On the Inference of Random Effects in Generalized Linear Mixed Models

In the first chapter, the problem of Bootstrap inference for the parameters of a GLMM is addressed. We formulate a bootstrapping strategy consisting on the random weighting of the contributions to the Joint Likelihood of Outcomes and Random Effects. Using the Laplace Approximation method for integrals on this function, yields a Random Weighted Log-Likelihood that produces the desired bootstrap replicates after optimization. In order to assess the properties of this procedure, that we name Random Weighted Likelihood Bootstrap (RWLB), we compare analytically their resulting EE to those of the Generalized Cluster Bootstrap for Gaussian LMM and conduct simulation studies both in a LMM and Mixed Logit regression contexts. The second chapter explores adaptations of the RWLB to the estimation of the uncertainty in prediction of random effects in a GLMM, as measured by the Mean Squared Error for the Predictors (MSEP)