Abstract
The pooled estimate of the average effect is of primary interest when fitting the random‐effects model for meta‐analysis. However estimates of study specific effects, for example those displayed on forest plots, are also often of interest. In this tutorial, we present the case, with the accompanying statistical theory, for estimating the study specific true effects using so called 'empirical Bayes estimates' or 'Best Unbiased Linear Predictions' under the random‐effects model. These estimates can be accompanied by prediction intervals that indicate a plausible range of study specific true effects. We coalesce and elucidate the available literature and illustrate the methodology using two published meta‐analyses as examples. We also perform a simulation study that reveals that coverage probability of study specific prediction intervals are substantially too low if the between‐study variance is small but not negligible. Researchers need to be aware of this defect when interpreting prediction intervals. We also show how empirical Bayes estimates, accompanied with study specific prediction intervals, can embellish forest plots. We hope that this tutorial will serve to provide a clear theoretical underpinning for this methodology and encourage its widespread adoption.
Highlights
The random-effects model[1,2,3,4,5,6] is routinely used in metaanalyses
We suggest continuing to use the term 'confidence interval' for the conventional intervals already shown on forest plots with the Yi, and using the alternative term 'study specific prediction interval' for those that accompany the θ^i
95% study specific prediction intervals using the Raudenbush variance formula have performed well. They have the defect that they provide under coverage in situations where τ2 is small but non-negligible. This is mainly caused by estimation of τ2, because coverage probabilities are close to the nominal coverage rate if the true τ2 rather than an estimate is used for creating the study specific prediction intervals
Summary
The random-effects model[1,2,3,4,5,6] is routinely used in metaanalyses. This model allows us to make inferences about the average effect whilst relaxing the usually strong, and so difficult to defend, assumption that all studies estimate a common treatment effect. The random-effects model achieves this by incorporating a between-study variance parameter. This parameter quantifies the variation in the studies' estimated effects that is not explained by withinstudy variation alone. A wide variety of estimators for the between-study variance, and corresponding methods for calculating confidence intervals, have been proposed.[7,8] Statistical tests for the absence of between-study
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