Parametric inference for mixed models defined by stochastic differential equations

¹ Paris-Sud University, Laboratoire de Mathématiques, Orsay, France; UMR CNRS 8145, University Paris 5, Paris, France; adeline.samson@univ-paris5.fr
² INSERM U738, Paris, France; University Paris 7, Paris, France; UMR CNRS 8145, University Paris 5, Paris, France

Received: 13 July 2006
Revised: 14 April 2007

Abstract

Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants. A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.