Allowing for correlations between correlations in random-effects meta-analysis of correlation matrices

Practical meta-analysis of correlation matrices generally ignores covariances (and hence correlations) between correlation estimates. The authors consider various methods for allowing for covariances, including generalized least squares, maximum marginal likelihood, and Bayesian approaches, illustrated using a 6-dimensional response in a series of psychological studies concerning prediction of exercise behavior change. Quantities of interest include the overall population mean correlation matrix, the contrast between the mean correlations, the predicted correlation matrix in a new study, and the conflict between the existing studies and a new correlation matrix. The authors conclude that accounting for correlations between correlations is unnecessary when interested in individual correlations but potentially important if concerned with a composite measure involving 2 or more correlations. A simulation study indicates the asymptotic normal assumption appears reasonable. Because of potential instability in the generalized least squares methods, they recommend a model-based approach, either the maximum marginal likelihood approach or a full Bayesian analysis.