Kendall's Tau-b

The correlation between two variables, $x$ and $y$, is:

$\tau _{b}={\frac {n_{c}-n_{d}}{\sqrt {(n_{t}-n_{x})(n_{t}-n_{y})}}}$

where

$n_{c}=\sum _{i=1}^{n}\sum _{j=1}^{n}w_{i}(I_{x_{i}>x_{j},y_{i}>y_{j}}+I_{x_{i}>x_{j},y_{i}>y_{j}})$,

$n_{d}=\sum _{i=1}^{n}\sum _{j=1}^{n}w_{i}(I_{x_{i}<x_{j},y_{i}>y_{j}}+I_{x_{i}>x_{j},y_{i}<y_{j}})$,

$n_{w}=\sum _{i=1}^{n}w_{i}$,

$n_{t}={\frac {n_{w}(n_{w}-1)}{2}}$,

$n_{x}=\sum _{j=1}^{t}\sum _{i=n}^{n}w_{i}I_{x_{i}=j}$,

$n_{y}=\sum _{j=1}^{r}\sum _{i=n}^{n}w_{i}I_{y_{i}=j}$,

$w_{i}$ is the Calibrated Weight for the $i$th of $n$ is the number of observations,

$x$ is a variable with $t$ unique values, categorised in the range ${1,2,..,t}$,

$y$ is a variable with $r$ unique values, categorised in the range ${1,2,..,r}$,

The tests statistic is:

$z={n_{c}-n_{d} \over {\sqrt {v}}}$

where

$v=(v_{0}-v_{x}-v_{y})/18+v_{1}+v_{2}$,

$v_{0}=n(n-1)(2n+5)$,

$v_{x}=\sum _{j}t_{xj}(t_{xj}-1)(2t_{xj}+5)$,

$v_{y}=\sum _{j}t_{yj}t_{yj}-1)(2t_{yj}+5)$,

$v_{1}=\sum _{j=1}^{r}t_{xj}(t_{xj}-1)(t_{xj}-2)$,

$v_{2}=\sum _{j=1}^{t}t_{yj}(t_{yj}-1)(t_{yj}-2)$,

$v_{3}=(v1v2)/(9n_{w}(n_{w}-1)(n_{w}-2))$,

$v_{4}=\sum _{j=1}^{r}t_{xj}(t_{xj}-1)$,

$v_{5}=\sum _{j=1}^{t}t_{yj}(t_{yj}-1)$,

$v_{6}=(v_{4}v_{5})/(2n_{w}(n_{w}-1))$,

${\hat {\sigma }}=(v_{0}-v_{x}-v_{y})/18+v3+v6$,

$z={\frac {n_{c}-n_{d}}{\hat {\sigma }}}$,

$p\approx 2(1-\Phi (|z|))$

See also

Correlations - Comparing Two Numeric Variables