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1“Ghost Chasing”:Demystifying Latent Variables and SEMAndrew AinsworthUniversity of California, Los Angeles2/20/2006 Latent Variable Models 2Topics“Ghost Chasing” and Latent VariablesWhat is SEM?SEM elements and JargonExample Latent VariablesSEM Limitations2/20/2006 Latent Variable Models 3“Ghost Chasing”Psychologists are in the business of Chasing “Ghosts”Measuring “Ghosts”“Ghost” diagnosesExchanging one “Ghost” for another “Ghost”Latent (AKA “Ghost”) VariablesAnything we can’t measure directlyWe must rely on measurable indicators2/20/2006 Latent Variable Models 4What is a Latent Variable?An operationalization of data as an abstract construct A data reduction method that uses “regression like” equationsTake many variables and explain them with a one or more “factors” Correlated variables are grouped together and separated from other variables with low or no correlation2/20/2006 Latent Variable Models 5Establishing Latent VariablesExploratory Factor AnalysisSummarizing data by grouping correlated variablesInvestigating sets of measured variables for underlying constructsOften done near the onset of research and/or scale construction2/20/2006 Latent Variable Models 6Establishing Latent VariablesConfirmatory Factor AnalysisTesting whether proposed constructs influence measured variablesWhen factor structure is known or at least theorizedOften done when relationships among variables are known2/20/2006 Latent Variable Models 7LatentVariableVariable1Variable3Variable2Variable4Conceptualizing Latent Variables2/20/2006 Latent Variable Models 8Conceptualizing Latent VariablesLatent variables –representation of the variance shared among the variables common variance without error or specific varianceTotalVarianceCommonVarianceUniqueVarianceSpecificVarianceRandomError2/20/2006 Latent Variable Models 9What is SEM?SEM – Structural Equation ModelingAlso Known AsCSA – Covariance Structure AnalysisCausal ModelsSimultaneous EquationsPath AnalysisConfirmatory Factor AnalysisLatent Variable Modeling2/20/2006 Latent Variable Models 10SEM in a nutshellCombination of factor analysis and regressionTests relationships variables Specify models that explain data with few parametersFlexible - Works with continuous and discrete variablesSignificance testing and model fit2/20/2006 Latent Variable Models 11Goals in SEMHypothesize a model that: Has a number of parameters less than the number of unique Variance/Covariance entries (i.e. (p*(p+1))/2)Has an implied covariance matrix that is not significantly different from the sample covariance matrixAllows us to estimate population parameters that make the sample data the most likely2/20/2006 Latent Variable Models 12Important Matricess matrix Sample CovariancesThe dataσ(θ) matrixModel Implied CovariancesResidual Covariance MatrixItem1Item2Item3Item4Item1σ211σ212σ213σ214Item2σ221σ222σ223σ224Item3σ231σ232σ233σ234Item4σ241σ242σ243σ244Item1Item2Item3Item4Item1s211s212s213s214Item2s221s222s223s224Item3s231s232s233s234Item4s241s242s243s2442/20/2006 Latent Variable Models 13SEM JargonMeasurement The part of the model that relates measured variables to latent factorsThe measurement model is the factor analytic part of SEMStructureThis is the part of the model that relates variable or factors to one another (prediction)If no factors are in the model then only path model exists between measured variables2/20/2006 Latent Variable Models 14SEM JargonModel SpecificationCreating a hypothesized model that you think explains the relationships among multiple variablesConverting the model to multiple equationsModel EstimationTechnique used to calculate parametersE.G. - Ordinary Least Squares (OLS), Maximum Likelihood (ML), etc.2/20/2006 Latent Variable Models 15SEM JargonModel IdentificationRules for whether a model can be estimatedFor example, For a single factor: At least 3 indicators with non-zero loadings no correlated errors Fix either the Factor Variance or one of the Factor Loadings to 12/20/2006 Latent Variable Models 16SEM JargonModel EvaluationTesting how well a model fits the dataJust like with other analyses (e.g. ANOVA) we look at squared differences SEM looks at the squared difference between the s and σ(θ) matrices While weighting the squared difference depending on the estimation method (e.g. OLS, ML, etc.)()()minpick a ( ) ( ) ' ( )Qs Wsσθσθσθ→ = − −2/20/2006 Latent Variable Models 17SEM JargonModel EvaluationEven with well fitting model you need to test significance of predictors Each parameter is divided by its SE to get a Z-score which can be evaluated SE values are calculated as part of the estimation procedure2/20/2006 Latent Variable Models 18Conventional SEM diagrams = measured variable { = latent variableÖ = regression weight or factor loadingÙ = covariance2/20/2006 Latent Variable Models 19Sample Variance/Covariance MatrixX1 X2 X3X1 1.8782 1.0824 1.1080X2 1.0824 2.3414 1.3409X3 1.1080 1.3409 2.60232/20/2006 Latent Variable Models 20Basic Tracing Rules for a Latent VariableOnce parameters are estimatedCalculating the Implied Covariance MatrixRules for Implied Variance Common Variance – trace a path from a variable back to itself, multiplying parametersAdd to it the unique variance of that DVRules for covariance between variablesTrace path from any variable to another, multiplying parameters2/20/2006 Latent Variable Models 21Implied Covariance MatrixX1 X3X212.9457(1)(.9457) .8944 .9838 1.8782Xσ==+=231.1445(1)(1.1716) 1.3409XXσ==13.9457(1)(1.1716) 1.1080XXσ==VariancesCovariances12.9457(1)(1.1445) 1.0824XXσ==221.1445(1)(1.1445) 1.3099 1.0314 2.3413Xσ==+=321.1716(1)(1.1716) 1.3726 1.2296 2.6022Xσ==+=1.17161.1445.9457LatentVariable(1.00)1.22961.0314.9838X1 X2 X3X1 1.8782 1.0824 1.1080X2 1.0824 2.3413 1.3409X3 1.1080 1.3409 2.60222/20/2006 Latent Variable Models 22Residual Matrix()1.8782 1.0824 1.1080 1.8782 1.0824 1.1080 0 0 01.0824 2.3414 1.3409 1.0824 2.3413 1.3409 0 .0001 01.1080 1.3409 2.6023 1.1080 1.3409 2.6022 0 0 .0001s residualσθ−=Function Min and Chi-Square''()( ( )) ( ( ))1.8782 1.0824 1.1080 1.8782 1.0824 1.1080 1 0 01.0824 2.3414 1.3409 1.0824 2.3413 1.3409 * 0 1 01.1080 1.3409 2.6023 1.1080 1.3409 2.6022 0 0 11.8782 1.0824 1.1080*1.0824 2.s WQs WsQσθσθ σθ=− − ==−2()21.8782 1.0824 1.10803414 1.3409 1.0824 2.3413 1.34091.1080 1.3409


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CSUN PSY 524 - Ghost Chasing

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