Graphical generalized residuals in fitting distributions: Applications to epidemiological follow‐up data

Three kinds of generalized residuals based on probability integral transformation are defined and their roles in graphical goodness of fit testing are discussed, and a simple method, based on non-parametric stratified residual plots, for selection of risk factors, is discussed.

Abstract

Three kinds of generalized residuals based on probability integral transformation are defined and their roles in graphical goodness of fit testing are discussed. Although they are useful in fitting parametric distributions, their value is questionable, however, in fitting Cox's proportional hazard rate (PHR) models. They almost always appear to give a good fit because of (a) almost non-parametric nature of the PHR models; (b) heavy right-hand censoring in epidemiological data; and (c) scale adjustment in graphical presentation. Thus, the apparent overall fit has little inferential meaning, but the residuals are useful in exploratory analysis. A simple method, based on non-parametric stratified residual plots, for selection of risk factors is discussed. It is, in fact, equivalent to comparison of several empirical survival functions, one for each set of values (stratum) of a covariable under consideration. It is also suggested that for the preliminary PHR model, the continuous variables should be discretized and indicator variables used in the model to check analytically the graphical perception of the possible functional form of the contribution of each covariable to the PHR model.