Multilevel latent class models for cross-classified categorical data: model definition and estimation through stochastic EM

We present an extension of the multilevel latent class model for dealing with multilevel cross-classified categorical data. Cross-classified data structures arise when observations are simultaneously nested within two or more groups, for example, children nested within both schools and neighborhoods. More specifically, we propose extending the standard hierarchical latent class model, which contains mixture components at two levels, say for children and schools, by including a separate set of mixture components for each of the higher-level crossed classifications, say for schools and neighborhoods. Because of the complex dependency structure arising from the cross-classified nature of the data, it is no longer possible to obtain maximum likelihood estimates of the model parameters, for example, using the EM algorithm. As a solution to the estimation problem, we propose an approximate estimation approach using a stochastic version of the EM algorithm. The performance of this approach, which resembles Gibbs sampling, was investigated through a set of simulation studies. Moreover, the application of the new model is illustrated using an Italian dataset on the quality of university experience at degree programme level, with degree programmes nested in both universities and fields of study.