ABSTRACT
Recently, technological developments in areas such as genomics and neuroimaging have led to situations where thousands or more hypotheses need to be tested simultaneously. The problem of such large-scale simultaneous testing is naturally expressed as a prediction problem for finding the true results in an optimal way (Lee and Bjørnstad, 2013). In this chapter, we formulate hypothesis testing as an HGLM with discrete random effects. Up to now, we have investigated models with continuous random effects, where the choice for the scale of random effects is crucial in defining the h-likelihood. From the transformation of continuous random effects in Chapter 4, we see that the Jacobian term appears in the extended likelihood, so that the mode of extended likelihood is not invariant with respect to the transformation of random effects. However, for the transformation of discrete random effects, such a Jacobian term does not appear in the extended likelihood. Thus, any extended likelihood defined on any scale of the discrete random effect becomes the h-likelihood. Until now, we have studied the estimation of parameters and prediction of unobservables by using h-likelihood. In this chapter we study how to use the h-likelihood for the testing of hypotheses by using discrete random effects. For this, we first review the classical single hypothesis testing.
