sem12ersh8750_lecture12

7/29/2019 sem12ersh8750_lecture12

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StructuralEquationModels:

PathAnalysiswithLatentVariables

IntroductiontoStructuralEquationModeling

Lecture#12(andlast)

April25,2012


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TodaysClass

Puttingitalltogether:

PathAnalysis

Observedvariables

ConfirmatoryFactorAnalysis/MeasurementModels

Latentvariables

Concernsinbuildingstructuralequationmodels Modelpredictedcovariancematricesforpathanalysiswith

observedandlatentvariables

ExamplesofSEMuses


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UNDERLYINGTHEORYOFSTRUCTURALEQUATIONMODELS


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StructuralEquationModels

AlthoughthetermSEMcanbeappliedtomanysettings,I

viewthelabelasbeingusedtodescribeanalyseswith

observedandlatentvariables

Astructuralequationmodelconsistsoftwoparts:

Measurementmodel(s)foreachlatentvariable Pathanalysisbetweenthelatentandobservedvariables

Uptothispoint,wehavecoveredbothinisolation todayweputthemtogethertoshowhowtheprocessworks

Youwillseethisextrastepisprettystraightforward


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REVIEWOFPATHANALYSIS


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TypesofVariablesintheAnalysis

AnimportantdistinctioninpathanalysisandSEMis

betweenendogenousandexogenousvariables

Endogenousvariable(s): variableswhosevariabilityisexplainedbyoneormorevariablesinamodel

Inlinearregression,thedependentvariableistheonlyendogenousvariableinananalysis

Exogenousvariable(s):variableswhosevariabilityisnotexplainedbyanyvariablesinamodel

Inlinearregression,theindependentvariable(s)arethe

exogenousvariablesintheanalysis

ERSH8750:Lecture#4 6


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DirectandIndirectEffectsofHSLonMSE


Mathematics

SelfEfficacy

CollegeMath

Experience

HighSchool

Math

Experience

DirectEffect

Residual(Endogenous)

Variance

ExogenousVariances

ExogenousCovariances

4.1590.463

0.696

0.258

0.363

0.105

33.7972.712

103.1288.508

1.7430.140

NotShownOnPathDiagram:

6.9041.304

49.3132.412

4.9120.074

IndirectEffect


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PathAnalysisinMatrixForm

Ourpathmodelsimultaneousequationswere:

2 endogenousvariables 1 exogenousvariable

Alternatively,wecouldrephrasethisinmatrixform:

Where:

(matrixofsize1 containingobservedexogenousvariables)

(matrixofsize1 containingobservedendogenousvariables)

Then:

(matrixofsize1 containinginterceptsforendogenousvariables)

0 0

0

(amatrixofcoefficientsrelatingtheendogenousvariablestothemselves)

(matrixofsize relatingexogenousvariablestoendogenousvariable(s))

, (where isthe residualcovariancematrix)

Here, willbediagonal(nocovariance)aswedonothaveanymoredegreesoffreedom



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PathAnalysisinMatrixForm

Theequationsfromthepreviousslidearecalledthestructuralform ofthepathmodel

Anotherformthatexistsinliteratureisthereducedform,whereallendogenousvariablesareonthelefthandside

Where

Thereducedformisnotasfrequentlyusedinpractice,butdoesariseinsomeresearchareasandinidentification



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PathAnalysiswithMatrices

Althoughnotexplainedbyourmodel,wecouldstatethat

themeanvectorofexogenousvariableswas:

Likewise,wecanstatethatthecovariancematrixofthe

exogenousvariablesis

Wewillusethesetermsinourmatrixversionofthemodelpredictedmeanandcovariancematrix



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ModelPredictedMeanVectorand

CovarianceMatrix

Theunconditionalmeanoftheendogenousvariablesis:

Thecovariancematrixoftheexogenousandendogenousvariablesisthen:

,

Thepoint:thatmodelspecificationshavedirectimplicationsfortheparametersofthemultivariatenormaldistribution



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MatchingMatriceswithResults

Tomorespecificallylinkourresultstothematricesfrom

thepreviouspage:


Name Matrix ModelEstimates

ResidualCovariance Matrix 33.797 0

0 103.128

Regression WeightsofExogenousonto

Endogenous

0.696

4.159

CovarianceMatrixofExogenous Variables 1.743

MeanVectorof ExogenousVariables 4.912

VectorofEndogenous VariableIntercepts 6.904

49.313

MatrixofEndogenousRegression Weights 0 0

0.363 0

Inverse matrixusedincalculations 1 0

0.363 1


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CovarianceMatrix

Theestimatedconditionalmeanoftheendogenousvariablesis:

Thesevaluescorrespondexactly(saturatedmodel)

Theestimatedcovariancematrixoftheexogenousand

endogenousvariablesis:

Thesearemostlyexact smalldifferences



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REVIEWOFCONFIRMATORY

FACTORANALYSIS


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OneFactorModelofFiveGRIItems

TheCFAmodelforthefiveGRIitems:

Here: responseofsubject onitem interceptofitem (listedasameanasthisistypicallywhatitbecomes) factorloadingofitem onfactor1(onlyonefactortoday) latentfactorscoreforsubject (sameforallitems)tofactor1(onlyonetoday) regressionlikeresidualforsubject onitem

Weassume 0, ;

iscalledtheuniquevariance ofitem Wealsoassume and areindependent

Also,wewillassume ,

Typically 0 (butnotalways) Factorvariancecanbeestimatedorfixed(moreonbothinidentification)



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OurCFAModelPathDiagram

MeasurementModel:

s=factorloadings

es=errorvariances

s=itemintercepts

StructuralModel:

=factorvariance

F1 =factormean

(Someofthesevalueswillhavetobe

restrictedforthemodeltobeidentified)

ERSH8750:Lecture#5

e1 e2 e3 e4 e5

Y4 Y5Y3Y2Y1

F1

1121 31 41

51

1

2

3

4 5

F1

1

16


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ModelPredictedMeanVector

Combiningacrossallitems,themeanvectorfortheitemsisgivenby:



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ModelImpliedCovarianceMatrix

Combiningacrossallitems,thecovariancematrixfortheitemsisgivenby:

Getusedtoseeingthis althoughyoualreadyhave(seetheregression

slides)

, , , ,

, , , ,

, , , ,

, , , ,

, , , ,

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0



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PUTTINGITTOGETHER:PATH

ANALYSISWITHLATENTVARIABLES


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ASmallSEMExample

TodemonstratehowSEMworks,wewilluseaverysmall

example:

Measurementmodel:threeGRIitemsformingonelatent

construct(gambling)

Note:withthreeitems,themeasurementmodelisjustidentified

(meaningperfectfit)

Pathmodel:ThepredictionofgamblingbytheSOGscore

Note:herewetreatSOGSscoreasbeingobservedwithouterror

GRI1

GRI3

GRI5

Gambling SOGS


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Step#1:BuildingtheMeasurementModel

Thefirststepinastructuralequationmodelistobuildthe

measurementmodel

Here,themeasurementmodelissimplifiedsoastoshowhow

SEMworks


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MeasurementModelFitAssessment

Ourthreeitemmeasurementmodelfitsperfectly


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MeasurementModelParameterEstimates


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MeasurementModel:

ImpliedCovarianceMatrix

Themeasurementmodelimpliedcovariancematrixis:

1.000

0.7260.996

0.407 1.000 0.726 0.996

0.648 0 0

0 0.535 0

0 0 0.546

1.055 0.295 0.405

0.295 0.749 0.294

0.405 0.297 0.950


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Step2:Estimatingthe

StructuralEquationModel

Oncethemeasurementmodelisfoundtofit,thenext

stepistoestimatethefullstructuralequationmodel

SOGSsum istreatedasanexogenousvariable

Alsocalledanindependentvariable

GAMBLING(andtheitemsmeasuringit)aretreatedasendogenousvariables

Alsocalleddependentvariables


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SEM:ModelIdentification

AsSEMintegratesbothmeasurementandpathmodels,theidentificationrulesforSEMborrowfromboth

Themeasurementmodel(foralllatentvariables)mustbelocallyidentified Includingrulesforsettingscaleoflatentfactor(s)

Thepathmodelmustbeidentified

Anecessarybutnotsufficientwayofensuringidentificationisthetrule(countingrule)

Thenumberofparametersmustbelessthanthetotalnumberofmeans+

variances/covariancesofall observedvariablesintheanalysis

Numberofobservedvariablesinouranalysis:4 Numberofvariances/covariances:4*(4+1)/2=10 Numberofmeans:4 Total:14

Numberofparametersinouranalysis 2factorloadings+1factorvariance+3uniquevariances+1directeffect+

3itemintercepts+1exogenousvariance=12


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SEM:MplusSyntax

TheMplussyntaxisacombinationofpathand

measurementmodels


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SEM:ModelFitAssessment

Becausewehavefewerparametersthanthetotal

possible,wemustnowassessourmodelfit


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SEM:ModelParameterOutput

Note:ourmeasurementmodelparametershavechanged

slightly(moreonwhyinamoment)


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SEM:StandardizedModelParameters

Here,weseethattheSOGShasacorrelationof.577with

theGAMBLINGlatentvariable


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SEM:ModelImpliedCovarianceMatrices

Noticeourmodelimpliedcovariancematrix:

Andtheresidualsfromthesaturatedmodel:

NEWWRINKLES: MEASUREMENTMODELDOESNOTFITSATURATEDMODEL

PERFECTLY OFFDIAGONALOFBLOCKCANCAUSEMODELMISFIT

E ti F f


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EquationFormof

OverallStructuralEquationModel

Ourstructuralequationmodelsimultaneousequations

were,forthemeasurementportion:

And,forthestructuralportion:

3 endogenousvariables

1 exogenousvariable


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MatrixFormofStructuralEquationModel

Inmatrices,themeasurementportionofthemodelisgivenby:

with

Further,thestructuralportionofthemodelisgivenby:

with

Intermsofourmodel: Therearenodirectexogenouspredictorsofourendogenous

measurementmodelparameters(so )

Therearenodirectpredictorsofourendogenouslatentvariablesbyotherlatentvariables(so )

Westandardizedourfactormeantozero(so

Putting Values into Matrices:


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PuttingValuesintoMatrices:

MeasurementModel

itemintercepts

factorloadingsforendogenousvariables

uniquevariancesof

endogenousvariables

Putting Values into Matrices:


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PuttingValuesintoMatrices:

StructuralModel

directregressioncoefficientofexogenous

variableontoendogenousfactor

residualvarianceofendogenousfactor

variance/covariancematrixforthe

exogenousvariables

Model Predicted Mean Vector and


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CovarianceMatrix

Thecovariancematrixoftheexogenousandendogenous

variablesisthen:

,

Thepoint:thestructuralequationmodelcanhave

significantmodelmisfitduetoboththemeasurementmodelandthestructuralmodel



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IssuesinBuildingStructuralEquationModels

BecauseofthemultiplewaysSEMscanexhibitmodelmisfit,theprocessofbuildingSEMscanbedifficult

Ingeneral,currentpracticestatesthatmeasurementmodelsshouldbebuiltfirst thenthefullSEM

Someresearchersofferquestionableadvice: Useonlyjustidentifiedmeasurementmodels

Why:fewerdegreesoffreedomwheremisfitcanhappen Badidea:poorreliabilityforlatentconstructs

BuildmeasurementmodelswithSEMssimultaneously Why:fullcalibrationcanleadtobetteroverallmodelfit Badidea:measurementshouldhappeninabsenceofexogenousvariables

UsetwostageanalysesforSEMs Why:measurementmodelthencannotchange Badidea:propagationofmeasurementerrorforsomefactorscoremethods


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SEMINPRACTICE:EXAMPLES

FROMREALWORLDANALYSES


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SEMinPractice

Todemonstratethepracticalsideofbuildingstructural

equationmodels,Iwillgooveracoupleexamplesfrom

realdataanalyses

Intheseexamples,themodelbuildingprocesswillbe

discussed,alongwithvaryingmethodsforanalysis

Thedatafortheseexamplesisnotavailable butthe

practiceshouldshowhowdecisionsaremadeabouthowSEMsareconstructedandinterpreted


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Example#1:EvaluationofAcademicProgress

Thisexamplecomesfromdatafromalargesoutheasternuniversity

Datainclude: PRE:scoresonapretestofmathematicsability,administered

tostudentswhentheyarriveattheuniversity Scoresarefromtotalnumbercorrect alphareliabilityof.81

POST:scoresonaposttestofmathematicsability(usingthesameitems),administeredtostudentsaftertwoyearsattheuniversity

Scoresarefromtotalnumbercorrect alphareliabilityof.81

CourseEnrollments: Ifastudenthadenrolledinoneof29coursesrelatedtomathand

scienceeducationattheuniversity Dataarebinary 0=didnotenroll;1=enrolled

l h i


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Example#1:ResearchQuestions

Theevaluationsoughttoanswerthefollowingquestions:

Didscoresimproveontheposttestwhencomparedwiththe

pretest?

Didcourseworksignificantlyaffecttheposttestscores?

Didthescoreonthepretestpredictthecourseworkstudents

took?

Didcourseworkmediatetherelationshipbetweenpretestand

posttest?

B ildi h SEM M d li I


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BuildingtheSEM:ModelingIssues

Becauseofthenatureofthedata,severalmodelingissuesmustbeconsideredwhenusingSEMtoanswertheresearchquestions

Becausepretestandposttestaresumscores(withaknown

reliability),eachcanbeusedasasingleindicator Inthiscase,theposttestsingleindicatorwillbeproblematicbecauseofthe

residualvariance(afterprediction)islessthantheoverallvariance Somustputsingleindicatormodelinlast

Eachofthecoursesisbinary(dichotomous),soincludingtheminthemodeldirectlyisnotanoption

Modelwouldtreatthemasnormallydistributedifnotcategorical Softwarewontallowcategoricalmediators

Couldusethemas: Countsforspecificcategories(thentreatcountasapproximatelynormal)

Whatwedid

Indicatorsofacourseworkfactor Hardtoenvision

M d li St t


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ModelingStrategy

Courses:

Createcountsofeachcoursecategory(3categoriestotal)

Treatcountsasapproximatelynormal(anduseMLR)

Useallvariablesinapathmodelwhere:

Pretestpredictscoursecountsandposttestscore Coursecountspredictposttestscore

Treatpretestandposttestassingleindicatorswherevariance

ofeachisweightedbythe.81reliabilityofeach Finalstepintheanalysis

I iti l S t F D i ti St ti ti


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InitialSyntax:ForDescriptiveStatistics

Initial Output Descriptive Statistics


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InitialOutput:DescriptiveStatistics

Model#1:PathModelw/o


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PosttestSingleIndicator

TheMplussyntax:

Modelfit:

Model #1: Relevant Output


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Model#1:RelevantOutput

Forbuildingasingleindicatoroutofposttest:

Model #2: Pre/Post Single Indicators


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Model#2:Pre/PostSingleIndicators

MplusSyntax:

Model #2: Model Fit Assessment


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Model#2:ModelFitAssessment

MplusOutput:

Normalizedresiduals:

Needforresidualcovariancesbetweencourseworksums

Model#3:SingleIndicatorswith


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ResidualCovariances

Mplussyntax:

Note:thismodelhasnodegreesoffreedomleft itis

justidentified Thereforemodelfitisperfect

Model #3: Results


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Model#3:Results

Model #3 Results


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Model#3Results

Model #3 Results


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Model#3Results

Model #3 Results


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Model#3Results

Model #3 Results


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Model#3Results

Example#1:ResearchQuestionsAnswered


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p Q

Theevaluationsoughttoanswerthefollowingquestions:

Didscoresimproveontheposttestwhencomparedwiththepretest?

Yes,posttestscoresimprovedby.654SDforeveryoneSDincreaseinthepretestscore(p


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p j

Thesecondexamplecomesfromarecentdissertation

titled:

RISKSANDBENEFITSOFSEEKINGANDRECEIVING

EMOTIONALSUPPORTDURINGTHEDIVORCEPROCESS:AN

EXAMINATIONOFDIVORCEEINDIVIDUALADJUSTMENT,

CLOSENESS,ANDRELATIONALSATISFACTIONWITH

MULTIPLEPARTNERSFROMASOCIALNETWORK

Thedissertationusedastructuralequationmodelto

investigatethepsychometricpropertiesofascaleand

theninvestigatedtheroleofthefactorsinpredicting

outcomesrelevanttotheresearch

Example#2:ResearchQuestions


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p Q

Thefollowinghypotheseswereexamined:H1:Initiatorstatusoffilingforthelegaldivorceisrelatedtohigheradjustmenttodivorce.

H2:Femaleshavehigheradjustmenttodivorce.

H3:Havingchildrenwiththeexspouseisrelatedtolowerlevels

ofadjustment.

H4:Thereisapositiverelationshipbetweenlengthoftimesincethedivorceandadjustmenttodivorce.

H5:Thereisanegativerelationshipbetweenageandadjustmenttodivorce.


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AnalysisofRiskinSeekingSupportScale


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y g pp

RiskinSeekingSupportScale:

6itemsinitially

Target exspouse:RMSEA=.12;CFI=.85

Target familymember:RMSEA=.06;CFI=.96 Target friend:RMSEA=.09;CFI=.91

Oneitemremoved: Target exspouse:RMSEA=.13;CFI=.87;Alpha=.69

Target familymember:RMSEA=.00;CFI=1.00;Alpha=.81

Target friend:RMSEA=.08;CFI=.95;Alpha=.75

AnalysisofEmotionalSupportProvision

Effectiveness Scale


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EffectivenessScale

EmotionalSupportProvisionEffectivenessScale

13items



Removed3reversecodeditems: 10items

Target exspouse:RMSEA=.09;CFI=.92;Alpha=.95

Target familymember:RMSEA=.11;CFI=.91;Alpha=.95


AnalysisofClosenessScale


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ClosenessScale:

18items



Removed7reversecodeditems 11items


Target familymember:RMSEA=.06;CFI=.95;Alpha=.90


AnalysisofRelationalSatisfactionScale


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Relationalsatisfactionscale:

7items


Target familymember:RMSEA=.05;CFI=.99;Alpha=.92 Target friend:RMSEA=.05;CFI=.98;Alpha=.88

InvarianceTesting


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Becausemanyofthescalesaskedaboutspecifictargets,

thesewereusedasmultiplegroupstoinspectfactorial

invariance

Modelspreviouslyreportedrepresentedtheconfigural models

RiskinSeekingSupportScale:

Metricinvarianceheld Partialscalarinvariance(3itemswithdifferentintercepts)

Partialresidualinvariance(2itemswithdifferentunique

variances)

InvarianceTesting


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Emotionalsupportprovisioneffectivenessscale Partialmetricinvariance(5itemsnotinvariant)

Partialscalarinvariance(5itemsnotinvariant)

Closenessscale

Partialmetricinvariance(1itemnotinvariant)

Partialscalarinvariance(5itemsnotinvariant)

RelationalSatisfactionScale

Partialmetricinvariance(3itemsnotinvariant) Partialscalarinvariance(3itemsnotinvariant)

SubsequentPathAnalysis


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FinalPathAnalysis


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CONCLUDINGREMARKS

WrappingUp


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Todaywasaboutputtingitalltogether:pathanalysisand

measurementmodels

TheSEMframeworkallowsforpowerfulinferential

analysestobeconductedinastatisticallyrigorousmanner

Butwiththepowercomesalotoffrustration datadonot

alwayscooperate

Youwillfindthatpeopletakegreatlibertieswithhowthey

conductSEManalyses Ihopethisclasshelpsyouunderstandhowpeopledo

SEMandhowtodoSEMyourself

UpNext


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Todayisourlastclassmeeting Nofinal;Nofinalhomework

Extracreditopportunity: Filloutcoursesurveyonline

https://portal.coe.uga.edu/apps/authorize/

EnsureyougetanAinthecourseItakethecomments

veryseriously!

sem12ersh8750_lecture12

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