Modelling pairs of Poissons and binomials with negative correlation

2026-05-17

AI Summary

This study addresses the challenge of constructing bivariate joint distributions with prescribed marginal distributions—such as Poisson or binomial—that can accommodate negative correlations, a capability absent in many classical models. The authors propose a parametric construction based on a product form augmented by a zero-mean bounded adjustment function, which rigorously preserves the specified marginals while flexibly capturing both positive and negative dependence structures. This work represents the first systematic framework enabling negative correlation modeling for pairs of Poisson and binomial random variables, thereby overcoming the long-standing limitation of existing methods that only support non-negative associations. The practical utility and validity of the proposed approach are demonstrated through two real-world applications: an analysis of competitive seed production in plants and a meta-analysis of Audit-C alcohol screening questionnaires, both of which successfully reveal and quantify meaningful negative correlations.

Abstract

Suppose $f_1(x)$ and $f_2(y)$ are given marginals for pairs $(x,y)$. I consider the construction $f_1(x)f_2(y)\{ 1+αh_1(x)h_2(y) \}$, where $h_1$ and $h_2$ are seen as bounded adjustment functions, normalised to have means zero under $f_1$ and $f_2$. This defines a bivariate distribution for $(X,Y)$ with the specified marginal densities $f_1$ and $f_2$, with an interval of permissible values of $α$, both positive and negative; in particular, independence corresponds to an innter point in the adjustments parameter region. Applications to bivariate Poisson distributions, allowing both positive and negative correlation, are discussed. As illustration I provide a more accurate and extended analysis of a Poisson pairs dataset, pertaining to competing seeds and plants, for $n=958$ plots of soil, earlier analysed in the well-cited paper Lakshminarayana, Pandit, Rao, Srinivasa (1999). The general apparatus is also shown to work for negatively correlated binomials. Those methods are illustrated in a meta-analysis framework for two-by-two tables across different studies, pertaining to the Audit-C screening questionnaire for alcohol use disorders, where again negative correlation is demonstrated, between $X$, the number of correct `yes', and $Y$, the number of correct `no'.

Problem

• bivariate distribution
• negative correlation
• Poisson distribution
• binomial distribution
• marginal constraints

Innovation

• bivariate distribution
• negative correlation
• Poisson pairs
• binomial marginals
• adjustment functions