Local Influence Diagnostics for Generalized Linear Mixed Models With Overdispersion

Abstract

Since the seminal paper by Cook and Weisberg (1982), local in- fluence, next to case deletion, has gained popularity as a tool to detect influential subjects and measurements for a variety of statistical models. For the linear mixed model the approach leads to easily interpretable and computationally convenient expressions, not only highlighting influential subjects, but also which aspect of their profile leads to undue influence on the model’s fit (Verbeke and Lesaffre 1998). Ouwens, Tan, and Berger (2001) applied the method to the Poisson-normal generalized linear mixed model (GLMM). Given the model’s non-linear structure, these authors did not derive interpretable components but rather focused on a graphical depiction of influence. In this paper, we consider GLMMs for binary, count, and time-to-event data, with the additional feature of accommodating overdispersion whenever necessary. For each situation, three approaches are considered, based on: (1) purely numerical derivations; (2) using a closed-form expression of the marginal likelihood function; and (3) using an integral representation of this likelihood. The methodology is illustrated in case studies of A Clinical Trial in Epileptic Patients.