Structural Equation Modeling SEM:  An Introduction

Structural Equation Modeling SEM: An Introduction PowerPoint PPT Presentation

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SEM. Advantages of SEMreduces measurement error by having multiple indicators of a latent variableability to test overall models and individual parametersability to statistically compare nested and non-nested modelsability to test models with multiple DVsability to model mediator variables (pro

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Structural Equation Modeling SEM: An Introduction

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1. Structural Equation Modeling (SEM): An Introduction Also referred to as covariance structure analysis, analysis of covariance structures, covariance structure modeling, LISREL modeling, causal modeling Merges the logic of factor analysis and multiple regression Goal is to develop a model that explains why observed variables are related i.e., explain the variance-covariance matrix (?) this is done through the simultaneous solution of equations representing your model

2. SEM Advantages of SEM reduces measurement error by having multiple indicators of a latent variable ability to test overall models and individual parameters ability to statistically compare nested and non-nested models ability to test models with multiple DVs ability to model mediator variables (processes) ability to model error terms ability to model relations across groups, across time

3. SEM 2 major approaches, each based on the researchers goals confirmatory factor analysis (CFA) structural equation modeling (SEM) For each major approach, one can take 1 of 3 additional methodological approaches strictly confirmatory alternative models model development

4. SEM Variables observed = measured, manifest, indicators can be items, subscales, or scales latent = theoretical constructs variables that are defined by the observed variables goal is to model the commonality in the observed variables sound like factor analysis? and then look at relations between latent variables sound like multiple regression?

5. Confirmatory Factor Analytic Model a priori measurement model is specified and tested direct relations between observed and latent variable(s) are modeled

6. Structural Model CFA plus a priori structural model is tested two step process (two-step modeling) establish the measurement model test the structural model direct relations among latent variables are modeled i.e., regression with latent variables see figure on next page, too big for here!

7. Structural Model continued

8. Structural Model continued Types of latent variables (LV) exogenous: LVs that only cause other LVs endogenous: LVs that are caused by other LVs pure DVs: are only caused mediators: are a cause and caused

9. The Process Model specification you write equations to specify each factor loading MV1 = ?(LV) + e1 e.g., BDI = ?11(Depression) + e1 relation among latent variables endogenous LV = ?(exogenous LV) + d2 Depression LV = ?21(Coping LV) + d2 these equations imply a model this model attempts to explain the variance-covariance matrix (?) i.e., relations among observed variables

10. The Process Estimating the model mathematically this is done with some heavy duty matrix algebra and tracing rules estimation procedures include maximum likelihood generalized least squares asymptotic distribution free methods etc. estimation produces a fit function tells us how well we have reproduced ? with our target model

11. The Process Determination of model fit done at two levels overall model fit individual parameter fit parameters = factor loadings, factor correlations, structural paths Overall model fit referred to as goodness of fit tells us if the model should be accepted or rejected

12. The Process if model is accepted, interpret model parameters if model is rejected, do not interpret model parameters Determining goodness of fit (see Hu & Bentler, 1999) the fit function from estimation is converted to a test statistic and numerous descriptive indices test statistic ?2 provides a statistical test of fit ?2 = ( fit function ) ( N 1) we want to this to be nonsignificant it never (rarely) is (large N)

13. The Process descriptive fit indices comparative fit index (CFI; Bentler, 1990) compares target model to a baseline model baseline model = null or independence model null model = no factors at all CFI values > .90 are good, .93 better, .95 great parsimony adjusted fit indices adjusts fit by weighting indices of fit by the number of parameters estimated root mean square error of approximation (RMSEA; Steiger, 1990) is best values less than .08 are good, .06 are better

14. The Process Fit of individual parameters we have statistical tests for each parameter referred to as critical ratios (CR) CR = parameter estimate / standard error these are distributed as z- (or t-)values we also typically evaluate standardized values for practical fit factor loadings > .30 ( or .45 in my world ) factor correlations & structural coefficients depend on your research literature

15. The Process What if my model and/or individual parameters do not fit? report that and stop, or go to the model modification phase SEM has what are called modification indices that tell you... whether or not there are parameters that you can add to improve fit via the LaGrange Multiplier test whether or not there are parameters that you can delete without hurting fit via the Wald test remember, however, that SEM is THEORY-DRIVEN!!!!!!!!

16. Practical Issues Assumptions of CFA large N: at least 200 but 400 is optimal, with the following restrictions number of cases to MVs 10 to 1 number of cases to estimated parameters 10 to 1 and variations on these if your data is non-normal number of cases to estimated parameters 15 to 1 or 20 to 1

17. Practical Issues Non-normal data typically evaluate multivariate non-normality assessed via Mardias coefficient scaled z-value Satorra-Bentler scaled ?2 corrects for non-normality in 2 ways overall ?2 and descriptive fit indices corrects for the overestimation (or inflation) of ?2 standard errors of parameters corrects for the underestimation of the standard errors

18. Practical Issues Measurement scale of MVs anything, but some assumptions/adjustments need to be made must assume that discrete/ordinal/interval data has an underlying normal distribution and/or use special correlations all binary variables = tetrachoric R binary and continuous = either polychoric polyserial correlations Number of observed variables per LV want a minimum of three (> 3 is better) Observed variables do NOT have to indicate a LV to be included in the model

19. The Process Summary of SEM propose a priori model(s) evaluate univariate and multivariate distribution of data for observed variables determine overall fit of measurement model by employing confirmatory factor analysis determine individual parameter fit of measurement model modify and re-establish fit if necessary or compare all a priori models

20. The Process Summary of SEM continued add structural model to measurement model determine overall fit of SEM determine individual parameter fit of primarily structural model including variance accounted for in each endogenous LV modify and re-establish fit if necessary or compare a priori models

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