[R-meta] Comparing dependent, overlapping correlation coefficients

Viechtbauer, Wolfgang (SP) wolfg@ng@viechtb@uer @ending from m@@@trichtuniver@ity@nl
Wed Aug 15 14:50:25 CEST 2018

```library(metafor)

source("https://gist.githubusercontent.com/wviechtb/700983ab0bde94bed7c645fce770f8e9/raw/5bb5601852b132af533aef41405d58a3ae04cf82/rmat.r")

study    var1 var2  ri  ni
1    X    Y    .20  50
1    X    Z    .30  50
1    Y    Z    .52  50
2    X    Y    .34  35
2    X    Z    .43  35
2    Y    Z    .44  35")

dat2 <- rmat(ri ~ var1 + var2 | study, n=c(50,35), data=dat)
dat2

res <- rma.mv(yi, dat2\$V, mods = ~ var1var2 - 1, random = ~ var1var2 | id, struct="UN", data=dat2\$dat)
res

### three contrasts
anova(res, L=c(1, -1, 0))
anova(res, L=c(1, 0, -1))
anova(res, L=c(0, 1, -1))

You are interested in the first of these three contrasts, that is, whether cor(X,Y) = cor(X,Z).

Actually, I would recommend to work with:

dat2 <- rmat(ri ~ var1 + var2 | study, n=c(50,35), data=dat, rtoz=TRUE)
dat2

This applies Fisher's r-to-z transformation.

Best,
Wolfgang

-----Original Message-----
From: Anna-Lena Schubert [mailto:anna-lena.schubert using psychologie.uni-heidelberg.de]
Sent: Wednesday, 15 August, 2018 11:08
To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] Comparing dependent, overlapping correlation coefficients

Hi Wolfgang,

thanks so much, I now believe I have sensible values in my V matrix.
They deviate slightly from manual calculations, but are really close.

I'm still lost on how to test for the interaction between X and Y then,
though. I tried

res <- rma.mv(yi, V, mods = ~ variable | studyID, data=dat,
method="ML"),

but the results don't correspond to the univariate meta-analyses I
conducted before. In addition, it tests (I believe) for a moderation of
all three correlations, while I'm only interested in the difference
between r_XY and r_XZ. Moreover, the manual says "In case the sampling
errors are correlated, then one can specify the entire
variance-covariance matrix of the sampling errors via the V argument",
but it seems I cannot simply leave "yi" out of the argument.

Best, Anna-Lena

Am 14.08.2018 um 22:18 schrieb Viechtbauer, Wolfgang (SP):
> You do not need escalc(). The rmat() function gives you the variances along the diagonal of the 'V' matrix.
>
> The variances should be (1 - ri^2)^2 / (ni - 1). You should be able to double-check that these values correspond to your data. Since ni should be the same for r_XY and r_XZ within a study, then it might be that the variances are roughly the same if the two correlations are not all that different. They should not be identical though (unless r_XY and r_XZ are the same).
>
> Best,
> Wolfgang
>
> -----Original Message-----
> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Anna-Lena Schubert
> Sent: Tuesday, 14 August, 2018 14:20
> To: James Pustejovsky
> Cc: r-sig-meta-analysis using r-project.org
> Subject: Re: [R-meta] Comparing dependent, overlapping correlation coefficients
>
> Hi James,
> I used Wolfgang's script on git to calculate the Cov(r_XY, r_XZ) by feeding it Cor(r_YZ). In the next step, I calculated Var(r_XY) and Var(r_XZ) by using the escalc function. However, Var(r_XY) always equals Var(r_XZ) for each study. Does this make sense?
> I nevertheless added all three measures per study into a variance-covariance matrix such as:
>                 r_XY r_XZ    r_XY r_XZ
> r_XY        0.004    0.0001    0    0
> r_XZ        0.0001    0.004    0    0
> r_XY        0    0    0.008    0.002
> r_XZ        0    0    0.002    0.008
> Then, I tried to feed everything into a multivariate meta-analysis:
>     res <- rma.mv(yi, V, mods = ~ variableType - 1, random = ~ variableType | studyNum, struct="UN", data=dat, method="ML")
> The estimates I get for both of the correlation coefficients correspond closely to those I get when only meta-analyzing one of the variable types, which seems great. However, I'm still somewhat concerned that Var(r_XY) = Var(r_XZ). Do you think there may have been some mistake in my code or does it make sense that these variances are equal?
> Best,
> Anna-Lena
>
> Am 10.08.2018 um 17:06 schrieb James Pustejovsky:
> Anna-Lena,
>
> The approach that you suggested (putting the data in "long" format and defining an indicator variable for whether Y or Z is the correlate) is just what I would recommend. However, there is a complication in that the estimates r_XY and r_XZ are correlated (correlated correlation coefficients...say that six times fast!), and the degree of correlation depends on r_YZ.
>
> 1) If you have extracted data on r_YZ then you could use this to compute Cov(r_XY, r_XZ) and then do a multivariate meta-analysis. See discussion here:
> https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-January/000483.html
> And this function for computing the required covariance matrices:
> https://gist.github.com/wviechtb/700983ab0bde94bed7c645fce770f8e9
> There are at least three further alternatives that might be simpler:
>
> 2) If you have r_YZ you could use it to compute the sampling variance of the difference between r_XY and r_XZ, that is:
>
> Var(r_XY - r_XZ) = Var(r_XY) + Var(r_XZ) - 2 * Cov(r_XY, r_XZ)
>
> You could then do a univariate meta-analysis on the difference between correlations.
>
> 3) If you do not have r_YZ then you won't be able to estimate Cov(r_XY, r_XZ) very well. You could make a guess about r_YZ and then follow approach (1) or (2) above, using cluster-robust variance estimation to account for the possibly mis-estimated sampling-variance covariance matrix.
>
> 4) Or you could ignore the covariance between r_XY and r_XZ entirely, fit the model to the long data as you describe above, and use cluster-robust variance estimation (clustering by sample) to account for the dependence between r_XY and r_XZ. This is the quickest and dirtiest approach, and the first thing I would try in practice before moving on to the more refined approaches above.
>
> James
>
> On Fri, Aug 10, 2018 at 9:21 AM Anna-Lena Schubert <anna-lena.schubert using psychologie.uni-heidelberg.de> wrote:
> Dear all,
>
> I want to run a meta-analysis that compares dependent, overlapping
> correlation coefficients (i.e., I want to see if X correlates more
> strongly with Y than it does with Z). I already ran a meta-analysis
> separately for both of these correlations and would now like to compare
> those two pooled effect sizes statistically. Confidence intervals of the
> two correlations do not overlap (r1 = .18 [.12; .24]; r2 = .32 [.25;
> .39]), but I wonder if there may be a more elegant way to compare these
> correlations than just based on CIs.
>
> I wonder, for example, if a factorial variable could be used to identify
> those correlations in a "long" data format style, and if I could test
> for a significant interaction between variable type (Y vs. Z) and the
> correlation in a meta-analysis:
>
>     Study    Variable    r
>     1    Y    .20
>     1    Z    .30
>     2    Y    .34
>     2    Z    .43
>
> I would greatly appreciate if anyone could tell me if that's a good idea
> or could recommend other approaches. Thanks in advance for any offers of
> help!
>
> Best,
> Anna-Lena
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