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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.graphical synthesis of the transitivity assumption

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?

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    $\begingroup$ See: Salanti, G. (2012). Indirect and mixed‐treatment comparison, network, or multiple‐treatments meta‐analysis: many names, many benefits, many concerns for the next generation evidence synthesis tool. Research synthesis methods, 3(2), 80-97. This addresses the issue you raise. $\endgroup$
    – dbwilson
    Commented Feb 21, 2019 at 14:23
  • $\begingroup$ @dbwilson I checked Salanti's paper but I think it is not really pertinent as it does not refer explicitly to correlation coefficient or Fisher z. $\endgroup$
    – Giuseppe Biondi-Zoccai
    Commented Feb 21, 2019 at 16:02
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    $\begingroup$ It doesn't matter what type of effect size is being used. The logic and math are the same -- you have an effect size (Fischer's Zr) and its associated inverse variance weight. From that point forward, meta-analysis, including network meta-analysis proceeds in the same way. As such, the issues related to direct and indirect effects and the various assumptions being made are the same. $\endgroup$
    – dbwilson
    Commented Feb 21, 2019 at 19:47
  • $\begingroup$ #dbwilson I understand your point in general, and indeed have performed NMAs of RCTs and DTA studies. However, there you have clear and universally unequivocal cases. In correlation this is not true, in the sense that a given subject may lead correlation between A and B, whereas another subject may lead correlation between B and C. Thus, you end up inferring inappropriately on the correlation between A and C. $\endgroup$
    – Giuseppe Biondi-Zoccai
    Commented Feb 22, 2019 at 9:00

2 Answers 2

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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

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Check "Exploring the use of network meta-analysis in education: examining the correlation between ORF and text complexity measures", this is closely correlated.

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    $\begingroup$ Can you edit this to explain what the link contains as if the link goes dead your answer will become valueless. $\endgroup$
    – mdewey
    Commented Mar 14, 2021 at 13:53
  • $\begingroup$ Welcome to Cross Validated! Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$
    – MarianD
    Commented Mar 14, 2021 at 14:11

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