Abstract
ABSTRACTSince the seminal paper by Cook and Weisberg [9], local influence, 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 [17]. Ouwens et al. [24] applied the method to the Poisson-normal generalized linear mixed model (GLMM). Given the model's nonlinear 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. Unlike when case deletion is used, this leads to interpretable components, allowing not only to identify influential subjects, but also to study the cause thereof. The methodology is illustrated in case studies that range over the three data types mentioned.
Highlights
To linear mixed models (LMM) for hierarchical Gaussian data [26], generalized linear mixed models (GLMM; [2, 19, 28]) have become a standard tool for the analysis of hierarchical data of a variety of data types
Using the extension proposed by Molenberghs, Verbeke, and Demetrio [20] and Molenberghs et al [21], we flexibly allow for overdispersion in the GLMM, by introducing conjugate random effects, in addition to normal ones
The data considered here are obtained from a randomized, double-blind, parallel group multi-center study for the comparison of placebo with a new anti-epileptic drug (AED), in combination with one or two other AED’s
Summary
To linear mixed models (LMM) for hierarchical Gaussian data [26], generalized linear mixed models (GLMM; [2, 19, 28]) have become a standard tool for the analysis of hierarchical data of a variety of data types. Residual analysis is not straightforward, given the presence of both fixed- and random-effects, so that even uniquely defining residuals is not possible For these and related reasons, Lesaffre and Verbeke [17] , chose local influence [1, 8] to examine influence in linear mixed models. Using the extension proposed by Molenberghs, Verbeke, and Demetrio [20] and Molenberghs et al [21], we flexibly allow for overdispersion in the GLMM, by introducing conjugate random effects, in addition to normal ones. This model is referred to as the combined model.
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