Testing multiple variance components in linear mixed-effects models

Testing zero variance components is one of the most challenging problems in the context of linear mixed-effects (LME) models. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under this null hypothesis is incorrect because the null is on the boundary of the parameter space. During the last two decades many tests have been proposed to overcome this difficulty, but these tests cannot be easily applied for testing multiple variance components, especially for testing a subset of them. We instead introduce a simple test statistic based on the variance least square estimator of variance components. With this comes a permutation procedure to approximate its finite sample distribution. The proposed test covers testing multiple variance components and any subset of them in LME models. Interestingly, our method does not depend on the distribution of the random effects and errors except for their mean and variance. We show, via simulations, that the proposed test has good operating characteristics with respect to Type I error and power. We conclude with an application of our process using real data from a study of the association of hyperglycemia and relative hyperinsulinemia.