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.