A hierarchical mixture model for clustering three-way data sets

Three-way data sets occur when various attributes are measured for a set of observational units in different situations. Examples are genotype by environment by attribute data obtained in a plant experiment, individual by time point by response data in a longitudinal study, and individual by brand by attribute data in a market research survey. Clustering observational units (genotypes/individuals) by means of a special type of the normal mixture model has been proposed. An implicit assumption of this approach is, however, that observational units are in the same cluster in all situations. An extension is presented that makes it possible to relax this assumption and that because of this may yield much simpler clustering solutions. The proposed extension—which includes the earlier model as a special case—is obtained by adapting the multilevel latent class model for categorical responses to the three-way situation, as well as to the situation in which responses include continuous variables. An efficient EM algorithm for parameter estimation by maximum likelihood is described and two empirical examples are provided.