Analysing the relationship between treatment effect and underlying risk in meta-analysis: comparison and development of approaches.

Three approaches for estimating the relationship between treatment effect and underlying risk in a meta-analysis of clinical trials have recently been published. The aim of each is to overcome the bias inherent in conventional regressions of treatment effect on control group risk, which arises from the measurement error in the observed control group risks in different trials. Here we describe these published approaches, and compare them with respect to their underlying models and methods of implementation. The underlying model for one of them is shown to be seriously flawed, while the other two are both statistically more appropriate than the conventional approaches, and differ from each other in only two assumptions. Both may be implemented using the Gibbs sampling algorithm in BUGS, and are exemplified here using a meta-analysis of mortality and bleeding data in trials of sclerotherapy for patients with cirrhosis. One approach is developed further; for the illustrative example considered, it is shown to be robust to different choices of prior distributions for the model parameters, and to the assumption of a linear relationship on a log-odds scale. It can also be used to estimate the level of underlying risk (and its standard error) at which the treatment effect crosses from benefit to harm, and other trial-level covariates may be included in the model as confounders. The BUGS code is provided in an Appendix, to enable applied researchers to perform the various analyses described.