Tests for variance components in varying coefficient mixed models

We consider a general class of varying coefficient mixed models where random effects are introduced to account for between-subject variation. To address the question of whether a varying coefficient mixed model can be reduced to a simpler varying coefficient model, we develop one-sided tests for the null hypothesis that all the variance components are zero. In addition to the purely null-based standard quasi-score test (SQT), we propose an extended quasi-score test (EQT) by constructing estimators that are consistent under both the null and alternative hypotheses. No assumptions are required for the distributions of random effects and random errors. Both SQT and EQT are consistent for global alternatives and local alternatives distinct at certain rates from the null. Furthermore, the asymptotic null distributions are simple and easy to use in practice. For comparison, we also adapt the one-sided score test (SST) in Silvapulle and Silvapulle (1995) and the likelihood ratio test (LRT) in Fan, Zhang, and Zhang (2001). Extensive simulations indicate that all proposed tests perform well and the EQT is more powerful than SQT, SST, and LRT. A data example is analyzed for illustration.