Concordance correlation coefficient explained

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

Definition

The form of the concordance correlation coefficient

\rho_c

as^[1]

\rho_c=

2\rho\sigma_x\sigma_y

\sigma

	2
\sigma
	y

+(\mu_x-

	2
\mu
	y)

where

\mu_x

and

\mu_y

are the means for the two variables and

	2
\sigma
	x

and

	2
\sigma
	y

are the corresponding variances.

\rho

is the correlation coefficient between the two variables.

This follows from its definition^[1] as

\rho_c=1-

	{\rmExpected orthogonal squared distance from the diagonal
	x=y} {{\rm

Expected orthogonal squared distance from the diagonal }x=y{\rm assuming independence}}.

When the concordance correlation coefficient is computed on a

-length data set (i.e.,

paired data values

(x_n,y_n)

, for

n=1,...,N

), the form is

\hat{\rho}_c=

2s_xy

	2
s
	y

+(\bar{x

-\bar{y})^2},

where the mean is computed as

\bar{x}=

	1
	N

	N
\sum
	n=1

x_n

and the variance

	2
s
	x

	1
	N

	N
\sum
	n=1

(x_n-\bar{x})²

and the covariance

s_xy=

	1
	N

	N
\sum
	n=1

(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 differently between implementations.

Relation to other measures of correlation

The concordance correlation coefficient is nearly identical to some of the measures called intra-class correlations. Comparisons of the concordance correlation coefficient with an "ordinary" intraclass correlation on different data sets found only small differences 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.

References

For a small Excel and VBA implementation by Peter Urbani see here

Notes and References

Lawrence I-Kuei Lin . Lawrence I-Kuei Lin . A concordance correlation coefficient to evaluate reproducibility . . 45 . 1 . 255 - 268 . March 1989 . 2720055 . 10.2307/2532051 . 2532051 .
Carol A. E. Nickerson . Carol A. E. Nickerson . A Note on "A Concordance Correlation Coefficient to Evaluate Reproducibility . . 53 . 4 . 1503 - 1507 . December 1997 . 10.2307/2533516 . 2533516 .
Reinhold Müller . Petra Büttner . A critical discussion of intraclass correlation coefficients . . December 1994 . 13 . 23–24 . 2465 - 2476 . 7701147 . 10.1002/sim.4780132310.
Klaus Krippendorff . Bivariate Agreement Coefficients for Reliability of Data . Klaus Krippendorff . E. F. Borgatta . Sociological Methodology . 2 . 139 - 150 . . . 1970 . 10.2307/270787. Sociological Methodology . 270787 .
Lawrence I-Kuei Lin . A Note on the Concordance Correlation Coefficient . . 56 . 324 - 325 . March 2000 . 10.1111/j.0006-341X.2000.00324.x. free .