Goodman and Kruskal's Gamma Coefficient for Ordinalized Bivariate Normal Distributions

We consider a bivariate normal distribution with linear correlation [Formula: see text] whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal's gamma coefficient, [Formula: see text], which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson's [Formula: see text] and Kendall's rank correlation [Formula: see text] for the bivariate normal distribution, and since in the continuous case, Kendall's [Formula: see text] coincides with Goodman and Kruskal's [Formula: see text], the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall's rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal's [Formula: see text] by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided.