A NOVEL BIVARIATE BETA DISTRIBUTION CONSTRUCTED FROM THE DIRICHLET DISTRIBUTION

Purpose: The beta distribution serves as a prior distribution for a binomial probability. In the case of a bivariate binomial distribution with two probabilities the corresponding prior can be a bivariate beta distribution on the unit square. This distribution can serve as a proper prior for correlated binomial responses. For example, in a Bayesian setting it can be used to model the sensitivities (specificities) of two index tests, based on the cross-classification of test results in a group of people with (without) disease. 

Method: We seek a prior distribution for the bivariate binomial distribution that (i) has beta marginal distributions, (ii) has support on the unit square, and (iii) allows positive and negative correlations throughout the whole range (-1, 1). Existing bivariate beta distribution families such as the Farlie–Gumbel–Morgenstern and Sarmanov families only allow for narrower correlation ranges.  

We use an additive construction scheme: Let U₁₁, U_10, U₀₁ have a Dirichlet distribution. We define the additive version X= U₁₁+U₁₀ and Y=U₀₁+U₀₁; then X, Y have a bivariate beta distribution. We derive the joint density of (X, Y), show that it satisfies (i)-(iii), and derive central moments. We present algorithms for fitting the joint density to data and for Bayesian inference.

Result: The proposed distribution is defined by 4 positive parameters. Shown are example densities when the parameters are (1, 1, 1, 1); (4, 2, 4, 1); and (2, 2, 2, 0.5), respectively:

We demonstrate use by modeling the sensitivities (specificities) of two ultrasonographic markers for detecting trisomy 21 in liveborn infants. Extension to k dimensions follows the same construction and the k-dimensional joint distribution is defined by 2^k parameters.

Conclusion: We provide a bivariate beta distribution using an additive construction scheme that allows correlations in the full range (-1, 1). This is an alternative to the Farlie–Gumbel–Morgenstern, Plackett, Mardia, and Sarmanov bivariate beta distribution families, which can have a much more restrictive correlation range.