Log-likelihood approximation in stochastic EM for multilevel latent class models

Multilevel cross-classified Latent Class Models are an extension of standard latent class for handling data in which each observation is simultaneously nested within two groups. The likelihood associated to the model is untractable and approximation methods such as stochastic versions of the EM can be applied. The knowledge of a final estimate of the log-likelihood can be helpful in the evaluation of parameter estimates and for selection purposes. We propose two alternative log-likelihood approximation procedures and test their performances in the Hierarchical Multilevel Latent Class Model for which a finite estimate of the likelihood is provided through a special version of the EM.