Directional nature of Goodman–Kruskal gamma and some consequences: identity of Goodman–Kruskal gamma and Somers delta, and their connection to Jonckheere–Terpstra test statistic

Abstract Although usually taken as a symmetric measure, G is shown to be directional coefficient of association. The direction in not related rows or columns the cross-table nor identity variables predictor criterion variable but, instead, number categories scales. Under conditions where there are no tied pairs dataset, equals Somers’ D so directed that with wider scale ( X ) explains response pattern narrower g ), is, │ ). Hence, = but ≠ and (symmetric). If pairs, estimates by more liberal comparison those Algebraic relation Jonckheere–Terpstra test statistic JT derived. Because connection , indicate proportion logically ordered test-takers item after they score. It strongly recommendable gamma should used it directionally only when willing explain behaviour scale. This fits well measurement modelling settings.