Models for potentially biased evidence in meta-analysis using empirically based priors

We present models for the combined analysis of evidence from randomized controlled trials categorized as being at either low or high risk of bias due to a flaw in their conduct. We formulate a bias model that incorporates between-study and between-meta-analysis heterogeneity in bias, and uncertainty in overall mean bias. We obtain algebraic expressions for the posterior distribution of the bias-adjusted treatment effect, which provide limiting values for the information that can be obtained from studies at high risk of bias. The parameters of the bias model can be estimated from collections of previously published meta-analyses. We explore alternative models for such data, and alternative methods for introducing prior information on the bias parameters into a new meta-analysis. Results from an illustrative example show that the bias-adjusted treatment effect estimates are sensitive to the way in which the meta-epidemiological data are modelled, but that using point estimates for bias parameters provides an adequate approximation to using a full joint prior distribution. A sensitivity analysis shows that the gain in precision from including studies at high risk of bias is likely to be low, however numerous or large their size, and that little is gained by incorporating such studies, unless the information from studies at low risk of bias is limited. We discuss approaches that might increase the value of including studies at high risk of bias, and the acceptability of the methods in the evaluation of health care interventions.