How to compare correlation coeffecients across studies (meta-analysis, simulation-analysis)? I have one metric dependent variable (grain yield) and three metric traits, A, B and C. Grain yield and all three traits were measured in several studies and each time a Pearson correlation between trait and grain-yield was reported.
So I have correlation-coeffecients for each combination:
grain yield ~ trait A (around 50 correlation-coeffecient values)
grain-yield ~ trait B (around 50 correlation-coeffecient values)
grain-yield ~ trait C (around 50 correlation-coeffecient values)
What statistical test would I use to compare these three sets of correlation coeffecients to see which trait is a better predictor for grain yield?
(One could also think of a scenario where I am doing a simulation study and I look what experimental design allows to predict true values best, in fact this is what I am doing but the example above is easier to explain and understand)
 A: You cannot just conduct a one-way ANOVA on the correlations because they are not independent. Within each study, you need to construct the variance-covariance matrix of the correlations (or the r-to-z transformed values). For this, you need to know the correlation between A and B, between A and C, and between B and C. You can find equations for the covariance between correlations in:
Olkin, I., & Finn, J. D. (1990). Testing correlated correlations. Psychological Bulletin, 108(2), 330–333. https://doi.org/10.1037/0033-2909.108.2.330
Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87(2), 245–251. https://doi.org/10.1037/0033-2909.87.2.245
So, for each study you then have 3 correlations (between grain yield and the three traits) and a 3x3 var-cov matrix. This can be used as input for a multivariate meta-analysis. See, for example:
Kalaian, H. A., & Raudenbush, S. W. (1996). A multivariate mixed linear model for meta-analysis. Psychological Methods, 1(3), 227-235. https://doi.org/10.1037/1082-989X.1.3.227
(this article is not focused on correlations, but the methods are the same). Chapter 16 from The Handbook of Research Synthesis and Meta-Analysis by Cooper et al. (2019) also discusses the required methods and is focused specifically on correlation coefficients. You can conduct such an analysis using the metafor package (for the code corresponding to chapter 16 above, see here) or the metaSEM package.
