EM algorithms for nonlinear mixed effects models

Implementing the Monte Carlo EM algorithm (MCEM) algorithm for finding maximum likelihood estimates (MLEs) in the nonlinear mixed effects model (NLMM) has encountered a great deal of difficulty in obtaining samples used for estimating the E step due to the intractability of the target distribution. Sampling methods such as Markov chain techniques and importance sampling have been used to alleviate such difficulty. The advantage of Markov chains is that they are applicable to a wider range of distributions than the approaches based on independent samples. However, in many cases the computational cost of Markov chains is significantly greater than that of independent samplers. The MCEM algorithms based on independent samples allow for straightforward assessment of Monte Carlo error and can be considerably more efficient than those based on Markov chains when an efficient candidate distribution is chosen, which forms the motivation of this paper. The proposed MCEM algorithm in this paper uses samples obtained from an easy-to-simulate and efficient importance distribution so that the computational intensity and complexity is much reduced. Moreover, the proposed MCEM algorithm preserves the flexibility introduced by independent samples in gauging Monte Carlo error and thus allows the Monte Carlo sample size to increase with the number of EM iterations. We also introduce an EM algorithm using Gaussian quadrature approximations (GQEM) for the E step. In low-dimensional cases, the GQEM algorithm is more efficient than the proposed MCEM algorithm and thus can be used as an alternative. The performances of the proposed EM methods are compared to the existing ML estimators using real data examples and simulations.