# Concordance correlation coefficient

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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[1]

$\rho_c = \frac{2\rho\sigma_x\sigma_y}{\sigma_x^2 + \sigma_y^2 + (\mu_x - \mu_y)^2},$

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,[1] while in another article Nickerson appears to have used the 1/(N-1),[2] 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.[2]

It has also been stated[3] that the ideas for concordance correlation coefficient "are quite similar to results already published by Krippendorff[4] in 1970".

In the original article[1] Lin suggested a form for multiple classes (not just 2). Over ten years later a correction to this form was issued.[5]

One example of the use of the concordance correlation coefficient is in a comparison of analysis method for functional magnetic resonance imaging brain scans.[6]

## References Edit

1. 1.0 1.1 1.2 Lawrence I-Kuei Lin (March 1989). A concordance correlation coefficient to evaluate reproducibility. Biometrics 45 (1): 255–268.
2. 2.0 2.1 Carol A. E. Nickerson (December 1997). A Note on "A Concordance Correlation Coefficient to Evaluate Reproducibility. Biometrics 53 (4): 1503–1507.
3. Reinhold Müller & Petra Büttner (December 1994). A critical discussion of intraclass correlation coefficients. Statistics in Medicine 13 (23-24): 2465–2476.
4. Klaus Krippendorff (1970). "Bivariate agreement coefficients for reliability of data" E. F. Borgatta Sociological Methodology, 139–150, San Francisco: Jossey-Bass.
5. Lawrence I-Kuei Lin (March 2000). A Note on the Concordance Correlation Coefficient. Biometrics 56: 324–325.
6. Nicholas Lange, Stephen C. Strother, J. R. Anderson, Finn Årup Nielsen, Andrew P. Holmes, Thomas Kolenda, Robert L. Savoy and Lars Kai Hansen (September 1999). Plurality and resemblance in fMRI data analysis. NeuroImage 10 (3 Part 1): 282–303.
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