A Bayesian Hierarchical Model for 2-by-2 Tables with Structural Zeros

Correlated binary data in a 2x2 table has been analyzed from both the frequentist and Bayesian perspectives, but a fully Bayesian hierarchical model has not been proposed. This is a commonly used model for correlated proportions when considering, for example, a diagnostic test performance where negative subjects are tested a second time. We consider a new hierarchical Bayesian model for the parameters resulting from a 2x2 table with a structural zero. We investigate the performance of the hierarchical model via simulation. We then illustrate the usefulness of the model by showing how a set of historical studies can be used to build a predictive distribution for a new study that can be used as a prior distribution for both the rate ratio and marginal probability of a positive test. We then show how the prior based on historical 2x2 tables can be used to power a future study that accounts for pre-experimental uncertainty. High quality prior information can lead to better decision making by improving precision in estimation and by providing realistic numbers to power studies.