A simple test for random effects in regression models

Testing that random effects are zero is difficult, because the null hypothesis restricts the corresponding
variance parameter to the edge of the feasible parameter space. In the context of generalized linear mixed
models, this paper exploits the link between random effects and penalized regression to develop a simple
test for a zero effect. The idea is to treat the variance components not being tested as fixed at their estimates
and then to express the likelihood ratio as a readily computed quadratic form in the predicted values of
the random effects. Under the null hypothesis this has the distribution of a weighted sum of squares of
independent standard normal random variables. The test can be used with generalized linear mixed models,
including those estimated by penalized quasilikelihood.