It is an inference on the measure of correlation among two ordered qualitative variables. Measuring strength of association of cross tabulated data. It was proposed by Leo Goodman and William Kruskal across a series of papers (1954, 1959, 1963 and 1972).

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 (Goktas and Qznur, 2011).

If C > D the variables have a positive association, but if C < D then the variables have a negative association. C and D are, respectivelly, the total number of concordances and discordances.

The ratio (C - D)/(C + D) is the Goodman-Kruskal´s gamma statistic (g). It can lie between -1 and 1. Sign indicates if ther is a negative or positive association. The magnitude indicates strength of association.

The asymtotic Goodman-Kruskal's gamma variance is calculated as,

varg = 4/(C + D)^2*(Sum_i Sum_j n_ij*((A_ij - B_ij)^2 - (C - D)^2/n)

where,
A_ij = Sum_k<i Sum_l<j a_kl + Sum_k>i Sum_l>j a_kl

B_ij = Sum_k>i Sum_l<j a_kl + Sum_k<i Sum_l>j a_kl

a_ij denote the observed frequency in cell (i,j) in an I x J contingency table.

Concordances, C is: C = Sum_i Sum_j a_ij*A_ij

Discordances, D is: D = Sum_i Sum_j a_ij*B_ij

The z-value = g/sqrt(varg)

Null hypothesis: no association (no relationship or independence) corresponding gamma = 0. Alternative hypothesis (one-sided) gamma < 0 or gamma > 0.

Syntax: function gkgammatst(x,alpha,tail)

Inputs:
x – rxc data matrix
alpha - significance value (default=0.05)
tail - two-sided=0; one-sided~=0

Output:
- Summary statistics from the analysis