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 IdentificationOnly identified models can be estimated in SEMA model is said to be identified if there is a unique solution for every estimateY = 10Y = One of theme needs to fixed in order for there to be a unique solutionsBottom line: some parts of a model need to be fixed in order for the model to be identifiedThis is especially true for complex modelsModel Identification:Step 1OveridentificationMore data points than parametersThis is a necessary but not sufficient condition for identificationJust IdentifiedData points equal number of parametersCan not test model adequacyUnderidentifiedThere are more parameters than data pointsCan’t do anything; no estimationParameters can be fixed to free DFsModel Identification:Step 2aThe 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 indicatorIf the factor is exogenous either is fineIf the factor is endogenous most set the factor to 1Model Identification:Step 2bFactors are identified:If there is only one factor then: at least 3 indicators with non-zero loadingsno correlated errorsIf there is more than one factor and 3 indicators with non-zero loadings per factor then:No correlated errorsNo complex loadingsFactors covaryModel Identification:Step 2bFactors 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 errorsNo complex loadingsNone of the variances or covariances among factors are zeroModel Identification:Step 3Relationships among the factors should either be orthogonal or recursive to be identifiedRecursive models have no feedback loops or correlated disturbancesNon-recursive models contain feedback loops or correlated disturbancesNon-recursive models can be identified but they are difficultModel EstimationAfter model specification:The population parameter are estimated with the goal of minimizing the difference between the estimated covariance matrix and the sample covariance matrixThis 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 EstimationIn factor analysis we compared the covariance matrix and the reproduced covariance matrix to assess fitIn SEM this is extended into an actual testIf the W matrix is selected correctly than (N – 1) * Q is Chi-square distributedThe difficult part of estimation is choosing the correct W matrixModel Estimation ProceduresModel Estimation Procedures differ in the choice of the weight matrixRoughly 6 widely used proceduresULS (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 FitHow well does the model fit the data?This can be answered by the Chi-square statistic but this test has many problemsIt is sample size dependent, so with large sample sizes trivial differences will be significantThere are basic underlying assumptions are violated the probabilities are inaccurateAssessing Model FitFit indicesRead 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 mentionedWhich do you choose?Different researchers have different preferences and different cutoff criterion for each indexWe will just focus on two fit indicesCFIRMSEAAssessing Model FitAssessing Model FitFit IndicesComparative 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 ApproximationCompares the estimated model to a saturated or perfect model0model2model model0 0 whichever is smaller and positiveNFRMSEAdfdfwhere F orc=-=))Model ModificationChi-square difference testNested 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 ModificationLangrange Multiplier testThis tests fixed paths (usually fixed to zero or left out) to see if including the path would improve the modelIf path is included would it give you better fitIt does this both univariately and multivariatelyWald TestThis tests free paths to see if removing them would hurt the modelLeads to a more parsimonious
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