The relation between Mann--Whitney's statistic and Kendall's correlation coefficient~$\tau$

M. G. Gel'berg

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 1, pp. 211-213

Résumé

It is shown that Kendall's correlation coefficient may be expressed as follows: $$ \tau=\frac{4\sum_{s=1}^k\sum_{f=1}^{s-1}U_{sf}-N^2+\sum_{s=1}^kn_{s\cdot}^2}{\sqrt{(N^2-\sum n_{s\cdot}^2)(N^2-\sum n_{\cdot t}^2)}} $$ where $N$ is the sample size $U_sf$ is Mann–Whitney's statistic for the conditional distributions of $Y$ given $X_s$ and $X_f\cdot$. For $k=l$, $n_{s\cdot}=n_{\cdot t}=N/k$ for all $s$ and $t$, put $\widehat p_{st}=U_{st}/n_{s\cdot}n_{f\cdot}$; then $$ \tau=2\Biggl[\frac{\sum_{s=1}^k\sum_{t=1}^{s-1}\widehat p_{st}}{1/2k(k-1)}-\frac12\Biggr]. $$ The first term in the brackets is the mean value of the normalized Mann–Whitney's statistic over all paired comparisons of conditional distributions of $Y$.