Comparing three partitions: an inferential approach based on multi-way contingency tables

Abstract

A number of indices have been suggested in the classification and in the psychometric literature for the purpose of comparing partitions. However, interest has mainly focused on the two-dimensional case. In this paper we tackle the problem of comparing three partitions. We follow the approach developed by Hubert and Arabie (1985) and exploit the similarity between partition comparison and the analysis of multi-way contingency tables. We adopt an inferential point of view and consider several hypotheses of independence among clusterings, including less "extreme" null models (e.g. conditional independence) which can be useful in practice. As is customary in this setting, we condition on both the number of clusters and the number of objects in each cluster, and we derive some exact results for our statistics. An illustrative application of the method is also presented.

Keywords:

• agreement
• cluster analysis
• Gibbs sampling
• Rand index
• randomization tests
• three-way tables