Avoiding zero between-study variance estimates in random-effects meta-analysis.

Fixed-effects meta-analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random-effects meta-analysis and meta-regression are therefore typically used to accommodate explained and unexplained between-study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between-study standard deviation, resulting in fixed-effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between-study standard deviation. When no prior information is available regarding the magnitude of the between-study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta-analysis and meta-regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between-study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity.