Random‐effects meta‐analysis of correlations: Monte Carlo evaluation of mean estimators

Several authors have cautioned against using Fisher's z‐transformation in random‐effects meta‐analysis of correlations, which seems to perform poorly in some situations, especially with substantial inter‐study heterogeneity. Attributing this performance largely to the direct z‐to‐r transformation (DZRT) of Fisher z results (e.g. point estimate of mean correlation), in a previous paper Hafdahl (2009) proposed point and interval estimators of the mean Pearson r correlation that instead use an integral z‐to‐r transformation (IZRT). The present Monte Carlo study of these IZRT Fisher z estimators includes comparisons with their DZRT counterparts and with estimators based on Pearson r correlations. The IZRT point estimator was usually more accurate and efficient than its DZRT counterpart and comparable to the two Pearson r point estimators – better in some conditions but worse in others. Coverage probability for the IZRT confidence intervals (CIs) was often near nominal, much better than for the DZRT CIs, and comparable to coverage for the Pearson r CIs; every approach's CI fell markedly below nominal in some conditions. The IZRT estimators contradict warnings about Fisher z estimators' poor performance. Recommendations for practising research synthesists are offered, and an Appendix provides computing code to implement the IZRT as in the real‐data example.