Maximal Interaction Two-Mode Clustering

Most classical approaches for two-mode clustering of a data matrix are designed to attain homogeneous row by column clusters (blocks, biclusters), that is, biclusters with a small variation of data values within the blocks. In contrast, this article deals with methods that look for a biclustering with a large interaction between row and column clusters. Thereby an aggregated, condensed representation of the existing interaction structure is obtained, together with corresponding row and column clusters, which both allow a parsimonious visualization and interpretation. In this paper we provide a statistical justification, in terms of a probabilistic model, for a two-mode interaction clustering criterion that has been proposed by Bock (1980). Furthermore, we show that maximization of this criterion is equivalent to minimizing the classical least-squares two-mode partitioning criterion for the double-centered version of the data matrix. The latter implies that the interaction clustering criterion can be optimized by applying classical two-mode partitioning algorithms. We illustrate the usefulness of our approach for the case of an empirical data set from personality psychology and we compare this method with other biclustering approaches where interactions play a role.