A proportional-reduction-in-impurity measure of association for categorical variables

This paper presents a proportional-reduction-in-impurity (PRI) measure for categorical association, that employs application-dependent loss functions which make the measure widely applicable. The well-known proportional -reduction-in-error (PRE) measure is shown to be a special case of the new PRI measure. Moreover, the asymptotic variance of the maximum likelihood estimator (MLE) of the measure is derived to facilitate its use for statistical inference. An extension of the PRI measure to compositional association is made to show that it can have a variety of applications. Selected loss functions are treated to illustrate the derivation of the measure.