Meta‐analyzing partial correlation coefficients using Fisher's z transformation

• Published in: Research synthesis methods Vol. 14; no. 5; pp. 768 - 773
• Main Author: van Aert, Robbie C. M.
• Format: Journal Article
• Language: English
• Published: England Wiley 01.09.2023; Wiley Subscription Services, Inc
• Subjects: Correlation; Correlation coefficients; Error analysis; Error of Measurement; Evaluation Methods; Meta Analysis; Normal distribution; Robustness (Statistics); Sampling; Simulation; Transformations; Variance; meta-analysis; sampling variance; Fisher's z transformation; partial correlation coefficient
• ISSN: 1759-2879; 1759-2887; 1759-2887
• DOI: 10.1002/jrsm.1654

The partial correlation coefficient (PCC) is used to quantify the linear relationship between two variables while taking into account/controlling for other variables. Researchers frequently synthesize PCCs in a meta‐analysis, but two of the assumptions of the common equal‐effect and random‐effects meta‐analysis model are by definition violated. First, the sampling variance of the PCC cannot assumed to be known, because the sampling variance is a function of the PCC. Second, the sampling distribution of each primary study's PCC is not normal since PCCs are bounded between −1 and 1. I advocate applying the Fisher's z transformation analogous to applying Fisher's z transformation for Pearson correlation coefficients, because the Fisher's z transformed PCC is independent of the sampling variance and its sampling distribution more closely follows a normal distribution. Reproducing a simulation study by Stanley and Doucouliagos and adding meta‐analyses based on Fisher's z transformed PCCs shows that the meta‐analysis based on Fisher's z transformed PCCs had lower bias and root mean square error than meta‐analyzing PCCs. Hence, meta‐analyzing Fisher's z transformed PCCs is a viable alternative to meta‐analyzing PCCs, and I recommend to accompany any meta‐analysis based on PCCs with one using Fisher's z transformed PCCs to assess the robustness of the results.