Modelling pairs of Poissons and binomials with negative correlation

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’.