Approximation techniques for maximizing likelihood functions of generalized linear mixed models for binary response data
Keywords:Generalized Linear Mixed Models, Adaptive Gauss-Hermit Quadrature, Likelihood Function, Binary Response, Medicine.
Evaluating Maximum likelihood estimates in Generalized Linear Mixed Models (GLMMs) has been a serious challenge due to some integral complexities encountered in maximizing its likelihood functions. It is computationally difficult to establish analytical solutions for the integrals. In view of this, approximation techniques would be needed. In this paper, various approximation techniques were examined including Laplace approximation (LA), Penalized Quasi likelihood (PQL) and Adaptive Gauss-Hermite Quadrature (AGQ) techniques. The performances of these methods were evaluated through both simulated and real-life data in medicine. The simulation results showed that the Adaptive Gauss-Hermit Quadrature approach produced better estimates when compared with PQL and LA estimation techniques based on some model selection criteria.
 Y. Lee and J. A. Nelder, Hierarchical generalised linear models: a synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika, (2001)88, 987â€“1006. https://doi.org/10.1093/biomet/88.4.987.
 N.E. Breslow, Extra-poisson variation in log-linear models. Applied Statistics, 33, (1984), 38â€“44. https://doi.org/10.2307/2347661.
 B.M. Bolker, C.J. Brooks, S.W. Clark, Geange, J. R, M.H.H. Poulsen, Stevens, J.S. White, Generalized linear mixed models: a practical guide for ecology and evolution, Trends in Ecology & Evolution, 24, (2009),127â€“135. https://doi.org/10.1016/j.tree.2008.10.008.
 M. Casals, K. Langohr, J.L. Carrasco1, L. Ronneg, Parameter Estimation of Poisson Generalized Linear Mixed Models Based on Three Different Statistical Principles: a Simulation Study. SORT 39 (2), (2015), 1-28.
 C. E. McCulloch S.R. Searle Generalized, Linear, and Mixed Models, New York: Willey, (2001).
 R Core Team, A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org, (2018).
 D.Gamerman, Sampling from the Posterior Distribution in Generalized Linear Mixed Models. Statistics and Computing, 7, (1997), 57-68. https://doi.org/10.1023/A:1018509429360.
 O.F. Christensen, R. Waagepetersen, Bayesian Prediction of Spatial Count Data Using Generalized Linear Mixed Models, JSTOR Biometrics, 58 (2) (2002), 280-286.
 Y. Zhao, J. Staudenmayer, B.A. Coull, M.P. Wand, General Design Bayesian Generalized Linear Mixed Models. Mathematics and statistics online, Statist. Sci. Volume 21 (1), (2006), 35-51. https://doi.org/10.1214/088342306000000015.
 Y. Fong, H. Rue, J. Wakefield, Bayesian inference for generalized linear mixed models. Biostatistics, 11, (2010) 397â€“412. https://doi.org/10.1093/biostatistics/kxp053.
 O.S. Adesina, D.A. Agunbiade, O.S. Osundina, Bayesian Regression Model for Counts in Scholarship, Journal of Mathematical Theory and Modeling. 7, 9, (2017) 46-57.
 B. Efron, B. Double Exponential Families and Their Use in Generalised Linear Regression. Journal of the American Statistical Association, 81(395), (1986) 709-721: URL https://doi.org/10.1080/01621459.1986.10478327.
 O.S. Adesina, T.O. Olatayo, O.O. Agboola, P.E. Oguntunde, Bayesian Dirichet Process Prior for Count Data. International Journal of Mechanical Engineering and Technology (IJMET), 9, 12, (2018) 630-646.
 M.E. Brooks, K. Kristensen, K.J. Benthem, A. Magnusson, C.W. Berg, A. Nielsen, H.J. Skaug, M. Maechler, M. Bolker, Modelling Zero-Inflated Count Data With glmmTMB. (2017) bioRxiv preprint bioRxiv: 132753; https://doi.org/10.1101/132753.
 P. Jong, G.Z. Heller, Generalized Linear Models for Insurance Data, (1st ed., Cambridge University Press, New York. (2008) ISBN-13 978-0-511-38677-0). https://doi.org/10.1017/CBO9780511755408.