Improved Fisher z estimators for univariate random-effects meta-analysis of correlations

Several authors have studied or used the following estimation strategy for meta-analysing correlations: obtain a point estimate or confidence interval for the mean Fisher z correlation, and transform this estimate to the Pearson r metric. Using the relationship between Fisher z and Pearson r random variables, I demonstrate the potential discrepancy induced by directly z-to-r transforming a mean correlation parameter. Point and interval estimators based on an alternative integral z-to-r transformation are proposed. Analytic expressions for the expectation and variance of certain meta-analytic point estimators are also provided, as are selected moments of correlation parameters; numerical examples are included. In an application of these analytic results, the proposed point estimator outperformed its usual direct z-to-r counterpart and compared favourably with an estimator based on Pearson r correlations. Practical implications, extensions of the proposed estimators, and uses for the analytic results are discussed.