# [R-meta] Comparing dependent, overlapping correlation coefficients

James Pustejovsky jepu@to @ending from gm@il@com
Fri Aug 10 17:06:48 CEST 2018

```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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>
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