Constrained Latent Class Analysis of Three-Way Three-Mode Data

The latent class model for two-way two-mode data can easily be extended to the case of three-way three-mode data as a tool to cluster the elements of one mode on the basis of two other modes simultaneously. However, as the number of manifest variables is typically very large in this type of analysis, the number of conditional probabilities rapidly increases with the number of latent classes, which may lead to an overparameter-ized model. To solve this problem, we introduce a class of constrained latent class models in which the conditional probabilities are a nonlinear function of basic parameters that pertain to each of the two modes. The models can be regarded as a probabilistic extension of related deterministic models or as a generalization of related probabilistic models. For parameter estimation, an EM algorithm can be used to locate the posterior mode, and a Gibbs sampling algorithm can be used to compute a sample of the posterior distribution. Furthermore, model selection criteria and measures to check the fit of the model are discussed. Finally the models are applied to study the types of reactions that occur when one is angry at a person in a certain situation.