The correlation between two variables, \(x\) and \(y\), is:
\[\begin{align} \tau_b = \frac{n_c-n_d}{\sqrt{(n_t-n_x)(n_t-n_y)}} \end{align}\]
where
\(n_c = \sum^n_{i=1} \sum^n_{j=1} w_i (I_{x_i\gt x_j,y_i\gt y_j}+I_{x_i\gt x_j,y_i\gt y_j} )\),
\(n_d = \sum^n_{i=1} \sum^n_{j=1} w_i ( I_{x_i\lt x_j,y_i\gt y_j}+I_{x_i\gt x_j,y_i\lt y_j})\),
\(n_w = \sum^n_{i=1} w_i \),
\( n_t =\frac{n_w(n_w-1)}{2}\),
\(n_x = \sum^t_{j=1} \sum^n_{i=n} w_i I_{x_i=j}\),
\(n_y = \sum^r_{j=1} \sum^n_{i=n} w_i I_{y_i=j}\),
\(w_i\) is the Calibrated Weight for the \(i\)th of \(n\) is the number of observations,
\(x\) is a variable with \(t\) unique values, categorized in the range \({{1,2,..,t}}\),
\(y\) is a variable with \(r\) unique values, categorised in the range \({{1,2,..,r}}\),
the test statistic is
\(z = {n_c - n_d \over \sqrt{ v } } \)
where
\(v = (v_0 - v_x - v_y)/18 + v_1 + v_2 \),
\( v_0 = n (n-1) (2n+5)\),
\(v_x = \sum_j t_{xj} (t_{xj}-1) (2 t_{xj} + 5) \),
\(v_y = \sum_j t_{yj} t_{yj}-1) (2 t_{yj} + 5) \),
\(v_1 = \sum^r_{j=1} t_{xj}(t_{xj}-1)(t_{xj}-2)\),
\(v_2 = \sum^t_{j=1} t_{yj}(t_{yj}-1)(t_{yj}-2) \),
\(v_3 = (v1 v2) / (9 n_w (n_w - 1) (n_w - 2)) \),
\(v_4 = \sum^r_{j=1} t_{xj}(t_{xj}-1)\),
\(v_5 = \sum^t_{j=1} t_{yj}(t_{yj}-1)\),
\(v_6 = (v_4 v_5) / (2 n_w (n_w - 1))\),
\(\hat{\sigma} = (v_0 - v_x - v_y) / 18 + v3 + v6\),
\(z = \frac{n_c - n_d}{\hat{\sigma}}\),
\(p \approx 2(1-\Phi(|z|))\)