Structural Equation Models: Mixture Models

This article discusses a modelling framework that links two well-known statistical methods: structural equation modelling (SEM) and latent class or finite mixture modelling. This hybrid approach was proposed independently by Arminger and Stein [1], Dolan and Van der Maas [4], and Jedidi, Jagpal and DeSarbo [5]. Here, we refer to this approach as mixture SEM or latent class SEM. There are two different ways to view mixture SEM. One way is as a refinement of multivariate normal (MVN) mixtures, where the within-class covariance matrices are smoothed according to a postulated SEM structure. MVN mixtures have become a popular tool for cluster analysis [6] [10], where each cluster corresponds to a latent (unobservable) class. Names that are used when referring to such a use of mixture models are latent profile analysis, mixture-model clustering, model-based clustering, probabilistic clustering, Bayesian classification, and latent class clustering. Mixture SEM restricts the form of such latent class clustering, by subjecting the class-specific mean vectors and covariance matrices to a postulated SEM structure such as a onefactor, a latent-growth, or an autoregressive model. This results in MVNmixtures that are more parsimonious and stable than models with unrestricted covariance structures. The other way to look at mixture SEM is as an extension to standard SEM similar to multiple group analysis. However, an important difference between this and standard multiple group analysis is that in mixture SEM group membership is not observed. By incorporating latent classes into a SEM model, various forms of unobserved heterogeneity can be detected. For example, groups that have identical (unstandardized) factor loadings but different error variances on the items in a factor analysis or groups that show