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<p>Hi James,</p>
<p>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? <br>
</p>
<p>I nevertheless added all three measures per study into a
variance-covariance matrix such as:</p>
<p> r_XY r_XZ r_XY r_XZ</p>
<p>r_XY 0.004 0.0001 0 0<br>
</p>
<p>r_XZ 0.0001 0.004 0 0</p>
<p>r_XY 0 0 0.008 0.002</p>
<p>r_XZ 0 0 0.002 0.008</p>
<p>Then, I tried to feed everything into a multivariate
meta-analysis: <br>
</p>
<p> res <- rma.mv(yi, V, mods = ~ variableType - 1, random = ~
variableType | studyNum, struct="UN", data=dat, method="ML")</p>
<p>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? <br>
</p>
<p>Best,</p>
<p>Anna-Lena<br>
</p>
<br>
<div class="moz-cite-prefix">Am 10.08.2018 um 17:06 schrieb James
Pustejovsky:<br>
</div>
<blockquote type="cite"
cite="mid:CAFUVuJyjqd3GV+JUsD0rAoa4KQK0-oGRGwfFoGW6SsZnMTZA-Q@mail.gmail.com">
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<div dir="ltr">Anna-Lena,
<div><br>
</div>
<div>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. </div>
<div><br>
</div>
<div>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:</div>
<div><a
href="https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-January/000483.html"
moz-do-not-send="true">https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-January/000483.html</a><br>
</div>
<div>And this function for computing the required covariance
matrices:</div>
<div><a
href="https://gist.github.com/wviechtb/700983ab0bde94bed7c645fce770f8e9"
moz-do-not-send="true">https://gist.github.com/wviechtb/700983ab0bde94bed7c645fce770f8e9</a><br>
</div>
<div>There are at least three further alternatives that might be
simpler:<br>
</div>
<div><br>
</div>
<div>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:</div>
<div><br>
</div>
<div>Var(r_XY - r_XZ) = Var(r_XY) + Var(r_XZ) - 2 * Cov(r_XY,
r_XZ)</div>
<div><br>
</div>
<div>You could then do a univariate meta-analysis on the
difference between correlations.</div>
<div><br>
</div>
<div>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. </div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<div>James</div>
<div> </div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr">On Fri, Aug 10, 2018 at 9:21 AM Anna-Lena
Schubert <<a
href="mailto:anna-lena.schubert@psychologie.uni-heidelberg.de"
target="_blank" moz-do-not-send="true">anna-lena.schubert@psychologie.uni-heidelberg.de</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">Dear all,<br>
<br>
I want to run a meta-analysis that compares dependent,
overlapping<br>
correlation coefficients (i.e., I want to see if X correlates
more<br>
strongly with Y than it does with Z). I already ran a
meta-analysis<br>
separately for both of these correlations and would now like
to compare<br>
those two pooled effect sizes statistically. Confidence
intervals of the<br>
two correlations do not overlap (r1 = .18 [.12; .24]; r2 = .32
[.25;<br>
.39]), but I wonder if there may be a more elegant way to
compare these<br>
correlations than just based on CIs.<br>
<br>
I wonder, for example, if a factorial variable could be used
to identify<br>
those correlations in a "long" data format style, and if I
could test<br>
for a significant interaction between variable type (Y vs. Z)
and the<br>
correlation in a meta-analysis:<br>
<br>
Study Variable r<br>
1 Y .20<br>
1 Z .30<br>
2 Y .34<br>
2 Z .43<br>
<br>
<br>
I would greatly appreciate if anyone could tell me if that's a
good idea<br>
or could recommend other approaches. Thanks in advance for any
offers of<br>
help!<br>
<br>
Best,<br>
Anna-Lena<br>
<br>
<br>
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<h3 class="auto-style2">Dr. Anna-Lena Schubert</h3>
<p style="line-height: 1;"><img
src="cid:part6.B22042F0.0A46DB70@psychologie.uni-heidelberg.de"
data-filename="Logo_Diff.png" style="width: 215.5px; float:
right; height: 49.9136px;"><font class="auto-style3"><span
class="auto-style1"><span class="auto-style4">Postdoc at
Section of Personality</span><br class="auto-style4">
<span class="auto-style4">Heidelberg University - Institute
of Psychology</span></span></font></p>
<pre style="line-height: 1;">Hauptstraße 47-51
<span style="line-height: 1;">D-69117 Heidelberg
</span><span style="line-height: 1;">Germany</span></pre>
<pre style="line-height: 1;">Phone: +49 6221 54 7746
Mail: <a class="moz-txt-link-abbreviated" href="mailto:anna-lena.schubert@psychologie.uni-heidelberg.de">anna-lena.schubert@psychologie.uni-heidelberg.de</a>
Web: <a class="moz-txt-link-freetext" href="http://www.psychologie.uni-heidelberg.de/ae/diff/diff/people-schubert.html">http://www.psychologie.uni-heidelberg.de/ae/diff/diff/people-schubert.html</a></pre>
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