Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Roberto Fontana, Elisa Perrone, Fabio Rapallo

TL;DR

The paper addresses the non-uniqueness of joint distributions for $d$ binary variables given only pairwise correlations by introducing a geometric framework that characterizes the full set of joint distributions with uniform margins. It shows the feasible space forms a convex polytope $\mathcal{F}(H_d)$ generated by extreme rays of the cone $\mathcal{C}(H_d)$ and develops symmetry properties that pair extreme rays, enabling exploration of higher-order dependence while preserving pairwise information. The contributions include a precise polyhedral characterization, algorithms for computing extreme pmfs using tools like $4ti2$, and demonstrations on a four-way table and a rater-agreement dataset to illustrate how higher-order interactions can vary within the admissible space. This framework provides a flexible, principled basis for simulation and model selection when only pairwise dependencies are known, with practical implications for avoiding unintended higher-order constraints in contingency-table analyses.

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.

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

TL;DR

The paper addresses the non-uniqueness of joint distributions for binary variables given only pairwise correlations by introducing a geometric framework that characterizes the full set of joint distributions with uniform margins. It shows the feasible space forms a convex polytope generated by extreme rays of the cone and develops symmetry properties that pair extreme rays, enabling exploration of higher-order dependence while preserving pairwise information. The contributions include a precise polyhedral characterization, algorithms for computing extreme pmfs using tools like , and demonstrations on a four-way table and a rater-agreement dataset to illustrate how higher-order interactions can vary within the admissible space. This framework provides a flexible, principled basis for simulation and model selection when only pairwise dependencies are known, with practical implications for avoiding unintended higher-order constraints in contingency-table analyses.

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.
Paper Structure (10 sections, 3 theorems, 45 equations, 10 tables)

This paper contains 10 sections, 3 theorems, 45 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Two-way binary tables
  4. Multi-way binary tables
  5. Algorithms for sampling from correlated binary random variables
  6. Geometric characterization of multi-way binary tables with uniform margins and fixed correlations
  7. Illustrative examples and applications
  8. Four-way binary table: geometric structure and extreme rays
  9. Empirical application on a rater agreement problem
  10. Discussion and future work

Key Result

Proposition 1

Let $(X_1,\ldots,X_d)$ be a multivariate Bernoulli random variable with distribution $p=(p_\alpha, \alpha \in \mathcal{D})$, uniform margins and given second-order moments $\mu_{12},\mu_{13},\ldots,\mu_{d-1,d}$. Then, the multivariate Bernoulli random variable $(X'_1,\ldots,X'_d)$ with distribution

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Example 1