Note on Rank-Biserial Correlation when There Are Ties

作     者：José Moral de la Rubia José Moral de la Rubia

作者机构：School of Psychology Universidad Autónoma de Nuevo León Monterrey México 

出 版 物：《Open Journal of Statistics》 (统计学期刊（英文）)

年 卷 期：2022年第12卷第5期

页      面：597-622页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Ordinal Variable Dichotomy Linear Association Nonparametric Statistics Descriptive Statistics 

摘      要：The objective of this article is to demonstrate with examples that the two-sided tie correction does not work well. This correction was developed by Cureton so that Kendall’s tau-type and Spearman’s rho-type formulas for rank-biserial correlation yield the same result when ties are present. However, a correction based on the bracket ties achieves the desired goal, which is demonstrated algebraically and checked with three examples. On the one hand, the 10-element random sample given by Cureton, in which the two-sided tie correction performs well, is taken up. On the other hand, two other examples are given, one with a 7-element random sample and the other with a clinical random sample of 31 participants, in which the two-sided tie correction does not work, but the new correction does. It is concluded that the new corrected formulas coincide with Goodman-Kruskal’s gamma as compared to Glass’ formula that matches Somers’ dY|X or asymmetric measure of association of Y ranking with respect to X dichotomy. The use of this underreported coefficient is suggested, which is very easy to calculate from its equivalence with Kruskal-Wallis’ gamma and Somers’ dY|X.