# [R-meta] confusion point: the various 'correlation' (rho, ρ) in multivariate meta-analytic model

Yefeng Yang ye|eng@y@ng1 @end|ng |rom un@w@edu@@u
Tue May 3 16:02:19 CEST 2022

```Hi subscribers

I am writing to ask one question regarding the various 'correlation' (rho) in multivariate models (implemented in rma.mv() function).  I would be grateful if would like to clarify my confusion.  I am hoping someone who is familiar with meta-analytic models can address my questions. Wolfgang definitely knows the answers, but I guess he is so busy that he has no time to answer my questions. Anyway, I briefly describe my question below.

Assume I am conducting a bivariate meta-analysis to estimate (1) the overall effects of two outcomes (cognition and anxiety), and (2) the correlation between the two outcomes. I am confused with the two types of correlations (probably three types) involved in this bivariate meta-analysis.

(i) rho in the variance-covariance matrix of dependent effect size's sampling errors:

when constructing (or approximately imputing) variance-covariance matrix of dependent effect size's sampling errors (via vcalc() function), we can use the argument rho (��) to specify the correlation of observed effect sizes or outcomes measured concurrently:

### construct the variance-covariance matrix assuming rho = 0.66 for effect sizes corresponding to the 'verbal' and 'math' outcome types

V <- vcalc(vi, cluster=studyID, type=outcome, data=dat, rho=0.66)

(ii) rho in the variance-covariance matrix of random effects structure:

say I am using ~ inner | outer to define the random effects structure ~ outcome | studyID. For any of the variance structures (e.g., compound symmetric structure [CS], heteroscedastic compound symmetric structure [HCS], UN [unstructured]), there is a correlation coefficient rho (��) denoting the correlation between the different levels of inner variable (in the our case, outcome). Then we can fit the bivariate random-effects model using rma.mv(), for example:

rma.mv(yi, vi, mods = ~ outcome, random = ~ group | studyID, struct="UN", data=mydata)

(iii) correlation between true effects size/outcomes

My question is,

1.   what are the differences and relationships between �� (sampling correlation; scenario i) in V matrix and �� in the random effects structure (scenario ii) in the context of my example and a more general condition?
2.  whether the second �� (correlation in the random effects structure; scenario ii) is exactly the third �� (correlation between the underlying true effects size/outcomes; scenario iii).
3.  Is �� (sampling correlation; scenario i) meaning the correlation between the observed effects/outcomes and �� in the random effects structure (scenario ii) is the correlation between the true effects/outcomes.
4.  Is there any other �� in multivariate or multilevel meta-analytic models?

Best,

Yefeng Yang PhD