All these indexes are based on concordant and discordant pairs.

A pair of observations is concordant if the subject who is higher on one variable also is higher on the other variable, and a pair of observations is discordant if the subject who is higher on one variable is lower on the other variable.

Let $C$ be the total number of concordant pairs (concordances) and $D$ the total number of discordant pairs (discordances) . If $C > D$ the variables have a positive association, but if $C < D$ then the variables have a negative association.

In symbols, given an $I \times J$ contingency table the concordant pairs with cell $i,j$ are

\[ a_{ij} = \sum_{k<i} \sum_{l<j} n_{kl} + \sum_{k>i} \sum_{l>j} n_{kl} \] the number of discordant pairs is \[ b_{ij} = \sum_{k>i} \sum_{l<j} n_{kl} + \sum_{k<i} \sum_{l>j} n_{kl} \] Twice the number of concordances, $C$ is given by: \[ 2 \times C = \sum_{i=1}^I \sum_{j=1}^J n_{ij} a_{ij} \] Twice the number of discordances, $D$ is given by: \[ 2 \times D = \sum_{i=1}^I \sum_{j=1}^J n_{ij} b_{ij} \] Goodman-Kruskal's $\gamma$ statistic is equal to the ratio: \[ \gamma= \frac{C-D}{C+D} \]

$\tau_a$ is equal to concordant minus discordant pairs, divided by a factor which takes into account the total number of pairs.

\[ \tau_a= \frac{C-D}{0.5 n(n-1)} \]

$\tau_b$ is equal to concordant minus discordant pairs divided by a term representing the geometric mean between the number of pairs not tied on x and the number not tied on y.

More precisely:

\[ \tau_b= \frac{C-D}{\sqrt{ (0.5 n(n-1)-T_x)(0.5 n(n-1)-T_y)}} \]

where $T_x= \sum_{i=1}^I 0.5 n_{i.}(n_{i.}-1)$ and $T_y=\sum_{j=1}^J 0.5 n_{.j}(n_{.j}-1)$ Note that $\tau_b \leq \gamma$.

$\tau_c$ is equal to concordant minus discordant pairs multiplied by a factor that adjusts for table size.

\[ \tau_c= \frac{C-D}{ n^2(m-1)/(2m)} \]

where $m= min(I,J)$;

Somers' $d_{y|x}$ is an asymmetric extension of $\gamma$ that differs only in the inclusion of the number of pairs not tied on the independent variable. More precisely

\[ d_{y|x} = \frac{C-D}{0.5 n(n-1)-T_x} \]

Null hypothesis: corresponding index = 0. Alternative hypothesis (one-sided) index < 0 or index > 0.

In order to compute confidence intervals and test hypotheses, this routine computes the standard error of the various indexes.

Note that the expression of the standard errors which is used to compute the confidence intervals is different from the expression which is used to test the null hypothesis of no association (no relationship or independence) between the two variables.

As concerns the Goodman-Kruskal's $\gamma$ index we have that:

\[ var(\gamma) = \frac{4}{(C + D)^4} \sum_{i=1}^I \sum_{j=1}^J n_{ij} (D a_{ij} - C b_{ij} )^2 \] where \[ d_{ij}=a_{ij}- b_{ij} \] The variance of $\gamma$ assuming the independence hypothesis is: \[ var_0(\gamma) =\frac{1}{(C + D)^2} \left( \sum_{i=1}^I \sum_{j=1}^J n_{ij} d_{ij}^2 -4(C-D)^2/n \right) \] As concerns $\tau_a$ we have that: \[ var(\tau_a)= \frac{2}{n(n-1)} \left\{ \frac{2(n-2)}{n(n-1)^2} \sum_{i=1}^I \sum_{j=1}^J (d_{ij} - \overline d)^2 + 1 - \tau_a^2 \right\} \qquad \mbox{with $i,j$ such that $N(i,j)>0$} \] where \[ \overline d = \sum_{i=1}^I \sum_{j=1}^J d_{ij} /n \qquad \mbox{with $i,j$ such that $N(i,j)>0$} \] The variance of $\tau_a$ assuming the independence hypothesis is: \[ var_0(\tau_a) =\frac{2 (2n+5)}{9n(n-1) } \] As concerns $\tau_b$ we have that: \[ var(\tau_b)= \frac{n}{w^4} \left\{ n \sum_{i=1}^I \sum_{j=1}^J n_{ij} \tau_{ij}^2 - \left( \sum_{i=1}^I \sum_{j=1}^J n_{ij}\tau_{ij}\right)^2 \right\} \] where \[ \tau_{ij} = 2n d_{ij} +2(C-D) n_{.j} w /n^3+2(C-D) (n_{i.}/n) \sqrt{ w_c/w_r} \qquad \mbox{and} \qquad w= \sqrt{w_rw_c} \] The variance of $\tau_b$ assuming the independence hypothesis is: \[ var_0(\tau_b) =\frac{4}{w_r w_c} \left\{ \sum_{i=1}^I \sum_{j=1}^J n_{ij} d_{ij} ^2 -4(C-D)^2/n \right\} \] As concerns Stuart's $\tau_c$ we have that: \[ var(\tau_c)= \frac{4m^2}{(m-1)^2 n^4} \left\{ \sum_{i=1}^I \sum_{j=1}^J n_{ij} d_{ij} ^2 -4(C-D)^2/n \right\} \] The variance of $\tau_c$ assuming the independence hypothesis is: \[ var_0(\tau_c) =var(\tau_c) \] As concerns $d_{y|x}$ we have that: \[ var( d_{y|x})= \frac{4}{w_r^4} \left\{ \sum_{i=1}^I \sum_{j=1}^J n_{ij} (w_r d_{ij} -2(C-D) (n-n_{i.}) \right\}^2 \] where \[ w_r= n^2- \sum_{i=1}^I n_{i.}^2 \] The variance of $d_{y|x}$ assuming the independence hypothesis is: \[ var_0(d_{y|x}) = \frac{4}{w_r^2} \left\{ \sum_{i=1}^I \sum_{j=1}^J n_{ij} d_{ij} ^2 -4(C-D)^2/n \right\} \]

From the theoretical point of view, Simon (1978) showed that all sample measures having the same numerator $(C-D)$ have the same efficacy and hence the same local power, for testing independence.