The exact distribution of Cochran's heterogeneity statistic in one-way random effects meta-analysis

The presence and impact of heterogeneity in the standard one-way random effects model in meta-analysis are often assessed using the Q statistic due to Cochran. We derive the exact distribution of this statistic under the assumptions of the random effects model, and also suggest two moment-based approximations and a saddlepoint approximation for Q. The exact and approximate distributions are then applied to obtain the corresponding distributions of the recently proposed heterogeneity measures I(2) and H(M)(2), the power of the standard test for the presence of heterogeneity and confidence intervals for the between-study variance parameter when the DerSimonian-Laird or the Hartung-Makambi estimator is used. The methodology is illustrated by revisiting a recent simulation study concerning the heterogeneity measures and applying all the proposed methods to four published meta-analyses.