Generalized mixed-effects modelling with a comparison of numerical estimation techniques

Abstract

One of the fundamental assumptions made in statistical modelling is independence between observations. However, nested datasets are omnipresent in today's society, where observations will be grouped within clusters. This clustering of data yields dependence between observations, for which generalized linear models lead to erroneous inferences. In light of this, we study models which cater for the dependence within the data in a binary and ordinal response framework. These models are used to analyze a clustered dataset relating to EU elections, where EU residents are nested within the EU member states. The main goal here is to study which society sectors are most interested in EU elections and how this interest varies between EU member states. The marginal likelihood of these models does not have a closed-form solution and so we resort to numerical estimation techniques. In particular, the likelihood is first approximated using a numerical approximation method and is then combined with a numerical optimization algorithm to obtain parameter estimates. This dissertation mainly focuses on an adaptive Gauss-Hermit quadrature approximation augmented with a modified Newton-Raphson optimization, although ordinary Gauss-Hermite quadrature is also studied in parallel to this. In addition, a Laplacian approximation augmented with a BFGS quasi-Newton optimization as well as quasi-likelihood methods, mainly marginal quasi-likelihood and penalized quasi-likelihood using generalized least squares, are also studied and compared directly to the adaptive Gauss-Hermite quadrature.