A Class of Regression Association Measures based on Concordance

Measures of regression association aiming at predictability of a dependent variable Y from an independent variable X have received considerable attention recently. In this article, we provide a unified discussion of some existing measures, including their rationale, properties, and estimation. Motivated by these measures, we consider a general class of regression association measures which views the regression association of Y from X as the association of two independent replications from the conditional distribution of Y given X. We illustrate that the so-called Markov product copulas can be employed as a neat and convenient building block for this class of measures, and the measures so constructed can be expressed as a common form of the proportion of the variance of some function of Y that can be explained by X, rendering the measures a direct interpretation in terms of predictability. Also, the notion of two independent replications from the conditional distribution leads to a simple nonparametric estimation method based on the induced order statistics, hence, no smoothing techniques are required. Under the considered general framework, the performances and utilities of the regression association measures are examined through simulations and real data applications.