Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.