UMMS Affiliation

Department of Population and Quantitative Health Sciences

Publication Date


Document Type



Biostatistics | Epidemiology | Quantitative, Qualitative, Comparative, and Historical Methodologies | Statistics and Probability


For meta-analysis of studies that report outcomes as binomial proportions, the most popular measure of effect is the odds ratio (OR), usually analyzed as log(OR). Many meta-analyses use the risk ratio (RR) and its logarithm because of its simpler interpretation. Although log(OR) and log(RR) are both unbounded, use of log(RR) must ensure that estimates are compatible with study-level event rates in the interval (0, 1). These complications pose a particular challenge for random-effects models, both in applications and in generating data for simulations. As background, we review the conventional random-effects model and then binomial generalized linear mixed models (GLMMs) with the logit link function, which do not have these complications. We then focus on log-binomial models and explore implications of using them; theoretical calculations and simulation show evidence of biases. The main competitors to the binomial GLMMs use the beta-binomial (BB) distribution, either in BB regression or by maximizing a BB likelihood; a simulation produces mixed results. Two examples and an examination of Cochrane meta-analyses that used RR suggest bias in the results from the conventional inverse-variance-weighted approach. Finally, we comment on other measures of effect that have range restrictions, including risk difference, and outline further research.


beta-binomial model, log-binomial model, relative risk, response ratio, risk difference

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© 2019 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI of Published Version



Res Synth Methods. 2019 Mar 10. doi: 10.1002/jrsm.1347. [Epub ahead of print] Link to article on publisher's site

Journal/Book/Conference Title

Research synthesis methods

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Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.