Computational methods for mixed-effects models

In this dissertation we develop the theory of likelihood estimation for linear and generalized linear mixed-effects models. In general we cannot derive a closed form solution to these estimates and we must determine them by using iterative algorithms such as the EM algorithm or a general nonlinear optimizer. For linear mixed-effects models we recommend using a moderate number of EM iterations followed by general nonlinear optimization of a profiled log-likelihood. We develop a modified EM (ECME) algorithm for this optimization and provide an efficient implementation based on matrix decompositions. Furthermore, we show that these same decompositions provide an efficient evaluation of the gradient of the profiled log-likelihood for use by general optimizers. For generalized linear mixed models, we propose using the penalized quasi-likelihood approximation to refine the starting estimates and then use more expensive methods such as Laplace approximation and adaptive Gauss-Hermite quadrature to better approximate the likelihood. We describe the design and implementation of an R package that implements these estimation methods. We also provide several examples that illustrate the behavior of this package.