Regression Diagnostics

TLDR

Robustness of regression estimates is crucial in small‑sample analyses. The study reviews recent developments in detecting and analyzing outliers and influential cases in multivariate studies, illustrated by two empirical applications. We review five diagnostic procedures—partial regression plots, the hat matrix, studentized residuals, DFITS, and DFBETAS—and demonstrate their application in two empirical studies. The applications show that diagnostics must be interpreted with substantive context, guiding targeted remedies rather than routine case deletion, and can identify issues such as sample composition, specification error, and measurement errors.

Abstract

Gauging the robustness of regression estimates is especially important in small-sample analyses. Here, we examine recent developments in the detection and analysis of outliers and influential cases in multivariate studies. Specifically, we review five diagnostic procedures: partial regression plots, the “hat” matrix, studentized residuals, DFITS i , and DFBETAS ij . The main part of the article presents two empirical applications (drawn from recent cross-national studies) that show (a) how the diagnostic procedures can be incorporated into the research process, and (b) what we can learn from them. These applications serve to underscore the point that the diagnostics cannot be employed mechanically. Instead, once a case is diagnosed as influential, remedial action requires a firm substantive grounding. Although case deletion may be warranted in some circumstances, it is an extreme remedy of last resort that should not be routinely followed. The more fruitful approach is to ask why a given case is influential. As our applications indicate, the diagnostics can be helpful in isolating such problems as sample composition, specification error, and errors in measurement.