Conventional meta-analyses (both fixed and random effects) of correlations are biased due to the correlation between the estimated correlation and its standard error. Simulations that are closely calibrated to match actual research conditions widely seen across correlational studies in psychology corroborate these biases and suggest two solutions: UWLS+3 and HS. UWLS+3 is a simple inverse-variance weighted average (the unrestricted weighted least squares) that adjusts the degrees of freedom and thereby reduces small-sample bias to scientific negligibility. UWLS+3 as well as the Hunter and Schmidt approach (HS) are less biased than conventional random-effects estimates of correlations and Fisher’s z, whether or not there is publication selection bias. However, publication selection bias remains a ubiquitous source of bias and false positive findings. Despite the correlation between the estimated correlation and its standard error even in the absence of any selective reporting, the precision-effect test/precision-effect estimate with standard error (PET-PEESE) nearly eradicates publication selection bias. Surprisingly, PET-PEESE keeps the rate of false positives (i.e., type I errors) within their nominal levels under the typical conditions widely seen across psychological research whether there is publication selection bias, or not.

Reference: T. D. Stanley, Hristos Doucouliagos, and Tomas Havranek (2024): "Reducing the biases of the conventional meta-analysis of correlations." Available at meta-analysis.cz/correlations.