Statistical Inference in Generalized Linear Mixed Models by Joint Modelling Mean and Covariance of Non-Normal Random Effects
Affiliation(s)
1 School of Insurance, University of International Business and Economics, Beijing, China. 2 School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, China. 3 School of Mathematics, University of Manchester, Manchester, UK.
1 School of Insurance, University of International Business and Economics, Beijing, China. 2 School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, China. 3 School of Mathematics, University of Manchester, Manchester, UK.
ABSTRACT
Generalized linear mixed models (GLMMs) are typically constructed by incorporating random effects into the linear predictor. The random effects are usually assumed to be normally distributed with mean zero and variance-covariance identity matrix. In this paper, we propose to release random effects to non-normal distributions and discuss how to model the mean and covariance structures in GLMMs simultaneously. Parameter estimation is solved by using Quasi-Monte Carlo (QMC) method through iterative Newton-Raphson (NR) algorithm very well in terms of accuracy and stabilization, which is demonstrated by real binary salamander mating data analysis and simulation studies.
KEYWORDS
Generalized Linear Mixed Models, Multivariate t Distribution, Multivariate Mixture Normal Distribution, Quasi-Monte Carlo, Newton-Raphson, Joint Modelling of Mean and Covariance
Generalized Linear Mixed Models, Multivariate t Distribution, Multivariate Mixture Normal Distribution, Quasi-Monte Carlo, Newton-Raphson, Joint Modelling of Mean and Covariance
Cite this paper
Chen, Y. , Fei, Y. and Pan, J. (2015) Statistical Inference in Generalized Linear Mixed Models by Joint Modelling Mean and Covariance of Non-Normal Random Effects. Open Journal of Statistics, 5, 568-584. doi: 10.4236/ojs.2015.56059.
Chen, Y. , Fei, Y. and Pan, J. (2015) Statistical Inference in Generalized Linear Mixed Models by Joint Modelling Mean and Covariance of Non-Normal Random Effects. Open Journal of Statistics, 5, 568-584. doi: 10.4236/ojs.2015.56059.
References
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[1] Breslow, N.E. and Clayton, D.G. (1993) Approximate Inference in Generalized Linear Mixed Models. Journal of the American Statistical Association, 88, 9-25. [2] Lin, X. and Breslow, N.E. (1996) Bias Correction in Generalized Linear 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 [3] 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 [4] 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 [5] 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, 61, 265-285. http://dx.doi.org/10.1111/1467-9868.00176 [6] Pan, J. and Thompson, R. (2003) Gauss-Hermite Quadrature Approximation for Estimation in Generalized Linear Mixed Models. Computational Statistics, 18, 57-78. [7] 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 [8] Al-Eleid, E.M.O. (2007) Parameter Estimation in Generalized Linear Mixed Models Using Quasi-Monte Carlo Methods. PhD Dissertation, The University of Manchester, Manchester. [9] McCullouch, C.E. (2003) Generalized Linear Mixed Models. NSF-CBMS Regional Conference Series in Probability and Statistics Volume 7. Institution of Mathematical Statistics and the American Statistical Association, San Francisco. [10] 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 [11] Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970081 [12] 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 [13] 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 [14] Pourahmadi, M. (2000) Maximum Likelihood Estimation of Generalized Linear Models for Multivariate Normal Covariance Matrix. Biometrika, 87, 425-435. http://dx.doi.org/10.1093/biomet/87.2.425 [15] 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 [16] 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