[R-meta] Comparing dependent, overlapping correlation coefficients

[Anna-Lena Schubert]

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
> <mailto: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
>
>
>             [[alternative HTML version deleted]]
>
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-- 
Signatur


      Dr. Anna-Lena Schubert

Postdoc at Section of Personality
Heidelberg University - Institute of Psychology

Hauptstraße 47-51
D-69117 Heidelberg Germany

Phone: +49 6221 54 7746
Mail: anna-lena.schubert using psychologie.uni-heidelberg.de
Web: http://www.psychologie.uni-heidelberg.de/ae/diff/diff/people-schubert.html


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