Abstract
In meta‐analysis of odds ratios (ORs), heterogeneity between the studies is usually modelled via the additive random effects model (REM). An alternative, multiplicative REM for ORs 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 ORs 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 ORs 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⩾100. The Mantel‐Haenszel‐based estimator of OR is very biased and is not recommended. The inverse‐variance approach is also somewhat biased for ORs≠1, 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 ORs. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.
Highlights
Meta-analysis aims to combine effects estimated from a number of studies to achieve greater precision of the conclusions
We developed theory of meta-analysis of odds ratios (ORs) based on the BB model
This model is a natural alternative to the standard random effects model (REM) based on normality of random effects
Summary
Meta-analysis aims to combine effects estimated from a number of studies to achieve greater precision of the conclusions. Standard models of meta-analysis are the fixed effect model (FEM) and the random effects model (REM) The former assumes that the LORs θj, j = 1, · · · , K, do not differ across the studies, that is, θj ≡ θ; the latter assumes that the LORs themselves are a random sample from, usually, a normal distribution, θj ∼ N(θ, τ2) with the between-studies variance τ2. The shortcomings of the inverse-variance method, as described earlier, in meta-analysis in general and in its application to the LORs are well known They include the bias in estimation of the combined effect, underestimation of its variance, and poor coverage of the obtained confidence intervals, especially for sparse data and/or small sample sizes, see [2] for discussion and further references. To obtain the combined effect, we study the standard inverse-variance method and a version of the Mantel-Haenszel (MH) method adjusted for clustering, [16] Both methods require estimation of the ICC ρ.
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