2003
Verona, Italy
Maximum likelihood estimation in nonlinear mixed effects models
Estelle Kuhn and Marc Lavielle
University Paris-Sud, France
Objectives : Our aim is to estimate parameters in non linear mixed effects models by maximum likelihood. Some well-knomw used methods are based on linearization of the likelihood. The properties of these estimator depend inter alia on the number N of observed subjects and on the number n_i of observations for the ith subject. For example, FO ensures no convergence of its estimator and FOCE converges only under the assumptions that n_i is equal to infinity for all subjects. We propose a method based on a stochastic approximation version of the EM algorithm, denoted SAEM, for which these restrictive condition mustn't be checked : the produced estimator converge toward a local maximum of the likelihood under very general regularity conditions on the model. SAEM also yields an estimation of the likelihood of the observations and a confidence interval for the estimated parameters.
Method : The expectation-maximization (EM) algorithm is a broadly applicable approach for the iterative computation of maximum likelihood estimates, useful in a variety of incomplete-data (or partially-observed-data) statistical problems. In a mixed effects model, the random effects can be considered as the non observed data. Then, EM can be used for estimating the parameters of a linear model, but the E-step becomes untractable whenever the model is non linear. SAEM replaces the E-step by a simulation step: at each iteration, the random effects are drawn with the conditional distribution and the parameters are updated using these simulated data. Furthermore, the simulated random effects allow to estimate the likelihood of the observations as well as the Fisher information matrix.
Results : This algorithm converges very quickly to the maximum likelihood estimator. We implemented SAEM on the pharmacodynamic simulated example used by Walker on 50 replications [1] : the relative root means square errors (RMSER) are lower than those obtained by FOCE, Laplacian methods and also the Monte Carlo EM proposed by Walker. We also compared the results of SAEM with FOCE on a pharmacokinetic simulated example used by Concordet [2] on 20 replications: the RMSER of the SAEM algorithm are also lower than with FOCE, particularly for the variability.