ABSTRACT
This chapter presents the use of random-effect models for variable selection and hypothesis testing, covering a recent development in multiple testing. It presents a study of an HGLM with discrete random effects. The chapter describes how to use the h-likelihood for the inferences about discrete random effects; it focuses on hypothesis testing, especially multiple testing of recent interest. The chapter also discusses a general approach to simultaneously perform variable selection and estimation of the regression coefficients via random-effect models. Here all the extended likelihoods, defined in any scale of the discrete random effect, are the h-likelihood. Many classical subset selection methods have been proposed: stepwise forward-selection, stepwise backward-elimination or best-subset selection. In many regression problems, the explanatory variables often possess a natural structure, so that we need variable selection, respecting these structures. Interaction terms in regression models form a natural hierarchy with the main effects, so their selection requires special considerations.
