Explained randomness in proportional hazards models

Abstract

A coefficient of explained randomness, analogous to explained variation but for non-linear models, was presented by Kent. The construct hinges upon the notion of Kullback-Leibler information gain. Kent and O'Quigley developed these ideas, obtaining simple, multiple and partial coefficients for the situation of proportional hazards regression. Their approach was based upon the idea of transforming a general proportional hazards model to a specific one of Weibull form. Xu and O'Quigley developed a more direct approach, more in harmony with the semi-parametric nature of the proportional hazards model thereby simplifying inference and allowing, for instance, the use of time dependent covariates. A potential drawback to the coefficient of Xu and O'Quigley is its interpretation as explained randomness in the covariate given time. An investigator might feel that the interpretation of the Kent and O'Quigley coefficient, as a proportion of explained randomness of time given the covariate, is preferable. One purpose of this note is to indicate that, under an independent censoring assumption, the two population coefficients coincide. Thus the simpler inferential setting for Xu and O'Quigley can also be applied to the coefficient of Kent and O'Quigley. Our second purpose is to point out that a sample-based coefficient in common use in the SAS statistical package can be interpreted as an estimate of explained randomness when there is no censoring. When there is censoring the SAS coefficient would not seem satisfactory in that its population counterpart depends on an independent censoring mechanism. However there is a quick fix and we argue in favour of its use. Copyright © 2004 John Wiley & Sons, Ltd.