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I am writing a meta-analysis looking at the correlation between different tests for assessing body composition.

The results of the studies that I have included use different ways to calculate the correlation, including linear regression ($R^{2}$), Pearson's $r$, concordance correlation coefficient (CCC), and mean difference (SD) + 95% CI.

I would like to compare these results. I have found that I can take $\sqrt{R^{2}}$ to get $r$. Is there also a way to calculate $r$ from CCC? Or is CCC so similar to Pearson's $r$, that I can just call it $r$ and throw everything on one pile?

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  • $\begingroup$ R-square and correlation are two versions of the same idea for a bivariate relationship; that's not a discovery, but a consequence of definitions. Otherwise these are different measures answering different questions. It is quite possible to get Pearson correlation and concordance correlation that are almost completely different. For example if $y \approx bx$ where $b \gg 0$ then correlation will be close to 1 but concordance correlation will be close to 0. $\endgroup$
    – Nick Cox
    Commented Oct 17, 2018 at 16:26

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Pearson's r measures linearity, while CCC measures agreement. Imagine a scatterplot between the two measures. High agreement implies that the scatterplot points are close to the 45 degrees line of perfect concordance which runs diagonally to the scatterplot, whereas a high Pearson's r implies that the scatterplot points are close to any straight line.

In practice, 

CCC = r * C_b

where r is Pearson's r, and C_b is a bias correction factor.

Therefore, CCC cannot be compared to Pearson's r. To calculate Pearson's r from CCC for a direct comparison you will need to divide CCC by C_b.

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