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Statistica Sinica 22 (2012), 123-148

doi:http://dx.doi.org/10.5705/ss.2009.229





TESTS FOR VARIANCE COMPONENTS IN VARYING

COEFFICIENT MIXED MODELS


Zaixing Li, Yuedong Wang, Ping Wu, Wangli Xu and Lixing Zhu


China University of Mining $\&$ Technology, Beijing,
University of California, Santa Barbara, East China Normal University,
RenMin University of China and Hong Kong Baptist University


Abstract: 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.



Key words and phrases: Extended quasi-likelihood, likelihood ratio test, longitudinal data, random effects, score test, smoothing spline, variance components, varying coefficient models.

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