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