Kendall's concordance coefficient

The Kendall'sche Concordance analysis (by Maurice Kendall ) is a non-parametric statistical technique for quantifying the agreement between multiple evaluators (raters). The Kendall concordance coefficient W thus provides an alternative

• Kappa statistics - intended for nominally scaled data - and
• Rank correlation coefficients for ordinal data (such as Spearman's and Kendall's ) - intended primarily for two judges - ${\displaystyle \rho }$ ${\displaystyle \tau }$

represent.

The concordance coefficient W is similar to the Cronbach's alpha for determining the reliability z. B. a test procedure. It takes values ​​between 0 and 1.

formula

When assessors rank the cases (= objects of observation, people, characteristics) in a ranking, each case is given a ranking by each assessor ; the sum of all rankings assigned for a case is then: ${\displaystyle j=1,2,3,...m}$ ${\displaystyle i=1,2,...N}$ ${\displaystyle R_{ij}}$ ${\displaystyle i}$

${\displaystyle \!\,T_{i}=\sum _{j}R_{ij}}$.

If an assessor does not assign a clear ranking (1,2,3, ... N) to a case, but B. If several cases have to share a ranking position, one speaks of "ranking". The total number of cases that share a specific rank with an appraiser is called the rank commitment length . ${\displaystyle j}$ ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle t_{jk}}$

Of course, several rank ties can also occur with an assessor if cases are assessed the same way. The number of tied ranks for an appraiser is: ${\displaystyle j}$

${\displaystyle \!\,s_{j}=\sum _{k}t_{jk}}$.

From this, Kendall's W is calculated as follows:

${\displaystyle W={\frac {12\cdot \sum _{i=1}^{N}(T_{i}-{\overline {T}})^{2}}{m^{2}(N^{3}-N)-m\cdot t}}}$

in which

${\displaystyle \!\,{\overline {T}}={\frac {1}{N}}\sum _{i}T_{i}}$

and

${\displaystyle t=\sum _{j=1}^{m}\sum _{k=1}^{s_{j}}(t_{jk}^{3}-t_{jk})}$.

W is directly related to Friedman's coefficient and Spearman's rank correlation coefficient : ${\displaystyle \chi ^{2}}$ ${\displaystyle \rho }$

${\displaystyle \chi ^{2}=m\cdot (N-1)\cdot W}$ and

${\displaystyle {\bar {\rho }}={\frac {m\cdot (W-{\frac {1}{m}})}{(m-1)}}}$,

where represents the mean of all rank correlations between the possible combinations of 2 assessors . ${\displaystyle {\bar {\rho }}}$ ${\displaystyle {\bar {\rho }}={\begin{pmatrix}m\\2\end{pmatrix}}^{-1}\sum \sum \rho }$

Literature and Sources

• J. Bortz, GA Lienert, K. Boehnke: Distribution-free methods in biostatistics. 3. Edition. Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-74706-2 , chap. 9. doi : 10.1007 / 978-3-540-74707-9_9
• MG Kendall, B. Babington Smith: The Problem of m Rankings . In: The Annals of Mathematical Statistics . tape 10 , no. 3 , September 1939, p. 275-287 , doi : 10.1214 / aoms / 1177732186 , JSTOR : 2235668 .