Constructing appropriate models for meta-analyses

Meta-analysis combines individual studies or trials to achieve one overall treatment effect estimate and has come a long way since first appearing within medical literature 30 years ago. Most articles examine how best to combine the individual trials and measure the combined estimate. A lot of articles also examine the different sources of variation, between study variation and within study variation, which occur when performing a meta-analysis, and how 'best' to account for the between study variation, the heterogeneity. Very little information however, has been published on the relationship which occurs between the treatment effect estimate and the heterogeneity. Most publications examine these two measures individually, assuming they are independent, however further examination of this relationship brings this assumption of independence into question. We have examined the relationship of the treatment effect estimates and their corresponding heterogeneities for 125 independent meta-analyses using the frequentist approach and note that the results indicate a relationship is present. This relationship will have a resulting effect on how one measures the treatment effect estimates and their corresponding heterogeneity and is something that is considered here, using a Bayesian approach, along with a few other Bayesian modeling approaches. Building on these Bayesian approaches, we consider whether a hierarchical model which would allow a meta-analysis of meta-analyses can be produced