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5. Finite Normal Mixture SEM Analysis by Fitting Multiple Conventional SEM Models

This paper proposes a two-stage maximum likelihood (ML) approach to normal mixture structural equation modeling (SEM) and develops a statistical inference that allows distributional mis-specification. Saturated means and covariances are estimated at stage 1 together with a sandwich-type covariance matrix. These are used to evaluate structural models at stage 2. Techniques accumulated in the conventional SEM literature for model diagnosis and evaluation can be used to study the model structure for each component. Examples show that the two-stage ML approach leads to correct or nearly correct models even when the normal mixture assumptions are violated and initial models are misspecified. Compared to single-stage ML, two-stage ML avoids the confounding effect of model specification and the number of components, and it is computationally more efficient. Monte Carlo results indicate that two-stage ML loses only minimal efficiency under the condition where single-stage ML performs best. Monte Carlo results also indicate that the commonly used model selection criterion BIC is more robust to distribution violations for the saturated model than that for a structural model at moderate sample sizes. The proposed two-stage ML approach is also extremely flexible in modeling different components with different models. Potential new developments in the mixture modeling literature can be easily adapted to study issues with normal mixture SEM.