Variation in baseline risk as an explanation of heterogeneity in meta-analysis

A relationship of baseline risk to treatment effect size has been suggested as a possible explanation of between-study heterogeneity in meta-analyses. To address this question, we develop regression models to examine the relationship between the logits (or other response measure) in the intervention and control groups. A weighted least squares (WLS) approach is described that allows for the heterogeneous sampling variation in the two groups, together with a correction of the coefficients for sampling error. Two approximate maximum likelihood (ML) solutions are also obtained, with or without an assumption of equal variances between groups within studies. A closed form ML solution exists with the assumption of equal variances. Both methods appear preferable to a previously suggested regression model of the log odds ratio on the control event rate; the methods proposed here use the same scale of measurement for both study groups, and eliminate an artifactual correlation in the regression error structure. The ML approach may be preferable because of its symmetric treatment of study groups, but WLS is more easily implemented with standard software. The methods are illustrated with data from meta-analyses on pre-term delivery and on therapies to lower serum cholesterol.