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

[Viechtbauer, Wolfgang (NP)]

Dear Yefeng,

Please see below for my responses.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>Behalf Of Yefeng Yang
>Sent: Thursday, 05 May, 2022 9:57
>To: r-sig-meta-analysis using r-project.org
>Subject: [R-meta] A further question about 'correlations' in multivariate model:
>correlation in observed effects vs. correlation in true effects
>
>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.

This is the within-study correlation I talked about.

>  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)

There is no inherent relationship between the two. It could be that the within-study correlations are positive but that the correlation in the underlying true effects are negative or vice-versa, or that the true effects are not correlated at all, and so on.

>  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)

I don't understand the question.

>  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")

Some might call it the between-study correlation, but to me this phrasing is a bit odd, because it sounds a bit like the correlation denotes how something in study A correlates with something in study B, but that is not what it denotes (it denotes how the true effects correlate within a study, so in that sense, it is also a within-study correlation, but let's not call it that to avoid confusion).

>  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)?

Yes - see 4.

>I would be grateful if would like to take the time to clarify my confusion.
>
>Best,
>
>Yefeng Yang PhD