Network meta-analysis of correlation coefficients: a logical paradox? A colleague of mine wishes to conduct a network meta-analysis of correlation coefficients for a set of imaging tests (eg ultrasound, computed tomography, magnetic resonance imaging, and so forth). 
He has collected correlation coefficients (R), R-squared, and Fisher z. He pushes me to embark in a formal network meta-analysis (eg a frequentist one with the netmeta R package).
However, after some thoughts, I think it is a logical paradox, in the sense that in my opinion the transitivity assumption (see figure below) does not apply to correlation coefficients in the same fashion it applies to clinical trials with unambiguous endpoint definitions or to diagnostic test accuracy studies with clear labeling of healthy and diseased subjects. 
Indeed, I fear that the correlation obtained when comparing test A and B, and that obtained when comparing B and C, cannot inform on the comparison between A and C, as the latter may depend on completely different cases.
I have also searched in Google and PubMed for "network meta-analysis" and "correlation", but did not find any meaningful reference.
According, my recommendation would simply to stick to a univariate meta-analysis approach, or otherwise use a multivariate meta-analysis one, for instance with mvmeta in Stata or R.
Am I correct?
 A: There are two points to comment on. Note that correlation coefficients do not measure differences, but agreement.
It is also right that correlation coefficients per se are not transitive. Of course it is possible that cor(x,y) > 0, cor(y,z) > 0, but cor(x,z) < 0. Set, for example, in R
x <- c(1,2,1,3,2,1,3,2,3,1,2,3);
y <- c(0,1,1,2,2,2,3,3,4,3,4,5);
z <- c(2,1,3,0,2,4,1,3,2,5,4,3);
cor(x,y);
cor(y,z);
cov(x,z);
Most important, we have to distinguish between meta-analysis of one-sample measures and meta-analysis of two-sample comparisons.
A pairwise meta-analysis of treatments effects (using effect measures such as  the mean difference, the relative risk, or the odds ratio) involves the comparison of two groups, and therefore the natural extension is network meta-analysis involving more than two groups. Traditionally, this has been based on a contrast-based approach. The consistency assumption says that the direct evidence and the indirect evidence agree, formally:
d(A-C) = d(A-B) + d(B-C)
By contrast to this, meta-analysis of one-sample measures (such as means, proportions, or incidence rates, or correlation coefficients) does not involve a comparison of two groups. I do not see any meaningful extension of one-sample meta-analysis to network meta-analysis.
A different thing would it be to compare correlation coefficients of two or more different studies by looking at their (transformed) differences, in analogy to comparing means, proportions or incidence rates. For such differences, a consistency assumption could be made (though I doubt that this would make much sense).
Gerta Rücker
A: Check "Exploring the use of network meta-analysis in education: examining the correlation between ORF and text complexity measures", this is closely correlated.
