Capturing the underlying distribution in meta-analysis: Credibility and tolerance intervals.

Tolerance intervals provide a bracket intended to contain a percentage (e.g., 80%) of a population distribution given sample estimates of the mean and variance. In random-effects meta-analysis, tolerance intervals should contain researcher-specified proportions of underlying population effect sizes. Using Monte Carlo simulation, we investigated the coverage for five relevant tolerance interval estimators: the Schmidt-Hunter credibility intervals, a prediction interval, two content tolerance intervals adapted to meta-analysis, and a bootstrap tolerance interval. None of the intervals contained the desired percentage of coverage at the nominal rates in all conditions. However, the prediction worked well unless the number of primary studies was small (<30), and one of the content tolerance intervals approached nominal levels with small numbers (<20) of primary studies. The bootstrap tolerance interval achieved near nominal coverage if there were sufficient numbers of primary studies (30+) and large enough sample sizes (N ≅ 70) in the included primary studies, although it slightly exceeded nominal coverage with large numbers of large-sample primary studies. Next, we showed the results of applying the intervals to real data using a set of previously published analyses and provided suggestions for practice. Tolerance intervals incorporate error of estimation into the construction of proper brackets for fractions of population true effects. In many contexts, such intervals approach the desired nominal levels of coverage.