Latent class and finite mixture models for multilevel data sets

An extension of latent class (LC) and finite mixture models is described for the analysis of hierarchical data sets. As is typical in multilevel analysis, the dependence between lower-level units within higher-level units is dealt with by assuming that certain model parameters differ randomly across higher-level observations. One of the special cases is an LC model in which group-level differences in the logit of belonging to a particular LC are captured with continuous random effects. Other variants are obtained by including random effects in the model for the response variables rather than for the LCs. The variant that receives most attention in this article is an LC model with discrete random effects: higher-level units are clustered based on the likelihood of their members belonging to the various LCs. This yields a model with mixture distributions at two levels, namely at the group and the subject level. This model is illustrated with three rather different empirical examples. The appendix describes an adapted version of the expectation-maximization algorithm that can be used for maximum likelihood estimation, as well as providing setups for estimating the multilevel LC model with generally available software.