MEASUREMENTBIAS
Researchers sometimesbecome concerned that survey responses arebiased based on the
way the questions are asked. For instance, it could be argued that the order in which
questionsareaskedcouldbe responsible for thecovarianceamong items thataregrouped
closetogether.Ifso,anuisancefactorbasedonthephysicalproximityofscaleitemsmaybeexplainingsomeoftheinteritemcovariance.
Similarly,researchersoftenarefacedwithresolvingthequestionofconstantmethodsbias.
Constantmethodsbiaswouldimplythatthecovarianceamongmeasureditemsisdrivenbythe
factthatsomeoralloftheresponsesarecollectedwiththesametypeofscale.Aquestionnaire
usingonlysemanticdifferentialscales,forinstance,maybebiasedbecausetheopposingterms
response form becomes responsible for covariance among the items. Thus, the covariance
couldbeexplainedbythewayrespondentsuseacertainscaletypeinadditiontoorinsteadof
thecontentofthescaleitems.Here,asimpleillustrationisprovidedusingtheHBATexample.It
showshowaCFAmodel canbeused toexamine thepossibilityofmeasurementbias in the
formofanuisancefactor.
The HBAT employee questionnaire consists of several different types of rating scales.
Although it could be argued that respondents prefer a single format on any questionnaire,
severaladvantagescomewithusingasmallnumberofdifferentformats.Oneadvantageisthat
theextenttowhichanyparticularscaletypeisbiasingtheresultscanbeassessedusingCFA.
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In this case, HBAT is concerned that the semantic differential items are causing
measurementbias.Theanalystarguesthatrespondentshaveconsistentpatternsofresponses
tosemanticdifferentialscalesnomatterwhatthesubjectoftheitemis.Therefore,asemantic
differentialfactormayhelpexplainresults.ACFAmodelcanbeusedtotestthisproposition.
Onewaytodoso istocreateanadditionalconstructthat isalsohypothesizedascausingthe
semanticdifferential items. In thiscase, itemsEP4, JS2, JS3,AC2,andSI4aremeasuredwithsemantic differential scales. Thus, the model needs to estimate paths between this new
constructandthesemeasureditems.Theadditionofanuisancefactorofthistypeviolatesthe
principlesofgoodmeasurementandsothenewmodelwillnothavecongenericmeasurement
properties.
We will modify the original HBAT CFA model shown in the text. A sixth construct is
introduced(6).Next,dependencepaths(causal inthiscase)wouldbeestimated(drawn if
using apathdiagram) from6 to EP4, JS2, JS3,AC2,and SI4. Thus, the factorpatternno
longer exhibits simple structure because each of these measured variables is now
determinedbothbyitsconceptualfactorandbythenewconstruct6.
Theanalystteststhismodelandobservesthefollowingfitstatistics.The2=232.6with
174degreesof freedomandtheRMSEA,PNFI,andCFIare .028, .80,and .99,respectively.
The added pathshave not provided a pooroverall fit although theRMSEA has increased
slightlyandthePNFIhasdecreased.However, the 2=4.0 (236.6232.6)with5 (179
174)degreesoffreedom,is insignificant. Inaddition,noneoftheestimatesassociatedwith
thebiasfactor(6)aresignificant.Thecompletelystandardizedestimatesof lambda(factor
loadings)andassociatedtvaluesareshownhere:
ParameterEstimate tValue
x2,6 0.14 1.19
x3,60.01 0.08
x17,60.16 1.32
x19,60.07 0.84
x21,60.20 1.48
Also,thevaluesfortheoriginalparameterestimatesremainvirtuallyunchangedas
well.Thus,basedonthemodelfitcomparisons,theinsignificantparameterestimates,and
theparameterstability,noevidencesupportsthepropositionthatresponsestosemanticdifferentialitemsarebiasingresults.TheHBATanalystconcludes,therefore,thatthiscaseis
notsubjecttomeasurementbias.Anotherfactorcouldbeaddedtoactasapotential
nuisancecausefortheitemsrepresentinganotherscaletype,suchasallLikertitems.The
testwouldproceedinmuchthesameway.Theendresultofallofthesetestsisthatthe
researchercanproceedtotestmorespecifichypothesesaboutemployeeretentionand
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relatedconstructs.
CONTINUOUSVARIABLEINTERACTIONS
Anapproachforhandlingacontinuousmoderatorwhichdoesnotinvolvecreatinggroupsfrom
thecontinuousmoderatoristocreateaninteractionbetweenthemoderatorandthepredictor.
Singlevariable interactions were treated in the text, so we focus here on a moderating
construct that would be measured by multiple indicators. Consider a SEM model with two
exogenousconstructspredictingasingleendogenousconstruct.Eachconstructisindicatedby
fourmeasured items. Ifthefirstconstruct (1) ishypothesizedasthepredictorconstructand
thesecondconstruct(2)ishypothesizedasamoderator,thenaninteractionconstructcanbe
createdtorepresentthemoderatingeffectbymultiplyingthe indicatorsofthepredictorand
moderator constructs together. Using this rationale, the indicators for the third interaction
construct(3)canbecomputedasfollows:
x9 = x1 x5x10
= x2 x6
x11= x
3 x
7
x12= x4 x8
Thesecomputedvariablescanthenbeaddedtotheactualdatacontaining12measured
variablesandthecovariancetermsbetweenthesecomputedvariablesandtheotherscanbe
calculated.Now,thecovariancematrixforthismodelwouldchangefrom1212to1616.
Thiscanbeshowninapathmodelformbythefollowingspecification.
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Estimatingthismodeliscomplicatedbyseveralfactors.Thesefactorsincludethefactthat
theassumptionthaterrortermsareuncorrelatedisnolongerfeasiblebecausetheloadingsfor
thethirdconstruct(3)areamathematicalfunctionofthoseforconstructs1and2(1and2).
ThisfactleadstoaquitecomplexSEMmodelsetupthatisrecommendedforadvancedusers
only.Soitisonlybrieflydescribedhere.Thissetuprequiresthattheintercepttermsforthe
measureditems(x)beestimatedasdescribedearlier.Theexogenousfactorloadingpattern
cannolongerexhibitsimplestructure.Eventhoughtheloadingestimatesforthethird
constructcanbecomputedbymultiplyingtheloadingestimatescorrespondingtothevariables
thatcreatedeachinteractionindicator,crossconstructloadingsalsoexistfortheinteractionterm.Theyarecomputedbycrossingthematchingxtermswithloadingestimates.Again,this
processisrathercomplextofollow,butasanexample,the12throwofxwouldendupas:
44 88 48
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Inaddition,theerrorvariancecovariancematrixforthexvariables()mustnowinclude
termsfortheappropriateerrorcovarianceitemsthatexistduetothecomputationalnatureof
theinterceptconstruct.Theseitemsneednotbeestimatedbecausetheyaredetermined
mathematicallyastheintercepttermforthemeasureditemusedtocomputetheinteraction
indicatortimestheerrorvarianceforaconstruct.Thisconceptismoreeasilyillustratedbyan
example.9istheerrorvariancetermforx9,thefirstindicatorforthemoderatorconstruct.Becauseitiscomputedasx1timesx5,anerrorcovariancetermisneededforboth 9,1and
9,5.Thevalueswouldbesetas 1times 1,1and 5times 5,5,respectively.
Afterfinishingasetupfollowingalongtheselines,themodelcanbeestimatedspecifying
onlythestructuralpathbetweentheinteractionconstructandtheoutcome.Ifmoderationis
supported,thecorrespondingestimate,3,1inthiscase,wouldbesignificant.Realizethatthe
effectsof1and2on1areofquestionablevalidityinthepresenceofasignificantinteraction.
Therefore,theyshouldbeestimatedandinterpretedonlyifthestructuralinteractionterm(3,1)
isinsignificant.
Interactiontermssometimescauseproblemswithmodelconvergenceanddistortionofthe
standarderrors.Therefore,largersamplesareoftenrequiredtominimizethedistortion.Anabsoluteminimumsamplesizewouldbe300forthistypeofanalysiswithasamplesizeofmore
than500recommended.