Concordance correlation coefficient

Definition

The form of the concordance correlation coefficient ${\displaystyle {\rho }_c}$ as

${\displaystyle {\rho }_c=\frac{2\rho {\sigma }_x{\sigma }_y}{{\sigma }_x^2+{\sigma }_y^2+({\mu }_x-{\mu }_y{)}^2},}$

where ${\displaystyle {\mu }_x}$ and ${\displaystyle {\mu }_y}$ are the means for the two variables and ${\displaystyle {\sigma }_x^2}$ and ${\displaystyle {\sigma }_y^2}$ are the corresponding variances. ${\displaystyle \rho }$ is Pearson's correlation coefficient between the two variables.

This follows from its definition, as:

${\displaystyle {\rho }_c=1-\frac{\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}x=y}{\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}x=y\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}.}$

When the concordance correlation coefficient is computed on a length-${\displaystyle N}$ data set (i.e., ${\displaystyle N}$ paired values ${\displaystyle (x_n,y_n)}$, for ${\displaystyle n=1,...,N}$), the form is

${\displaystyle {\hat{\rho }}_c=\frac{2s_{xy}}{s_x^2+s_y^2+(\overline{x}-\overline{y}{)}^2},}$

where the mean is computed as

${\displaystyle \overline{x}=\frac{1}{N}\sum _{n=1}^N{x}_n}$

and the variance

${\displaystyle s_x^2=\frac{1}{N}\sum _{n=1}^N(x_n-\overline{x}{)}^2}$

and the covariance

${\displaystyle s_{xy}=\frac{1}{N}\sum _{n=1}^N(x_n-\overline{x})(y_n-\overline{y}).}$

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 normalizing with ${\displaystyle 1/N}$ normalization, while in another article Nickerson appears to have used ${\displaystyle 1/(N-1)}$, i.e., the concordance correlation coefficient may be computed slightly differently between implementations.