Relationship between Kendall’s tau and Goodman and Kruskal’s gamma after ordinalizing a bivariate normal random variable

Goodman and Kruskal’s gamma is a measure of association between two ordinal variables, based on probabilities of concordance and discordance, which can be seen as an adjustment to the discrete case of Kendall’s rank correlation tau between two continuous random variables. By considering a standard bivariate normal random variable acting as a latent underlying distribution, we examine the relationship between its value of Kendall’s tau (which is related to Pearson’s correlation coefficient through a well-known analytic formula) and the value of gamma for a bivariate ordinalized distribution, by varying its margins and examining in particular uniform, unimodal symmetric, and triangular distributions. Based on this study, a procedure for finding the value of tau inducing a target value of gamma after ordinalization of a standard bivariate normal random variable is devised.