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

Viechtbauer, Wolfgang (SP) wolfg@ng@viechtb@uer @ending from m@@@trichtuniver@ity@nl
Fri Oct 5 17:48:57 CEST 2018

```The first output is more difficult to interpret, so I'll go with the second.

You want to test:

H0: X.Z:M0 - Y.Z:M0 = X.Z:M1 - Y.Z:M1,

which is equivalent to:

H0: X.Z:M0 - Y.Z:M0 - X.Z:M1 + Y.Z:M1 = 0

So, you want to use:

anova(res(0,1,-1,0,-1,1))

And indeed, this is testing if the difference between cor(X,Y) and cor(X,Z) is moderated by M (i.e., is different for M0 vs M1).

Best,
Wolfgang

-----Original Message-----
From: Anna-Lena Schubert [mailto:anna-lena.schubert using psychologie.uni-heidelberg.de]
Sent: Monday, 01 October, 2018 14:47
To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] Comparing dependent, overlapping correlation coefficients

I have recently tried to finish all analyses and have realized that I still haven't fully understood how moderation works here. In the example below, I'm interested in whether the difference between cor(X,Y) and cor(X,Z) is moderated by some factor. From my reading of the metafor examples, var1var2X.Z:M should indicate whether the cor(X,Z) is moderated by M. However, Wolfgang says that term instead describes a moderation of the difference between correlations. I'm perfectly happy to believe him, but I would really like to understand why this is the case.

In my empirical example the results look as follows:

estimate      se     zval    pval    ci.lb    ci.ub
var1var2X.Y       0.6580  0.0218  30.1818  <.0001   0.6152   0.7007  ***
var1var2X.Z      -0.1282  0.0468  -2.7420  0.0061  -0.2199  -0.0366   **
var1var2Y.Z      -0.2896  0.0480  -6.0315  <.0001  -0.3837  -0.1955  ***
M1                0.0736  0.0301   2.4474  0.0144   0.0147   0.1326    *
var1var2X.Z:M1   -0.1648  0.0786  -2.0969  0.0360  -0.3188  -0.0108    *
var1var2Y.Z:M1   -0.1266  0.0845  -1.4983  0.1340  -0.2922   0.0390

Here I'm interested in the difference between cor(x,z) and cor(y,z).  Which of these interaction terms is critical to the moderation of this correlation, and why? Also, how can I interpret the direction of the effect?

If I look at another display of the same results, I get an understanding of the moderation, but I'm not sure about a clear test here. Would anova(res(0,1,-1,0,1,-1)) be a sensible test? What I'd really want to test would be if X.Z:M0 - Y.Z:M0 = X.Z:M1 - Y.Z:M1

estimate      se     zval    pval    ci.lb    ci.ub
var1var2X.Y:M0    0.6580  0.0218  30.1818  <.0001   0.6152   0.7007  ***
var1var2X.Z:M0   -0.1282  0.0468  -2.7420  0.0061  -0.2199  -0.0366   **
var1var2Y.Z:M0   -0.2896  0.0480  -6.0315  <.0001  -0.3837  -0.1955  ***
var1var2X.Y:M1    0.7316  0.0207  35.3007  <.0001   0.6910   0.7722  ***
var1var2X.Z:M1   -0.2194  0.0483  -4.5437  <.0001  -0.3140  -0.1248  ***
var1var2Y.Z:M1   -0.3426  0.0491  -6.9767  <.0001  -0.4388  -0.2463  ***
```