Diagnostics for Use with Regression Recursive Residuals

Abstract

Conventional regression case diagnostics use the fit of the regression model to all cases to assess the deviation of individual cases from the full sample. Unfortunately, as few as two bad cases can make these diagnostics completely unreliable. An alternative approach is that based on recursive fitting, in which the sample is stripped away case by case and the regression refitted after each deletion until there are only as many cases as parameters. Recursive residuals are residuals obtained from these successive recursive fits. They are particularly effective for diagnosis when the assumptions of the regression model do not hold for the full data set but do hold after some “bad” cases are stripped away. Conventional regression case diagnostics provide not only residuals but also measures of leverage and influence, and in this article, I propose analogous single-case diagnostics for use with recursive fitting. Furthermore, since recursive fitting focuses on the compatibility of various subregressions with the full regression, it is also important to have cumulative measures that assess the leverage, mean compatibility, and influence of all excluded cases; measures of these quantities are also defined. The diagnostics are applied successfully to one artificially constructed data set and to another data set from the literature, and it is shown how they provide useful insights into the fit of the model.