Explained randomness

We recall the main ideas concerning information gain and how it can be used to obtain a measure of dependence. This measure can be interpreted as a measure of explained randomness. We recall the work of Kent and O'Quigley (1988) in adapting such measures to proportional hazards regression. An alternative to the measure of Kent and O'Quigley, in which the order of conditioning is reversed, is presented. This alternative measure (Xu and O'Quigley 2000) fits in more naturally with the inference structure for the proportional hazards model and, in particular, makes a direct appeal to the main theorem of proportional hazards regression (Section 7.4). Among the several advantages of the Xu and O'Quigley measure is that it readily accommodates time-dependent covariates. Extensions to multiple covariates are immediate. We also indicate the straightforward extension to partial coefficients. In practice we expect a measure of explained randomness and one of explained variation to agree. We take the position that any purpose to which we may wish to put a coefficient of explained variation will be, for the most part, equally well addressed by a coefficient of explained randomness.