Quasi-Monte Carlo Estimation in Generalized Linear Mixed Model with Correlated Random Effects

Yin Chen; Yu Fei; Jianxin Pan

doi:10.4236/oalib.1102002

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Open Access Library Journal > Vol.2 No.10, October 2015

Quasi-Monte Carlo Estimation in Generalized Linear Mixed Model with Correlated Random Effects ()

¹School of Insurance, University of International Business and Economics, Beijing, China.
²School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, China.
³School of Mathematics, University of Manchester, Manchester, UK.

DOI: 10.4236/oalib.1102002 PDF HTML XML 705 Downloads 1,111 Views Citations

Abstract

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.

Keywords

Generalized Linear Mixed Models, Correlated Random Effects, Quasi-Monte Carlo, Mean-Covariance Modelling, High-Dimensional Integrals, Newton-Raphson Method

Share and Cite:

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, 1-16. doi: 10.4236/oalib.1102002.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	McCulloch, P. and Nelder, J.A. (1989) Generalized Linear Models. 2nd Edition, Chapman and Hall, London. http://dx.doi.org/10.1007/978-1-4899-3242-6
[2]	Breslow, N.E. and Clayton, D.G. (1993) Approximate Inference Ingeneralized Linear Mixed Models. Journal of the American Statistical Association, 88, 9-25.
[3]	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
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	Pan, J. and Thompson, R. (2003) Gauss-Hermite Quadrature Approximation for Estimation in Generalised Linear Mixed Models. Computational Statistics, 18, 57-78.
[10]	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
[11]	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
[12]	Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970081
[13]	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
[14]	Traub, J.F. and Wozniakowski, H. (1992) The Monte Carlo Algorithm with Pseudorandom Generator. Mathematics of Computation, 58, 323-339. http://dx.doi.org/10.1090/S0025-5718-1992-1106984-4
[15]	Fang, K.T. and Wang, Y. (1994) Number-Theoretic Methods in Statistics. Chapman and Hall, London. http://dx.doi.org/10.1007/978-1-4899-3095-8
[16]	Morokoff, W.J. and Caflisch, R.E. (1995) Quasi-Monte Carlo Integration. Journal of Computational Physics, 122, 218-230. http://dx.doi.org/10.1006/jcph.1995.1209
[17]	Pourahmadi, M. (1999) Joint Mean-Covariance Models with Applications to Longitudinal Data: Unconstrained Parameterization. Biometrika, 86, 677-690. http://dx.doi.org/10.1093/biomet/86.3.677
[18]	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
[19]	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
[20]	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
[21]	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
[22]	Pan, J. and MacKenzie, G. (2003) On Modelling Mean-Covariance Structures in Longitudinal Studies. Biometrika, 90, 239-244. http://dx.doi.org/10.1093/biomet/90.1.239
[23]	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
[24]	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
[25]	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
[26]	Aitken, M. (1999) A General Maximum Likelihood Analysis of Variance Components in Generalized Linear Models. Biometrics, 55, 117-128. http://dx.doi.org/10.1111/j.0006-341X.1999.00117.x