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

Abstract

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.

Re-expressing coefficients from regression models for inclusion in a meta-analysis

“…To provide a context for interpretation of our results we have designated an amount of relative bias that we considered important. We note that an acceptable magnitude of bias is often not quantitated in reports like ours (e.g., [ 12 , 13 ]). Reluctance to define general-use cutoffs for acceptable bias is also reflected throughout the epidemiologic literature; for example see ROBINS-E material on confounding [ 14 ].…”

“…How best to synthesize the results of the log-transformed and absolute exposure evidence streams remains an open question and may depend on the scientific discipline, scale of the outcome, and other considerations. In fields such as economics and psychology, meta-analysis of correlation coefficients is a well-recognized approach that could be applied to the evidence synthesis problem discussed here [ 12 ]. Regression coefficients would need to first be re-expressed as correlation coefficients [ 13 ].…”

Multi-View and Multimodal Graph Convolutional Neural Network for Autism Spectrum Disorder Diagnosis

“…The above equation represents the linear correlation between two time series, x and y, of length T. Here, x and y represent the mean time series. After computing the Pearson correlation coefficient between ROIs, the Fisher z transformation [24] is applied to enhance the variance consistency, thus obtaining a functional connectivity matrix for each participant. The same process is applied to the other views of the participant.…”