Adeniyi, Isaac Adeola
(2020):
*Bayesian Generalized Linear Mixed Effects Models Using Normal-Independent Distributions: Formulation and Applications.*
Forthcoming in: AStA-Advances in Statistical Analysis

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## Abstract

A standard assumption is that the random effects of Generalized Linear Mixed Effects Models (GLMMs) follow the normal distribution. However, this assumption has been found to be quite unrealistic and sometimes too restrictive as revealed in many real-life situations. A common case of departures from normality includes the presence of outliers leading to heavy-tailed distributed random effects. This work, therefore, aims to develop a robust GLMM framework by replacing the normality assumption on the random effects by the distributions belonging to the Normal-Independent (NI) class. The resulting models are called the Normal-Independent GLMM (NI-GLMM). The four special cases of the NI class considered in these models’ formulations include the normal, Student-t, Slash and contaminated normal distributions. A full Bayesian technique was adopted for estimation and inference. A real-life data set on cotton bolls was used to demonstrate the performance of the proposed NI-GLMM methodology.

Item Type: | MPRA Paper |
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Original Title: | Bayesian Generalized Linear Mixed Effects Models Using Normal-Independent Distributions: Formulation and Applications |

English Title: | Bayesian Generalized Linear Mixed Effects Models Using Normal-Independent Distributions: Formulation and Applications |

Language: | English |

Keywords: | Generalized Linear Mixed Effects Models, Normal-Independent class, Normal density, Student-t, Slash density, Bayesian Method. |

Subjects: | C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling |

Item ID: | 99165 |

Depositing User: | Professor Waheed Babatunde Yahya |

Date Deposited: | 23 Mar 2020 03:07 |

Last Modified: | 15 Sep 2024 19:19 |

References: | Adeniyi, I.A., Shobanke, D. A. and Edogbanya H. O. (2019). Re-parameterization of the COM-Poisson Distribution Using Spectral Algorithms. Pakistan Journal of Statistics and Operation Research, 15(3): 701-712. Adeniyi, I.A., Yahya, W. B. and Ezenweke C. P. (2018). A Note on Pharmacokinetics Modelling of Theophylline Concentration Data on Patients with Respiratory Diseases. Turkiye Klinikleri Journal of Biostatistics;10(1):27-45. Agresti, A., Caffo, B. and Ohman-Strickland, P. (2004). Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies. J. Comput. Graph. Statist., 47: 639–653. Azzalini A. and Capitanio A. (1999). Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society Series B, 61: 579–602. Azzalini A. and Dalla-Valle A. (1996). The multivariates skew-normal distribution. Biometrika, 8: 715–726. Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88: 9–25. Brooks, S. P. (1998) Markov chain Monte Carlo method and its application. The Statistician, 47: 69–100. Cameron, A.C. and Trivedi, P.K. (1998). Regression Analysis of Count Data. Cambridge University Press. Cambridge, UK. Casella, G. and George, E. (1992) Explaining the Gibbs sampler. American Statistician, 46: 167–74. Chen, R. and Huang, Y. (2016). Mixed-Effects Models with Skewed Distributions for Time-Varying Decay Rate in HIV Dynamics. Commun. Stat. Simul. Comput, 45(2): 737–757. Chen J., Zhang, D., and Davidian, M. (2002). A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution. Biostatistics, 3: 347–360. Chen, G. and Luo, S. (2016). Robust Bayesian hierarchical model using normal/independent distributions. Biometrical Journal, 58(4): 831–85. Conway, R. W. and Maxwell, W. L. (1962). A queuing model with state dependent service rates. Journal of Industrial Engineering, 12, 132–136. da Silva, A. M., Degrande, P. E., Fernandes, M. G., Suekane, R., and Zeviani, W. M. (2012). Impacto de diferentes níveis de desfolha artificial nos estádios fenológicos do algodoeiro. Revista de Ciências Agrárias, 35(1), 163-172. Davidian, M. Gilitinan D.M. (1995) Nonlinear models for repeated measurement data. London: Chapman & Hall. Fahrmeir, L. and Tutz, G.T. (2001). Multivariate Statistical Modelling Based on Generalized Linear Models, 2nd edition. Springer-Verlag, New York. Fernandez C. and Steel M. F. (1998). On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93:359–371. Gallant, A. R. and Nychka, D. W. (1987). Semi-nonparametric maximum likelihood estimation. Econometrica, 55: 363–390. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6: 721–741. Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2004). Bayesian Data Analysis, Second Edition. New York: Chapman and Hall. Gelman, A. and Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457-511. Ghosh P., Branco M., and Chakraborty H. (2007). Bivariate random effect model using skew normal distribution with application to HIV-RNA. Statistics in Medicine, 26: 1255–1267. Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996) Markov Chain Monte Carlo in Practice. Chapman & Hall, London. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57: 97–109. Heagerty, P. J. and Kurland, B. F. (2001). Misspecified maximum likelihood estimates and generalised linear mixed models. Biometrika, 88: 973–985 Huang, Y. Chen, R. and Dagne, G. (2011). Simultaneous Bayesian Inference for Linear, Nonlinear and Semiparametric Mixed-Effects Models with Skew-Normality and Measurement Errors in Covariates. The International Journal of Biostatistics, 7(1). Lachos V. H., Bandyopadhyay D. and Dey D.K. (2011). Linear and nonlinear mixed-effects models for censored HIV viral loads using normal/independent distributions. Biometrics. 2011; 67(4):1594–1604. Lachos, V. H., Castro, L. M., and Dey, D. K. (2013). Bayesian inference in nonlinear mixed–effects models using normal independent distributions. Computational Statistics & Data Analysis, 64: 237–252. Laird, N.M. and Ware, J.H. (1982). Random-effects models for longitudinal data. Biometrics, 38: 963-974. Lange, K. L., Little, R., and Taylor, J. (1989). Robust statistical modeling using t distribution. Journal of the American Statistical Association, 84: 881–896. Lange, K. L. and Sinsheimer, J. S. (1993). Normal/independent distributions and their applications in robust regression. Journal of Computational and Graphical Statistics, 2: 175–198. Li, Y., Brown, P., Rue, H., al Maini, M., and Fortin, P. (2012) Spatial modelling of lupus incidence over 40 years with changes in census areas. Journal of the Royal Statistical Society: Series C, 61 (1), 99-115. Lin, T. I. and Lee, J. C. (2007). Estimation and prediction in linear mixed models with skew-normal random effects for longitudinal data. Statistics in Medicine, 27: 1490-1507. Litiere, S., Alonso, A. and Molenberghs, G. (2007). Type I and type II error under random-effects misspecification in generalized linear mixed models. Biometrics, 63: 1038–1044. Litiere, S., Alonso, A. and Molenberghs, G. (2008). The impact of a misspecified random-effects distribution on the estimation and the performance of inferential procedures in generalized linear mixed models. Stat. Med., 27: 3125–3144. Liu, C. (1996). Bayesian robust multivariate linear regression with incomplete data. Journal of the American Statistical Association, 91. McCullagh, P. and Nelder, J. A. (1997). Generalized Linear Models, 2nd edition. Chapman & Hall/CRC. McCulloch, C.E. and Searle, S.R. (2001). Generalized, Linear, and Mixed Models. Wiley, New York. Meza, C., Osorio, F., and de la Cruz, R. (2012). Estimation in non-linear mixed-effects models using heavy-tailed distributions. Statistics and Computing, 22: 121–139. Neuhaus, J.M., Hauck, W.W. and Kalbfleisch, J.D. (1992). The effects of mixture distribution misspecification when fitting mixed-effects logistic models. Biometrika, 79: 755 – 762. Osiewalski, J. (1999). Bayesian analysis of nonlinear regression with equi-correlated elliptical errors. Test, 8: 339–344. Osiewalski, J. and Steel, M. F. J. (1993). Robust Bayesian-inference in elliptic regression models. Journal of Econometrics, 57, 345–363. Osorio, F., Paula, G. A., and Galea, M. (2007). Assessment of local influence in elliptical linear models with longitudinal structure. Computational Statistics and Data Analysis, 51: 4354–4368. Pinheiro, J. and Bates, D. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4: 12–35. Pinheiro, J. and Bates, D. (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York. R Core Team (2019). R: A language and environment for statistical computing. R Foundation for Statistical Computing.Vienna, Austria. URL: http://www.R-project.org/. Rodriguez, G. and Goldman, N. (1995). An assessment of estimation procedures for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 158: 73–89. Rosa, G. J. M., Padovani, C. R., and Gianola, D. (2003). Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation. Biometrical Journal, 45: 573–590. Rue, H., Martino, S., Chopin, N., 2009. Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the Royal Statistical Society Series B, 71 (2), 1-35. Sahu S., Dey D. K., and Branco M. D. (2003). A new class of multivariate skew distribution with application to Bayesian regression models. The Canadian Journal of Statistics, 31(2): 129-150. Samuels, M. L., Witmer, J. A., and Schaffner A. A. (2012). Statistics for the Life Sciences, 4th edition. Prentice Hall, Boston. Savalli, C., Paula, G. A., and Cysneiros, F. (2006). Assessment of variance components in elliptical linear mixed models. Statistical Modelling, 6: 59–76. Schall, R. (1991). Estimation in generalized linear models with random effects. Biometrika, 78: 717-727. Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S. and Boatwright, P. (2005). A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution. Applied Statistics, 54, 127–142. Spiegelhalter, D. J., Abrams, K. R. and Myles, J. P. (2004). Bayesian Approach to Clinical Trials and Health-care Evaluation. John Wiley & Sons, Ltd, Chichester, UK. Spiegelhalter, D. J. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology). 64(4): 583-639. Verbeke, G. and Lesaffre, E. (1996). A liner mixed-effects models with heterogeneity in the random effects population. Journal of the American Statistical Association, 433: 217–221. Zeger, S. L. and Karim, M. R. (1991). Generalized linear models with random effects: a Gibbs sampling approach. Journal of the American Statistical Association, 86:79–86. Zuur, A. F., Ieno, E. N., Walker, N. J., Saveliev, A. A., and Smith G. M. (2009). Mixed Effects Models and Extensions in Ecology with R. Springer, New York. Lunn, D.J., Thomas, A., Best, N., and Spiegelhalter, D. (2000) WinBUGS - a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing, 10, 325-337. Zeviani, W. M., Riberio Jr, P. J., Bonat, W. H., Shimakura, S. E., and Muniz, J. A. (2014) The Gamma-count distribution in the analysis of experimental under-dispersed data. Journal of Applied Statistics. 41, 2616-2626. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/99165 |