EXPLORING PARTIAL RESIDUAL PLOTS

Partial residual plots have a long history and, judging from their prominence in the literature, are frequently used. In this article, I explore the structure and usefulness of partial residual plots and augmented partial residual plots as basic tools for dealing with curvature as a function of selected covariates x(2) in regression problems in which the covariate vector x is partitioned as x(T) = (x(1)(T), X(2)(T) ). The usefulness of these plots for obtaining a good impression of curature can depend on the behavior of the covariates through the conditional expectation E(x(1)/x(2)). Partial residual plots seem to perform best under linear conditional expectations. Augmented partial residual plots allow E(x(1)/x(2)) to be a quadratic function of X(2). This development leads to a new class of plots, called CERES plots, that includes partial and augmented partial residual plots as special cases. CERES plots may be useful for obtaining an impression of curvature as a function of X(2) when the conditional expectations E(x(1)/x(2)) are neither linear nor quadratic. The relationship between these developments and generalized additive models is discussed as well.