Parameter estimation by maximizing the marginal likelihood function in
generalized linear mixed models (GLMMs) is highly challenging because it may
involve analytically intractable high-dimensional integrals. In this paper,
we propose to use Quasi-Monte Carlo (QMC) approximation through implementing Newton-Raphson
algorithm to address the estimation issue in GLMMs. The random effects release
to be correlated and joint mean-covariance modelling is considered. We demonstrate
the usefulness of the proposed QMC-based method in approximating high-dimensional
integrals and estimating the parameters in GLMMs through simulation studies.
For illustration, the proposed method is used to analyze the infamous
salamander mating binary data, of which the marginalized likelihood involves
six 20-dimensional integrals that are analytically intractable, showing that it
works well in practices.
Cite this paper
Chen, Y. , Fei, Y. and Pan, J. (2015). Quasi-Monte Carlo Estimation in Generalized Linear Mixed Model with Correlated Random Effects. Open Access Library Journal, 2, e2002. doi: http://dx.doi.org/10.4236/oalib.1102002.
Breslow, N.E.
and Clayton, D.G.
(1993) Approximate
Inference Ingeneralized Linear Mixed Models. Journal of the American
Statistical Association, 88, 9-25.
Lin, X.
and Breslow, N.E.
(1996) Bias
Correction in Generalized Lnear Mixed Models with Multiple Components of Dispersion. Journal of the American Statistical Association, 91, 1007-1016. http://dx.doi.org/10.1080/01621459.1996.10476971
Lee, Y. and
Nelder, J.A.
(2001) Hierrarchical
Generalized Linear Models: A Synthesis of Generalized Linear Models, Random
Effect Models and Structured Dispersions. Biometrika, 88, 987-1006. http://dx.doi.org/10.1093/biomet/88.4.987
Karim, M.R. and
Zeger, S.L.
(1992) Generalized
Linear Models with Random Effects; Salamnder Mating Revisited. Biometrics,
48, 681-694. http://dx.doi.org/10.2307/2532317
McCulloch, C.E.
(1994) Maximum
Likelihood Variance Components Estimation for Binary Data. Journal of the
American Statistical Association, 89, 330-335. http://dx.doi.org/10.1080/01621459.1994.10476474
McCulloch, C.E.
(1997) Maximum
Likelihood Algorithms for Generallized Linear Mixed Models. Journal of the
American Statistical Association, 92, 162-170. http://dx.doi.org/10.1080/01621459.1997.10473613
Chan, J.S.,
Kuk, A.Y. and Yam, C.H.
(2005)
Monte Carlo Approximation through Gibbs Output in Generalized Linear Mixed Models. Journal of Maltivariate Analysis, 94, 300-312. http://dx.doi.org/10.1016/j.jmva.2004.05.004
Pan, J. and
Thompson, R.
(2003)
Gauss-Hermite Quadrature Approximation for Estimation in Generalised Linear
Mixed Models. Computational Statistics, 18, 57-78.
Pan, J.
and Thompson, R. (2007) Quasi-Monte
Carlo Estimation in Generalized Linear Mixed Models. Computational Statistics
and Data Analysis, 51, 5765-5775. http://dx.doi.org/10.1016/j.csda.2006.10.003
Ye, H.
and Pan, J. (2006) Modelling Covariance Structures in Generalized Estimating Equations for
Longitudinal Data. Biometrika, 93, 927-941. http://dx.doi.org/10.1093/biomet/93.4.927
Spanier, J.
and Maize, E. (1994) Quasi-Random Methods for Estimating Integrals Using Relatively Small Samples. SIAM Review,
36, 19-44. http://dx.doi.org/10.1137/1036002
Pourahmadi, M. (2000) Maximum Likelihood Estimation of Generalised Linear Models for
Multivariate Normal Covariance Matrix. Biometrika, 87, 425-435. http://dx.doi.org/10.1093/biomet/87.2.425
Daniels, M.J.
and Pourahmadi, M. (2002) Bayesian Analysis of Covariance Matrices and Dynamic Models for Longitudinal
Data. Biometrika, 89, 553-566. http://dx.doi.org/10.1093/biomet/89.3.553
Daniels, M.J.
and Zhao, Y.D. (2003) Modelling the Random Effects Covariance Matrix in Longitudinal Data. Statistics
in Medicine, 22, 1631-1647. http://dx.doi.org/10.1002/sim.1470
Smith, M.
and Kohn, R. (2002) Parsimonious Covariance Matrix Estimation for Longitudinal Data. Journal
of the American Statistical Association, 97, 1141-1153. http://dx.doi.org/10.1198/016214502388618942
Shun, Z. (1997) Another Look at the Salamander Mating Data: A
Modified Laplace Approximation Approach. Journal of the American Statistical
Association, 92, 341-349. http://dx.doi.org/10.1080/01621459.1997.10473632
Breslow, N.E.
and Lin, X. (1995) Bias Correction in Generalised Linear Mixed Models with a
Single-Component of Dispersion. Biometrika, 82, 81-91. http://dx.doi.org/10.1093/biomet/82.1.81
Booth, J.G.
and Hobert, J.P. (1999) Maximizing Generalized Linear Mixed Model Likelihoods with an Automated
Monte Carlo EM Algorithm. Journal of the Royal Statistical Society, B, 61, 265-285. http://dx.doi.org/10.1111/1467-9868.00176
