We shed much needed light upon a critical assumption that is oft-overlooked---or not considered at all---in random-effects meta-analysis.Namely, that between-study variance is constant across \emph{all} studies which implies they are from the \emph{same} population. Yet it is not hard to imagine a situation where there are several and not merely one population of studies, perhaps differing in their between-study variance (i.e., heteroskedasticity). The objective is to then make inference, given that there are variations in heterogeneity. There is an immediate problem, however, in that modeling heterogeneous variance components is not straightforward to do in a general way. To this end, we propose novel methodology, termed Bayesian location-scale meta-analysis, that can accommodate moderators for both the overall effect (location) and the between-study variance (scale). After introducing the model, we then extend heterogeneity statistics, prediction intervals, and hierarchical shrinkage, all of which customarily assume constant heterogeneity, to include variations therein. With these new tools in hand, we go to work demonstrating that quite literally \emph{everything} changes when between-study variance is not constant across studies. The changes were not small and easily passed the interocular trauma test---the importance hits right between the eyes. Such examples include (but are not limited to) inference on the overall effect, a compromised predictive distribution, and improper shrinkage of the study-specific effects. Further, we provide an illustrative example where heterogeneity was not considered a mere nuisance to show that modeling variance for its own sake can provide unique inferences, in this case into discrimination across nine countries. The discussion includes several ideas for future research. We have implemented the proposed methodology in the {\tt R} package \textbf{blsmeta}.
Fixed effects and variance components estimation in three-level meta-analysis
