The impact of random-effects misspecification on maximum likelihood estimators in generalized linear mixed models

Abstract

... This thesis can be structurally divided into three parts. The first part presents a concise introduction to mental health, and to the key motivating study on schizophrenia, which is used throughout this work (Chapter 2). Chapter 3 provides a brief review of the generalized linear mixed model and a short discussion on some of the challenges involved in its application. The introductory part ends with Chapter 4, which describes an initial analysis of the case study using a logistic-normal model. The second part, consisting of 4 chapters, presents an overall picture of the effect of random-effects misspecification on maximum likelihood estimation. First, some important contributions by White (1982) on likelihood inferences under general model misspecification are summarized in Chapter 5. Chapter 6 focuses on random-effects misspecification in the special case of linear mixed models, whereas Chapter 7 undertakes the study of the impact of this type of misspecification on the maximum likelihood estimators in generalized linear mixed models. Guidelines are supplied to distinguish those situations in which the misspecification has a negligible impact, from those in which the misspecification can have serious consequences. Finally, Chapter 8 provides a theoretical result which states that whenever a subset of fixed-effects parameters, not included in the random-effects structure, equals zero, the corresponding maximum likelihood estimator will consistently estimate zero. This implies that under certain conditions a significant effect could be considered as a reliable result, even if the random-effects distribution is misspecified. The third part of this work comprises 5 additional chapters, which focus on remedial measures for random-effects misspecification. Following some ideas by White (1982), Chapters 9 and 10 introduce a set of diagnostic tools to detect misspecification. The availability of such a toolbox then naturally raises the issue of how to proceed in the presence of misspecification. When the number of subjects and the number of repeated measurements per subject are sufficiently large, it will be shown in Chapter 11 that the maximum likelihood estimators of the mean structure remain asymptotically robust, irrespective of the distribution of the random effects. However, when the available information is not sufficiently large to rely on asymptotic results, alternative approaches need to be considered. Since at the moment there does not seem to exist a model family which is generally robust against this type of misspecification, Chapter 12 proposes a sensitivity analysis, where different distributions are considered for the random effects. In some specific situations, robust alternative models can be found. For instance, the linear mixed model for normal responses is known to be robust against random-effects misspecification. In Chapter 13, it will be shown that another example is given by the Poisson-gamma model for repeated counts. Finally, Chapter 14 recapitulates some concluding remarks and offers a perspective on possible future research.