A taxonomy of latent structure assumptions for probability matrix decomposition models

A taxonomy of latent structure assumptions (LSAs) for probability matrix decomposition (PMD) models is proposed which includes the original PMD model (Maxis, De Boeck, & Van Mechelen, 1996) as well as a three-way extension of the multiple classification latent class model (Marls, 1999). It is shown that PMD models involving different LSAs axe actually restricted latent class models with latent variables that depend on some external variables. For parameter stimation a combined approach is proposed that uses both a mode-finding algorithm (EM) and a sampling-based approach (Gibbs sampling). A simulation study is conducted to investigate the extent o which information criteria, specific model checks, and checks for global goodness of fit may help to specify the basic assumptions of the different PMD models. Finally, an application is described with models involving different latent structure assumptions for data on hostile behavior in frustrating situations. Key words: discrete data, matrix decomposition, Bayesian analysis, data augmentation, posterior predic-tive check, psychometrics. PMD models were introduced by Maris, De Boeck, and Van Mechelen (1996) to analyze three-way three-mode binary data. The data typically represent associations between two types of elements that are repeatedly observed, for instance, persons who judge whether or not the