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 x₂ in regression problems in which the covariate vector x is partitioned as x^r = (x₁^T, x₂^T). The usefulness of these plots for obtaining a good impression of curvature can depend on the behavior of the covariates through the conditional expectation E(x_1/x₂). Partial residual plots seem to perform best under linear conditional expectations. Augmented partial residual plots allow E(x_1/x₂) to be a quadratic function of x₂. This de­velopment leads to a new class of plots, called CERES plots, that includes partial and augmented partial residual plots as spccial eases. CERES plots may be useful for obtaining an impression of curvature as a function of x₂ when the conditional expectations E(x_1/x₂) are neither linear nor quadratic. The relationship between these developments and generalized additive models is discussed as well. © 1993 American statistical association and the American society for quality control.