Model determination for categorical data with factor level merging

Summary.  We deal with contingency table data that are used to examine the relationships between a set of categorical variables or factors. We assume that such relationships can be adequately described by the cond`itional independence structure that is imposed by an undirected graphical model. If the contingency table is large, a desirable simplified interpretation can be achieved by combining some categories, or levels, of the factors. We introduce conditions under which such an operation does not alter the Markov properties of the graph. Implementation of these conditions leads to Bayesian model uncertainty procedures based on reversible jump Markov chain Monte Carlo methods. The methodology is illustrated on a 2×3×4 and up to a 4×5×5×2×2 contingency table.