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Zero patterns in multi-way binary contingency tables with uniform margins

Roberto Fontana, Elisa Perrone, Fabio Rapallo

TL;DR

The paper tackles the problem of transforming a $d$-dimensional binary contingency table into an equivalent table with uniform margins while preserving the dependence structure encoded by conditional odds ratios. It develops a theory linking discrete copulas, transportation polytopes, and IPFP, and provides zero-pattern–dependent existence criteria alongside methods (extreme rays and linear programming) to determine feasibility for arbitrary $d$-way tables. A key finding is that conditional odds ratios alone may be insufficient to determine the transformed table in the presence of zeros, necessitating higher-order odds-ratio constraints to guarantee uniqueness. The work offers practical computational routes for testing existence and constructing transformed tables, with implications for margin-free discrete dependence modeling and future algebraic-statistics developments.

Abstract

We study the problem of transforming a multi-way contingency table into an equivalent table with uniform margins and same dependence structure. This is an old question which relates to recent advances in copula modeling for discrete random vectors. In this work, we focus on multi-way binary tables and develop novel theory to show how the zero patterns affect the existence of the transformation as well as its statistical interpretability in terms of dependence structure. The implementation of the theory relies on combinatorial and linear programming techniques, which can also be applied to arbitrary multi-way tables. In addition, we investigate which odds ratios characterize the unique solution in relation to specific zero patterns. Several examples are described to illustrate the approach and point to interesting future research directions.

Zero patterns in multi-way binary contingency tables with uniform margins

TL;DR

The paper tackles the problem of transforming a -dimensional binary contingency table into an equivalent table with uniform margins while preserving the dependence structure encoded by conditional odds ratios. It develops a theory linking discrete copulas, transportation polytopes, and IPFP, and provides zero-pattern–dependent existence criteria alongside methods (extreme rays and linear programming) to determine feasibility for arbitrary -way tables. A key finding is that conditional odds ratios alone may be insufficient to determine the transformed table in the presence of zeros, necessitating higher-order odds-ratio constraints to guarantee uniqueness. The work offers practical computational routes for testing existence and constructing transformed tables, with implications for margin-free discrete dependence modeling and future algebraic-statistics developments.

Abstract

We study the problem of transforming a multi-way contingency table into an equivalent table with uniform margins and same dependence structure. This is an old question which relates to recent advances in copula modeling for discrete random vectors. In this work, we focus on multi-way binary tables and develop novel theory to show how the zero patterns affect the existence of the transformation as well as its statistical interpretability in terms of dependence structure. The implementation of the theory relies on combinatorial and linear programming techniques, which can also be applied to arbitrary multi-way tables. In addition, we investigate which odds ratios characterize the unique solution in relation to specific zero patterns. Several examples are described to illustrate the approach and point to interesting future research directions.

Paper Structure

This paper contains 10 sections, 5 theorems, 49 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background, notation, and relevance to the state-of-the-art
  3. Table transformations and discrete copulas
  4. Relations with the classical theory of log-linear models
  5. Impact of zeros in the existence of the transformed multi-way table
  6. $d$-way binary tables
  7. Examples in the $2^3$ case
  8. Extension of the approach to arbitrary $d$-way tables
  9. The role of conditional odds ratios in determining the unique transformed table
  10. Conclusions

Key Result

Theorem 2.1

Let $\mathbf{p}$ be in the set $\mathcal{P}_{R \times S}$ of all $(R \times S)$ probability distributions. We define $\text{Supp}(\mathbf{p})=\{(i,j) \in [R] \times [S] \text{ s.t. } p_{i,j}>0\}$, the support of $\mathbf{p}$, and $\text{N}(\mathbf{p})=\{(v_{X_1} \times v_{X_2}): v_{X_1} \subset [R],

Figures (4)

  • Figure 1: Bubble plot of original data (a) and transformed data (b) for the Sheffield smallpox epidemic data in Table \ref{['tab:id7']}. Part (c) represents a possible zero pattern in the observed table that motivates the investigations in this paper.
  • Figure 2: Two $2^4$ tables with zero patterns. The first table in the upper panel, the second table in the lower panel. The variables$X_1, X_2, X_3$ are the coordinates of the cubes, the variable $X_4$ divides the table into two cubes. Red dots denote zero probabilities. To ease readability, vertex labels are omitted.
  • Figure 3: Configurations with 1 or 2 zeros in the $2^3$ table. Red dots denote zero probabilities. (a): one zero; (b): two zeros on the same edge; (c): two zeros in diagonal position on a face; (d): two zeros in different faces.
  • Figure 4: The general probability table in the $2^3$ case with two zeros in the diagonal of a face.

Theorems & Definitions (10)

  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Example 3.5
  • Example 4.1
  • Example 4.2