# 2004

Uppsala, Sweden

**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 [1] proposed to combine SAEM, a stochastic
approximation version of the EM algorithm [2], 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 [3]. 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.

**References:**:

[1] Kuhn E. and Lavielle M. Coupling a stochastic approximation version of EM
with an MCMC procedure, ESAIM PS, to appear.

[2] Delyon B, Lavielle M. and Moulines E. Convergence of a stochastic
approximation version of the EM algorithm. Annals of Statistics. 27 (1999), 94-
128.

[3] 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.