Influence analysis for linear mixed‐effects models

Abstract

In this paper, we extend several regression diagnostic techniques commonly used in linear regression, such as leverage, infinitesimal influence, case deletion diagnostics, Cook's distance, and local influence to the linear mixed-effects model. In each case, the proposed new measure has a direct interpretation in terms of the effects on a parameter of interest, and collapses to the familiar linear regression measure when there are no random effects. The new measures are explicitly defined functions and do not necessitate re-estimation of the model, especially for cluster deletion diagnostics. The basis for both the cluster deletion diagnostics and Cook's distance is a generalization of Miller's simple update formula for case deletion for linear models. Pregibon's infinitesimal case deletion diagnostics is adapted to the linear mixed-effects model. A simple compact matrix formula is derived to assess the local influence of the fixed-effects regression coefficients. Finally, a link between the local influence approach and Cook's distance is established. These influence measures are applied to an analysis of 5-year Medicare reimbursements to colon cancer patients to identify the most influential observations and their effects on the fixed-effects coefficients.