Benutzer:JonskiC/Konkordanz-Korrelationskoeffizient

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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.

Definition

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 $\sigma _{x}^{2}$ and $\sigma _{y}^{2}$ 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

${\hat {\rho }}_{c}={\frac {2s_{xy}}{s_{x}^{2}+s_{y}^{2}+({\bar {x}}-{\bar {y}})^{2}}},$

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

$s_{xy}={\frac {1}{N}}\sum _{n=1}^{N}(x_{n}-{\bar {x}})(y_{n}-{\bar {y}}).$