Characterization of multi-way binary tables with uniform margins and fixed correlations

TL;DR

This paper develops a geometric framework to characterize all possible joint distributions of multi-way binary tables with fixed pairwise correlations and uniform margins, revealing the structure of higher-order dependencies.

Contribution

It introduces a convex polytope model for the admissible set of distributions, analyzing its symmetry and extremal points, which was not previously available.

Findings

The admissible set forms a convex polytope.

Extreme rays characterize fundamental higher-order dependencies.

Framework enables exploration of the full dependence space.

Abstract

In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint distribution, but instead define a family of admissible distributions that share the same pairwise dependence while potentially differing in higher-order interactions. In this paper, we introduce a geometric framework to describe the entire feasible set of such joint distributions with uniform margins. We show that this admissible set forms a convex polytope, analyze its symmetry properties, and characterize its extreme rays. These extremal distributions provide fundamental insights into how higher-order dependence structures may vary while preserving the prescribed pairwise information. Unlike traditional methods for table generation, which return a single table, our framework makes it possible to explore and understand the full admissible space of dependence structures, enabling more flexible choices for modeling and simulation. We illustrate the usefulness of our theoretical results through examples and a real case study on rater agreement.