Allowing for Correlations Between Correlations in Random-Effects Meta-Analysis of Correlation Matrices A. Toby Prevost University of Cambridge and MRC Biostatistics Unit Dan Mason University of Cambridge Simon Griffin MRC Epidemiology Unit Ann-Louise Kinmonth and Stephen Sutton University of Cambridge David Spiegelhalter MRC Biostatistics Unit Practical meta-analysis of correlation matrices generally ignores covariances (and hence correlations) between correlation estimates. The authors consider various methods for allow- ing 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 impor- tant 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 ap- proach, either the maximum marginal likelihood approach or a full Bayesian analysis. Keywords: correlations between correlations, Bayesian analysis, maximum marginal likeli- hood, conflict, theory of planned behavior Studies that evaluate explanatory models of psychological and behavioral outcomes often lead to a correlation matrix either as an endpoint or as a precursor to a summary of evidence. Researchers in disciplines such as health, educa- tional, and occupational psychology are often encouraged to report the full correlation matrix. If a number of similar studies have been carried out on a specific topic, it is natural to wish to combine their findings regarding a particular outcome in a meta-analysis as in, for example, Hagger, Chatzisarantis, and Biddle (2002); Hausenblas, Carron, and Mack (1997); and Podsakoff, Bommer, Podsakoff, and MacKenzie (2006). However, where correlation matrix information is avail- able across a set of similar studies, there is an opportunity to A. Toby Prevost, General Practice and Primary Care Research Unit, University of Cambridge, Institute of Public Health, Cam- bridge, United Kingdom, and Medical Research Council (MRC) Biostatistics Unit, Institute of Public Health, Cambridge, United Kingdom; Dan Mason, Ann-Louise Kinmonth, and Stephen Sut- ton, General Practice and Primary Care Research Unit, University of Cambridge, Institute of Public Health, Cambridge, United King- dom; Simon Griffin, MRC Epidemiology Unit, Institute of Meta- bolic Science, Cambridge, United Kingdom; David Spiegelhalter, MRC Biostatistics Unit, Institute of Public Health, Cambridge, United Kingdom. A. Toby Prevost was funded by the UK MRC (MRC Cooper- ative Grant G0200391, Development and Evaluation of Innovative Strategies for the Prevention of Chronic Disease in Primary Care) and the UK National Health Service Research and Development. Dan Mason was funded by UK Nuffield Foundation Open Door Project Grant RG40462. Software code to run analyses described in this article is avail- able from http://www.phpc.cam.ac.uk/cams/toby/metaanalysis. Correspondence concerning this article should be addressed to A. Toby Prevost, Institute of Public Health, Robinson Way, Cam- bridge CB2 0SR, United Kingdom. E-mail: toby.prevost @phpc.cam.ac.uk Psychological Methods 2007, Vol. 12, No. 4, 434 – 450 Copyright 2007 by the American Psychological Association 1082-989X/07/$12.00 DOI: 10.1037/1082-989X.12.4.434 434