On mathematical optimization for clustering categories in contingency tables

Many applications in data analysis study whether two categorical variables are independent using a function of the entries of their contingency table. Often, the categories of the variables, associated with the rows and columns of the table, are grouped, yielding a less granular representation of the categorical variables. The purpose of this is to attain reasonable sample sizes in the cells of the table and, more importantly, to incorporate expert knowledge on the allowable groupings. However, it is known that the conclusions on independence depend, in general, on the chosen granularity, as in the Simpson paradox. In this paper we propose a methodology to, for a given contingency table and a fixed granularity, find a clustered table with the highest \(\chi ^2\) statistic. Repeating this procedure for different values of the granularity, we can either identify an extreme grouping, namely the largest granularity for which the statistical dependence is still detected, or conclude that it does not exist and that the two variables are dependent regardless of the size of the clustered table. For this problem, we propose an assignment mathematical formulation and a set partitioning one. Our approach is flexible enough to include constraints on the desirable structure of the clusters, such as must-link or cannot-link constraints on the categories that can, or cannot, be merged together, and ensure reasonable sample sizes in the cells of the clustered table from which trustful statistical conclusions can be derived. We illustrate the usefulness of our methodology using a dataset of a medical study.