Mixed-effects models have become a popular approach for the analysis of grouped data that arise in many areas as diverse as clinical trials, epidemiology, and sociology. In the case of linear mixed-effects (LME) models, the likelihood function can be expressed in close form, with efficient computational algorithms having been proposed for maximum likelihood and restricted maximum likelihood estimations. For nonlinear mixed-effects (NLME) models and generalized linear mixed models (GLMMs), however, the likelihood function does not have a closed form. Different likelihood approximations, with varying degrees of accuracy and computational complexity, have been proposed for these models.

The lme4 package that have developed by prof. Bates from Univ. of Wisconsin at Madison, has many advantages relative to earlier methods. It provides method for LME, GLMMs, NLME and even generalized nonlinear mixed models (which have application for what is called item-response analysis in psychometrics). The approximation used in lme4 for models except LME is a Laplace approximation to the integral that defines the likelihood.

This project continues to complete the package lme4 under supervision of prof. Bates. Adaptive Gauss-Hermite quadrature (AGQ) method will be used to evaluate the integrals. This method can be implemented with arbitrary degrees of accuracy, leading to nearly unbiased estimates, while first-order Laplacian approximations have been reported to produce biased estimates under some distributional scenarios.

This project will carried out in R environment but written in C because its advantage of memory manipulation. I will continue prof. Bates' style in C writing and implement the following job in similar way of coding. More computational problems possibly will arise such as sparse matrices manipulation and high-order integral evaluation. In addition, further simulation and data set test study is necessary in debugging.