Concordance correlation coefficient

In statistics, the concordance correlation coefficient measures the agreement between two variables, e.g., to evaluate reproducibility or for inter-rater reliability.

Lawrence Lin has the form of the concordance correlation coefficient $$\rho_c$$ as
 * $$\rho_c = \frac{2\rho\sigma_x\sigma_y}{\sigma_x^2 + \sigma_y^2 + (\mu_x - \mu_y)^2},$$

where $$\mu_x$$ and $$\mu_y$$ are the means for the two variables and $$\sigma^2_x$$ and $$\sigma^2_y$$ are the corresponding variances. $$\rho$$ is the correlation coefficient between the two variables.

When the concordance correlation coefficient is computed on a N-length data set (i.e., two vectors of length N) the form is
 * $$r_c = \frac{2 s_{xy}}{s_x^2 + s_y^2 + (\bar{x} - \bar{y})^2},$$

where the mean is computed as
 * $$\bar{x} = \frac{1}{N} \sum_{n=1}^N x_n$$

and the variance
 * $$s_x^2 = \frac{1}{N} \sum_{n=1}^N (x_n - \bar{x})^2$$

and the covariance
 * $$s_{xy} = \frac{1}{N} \sum_{n=1}^N (x_n - \bar{x})(y_n - \bar{y})$$

Whereas the ordinary correlation coefficient (Pearson's) is immune to whether the biased or unbiased versions for estimation of the variance is used, the concordance correlation coefficient is not. In the original article Lin suggested the 1/N normalization, while in another article Nickerson appears to have used the 1/(N-1), i.e., the concordance correlation coefficient may be computed slightly different between implementations.

The concordance correlation coefficient is nearly identical to some of the measures called intra-class correlations, and comparisons of the concordance correlation coefficient with an "ordinary" intraclass correlation on different data sets found only small differencies between the two correlations, in one case on the third decimal.

It has also been stated that the ideas for concordance correlation coefficient "are quite similar to results already published by Krippendorff in 1970".

In the original article Lin suggested a form for multiple classes (not just 2). Over ten years later a correction to this form was issued.