[R-meta] A further question about 'correlations' in multivariate model: correlation in observed effects vs. correlation in true effects

[Yefeng Yang]

Hi subscribers,

Excited to be here again. Last time, I asked one question regarding the within-study correlation (used for the variance-covariance matrix of sampling errors of effect size estimates) and the correlation in *true* effects (estimated from the variance-covariance matrix of random effect). Please refer to my question here: https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2022-May/004026.html.

Regarding my question, Wolfgang gave me a fairly clear answer, which resolved my long-standing confusion. I appreciated this very much. See his answer here: https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2022-May/004029.html.

Now, I have further, albeit small, questions about the relationship between correlation in observed effects and that in true effects (these questions were inspired by "variance in the observed effect and that in the true effect"):


  1.   is there a concept of correlation in observed effects? I mean whether correlation in observed effects is meaningful?  If yes, what is its general meaning or implication? You can use an example of one study containing two outcomes (cognition and anxiety) to explain it to me.
  2.  what is the relationship between correlation in observed effects and that in true effects? I am expecting a quantitative relationship between the two (just like the variance of the observed effect =  variance of true effect + sampling variance)
  3.  any parameters in the meta-analytic model (see below for an example) can reflect or denote the correlation in the observed effect (if the corresponding concept exists; see question 1): rma.mv(yi, V, mods = ~ outcome, random = ~ outcome | study, struct="UN", data=mydata)
  4.  correlation in variance-covariance matrix of sampling variance can be formally called within-study correlation. This is fairly clear. I wonder what is the formal name or general name of correlation (rho) in the variance-covariance of a random effect (for example rho in the structure like "UN")
  5.  in contrast to the within-study correlation, there is also another correlation, named the between-study correlation involved in the multivariate meta-analytic model (see Riley R D, Abrams K R, Sutton A J, et al. Bivariate random-effects meta-analysis and the estimation of between-study correlation[J]. BMC Medical Research Methodology, 2007, 7(1): 1-15.). So what is the meaning of between-study correlation? Is between-study correlation exactly the correlation in true effects (as you explained in the meta-analytic context, we have multiple studies, not only one study for the within-study correlation)?

I would be grateful if would like to take the time to clarify my confusion.

Best,

Yefeng Yang PhD

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