# Maximum likelihood estimation in nonlinear mixed effects models with the SAEM algorithm

The statistical model for most population PK/PD analyses is the nonlinear-mixed effects model (NLMEM). As opposed to linear models, there are statistical issues to express the optimisation criteria for these nonlinear models so that first approximation methods (FO and FOCE) based on linearization of the model were proposed. It is well known that these methods have several methodological and theoretical drawbacks. They are also very sensitive to initial estimates which make lot’s of run to failed to converge with a waste of time for the modeller.

Population analyses are now used not only to provide mean estimates but also to make model selection, hypothesis testing, simulations and predictions based on all the estimated components: better estimation methods are therefore needed.

The SAEM (Stochastic Approximation EM) algorithm avoids any linearization and is based on recent statistical algorithms. This algorithm is a powerful tool for Maximum Likelihood Estimation (MLE) for very general incomplete data models. The convergence of this algorithm to the MLE and its good statistical properties have been proven.

This iterative procedure consists at each iteration, in successively simulating the random effects with the conditional distribution, and updating the unknown population parameters of the model. MCMC (Markov Chain Monte Carlo) is used for the simulation step. The observed likelihood and the Fisher Information matrix can also be estimated by using a stochastic approximation procedure.

The SAEM algorithm is implemented in the MONOLIX software for usual PK/PD models. Nevertheless, SAEM can handle very general non linear mixed effect models:

–          left-censored data,

–          models defined by stochastic differential equations,

–          multi-responses models,

–          mixtures of distributions,

–          inter-occasion variability,

–          missing covariates,

–          time to event data,

–          count data,

–          dropouts,

–          …

Indeed, all these models are statistical models that include a set of observations and a set of non observed data. SAEM requires the computation of the conditional distribution of these non observed data and their simulation at each iteration.

Some of these models are already implemented in version 2.1 of the MONOLIX software (http://www.math.u-psud.fr/~lavielle/monolix/monolix_software.html).

Delyon B., Lavielle M., and Moulines E., “Convergence of a stochastic approximation version of the EM algorithm”, ‘The Annals of Stat., vol. 27, no. 1, pp 94–128, 1999

Kuhn E., Lavielle M.,  “Coupling a stochastic approximation version of EM with a MCMC procedure” ESAIM P&S, vol. 8, pp 115–131, 2004

Kuhn E., Lavielle M.,  “Maximum likelihood estimation in nonlinear mixed effects models”  Computational Statistics and Data Analysis, vol. 49, No. 4, pp 1020–1038, 2005.

Lavielle M., Meza C.   “A Parameter Expansion version of the SAEM algorithm”  Statistics and Computing (to appear), 2007

Donnet S., Samson A..,  “Estimation of parameters in incomplete data models defined by dynamical systems”  Journal of Statistical Planning and Inference (to appear), 2007.

Samson A., Lavielle M., Mentré F.   “Extension of the SAEM algorithm to left-censored data in non-linear mixed-effects model: application to HIV dynamics model”  Computational Statistics and Data Analysis, vol. 51, pp. 1562–1574, 2006.

Lavielle M., Mentré F.   “Estimation of population pharmacokinetic parameters of saquinavir in HIV patients and covariate analysis with the SAEM algorithm implemented in MONOLIX”  Journal of Pharmacokinetics and Pharmacodynamics (to appear), 2007.