metafor: correlational output from multivariate meta-analysis question I have a quick question about the metafor package. I have been using the author's guide for stochastically dependent effect sizes (via github) to run a [multivariate meta-analysis][1] using multiple variables from the same treatment studies.
We have a special interest in looking at whether one variable decreases as the other(s) decrease. So we are very interested in the correlational output that metafor also supplies, while also controlling for some of the shared variance.
My question is how do you interpret the rho part of the output? Is this a simply a pearson correlation that attempts to estimate Rho at the population level or is this a rank-ordered Spearman's rho correlation? I need to test the significance of those correlations, hence I need to know how to interpret the rho columns.
This is the example output from the website above after running a 2-outcome multivariate meta-analysis:
         rho.math  rho.vrbl    math  vrbl 
math           1    -1.000       -    no 
verbal    -1.000         1       5     - 

An example of my output is:
                rho.a___  rho.d___  rho.e___  rho.em__  rho.i___    a___  d___  e___  em__  i___ 
a______                1     0.825     0.541     0.878     0.717       -    no    no    no    no 
d_________         0.825         1     0.921     0.935     0.822      57     -    no    no    no 
e___________       0.541     0.921         1     0.783     0.741      39    47     -    no    no 
em__________       0.878     0.935     0.783         1     0.611      10    10     4     -    no 
i_____________     0.717     0.822     0.741     0.611         1      37    45    22     2     - 

Any thoughts or help would be appreciated. Thanks!
[1]: https://wviechtb.github.io/meta_analysis_books/cooper2019.html#13)_Stochastically_Dependent_Effect_Sizes

The values in the table are the estimated correlations between the random effects, which are assumed to follow a multivariate normal distribution. So these are the ML/REML estimates of the correlations in that multivariate normal distribution.
How to call them in a matter of personal preference. I wouldn't call them 'Pearson correlations' because that might suggest that they are obtained by taking two observed variables and computing the Pearson product-moment correlation coefficient based on them, but that is not how they are obtained. They are definitely not 'rank-order' correlations.
If you need to test them, you cannot use standard methods that are used for testing Pearson product-moment correlation coefficients. Instead, you can do likelihood ratio tests, where you compare the fitted model with a model where you constrain one (or multiple) of the correlations to 0 (you can do this via the rho argument; see help(rma.mv) and especially the section on "Fixing Variance Components and/or Correlations").
