Mixture Model Clustering with Covariates Using Adjusted Three-Step Approaches

Abstract

When using mixture models, researchers may investigate the associations between cluster membership and covariates by introducing these variables in a (logistic) regression model for the prior class membership probabilities. However, a very popular alternative among applied researchers is a three-step approach in which after estimating the mixture model (step 1) and assigning subjects to clusters (step 2), the cluster assignments are regressed on covariates (step 3). For mixture models for categorical responses, (Bolck et al., Political Anal 12:3–27, 2004) and (Vermunt, Political Anal 18:450–469, 2010) showed this approach may severely downward bias covariate effects, and moreover showed how to adjust for this bias. This paper generalizes their corrections methods to be applicable also with mixture models for continuous responses, where the main complicating factor is that a complex multidimensional integral needs to be solved to obtain the classification errors needed for the corrections. We propose approximating this integral by a summation over the empirical distribution of the response variables. The simulation study showed that the approaches work well, except for the combination of very badly separated components and a small sample size.