Bayesian analysis on meta-analysis of case-control studies accounting for within-study correlation

In retrospective studies, odds ratio is often used as the measure of association. Under independent beta prior assumption, the exact posterior distribution of odds ratio given a single 2x2 table has been derived in the literature. However, independence between risks within the same study may be an oversimplified assumption because cases and controls in the same study are likely to share some common factors and thus to be correlated. Furthermore, in a meta-analysis of case-control studies, investigators usually have multiple 2x2 tables. In this article, we first extend the published results on a single 2x2 table to allow within study prior correlation while retaining the advantage of closed-form posterior formula, and then extend the results to multiple 2x2 tables and regression setting. The hyperparameters, including within study correlation, are estimated via an empirical Bayes approach. The overall odds ratio and the exact posterior distribution of the study-specific odds ratio are inferred based on the estimated hyperparameters. We conduct simulation studies to verify our exact posterior distribution formulas and investigate the finite sample properties of the inference for the overall odds ratio. The results are illustrated through a twin study for genetic heritability and a meta-analysis for the association between the N-acetyltransferase 2 (NAT2) acetylation status and colorectal cancer.