Partial Residual Plots Based on Robust Fits

Abstract

In seeking information concerning the functional relationship between additional predictors and a response, partial residual plots are useful graphical tools for identifying the unknown function. Usually these plots are based on least squares (LS) fitting. The sensitivity of LS fitting to outliers, though, can distort their partial residuals, making the identification of the unknown function difficult to impossible. Furthermore, in this setting the unknown function can be nonlinear. Points that are located in areas of extreme curvature can act as outliers resulting in similar distortion. In this article, partial residual plots based on highly efficient robust fits are discussed and their firstorder behavior is determined. Based on this discussion, robust partial residuals will be far less distorted than their LS counterparts in the presence of outliers and, hence, will be more powerful in identifying the unknown function. Furthermore, a measure of precision (efficiency) is developed to compare partial residual plots based on different fitting criteria. Three examples and a small Monte Carlo investigation show that the robust partial residual plots, unlike their LS counterparts, are not distorted by outliers or points that are located in areas of extreme curvature.

KEY WORDS:

• Huber estimate
• Linear model
• Outlier
• Regression diagnostics
• Rank-based methods
• Wilcoxon estimate