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)
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)
x – rxc data matrix
alpha - significance value (default=0.05)
tail - two-sided=0; one-sided~=0
- Summary statistics from the analysis
Antonio Trujillo-Ortiz (2021). gkgammatst (https://www.mathworks.com/matlabcentral/fileexchange/42645-gkgammatst), MATLAB Central File Exchange. Retrieved .
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