Can someone explain intuitively (or even mathematically) why Lin's concordance correlation coefficient (CCC) is always less than or equal to pearson's correlation coefficient (PCC). A related query exists as How to compare concordance correlation coefficient to Pearson's r? but there is no quantitative comparison between CCC and PCC. The reason why I say $CCC<=PCC$ is based on formula of CCC as $CCC = 2\times PCC\times \sigma_x \times \sigma_y /(\sigma_x^2+\sigma_y^2+(\mu_x-\mu_y)^2)$, where x and y are the 2 variables on which we evaluate correlation. Clearly, if $\mu_x = \mu_y $ and $\sigma_x = \sigma_y$, then $CCC=PCC$. For all other scenarios, $CCC<PCC$, based on arithmetic and geometric mean relationships.


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