Correlated binary data occur very frequently in statistical practice. In many applications, it is reasonable to assume that data from the same cluster are exchangeable. Such data are commonly encountered in cluster sample surveys, teratological experiments, ophthalmologic and otolaryngologic studies, and other clinical trials. The standard methods of analyzing these data include the use of beta-binomial models and generalized estimating equations with third and fourth moments specified by “working matrices.” The focus of these procedures is an estimation of the mean and variance parameters. More information can be obtained when data are exchangeable. By expressing the joint distribution of a set of exchangeable binary random variables in terms of the probability of similar response within cluster, this article introduces a procedure for obtaining maximum likelihood estimates of population parameters such as the marginal means, moments, and correlations of orders two and higher. Applications are made to data sets from a clinical trial and a developmental toxicity study.