SEM_Basics a Supplement to Multivariate Data Analysis

7/31/2019 SEM_Basics a Supplement to Multivariate Data Analysis

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MultivariateDataAnalysis,PearsonPrenticeHallPublishing Page1

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

C3.

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

Direct(Causal) Indirect(Causal) Indirect(Noncausal)

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C1C3 P3,1 P3,2xP2,1 None

C2C3 P3,2 None P3,1 xP2,1

C1C4 P4,1 P4,2xP2,1

P4,3xP3,1

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None

C3C4 P4,3 None P4,1xP3,1

P4,1xP2,1xP3,2P4,2xP3,2

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


13/35


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

ExogenousIndicator ExogenousConstruct ErrorX1

= x111 +

x122 +

x133 + 1

X2=

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 +


17/35


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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22/35


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.


24/35


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


25/35


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.


26/35


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


28/35


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.

Documents

SEM_Basics a Supplement to Multivariate Data Analysis