Modelling pairs of Poissons and binomials with negative correlation

TL;DR

This paper introduces a flexible method for constructing bivariate distributions with specified marginals, allowing for both positive and negative correlations, and applies it to Poisson and binomial data with real-world examples.

Contribution

It develops a new approach to model bivariate distributions with adjustable correlation, extending existing models to include negative correlation for Poisson and binomial variables.

Findings

The method accurately models negative and positive correlations in Poisson data.

Application to seed and plant data shows improved analysis over previous methods.

Meta-analysis demonstrates negative correlation in alcohol use disorder screening data.

Abstract

Suppose $f_1(x)$ and $f_2(y)$ are given marginals for pairs $(x,y)$. I consider the construction $f_1(x)f_2(y)\{ 1+\alpha h_1(x)h_2(y) \}$, 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 $\alpha$, 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'.