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TRA-2-3
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 *U*_{11},* U*_{10,}* U*_{01}
have a Dirichlet distribution. We define the additive
version *X= U*_{11}+*U*_{10} and *Y*=*U*_{01}+*U*_{01}*; *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.