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:

$z=\frac{n_c-n_d}{\sqrt{v}}$

where

$v=(v_0-v_x-v_y)/18+v_1+v_2$,

$v_0=n(n-1)(2n+5)$,

$v_x=\sum _j{t}_{xj}(t_{xj}-1)(2t_{xj}+5)$,

$v_y=\sum _j{t}_{yj}{t}_{yj}-1)(2t_{yj}+5)$,

$v_1=\sum _{j=1}^r{t}_{xj}(t_{xj}-1)(t_{xj}-2)$,

$v_2=\sum _{j=1}^t{t}_{yj}(t_{yj}-1)(t_{yj}-2)$,

$v_3=(v1v2)/(9n_w(n_w-1)(n_w-2))$,

$v_4=\sum _{j=1}^r{t}_{xj}(t_{xj}-1)$,

$v_5=\sum _{j=1}^t{t}_{yj}(t_{yj}-1)$,

$v_6=(v_4{v}_5)/(2n_w(n_w-1))$,

$\hat{\sigma }=(v_0-v_x-v_y)/18+v3+v6$,

$z=\frac{n_c-n_d}{\hat{\sigma }}$,

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

See also

Correlations - Comparing Two Numeric Variables