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 2 × 2 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 2 × 2 tables. In this article, we first extend the published results on a single 2 × 2 table to allow within study prior correlation while retaining the advantage of closed-form posterior formula, and then extend the results to multiple 2 × 2 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.
Bibliographical noteFunding Information:
Views expressed in this paper are the author's professional opinions and do not necessarily represent the official positions of the US Food and Drug Administration. Yong Chen's research was partially supported by a start-up fund from the University of Texas School of Public Health. Haitao Chu was supported in part by the US Department of Health and Human Services Agency for Healthcare Research and Quality Grant R03HS020666 and P01CA142538 from the U.S. National Cancer Institute. Sheng Luo's research was partially supported by two NIH/NINDS grants: U01 NS043127 and U01 NS43128. We gratefully acknowledge the editor, Dr. Brian Everitt, and an anonymous reviewer for constructive comments that greatly improved the manuscript. We also want to thank Peng Wei for the helpful comments.
- Sarmanov family
- bivariate beta-binomial model
- exact method
- hypergeometric function
- odds ratio