Beta-binomial model for meta-analysis of odds ratios

In meta-analysis of odds ratios (${\OR}s$), heterogeneity between the studies is usually modelled via the additive random effects model (REM). An alternative, multiplicative random effects model for ${\OR}s$ uses overdispersion. The multiplicative factor in this overdispersion model (ODM) can be interpreted as an intra-class correlation (ICC) parameter. This model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta-distributed, resulting in beta-binomial distributions. We propose two new estimators of the ICC for meta-analysis in this setting. One is based on the inverted Breslow-Day test, and the other on the improved gamma approximation by Kulinskaya and Dollinger (2015, p. 26) to the distribution of Cochran's $Q$. The performance of these and several other estimators of ICC on bias and coverage is studied by simulation. Additionally, the Mantel-Haenszel approach to estimation of odds ratios is extended to the beta-binomial model, and we study performance of various ICC estimators when used in the Mantel-Haenszel or the inverse-variance method to combine odds ratios in meta-analysis. The results of the simulations show that the improved gamma-based estimator of ICC is superior for small sample sizes, and the Breslow-Day-based estimator is the best for $n\geq100$. The Mantel-Haenszel-based estimator of ${\OR}$ is very biased and is not recommended. The inverse-variance approach is also somewhat biased for ${\OR}s\neq1$, but this bias is not very large in practical settings. Developed methods and R programs, provided in the Web Appendix, make the beta-binomial model a feasible alternative to the standard REM for meta-analysis of odds ratios.