On Testing Random Effects In Linear Mixed Models

We can approach problems involving random effects in linear mixed models directly through the random effects or through parameters such as that the variance components that describe the distributions of the random effects. Both approaches are useful but lead to different issues. For example, working with the random effects raises questions of how to estimate them and can mean dealing with a large number of random effects, while working with variance components leads to testing hypotheses on the boundary of the parameter space. In both cases, finding good approximations to the null distribution of the test statistic can be challenging so modern approaches often rely on simulation. In this talk, we re-examine the F-test based on linear combinations of the responses, for testing random effects in linear mixed models. We present a general derivation of the test, highlight its computation speed, its generality, and its exactness as a test, and report empirical studies into the finite sample performance of the test. We conclude the presentation by reporting our latest results from ongoing research that investigates connections between testing and model selection by discussing some tests of significance of random effects and exploring their relationship to model selection procedures.