Stochastic Approximation EM algorithm in nonlinear mixed effects models: an evaluation by simulation
Adeline Samson (1), Marc Lavielle (2) and France Mentré (1)
(1) INSERM E 0357, Bichat hospital, Paris; (2) University Paris-Sud, Bat. 425, Orsay, France
Context:: Maximum likelihood estimation in nonlinear mixed effects models cannot be directly performed as the likelihood has no close form. Most of algorithms implemented in software are based on a linearization of the model. These algorithms could produce inconsistent estimates and an increase of type I error of Likelihood Ratio Test. To avoid the linearization of the models, Kuhn and Lavielle  proposed to combine SAEM, a stochastic approximation version of the EM algorithm , with a Markov Chain Monte Carlo procedure.
Objectives:: To implement the SAEM algorithm in R and an evaluation of the likelihood without linearization. To evaluate by simulation the estimation properties of SAEM and of the FOCE method implemented in the nlme function of Splus.
Methods:: We evaluate the likelihood of the SAEM estimates by importance sampling. The instrumental distributions used in the importance sampling procedure are gaussian approximations of individual posterior distributions. We simulate 200 datasets from a biexponential model of the viral load decrease during anti-HIV treatment, 100 datasets with N=40 subjects and 100 with N=200 subjects. This model involves 4 fixed effects with exponential random effects. An additive error on the log viral load with an homoscedastic variance is assumed. We assume identical sampling times for all subjects taken at 1, 3, 7, 14, 28 and 56 days. Parameter values of simulation are taken from Ding and Wu . We evaluate the type I error of the Likelihood Ratio Test with both nlme and SAEM by testing a treatment effect on the first viral decay rate.
Results:: Some of the nlme estimates are significantly biased whereas they are not with SAEM. The RMSE are rather similar with SAEM and nlme. The type I error of the LRT with nlme is overestimated and is 13% when N=40 subjects and 18% when N=200 subjects for a nominal level of 5%. The type I errors of the LRT on the same example are respectively 6% and 5% with SAEM. SAEM is a powerful tool which provides maximum likelihood estimation in nonlinear mixed effects models without linearization.
 Kuhn E. and Lavielle M. Coupling a stochastic approximation version of EM with an MCMC procedure, ESAIM PS, to appear.
 Delyon B, Lavielle M. and Moulines E. Convergence of a stochastic approximation version of the EM algorithm. Annals of Statistics. 27 (1999), 94- 128.
 Ding A.A. and Wu H. Assesing antiviral potency of anti-HIV therapies in vivo by comparing viral decay rates in viral dynamics models. Biostatistics, 2 (2001), 1-18.