Structural equation modeling BY Abdul Rahim Chandio

Structural equation modeling is a multivariate statistical analysis
technique that is used to analyze structural relationships and this
technique is applied for the combination of factor analysis and
multiple regression analysis, and it is used to analyze
the structural relationship between Reflective measurement model happens when the indicators of a
construct are considered to be caused by that construct.
Whereas a formative measurement is measured variables which
are considered to be the cause of the latent variable and in a
formative construct, the indicators cause the construct, whereas in
a more conventional latent variables, sometimes called reflective
constructs, the indicators are caused by the latent variable.
Path diagram is considered for showing which variables cause
changes in other variables. They may also be given a narrower,
more specific interpretation.

2. Presentation Topic: Structural equation
modeling
BY
Abdul Rahim Chandio
Ahmad Nawaz
AsgharHaqvi
Under Supervision of
Professor Dr. Sharif Abbasi
Department of Public Administration
University of Sindh,
Jamshoro

4. Introduction
•Structuralequationmodelingisamultivariatestatisticalanalysis
techniquethatisusedtoanalyzestructuralrelationshipsandthis
techniqueisappliedforthecombinationoffactoranalysisand
multipleregressionanalysis,anditisusedtoanalyze
thestructuralrelationshipbetween
•Reflectivemeasurementmodelhappenswhentheindicatorsofa
constructareconsideredtobecausedbythatconstruct.
•Whereasaformativemeasurementismeasuredvariableswhich
areconsideredtobethecauseofthelatentvariableandina
formativeconstruct,theindicatorscausetheconstruct,whereasin
amoreconventionallatentvariables,sometimescalledreflective
constructs,theindicatorsarecausedbythelatentvariable.
•Pathdiagramisconsideredforshowingwhichvariablescause
changesinothervariables.Theymayalsobegivenanarrower,
morespecificinterpretation.

5. •compositereliabilityisalsocalledconstructreliabilityanditisameasureofinternal
consistencyinscaleitems,muchlikeCronbach’salpha.
•InSEMterms,thereliabilityofanindicatorisdefinedasthevarianceinthatindicatorthatis
notaccountedforbymeasurementerror.Itiscommonlyrepresentedbythesquaredmultiple
correlationcoefficient,whichrangesfrom0to1.
•ConvergentValidityimpliesthelowestvalueofAVEis0.688whichisabovecriticalvalueof
0.5orhigher.
•discriminantvaliditytestswhetherconceptsormeasurementsthatarenotsupposedtobe
relatedareactuallyunrelated.Asuccessfulevaluationofdiscriminantvalidityshowsthata
testofaconceptisnothighlycorrelatedwithothertestsdesignedtomeasuretheoretically
differentconcepts.
•Endogenousvariablesarevariablesinastatisticalmodelthatarechangedordeterminedby
theirrelationshipwithothervariables.
•latentvariableshighlightsimpliesvariablesthatarenotdirectlyobservedbutarerather
inferredfromothervariablesthatareobservedanddirectlymeasured.
•coefficientsarethedefaultvaluesreturnedbyallstatisticalprograms.Inshort,theyreflectthe
expected(linear)changeintheresponsewitheachunitchangeinthepredictor.Fora
coefficientvalueβ=0.5,forexample,a1unitchangeinxthereis,onaverage,an0.5unit
changeiny.
•Thegoodnessoffitofastatisticalmodeldescribeshowwellitfitsasetofobservations.
Measuresofgoodnessoffittypicallysummarizethediscrepancybetweenobservedvalues
andthevaluesexpectedunderthemodelinquestion.

6. 1. Importance of Structural Equation
Modelling in Analysis
•Structuralequationmodelingisamultivariatestatisticalanalysis
techniquethatisusedtoanalyzestructuralrelationships.
•Thistechniqueisthecombinationoffactoranalysisandmultiple
regressionanalysis,anditisusedtoanalyzethestructural
relationshipbetweenmeasuredvariablesandlatentconstructs.
•Itprovidesnumericalestimatesforeachoftheparameters(arrows)
inthemodeltoindicatethestrengthoftherelationships.Thus,in
additiontotestingtheoveralltheory,SEMthereforeallowsthe
researchertodiagnosewhichobservedvariablesaregoodindicators
ofthelatentvariables.
•Thismethodispreferredbytheresearcherbecauseitestimatesthe
multipleandinterrelateddependenceinasingleanalysis.Inthis
analysis,twotypesofvariablesareusedendogenousvariablesand
exogenousvariables.Endogenousvariablesareequivalentto
dependentvariablesandareequaltotheindependentvariable.

7. •Theory:Thiscanbethoughtofasasetofrelationshipsprovidingconsistencyand
comprehensiveexplanationsoftheactualphenomena.Therearetwotypesofmodels:
•Measurementmodel:Themeasurementmodelrepresentsthetheorythatspecifieshow
measuredvariablescometogethertorepresentthetheory..
•Structuralmodel:Representsthetheorythatshowshowconstructsarerelatedtoother
constructs.Structuralequationmodelingisalsocalledcasualmodelingbecauseitteststhe
proposedcasualrelationships.Thefollowingassumptionsareassumed:
•Multivariatenormaldistribution:Themaximumlikelihoodmethodisusedandassumed
formultivariatenormaldistribution.Smallchangesinmultivariatenormalitycanleadtoa
largedifferenceinthechi-squaretest.
•Linearity:Alinearrelationshipisassumedbetweenendogenousandexogenousvariables.
•Outlier:Datashouldbefreeofoutliers.Outliersaffectthemodelsignificance.
•Sequence:Thereshouldbeacauseandeffectrelationshipbetweenendogenousand
exogenousvariables,andacausehastooccurbeforetheevent.
•Non-spuriousrelationship:Observedcovariancemustbetrue.
•Modelidentification:Equationsmustbegreaterthantheestimatedparametersormodels
shouldbeoveridentifiedorexactidentified.Underidentifiedmodelsarenotconsidered.
•Samplesize:Mostoftheresearchersprefera200to400samplesizewith10to15
indicators.Asaruleofthumb,thatis10to20timesasmanycasesasvariables.
•Uncorrelatederrorterms:Errortermsareassumeduncorrelatedwithothervariableerror
terms.Data:Intervaldataisused.

8. •Structuralequationmodeling,orSEM,isaverygeneral,chieflylinear,chiefly
cross-sectionalstatisticalmodelingtechnique..SEMisalargelyconfirmatory,
ratherthanexploratory,technique.Thatis,aresearcheraremorelikelytouse
SEMtodeterminewhetheracertainmodelisvalid.,ratherthanusingSEMto
"find"asuitablemodel--althoughSEManalysesofteninvolveacertain
exploratoryelement.
•InSEM,interestusuallyfocusesonlatentconstructs--abstractpsychological
variableslike"intelligence"or"attitudetowardthebrand"--ratherthanonthe
manifestvariablesusedtomeasuretheseconstructs.Measurementis
recognizedasdifficultanderror-prone.
•Byexplicitlymodelingmeasurementerror,SEMusersseektoderiveunbiased
estimatesfortherelationsbetweenlatentconstructs.Tothisend,
•SEMallowsmultiplemeasurestobeassociatedwithasinglelatentconstruct.A
structuralequationmodelimpliesastructureofthecovariancematrixofthe
measures(henceanalternativenameforthisfield,"analysisofcovariance
structures").Anditcanbeconsideredaplausibleexplanationforrelations
betweenthemeasures.

9. Structural equation modeling, reflective and
formative measures, path diagram, AMOS.
•Structuralequationmodelingincludesadiversesetofmathematicalmodels,computer
algorithms,andstatisticalmethodsthatfitnetworksofconstructstodata.
•ThepurposeofSEMistoexamineasetofrelationshipsbetweenoneormoreIndependent
Variables(IV)andoneormoreDependentVariables(DV).
•StructuralEquationModeling(SEM)isapowerfultechniqueasitestimatesthecausal
relationshipbetweenmorethanonedependentvariableandmanyindependentvariables.
•Structuralequationmodellingisanewsophisticated,techniquethatallowsyoutotest
variousmodelsconcerningtheinter-relationshipsamongasetofvariables.Basedon
multipleregressionandfactoranalytictechniques,itallowsyoutoevaluatetheimportance
ofeachoftheindependentvariablesinthemodelandtotesttheoverallfitofthemodelto
yourdata.Italsoallowsyoutocomparealternativemodels.SPSSdoesnothaveastructural
equationmodellingmodule,butitdoessupportan‘addon’calledAMOS.

10. Goals of SEM
•Tounderstandthepatternsofcorrelation/covarianceamong
asetofvariables.
•Toexplainasmuchoftheirvarianceaspossiblewiththe
modelspecified.
•Thepurposeofstructuralequationmodeling(SEM)isto
defineatheoreticalcausalmodelconsistingofasetof
predictedcovariancesbetweenvariablesandthentest
whetheritisplausiblewhencomparedtotheobserveddata.
•Structuralequationmodeling(SEM)isaconceptto
combinethestatisticaltechniquessuchasexploratoryfactor
analysisandmultipleregression.
•Moreover,Itexaminesasetofrelationshipsbetweenoneor
moreIndependentVariablesandoneormoreDependent
Variables.

11. Types of SEM
•TherearetwotypesofSEM:Variance-based/PLS-SEMand
Covariance-basedSEM
Variance-basedSEM/PLS-
•Variance-based/PLS-SEMSEMisusefulinexploratoryresearch
anditworkswellwithsmallsampleofaround100.
Covariance-basedSEM
•Itisusedbyresearchersmostlytoconfirmtheresearch
studiesortheories.However,itdemandslargesample,
around300.

12. Steps to perform SEM
•Specify the Structural Model
•Specify the Measurement Models
•Data Collection & Examination
•PLS Path Model Estimation
•Assess the Results of Measurement Models
•Assess Results of the Structural Model
•Interpretation of Results & Drawing Conclusions

13. Path diagrams

14. Path Diagram Rules
•Latentvariablesareenclosedbyovals.
•Observable(manifest)variablesareenclosedbyrectangles.
•Errortermsarenotenclosed
oSometimesthearrowsfromtheerrortermsseemtocomefromnowhere.The
symbolfortheerrortermdoesnotappearinthepathdiagram.
oSometimestherearenoarrowsfortheerrortermsatall.Itisjustassumedthat
suchanarrowpointstoeachendogenousvariable.
•Straight,single-headedarrowspointfromeachvariableontherightsideof
anequationtotheendogenousvariableontheleftside.
oSometimesthecoefficientiswrittenonthearrow,butsometimesitisnot.
•Acurved,double-headedarrowbetweentwovariables(alwaysexogenous
variables)meanstheyhaveanon-zerocovariance.
oSometimesthesymbolforthecovarianceiswrittenonthecurvedarrow,but
sometimesitisnot.

15. •PathDiagramsplayafundamentalroleinstructural
modeling.Pathdiagramsarelikeflowcharts.Theyshow
variablesinterconnectedwithlinesthatareusedtoindicate
causalflow.
•Pathdiagramisconsideredasadeviceforshowingwhich
variablescausechangesinothervariables.However,path
diagramsneednotbethoughtofstrictlyinthisway.They
mayalsobegivenanarrower,morespecificinterpretation.
•Agraphicrepresentationofacausalmodelderivedfroma
correlationalstudy,showingthestrengthsofhypothesized
causaleffectsofcertainvariablesoncertainothervariables.

16. Reflective and Formative measures
Researchersalwaysargueanddebatetheoperationalizationof
formativeorareflectivemeasurementinPartialLeastSquares
StructuralEquationModeling(PLS-SEM).
Aformativemeasurementmodelhappenswhenthemeasured
variablesareconsideredtobethecauseofthelatentvariable.
Thevalueofacarisdeterminedbyitsage,condition,size,make,
etc.IfacarismorevaluableitdoesnotturnfromaMercedesintoa
Ford.Instead,beingaMercedesisapredictorofbeingmore
valuable,andvalueistheoutcome.
Inareflectivemeasurementmodel,weexpectthecovariances
betweentheindicatorstobezero,whenthelatentvariableis
partialledout-thatisthereasonthattwotestscorescorrelateis
becausetheyarecausedbythesamething.
Inaformativemeasurementmodel,wedon'thaveanythingtosay
aboutthecovariancesoftheitems,theycouldbezero,positiveor
negative.Formativemeasurementmodelsarehardertoestimate-
theyarenotidentifiedontheirown.

17. •Managementscholarsoftenidentifystructural
relationshipsamonglatent,unobservedconstructsby
statisticallyrelatingcovariationbetweenthelatent
constructsandtheobservedvariablesorindicatorsof
thelatentconstructs.
•ItarguethatifvariationinanindicatorXisassociated
withvariationinalatentconstructY,thenexogenous
interventionsthatchangeYcanbedetectedinthe
indicatorX.Mostscholarsassumethisrelationship
betweenconstructandindicatorisreflective.
•notherwords,thechangeinXreflectsthechangein
thelatentconstructY.Withreflective(oreffect)
measurementmodels,causalityflowsfromthelatent
constructtotheindicator.

18. distinction between formative
and reflective measures
•Conceptually,areflectivemeasurementmodelhappens
whentheindicatorsofaconstructareconsideredtobe
causedbythatconstruct.
•Aformativemeasurementmodelhappenswhenthe
measuredvariablesareconsideredtobethe
causeofthelatentvariable.

19. Reflective and Reformative model

20. •Formativemeasurementspecifiesthatthe
observableindicatorsareconsideredtocause
thelatentconstruct.
•Thus,formativeconstructsshouldbeassessed
basedonthestatisticalsignificanceandsizeof
theindicatorweightsaswellascollinearity
amongindicators.

21. AMOS (Analysis of Moment
Structures)
•Amos(AnalysisofMomentStructures)isanIBMSPSSStatisticsmodule
designedfortheanalysisofcovariancestructuremodels,including
structuralequationmodeling(SEM),pathanalysis,andconfirmatory
factoranalysis(CFA).
•Amosfeaturesaneasy-to-useinterfaceforbootstrappingmethods,which
canbeappliedtoparameterestimates,effectestimates,samplemeans,
samplevariancesandcovariances,correlations,modelcomparisons,and
comparisonsofestimationmethods.
•AMOSisstatisticalsoftwareanditstandsforanalysisofamoment
structures.
•AMOSisanaddedSPSSmodule,andisspeciallyusedforStructural
EquationModeling,pathanalysis,andconfirmatoryfactoranalysis.
•AMOSisavisualprogramforstructuralequationmodeling(SEM).In
AMOS,wecandrawmodelsgraphicallyusingsimpledrawingtools.
AMOSquicklyperformsthecomputationsforSEManddisplaysthe
results.

22. In the calculation of SEM coefficients, AMOS
uses the following methods
•Maximumlikelihood
•Unweightedleastsquares
•Generalizedleastsquares
•Browne’sasymptoticallydistribution-free
criterion.
•Scale-freeleastsquares

23. Construction of model in AMOS
•First,wehavetorunAMOS.
•Byclickingthe“start”menuandselectingthe“AMOSgraphic”
option,wecanruntheprogram.
•ThemomentAMOSstartsrunning,awindowappearscalledthe
“AMOSgraphic.”Inthiswindow,wecanmanuallydrawourSEM
model
1.AttachingData:Byselectingafilenamefromthedatafile
operation.WecanattachdatainAMOSforSEManalysis.This
optionalsoappearsifwewillclickonthe“selectdata”icon
2.ObservedVariable:Arectangleiconisusedtodrawthe
observedvariable.

24. 3.UnobservedVariable:Acircleiconisusedto
drawtheunobservedvariable.
4.CauseEffectRelationship:Asingleheaded
arrowinAMOSisusedtodrawthecauseeffect
relationshipbetweentheobservedandunobserved
variables
5.Covariance:Adoubleheadedarrowisusedto
drawthecovariancebetweenvariables.
6.ErrorTerm:InAMOS,theerrortermiconis
nexttotheunobservedvariableicon,anditisusedto
drawthelatentvariable.
7.NamingtheVariable:Whenwerightclickona
variableinagraphicalwindow,thefirstoption,“object
properties,”isusedtogivethenameofthevariablein
AMOS.

25. AMOS important output
•VariableSummary:toestimatethehowmanyvariablesandwhichvariables
areusedforSEManalysis.
•AccessingtheNormality:InSEMmodel,datashouldbenormallydistributed.
AMOSwillgivethetextoutput,andSkewness,KurtosisandMahalanobisd-
squaredtestwilltellusaboutthenormalityofthedata.
•Estimates:InAMOStextoutput,theestimateoptionwillgivetheresultfor
regressionweight,standardizedloadingforfactor,residual,correlation,
covariance,directeffect,totaleffectetc.
•ModificationIndex:InAMOStextoutput,themodificationindexresultshows
thereliabilityofthepathdrawnintheSEMmodel.IfMIindexvalueislarge,
thenwecanaddmorepathstotheSEMmodel.
•ModelFit:InAMOStextoutput,modelfitoptionwillgivetheresultfor
goodnessoffitindexes,suchasGFI,RMR,TLI,BIC,RMSER,etc.
•ErrorMessage:Ifthereisanyproblemduringtheprocessofdrawingthe
model(forexample,ifweforgettodrawtheerrortermorifwedrawthe
covariancebetweentwovariables,orifmissingdataispresent),thenAMOS
willeithernotcalculatetheresultoritwillgiveanerrormessage

26. SEM 2-step approach: composite reliability,
indicator reliability, convergent and discriminant
validity
TheevaluationofanSEMmodelshouldbedoneinasystematicorder.Firstweshouldassessmeasurement
modelsandthenthestructuralmodel.TheSEMconceptsthatwelearnedwillbeusedintheevaluation:
ReflectiveMeasurementModel
•CompositeReliability
•IndicatorReliability
•ConvergentValidity(AVE)
•DiscriminantValidity
FormativeMeasurementModel
•Convergentvalidity
•CollinearityStatistics(VIF)
•Outerweightsrelevance(bootstrapping)
StructuralModel
•Sizeofpathcoefficients.
•Significanceofpathcoefficients(T-Statistics).
•Coefficientsofdetermination(R2)
•Predictiverelevance(Q2)
•Sizeofpathcoefficients
•Significanceofpathcoefficients(T-Statistics)
•f2effectsizes
•q2effectsizes

27. Composite reliability
Thesmallestvaluesis0.869thatisabovethecriticalvalueof0.7.(CUSAis1asitisa
singleindicatorconstruct).
compositereliability(sometimescalledconstructreliability)isameasureofinternal
consistencyinscaleitems,muchlikeCronbach’salpha.
Itcanbethoughtofasbeingequaltothetotalamountoftruescorevariancerelativeto
thetotalscalescorevariance.Alternatively,it’san“indicatorofthesharedvariance
amongtheobservedvariablesusedasanindicatorofalatentconstruct”
Formula:
ConfirmatoryFactorAnalysisisonewaytomeasurecompositereliability,and
itiswidelyavailableinmanydifferentstatisticalsoftwarepackages.Byhand,the
calculationsarealittlecumbersome.Theformulais:

28. Indicator Reliability
•Reliabilityreferstowhetherdatacanbereplicatedfromoneobservertoanother
(interobserverreliability)orbythesameobserveronmorethanoneoccasion
(intraobserverreliability).
•Thereliabilityofaqualityindicatorrelatestoitslevelofmeasurementerror
•InSEMterms,thereliabilityofanindicatorisdefinedasthevarianceinthatindicator
thatisnotaccountedforbymeasurementerror.Itiscommonlyrepresentedbythe
squaredmultiplecorrelationcoefficient,whichrangesfrom0to1.
•ItischeckedtheloadingsofreflectiveindicatorsmodelinCaseStudyIwhichareall
above0.708
•aquantitativecharacterizationofthereliabilityoftechnicaldevices.Areliability
indicatormaybeindividualorcomposite,dependingonthenumberofpropertiesit
characterizes.Anindividualindicatorcorrespondstoasingleproperty,suchasthe
failurerate.
•Theindicatorsmostfrequentlyusedinpracticearetheaverageoperatingtimeto
failure,theprobabilityoffailure-freeoperationduringaspecifiedtimeinterval,the
meancyclesbetweenfailures,theaveragefailurerate,theoperationalreadiness,and
thetechnicalusefactor.
•Itdescribesthestate‐of‐the‐artoftheresearchonreliabilityindicatorsforeither
electronicormechanicalcomponents.Typicalcomponentfailurepatternsarepresented
anddiscussed.Theconceptofcomponentscreeningusingreliabilityindicatorsisthen
outlined.

29. Convergent Validity
•ThelowestvalueofAVEis0.688whichisabovecriticalvalueof0.5orhigher.
•ConvergentValidityisasub-typeofconstructvalidity.Constructvaliditymeansthatatest
designedtomeasureaparticularconstruct(i.e.intelligence)isactuallymeasuringthat
construct.Convergentvaliditytakestwomeasuresthataresupposedtobemeasuringthe
sameconstructandshowsthattheyarerelated.
•Convergentvalidity,aparameteroftenusedinsociology,psychology,andotherbehavioral
sciences,referstothedegreetowhichtwomeasuresofconstructsthattheoreticallyshould
berelated,areinfactrelated.Convergentvalidity,alongwithdiscriminantvalidity,isa
subtypeofconstructvalidity.
•ConvergentValidityisasub-typeofconstructvalidity.Constructvaliditymeansthatatest
designedtomeasureaparticularconstruct(i.e.intelligence)isactuallymeasuringthat
construct.Convergentvaliditytakestwomeasuresthataresupposedtobemeasuringthe
sameconstructandshowsthattheyarerelated.Conversely,discriminantvalidityshowsthat
twomeasuresthatarenotsupposedtoberelatedareinfact,unrelated.Bothtypesof
validityarearequirementforexcellentconstructvalidity.
•Forexample,inordertotesttheconvergentvalidityofameasureofself-esteem,a
researchermaywanttoshowthatmeasuresofsimilarconstructs,suchasself-worth,
confidence,socialskills,andself-appraisalarealsorelatedtoself-esteem,whereasnon-
overlappingfactors,suchasintelligence,shouldnotrelate.[4]

30. Discriminant Validity
•First,weshouldlookattheFornell-Larckercriterion,thesquarerootof
theAVEofeachreflectiveconstructshouldbelargerthanthe
correlationswiththeremainingconstructsinthemodel.
•Andthen,thecross-loadingofreflectiveindicatorsofeachlatent
variableshouldbehigherthanotherloadings.ForexampleATTRand
COMParehighlightedintablegivenbelow.
•Inpsychology,discriminantvaliditytestswhetherconceptsor
measurementsthatarenotsupposedtoberelatedareactuallyunrelated.
•Discriminantvaliditytestswhetherbelievedunrelatedconstructsare,in
fact,unrelated.Discriminantvaliditywouldensurethat,inthestudy,the
non-overlappingfactorsdonotoverlap.Forexample,selfesteemand
intelligenceshouldnotrelate(toomuch)inmostresearchprojects.
•Inordertoestablishdiscriminantvaliditythereisneedforanappropriate
AVE(AverageVarianceExtracted)analysis.InanAVEanalysis,wetest
toseeifthesquarerootofeveryAVEvaluebelongingtoeachlatent
constructismuchlargerthananycorrelationamonganypairoflatent
constructs.

31. R2 of endogenous (dependent) latent variable,
β coefficient, Goodness of fit (GoF)
R2ofendogenous
•Endogenousvariablesarevariablesinastatisticalmodelthatarechangedor
determinedbytheirrelationshipwithothervariables.
•Endogenousvariablesarevariablesinastatisticalmodelthatarechangedor
determinedbytheirrelationshipwithothervariables.
•Inotherwords,anendogenousvariableismatchedtosynonymouswitha
dependentvariable,meaningitcorrelateswithotherfactorswithinthesystembeing
studied.Therefore,itsvaluesmaybedeterminedbyothervariables.
•Endogenousvariablesaretheoppositeofexogenousvariables,whichare
independentvariables.
•Endogenousvariablesaredependentvariables,meaningtheycorrelatewithother
factors—althoughitcanbeapositiveornegativecorrelation.
•Endogenousvariablesareimportantineconomicmodelingbecausethey
showwhetheravariablecausesaparticulareffect.
•R-squaredisalwaysbetween0and100%:0%indicatesthatthemodelexplains
noneofthevariabilityoftheresponsedataarounditsmean.100%indicatesthatthe
modelexplainsallthevariabilityoftheresponsedataarounditsmean.

32. •Endogenousvariableshavevaluesthatshiftaspartofafunctionalrelationship
betweenothervariableswithinthemodel.Therelationshipisalsoreferredtoas
dependentandisseenaspredictableinnature.
•Endogenousvariablesarethedependentvariablesthatcorrelatewitheachother,
knowingtowhatextentexogenousvariablesimpactamodelisimportantto
consider.

33. Difference between endogenous and
exogenous variable
•Anexogenousvariableisonewhosevalueisdeterminedoutsidethemodel.
•Endogenousvariableisavariablewhosevalueisdeterminedbythemodel.
•Anexogenousvariableinthecontextofregressionanalysisisavariablewhichis
notaffectedbyothervariables.Onceintroducedintothesystemitisregardedas
fixed.
•Ontheotherhandavariableissaidtobeendogenousifitsvalueisdeterminedby
othervariables.
•Endogenousvariableisadependent
•exogenousvariableisindependent

34. Latent variable
•Instatisticslatentvariablesimplies“liehidden”,andthesearevariablesthatare
notdirectlyobservedbutareratherinferred(throughamathematicalmodel)from
othervariablesthatareobserved(directlymeasured).
•Thestandardsolutionthatpsychologiststaketomeasuringlatentvariablesisto
useaseriesofquestionsthatarealldesignedtomeasurethelatentvariable.This
isknownasamulti-itemscale,wherean“item”isaquestion,anda“scale”isthe
resultingestimateofthelatentvariable.
•Instructuralequationmodel(SEM)useslatentvariablestoaccountfor
measurementerror.LatentVariables.Alatentvariableisahypotheticalconstruct
thatisinvokedtoexplainobservedcovariationinbehavior.Examplesinpsychology
includeintelligence(a.k.a.cognitiveability),TypeApersonality,anddepression.
•Alatentvariableisavariablethatcannotbeobserved.Thepresenceoflatent
variables,however,canbedetectedbytheireffectsonvariablesthatare
observable.

35. •Alatentvariableisavariablethatcannotbeobserved.Thepresenceoflatent
variables,however,canbedetectedbytheireffectsonvariablesthatare
observable.Mostconstructsinresearcharelatentvariables.Considerthe
psychologicalconstructofanxiety,forexample.Anysingleobservablemeasure
ofanxiety,whetheritisaself-reportmeasureoranobservationalscale,cannot
provideapuremeasureofanxiety.Observablevariablesareaffectedby
measurementerror.Measurementerrorreferstothefactthatscoresoftenwill
notbeidenticalifthesamemeasureisgivenontwooccasionsorifequivalent
formsofthemeasurearegivenonasingleoccasion.Inaddition,most
observablevariablesareaffectedbymethodvariance,withtheresultsobtained
usingamethod.
•Theideaisthatthevalueofthelatentvariablecausedpeopletorespondasthey
didontheobservedindicators.
•Onatechnicalnote,estimationofalatentvariableisdonebyanalyzingthe
varianceandcovarianceoftheindicators.Themeasurementmodelofalatent
variablewitheffectindicatorsisthesetofrelationships(modeledasequations)
inwhichthelatentvariableissetasthepredictoroftheindicators.
•StructuralEquationModelingshowshowalllatentvariablesarerelatedtoeach
other.ThedependentlatentvariableinSEMwhichhaveone-wayarrows
pointingtothemiscalledendogenousvariablewhileothersareexogenous
variables.

36. •The opposite of an observed variable is alatent variable, also referred to as afactororconstruct.
•A latent variable is hidden, and therefore can’t be observed. While observed variables are the only
type of variable used inregression analysis, SEM can handle other types of variables including
latent, unobserved and theoretical variables.
•Observed variables are represented by rectangular nodes in SEM and latent variables are represented
by circles.
•An important difference between the two types of variables is that an observed variable usually has
ameasurement errorassociated with it, while a latent variable does not.
•

37. The idea is that the value of the latent variable caused people to respond as
they did on the observed indicators.

38. β coefficient
•Thebetacoefficientisthedegreeofchangeintheoutcomevariableforevery1-unitof
changeinthepredictorvariable.
•Ifthebetacoefficientispositive,theinterpretationisthatforevery1-unitincreaseinthe
predictorvariable,theoutcomevariablewillincreasebythebetacoefficientvalue.
•Theformulaforcalculatingbetaisthecovarianceofthereturnofanassetwiththereturn
ofthebenchmark,dividedbythevarianceofthereturnofthebenchmarkoveracertain
period.
•Betaisameasureusedinfundamentalanalysistodeterminethevolatilityofanassetor
portfolioinrelationtotheoverallmarket.
•Theoverallmarkethasabetaof1.0,andindividualstocksarerankedaccordingtohow
muchtheydeviatefromthemarket..
•Tocalculatethebetaofasecurity,thecovariancebetweenthereturnofthesecurityand
thereturnofthemarketmustbeknown,aswellasthevarianceofthemarketreturns.
Variance
Beta= ______
Covariance​
•Where:Covariance=Measureofastock’sreturnrelativetothatofthemarket
Variance=Measureofhowthemarketmovesrelativetoitsmean.

39. •Covariancemeasureshowtwostocksmovetogether.Apositivecovariancemeansthestockstendtomove
togetherwhentheirpricesgoupordown.Anegativecovariancemeansthestocksmoveoppositeofeach
other.
•Variance,ontheotherhand,referstohowfarastockmovesrelativetoitsmean.Forexample,varianceisused
inmeasuringthevolatilityofanindividualstock'spriceovertime.Covarianceisusedtomeasurethe
correlationinpricemovesoftwodifferentstocks.
•Theformulaforcalculatingbetaisthecovarianceofthereturnofanassetwiththereturnofthebenchmark,
dividedbythevarianceofthereturnofthebenchmarkoveracertainperiod.
•Betacouldbecalculatedbyfirstdividingthesecurity'sstandarddeviationofreturnsbythebenchmark's
standarddeviationofreturns.Theresultingvalueismultipliedbythecorrelationofthesecurity'sreturnsand
thebenchmark'sreturns.
•CalculatingtheBetaforApple:
•AninvestorislookingtocalculatethebetaofApple(AAPL)ascomparedtotheSPDRS&P500ETFTrust
(SPY).Basedonrecentfive-yeardata,thecorrelationbetweenAAPLandSPYis0.83.AAPLhasastandard
deviationofreturnsof23.42%andSPYhasastandarddeviationofreturnsof32.21%.
Beta of AAPL=0.83×(0.3221
_______
0.2342) =0.6035
•Inthiscase,AAPLwouldbeconsideredlessvolatilethanSPY,asitsbetaof0.6035indicatesthestock
theoreticallyexperiences40%lessvolatility.

40. Goodness of fit (GoF)
•Thegoodnessoffitofastatisticalmodeldescribeshowwellitfitsasetof
observations.Measuresofgoodnessoffittypicallysummarizethediscrepancy
betweenobservedvaluesandthevaluesexpectedunderthemodelinquestion.
•Thegoodnessoffittestisastatisticalhypothesistesttoseehowwellsampledata
fitadistributionfromapopulationwithanormaldistribution....Goodness-of-fit
establishesthediscrepancybetweentheobservedvaluesandthosethatwouldbe
expectedofthemodelinanormaldistributioncase.
•Whenthereisamatchbetweenthedemandsandexpectationsoftheenvironment
andthechild'stemperamentandabilities,thatisagoodfit.Thismakessuccessand
highself-esteemmorelikely.Whenthereisnotagoodfit,thereisagreaterriskfor
difficultiesforthechild.
•Tointerpretthetest,you'llneedtochooseanalphalevel(1%,5%and10%are
common).Thechi-squaretestwillreturnap-value.Ifthep-valueissmall(lessthan
thesignificancelevel),youcanrejectthenullhypothesisthatthedatacomesfrom
thespecifieddistribution.