Bayesian analysis for a single 2 x 2 table

The simple comparison of two binomial populations is frequently of interest in epidemiology when the domains are large. For small domains, however, there are no exact methods except Fisher\u27s exact test. A basic problem, therefore, is to compare two populations by assessing the difference between the proportions of individuals who possess a characteristic in the first and second populations. When there is prior information, we take the proportions to have independent conjugate beta distributions with known parameters, thereby facilitating a Bayesian analysis. We consider Bayesian inference on functions of the proportions, and the three most common scalar measures used in epidemiology and health services research, namely relative risk, odds ratio and attributable risk. We develop the highest density regions (both exact and approximate) for relative risk, odds ratio and attributable risk. In addition, we consider the Bayes factor for testing whether the model with a common proportion holds rather than one with distinct proportions. Using data from the population-based Worcester Heart Attack Study, we apply our methodology to study gender differences in the therapeutic management of patients with acute myocardial infarction (AMI) by selected demographic and clinical characteristics. The Bayes factor, the approximate and exact intervals generally suggest that there are no substantial differences in the pharmacologic management of males and females hospitalized with AMI