Who's Afraid of the Wallenius Distribution?

TL;DR

This paper explores the application of the Wallenius noncentral hypergeometric distribution for analyzing contingency tables with fixed margins, introducing new inference methods and a novel MCMC algorithm for Bayesian analysis.

Contribution

It presents likelihood-based and Bayesian inference approaches for Wallenius distribution analysis, including a new sphere walk Metropolis algorithm for Bayesian inference.

Findings

Analysis of 2x2 tables is straightforward with odds ratio interpretation.

Multigroup tables require nuanced numerical optimization techniques.

The sphere walk Metropolis algorithm enables efficient Bayesian inference.

Abstract

This paper is about the use of the Wallenius noncentral hypergeometric distribution for analysing contingency tables with two or more groups and two categories and with row margins and sample size, that is both margins, fixed. The parameters of the distribution are taken to be weights which are positive and sum to one and are thus defined on a regular simplex. The approach to analysis is presented for likelihood-based and Bayesian inference and is illustrated by example, with datasets taken from the literature and, in one case, used to generate semi-synthetic data. The analysis of two-by-two contingency tables using the univariate Wallenius distribution is shown to be straightforward, with the parameter a single weight which translates immediately to the requisite odds and the odds ratio. The analysis of contingency tables with more than two groups based on the multivariate Wallenius distribution was however more nuanced than that of the two-group tables. Specifically, some numerical subtleties were required in order to implement the necessary calculations. In particular, optimisation with respect to the weights was performed by transforming the weights to yield an unconstrained optimisation problem and likelihoods which are extremely small were scaled by an appropriate multiplying factor without compromising the elements of inference. Furthermore, a novel Markov chain Monte Carlo algorithm for Bayesian inference, termed the sphere walk Metropolis, was constructed. The proposal is implemented in Cartesian coordinates on the reference simplex and the Metropolis filter in barycentric coordinates on the regular simplex, with the transition between barycentric and Cartesian coordinates effected seamlessly.