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CSUN PSY 524 - Structural Equation Modeling 3

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Structural Equation Modeling 3Model IdentificationModel Identification: Step 1Model Identification: Step 2aModel Identification: Step 2bSlide 6Model Identification: Step 3Model EstimationSlide 9Model Estimation ProceduresSlide 11Assessing Model FitSlide 13Slide 14Slide 15Model ModificationSlide 17Structural Equation Modeling 3Psy 524 Andrew AinsworthModel IdentificationOnly identified models can be estimated in SEMA model is said to be identified if there is a unique solution for every estimateY = 10Y = One of theme needs to fixed in order for there to be a unique solutionsBottom line: some parts of a model need to be fixed in order for the model to be identifiedThis is especially true for complex modelsModel Identification:Step 1OveridentificationMore data points than parametersThis is a necessary but not sufficient condition for identificationJust IdentifiedData points equal number of parametersCan not test model adequacyUnderidentifiedThere are more parameters than data pointsCan’t do anything; no estimationParameters can be fixed to free DFsModel Identification:Step 2aThe factors in the measurement model need to be given a scale (latent factors don’t exist)You can either standardize the factor by setting the variance to 1 (perfectly fine)Or you can set the regression coefficient predicting one of the indicators to 1; this sets the scale to be equal to that of the indicator; best if it is a marker indicatorIf the factor is exogenous either is fineIf the factor is endogenous most set the factor to 1Model Identification:Step 2bFactors are identified:If there is only one factor then: at least 3 indicators with non-zero loadingsno correlated errorsIf there is more than one factor and 3 indicators with non-zero loadings per factor then:No correlated errorsNo complex loadingsFactors covaryModel Identification:Step 2bFactors are identified:If there is more than one factor and a factor with only 2 indicators with non-zero loadings per factor then:No correlated errorsNo complex loadingsNone of the variances or covariances among factors are zeroModel Identification:Step 3Relationships among the factors should either be orthogonal or recursive to be identifiedRecursive models have no feedback loops or correlated disturbancesNon-recursive models contain feedback loops or correlated disturbancesNon-recursive models can be identified but they are difficultModel EstimationAfter model specification:The population parameter are estimated with the goal of minimizing the difference between the estimated covariance matrix and the sample covariance matrixThis goal is accomplished by minimizing the Q function:Q = (s – ))’W(s – ))Where s is a vectorized sample covariance marix,  is a vectorized estimated matrix and  indicates that  is estimated from the parameters and W is a weight matrixModel EstimationIn factor analysis we compared the covariance matrix and the reproduced covariance matrix to assess fitIn SEM this is extended into an actual testIf the W matrix is selected correctly than (N – 1) * Q is Chi-square distributedThe difficult part of estimation is choosing the correct W matrixModel Estimation ProceduresModel Estimation Procedures differ in the choice of the weight matrixRoughly 6 widely used proceduresULS (unweighted least squares)GLS (generalized least squares)ML (maximum likelihood)EDT (elliptical distribution theory)ADF (asymptotically distribution free)Satorra-Bentler Scaled Chi-Square (corrected ML estimate for non-normality of data)Model Estimation ProceduresAssessing Model FitHow well does the model fit the data?This can be answered by the Chi-square statistic but this test has many problemsIt is sample size dependent, so with large sample sizes trivial differences will be significantThere are basic underlying assumptions are violated the probabilities are inaccurateAssessing Model FitFit indicesRead through the book and you’ll find that there are tons of fit indices and for everyone in the book there are 5 – 10 not mentionedWhich do you choose?Different researchers have different preferences and different cutoff criterion for each indexWe will just focus on two fit indicesCFIRMSEAAssessing Model FitAssessing Model FitFit IndicesComparative Fit Index (CFI) – compares the proposed model to an independence model (where nothing is related)est.modelindep.model2indep.model indep.model indep.model2est.model est.model est.model1where CFIdfand dfttt ct c= -= -= -Assessing Model Fit Root Mean Square Error of ApproximationCompares the estimated model to a saturated or perfect model0model2model model0 0 whichever is smaller and positiveNFRMSEAdfdfwhere F orc=-=))Model ModificationChi-square difference testNested models (models that are subsets of each other) can be tested for improvement by taking the difference between the two chi-square values and testing it at a DF that is equal to the difference between the DFs in the two models (more on this in lab)Model ModificationLangrange Multiplier testThis tests fixed paths (usually fixed to zero or left out) to see if including the path would improve the modelIf path is included would it give you better fitIt does this both univariately and multivariatelyWald TestThis tests free paths to see if removing them would hurt the modelLeads to a more parsimonious


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CSUN PSY 524 - Structural Equation Modeling 3

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