A new coefficient of separation

Abstract: A coefficient is introduced that quantifies the extent of separation of a random variable $Y$ relative to a number of variables $\mathbf{X} = (X_1, \dots, X_p)$ by skillfully assessing the sensitivity of the relative effects of the conditional distributions. The coefficient is as simple as classical dependence coefficients such as Kendall's tau, also requires no distributional assumptions, and consistently estimates an intuitive and easily interpretable measure, which is $0$ if and only if $Y$ is stochastically comparable relative to $\mathbf{X}$, that is, the values of $Y$ show no location effect relative to $\mathbf{X}$, and $1$ if and only if $Y$ is completely separated relative to $\mathbf{X}$. As a true generalization of the classical relative effect, in applications such as medicine and the social sciences the coefficient facilitates comparing the distributions of any number of treatment groups or categories. It hence avoids the sometimes artificial grouping of variable values such as patient's age into just a few categories, which is known to cause inaccuracy and bias in the data analysis. The mentioned benefits are exemplified using synthetic and real data sets.