INTRODUCTION

The development of mixture models dates back to the nineteenth century (Newcomb, 1886). In finite mixture models, it is assumed that the observations of a sample arise from two or more unobserved classes, of unknown proportions, that are mixed. The purpose is to unmix the sample and to identify the underlying classes. Mixture models present a model-based approach to clustering. They allow for hypothesis testing and estimation within the framework of standard statistical theory. The mixture model approach to clustering moreover presents an extremely flexible class of clustering algorithms that can be tailored to a very wide range of substantive problems. Mixture models are statistical models, which involve a specific form of the distribution function of the observations in each of the underlying populations (which is to be specified). The distribution function is used to describe the probabilities of occurrence of the observed values of the variable in question. The normal distribution, for example, is the most frequently used distribution for continuous variables that take values in the range of minus infinity to infinity. The binomial distribution describes the probabilities of occurrence of binary (0/1) variables, and the Poisson distribution the probabilities of occurrence of discrete (count) variables. Certain classes of mixture models based on the latter two distributions have become known in the literature as latent class models. Lazarsfeld and Henry (1968) provide one of the first extensive treatments of this topic.