Meta-analyzing correlation matrices in the presence of hierarchical effect size multiplicity

To synthesize evidence on the relations among multiple constructs, measures, or concepts, meta-analyzing correlation matrices across primary studies has become a crucial analytic approach. Common meta-analytic approaches employ univariate or multivariate models to estimate a pooled correlation matrix, which is subjected to further analyses, such as structural equation modeling. In practice, meta-analysts often extract multiple correlation matrices per study from various samples, study sites, labs, or countries, thus introducing hierarchical effect size multiplicity into the meta-analytic data. However, this feature has largely been ignored when pooling correlation matrices for meta-analysis. To contribute to the methodological development in this area, we describe a multilevel, multivariate, and random-effects modeling approach, which pools correlation matrices meta-analytically and, at the same time, addresses hierarchical effect size multiplicity. Specifically, it allows meta-analysts to test various assumptions on the dependencies among random effects, aiding the selection of a meta-analytic baseline model. We describe this approach, present four working models within it, and illustrate them with an example and the corresponding R code.