文摘
We consider a likelihood approximation in generalized linear mixed-effects models (GLMM) with multilevel nested random effects. Likelihood evaluation in such models is difficult, hindered by the need for high dimensional integration, where the dimension is proportional to the number of units per level and the number of random effects per unit. Various integration approaches have been proposed, including the penalized quasi-likelihood method, Laplace approximation, quadrature approximation, simulation, and MCMC algorithms. We propose a new quadrature approximation method, which is based on the spherical radial integration approach of Monahan and Genz (J Am Stat Assoc 92:664–674 1997), and at the same time takes advantage of the hierarchical structure of the integration. Our new hierarchical spherical radial method has a time complexity that is linear in the number of units per level and the number of random effects per unit, in contrast to the exponential complexity of the adaptive Gaussian quadrature method of Pinheiro and Chao (J Comput Graph Stat 15:58–81 2006) for the same problem. Using a spline approximation, the generalized additive mixed models (GAMM) are GLMMs with two levels of nested random effects. We apply our method to estimation of GAMMs. We compare it with competing methods through simulations and apply our method to analyze virologic and immunologic responses in an AIDS clinical trial. An R package is written and available at http://users.wpi.edu/~jgagnon/software.html. Keywords AIDS Nonparametric random effects Likelihood approximation Generalized linear mixed models Generalized additive mixed models Laplace Quadrature Spherical radial Nested random effects