Abstract
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.
Highlights
For meta-analysis of studies that report binary outcomes, the most popular measure of effect is the odds ratio (OR), usually analyzed in the log scale, as the difference in log-odds between the two groups
We have focused on the risk ratio (RR), but other measures of effect have range restrictions, including risk difference, re√sponse ratio, and arcsin( p) for binomial proportions; methods for these need further research
We find the log-binomial linear mixed models (LMMs) with fixed αj not suitable for modeling the RR
Summary
For meta-analysis of studies that report binary outcomes (usually summarized as the number of subjects who had an event and the number who had no event, in a treatment group and a control group), the most popular measure of effect is the odds ratio (OR), usually analyzed in the log scale, as the difference in log-odds between the two groups. We discuss the role of the restrictions in models for fixed-effect and, especially, random-effects meta-analysis, examine their impact on generation of data for simulation studies and on the results, and deduce their likely contribution to bias in examples and in a sizable number of Cochrane reviews. Using a collection of 1286 meta-analyses of RR, in a 2004 Cochrane Library issue, we explore (in Section 6) several practical implications of the restriction on the range of the binomial rates (represented by truncation of the distribution of random effects). We have focused on the RR, but other measures of effect have range restrictions, including risk difference, re√sponse ratio (ie, the log of the ratio of means), and arcsin( p) for binomial proportions; methods for these need further research
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