Deviance residuals and normal scores plots

SUMMARY We discuss the use of normal order statistics plots, based on deviance residuals, to check distributional assumptions in regression models. Continuous and discrete error distributions are considered, as are censored data. Misspecified error distributions and discrimination between competing models are discussed, with an example. Residual plots to detect inadequacies in normal linear regression models have a long history. An example is the normal scores or rankit plot: a plot of ordered residuals against normal order statistics, which is used to detect outliers and to check distributional assumptions. Under- or overdispersion in such a plot may also indicate a misspecified systematic component of the model. The purpose of this paper is to discuss such plots for regressions with nonnormal errors, such as generalized linear models, for which deviance residuals are commonly used. Deviance residuals are known to be approximately normal in many cases. In this paper we briefly describe the properties of rankit plots based on them for continuous distributions, outline analogous results for discrete and censored data, and describe a method for discriminating between competing models with different error distributions. Most roads lead to Rome for the normal distribution in the sense that many definitions of residuals are functions of (y - jt)/ o. Not so for other distributions, for which various