Posterior Invariance of Multiplicative Contrasts under Margin Constraints in Contingency Tables

TL;DR

This paper investigates when posterior inference for generalized odds ratios in contingency tables remains unaffected by fixing margins, revealing invariance conditions based on contrast coefficients.

Contribution

It provides a precise characterization of invariance conditions for posterior distributions of multiplicative contrasts under different margin constraints in Bayesian analysis.

Findings

Posterior distribution of a generalized odds ratio is invariant if contrast coefficients sum to zero within the margin.

Invariance holds under mild assumptions about prior independence and model structure.

The results generalize classical invariance properties of odds ratios to broader contrast classes.

Abstract

Measures of association in contingency tables, such as odds ratios and their generalizations, are often studied under different sampling schemes that either fix or leave random the margins of the table. While classical results show that certain odds ratios are unaffected by constraining the margins, it is less clear when this invariance holds more generally. This paper studies posterior inference for a broad class of multiplicative contrasts of multinomial cell probabilities, which we refer to as generalized odds ratios, and addresses exactly when fixing a margin alters inference about them. We consider Bayesian inference under multinomial sampling and under models in which partition sums of the table are fixed in advance, and assume that the marginal and conditional parameters are independent a priori. Under additional mild assumptions, we show that the posterior distribution of a generalized odds ratio is invariant to fixing a margin if and only if the coefficients defining the contrast sum to zero within the margin.