A full likelihood procedure for analysing exchangeable binary data

A full-likelihood procedure is proposed for analyzing correlated binary data under the assumption of exchangeability. The binomial and beta-binomial models are shown to occur as special cases correspondingly, respectively, to the choice of degenerate and beta-mixing distributions. For a finite exchangeable binary sequence of random variables, expressions for the joint distribution, moments, and correlations of all orders are derived. Maximum likelihood estimates of the moments of all orders are computed and used to estimate correlations and the distribution of the number of responses in a cluster. In an application to developmental toxicology data analysis, the procedure introduced is compared with a beta-binomial and a generalized estimating equation procedure in which mean response and intralitter correlation are linked to dose.