| Summary: | The likelihood for the parameters of a generalised linear mixed model involves an integral which may be of very high dimension. Because of this apparent intractability, many alternative methods have been proposed for inference in these models, but it is shown that all can fail when the model is sparse, in that there is only a small amount of information available on each random effect. The sequential reduction method developed in this thesis seeks to fill in this gap, by exploiting the dependence structure of the posterior distribution of the random effects to reduce dramatically the cost of approximating the likelihood in models with sparse structure. Examples are given to demonstrate the high quality of the new approximation relative to the available alternatives. Finally, robustness of various estimators to misspecification of the random effect distribution is considered. It is found that certain marginal composite likelihood estimators are not robust to such misspecification in situations in which the full maximum likelihood estimator is robust, providing a counterexample to the notion that composite likelihood estimators will always be at least as robust as the maximum likelihood estimator under model misspecification.
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