The Standardized Influence Matrix and its Applications

Abstract

In this article we introduce the standardized influence matrix (SIM) of parameter estimators as a conjugate of the sample covariance matrix of standardized influence functions (SIFs) evaluated at the data points. We propose principal component analysis for the SIM and its complement (SIM analysis) as a diagnostic tool for regression or other statistical inference, and provide theoretical insight to the local influence defined by Cook. We show that SIM analysis reveals the multivariate structure of outlying and/or influential points. Specifically, SIM analysis uncovers hidden structures of influence, such as clustering, that cannot be identified by the lengths of the standardized influences. Finally, examples in linear regression show that a diagnostic method using SIM is more effective if robust parameter estimates are used in calculating the sample SIM.