Parameter estimation of complex mixed models based on meta-model approach

Complex biological processes are usually experimented along time among a collection of individuals, longitudinal data are then available. The statistical challenge is to better understand the underlying biological mechanisms. A standard statistical approach is mixed-effects model where the regression function is highly-developed to describe precisely the biological processes (solutions of multi-dimensional ordinary differential equations or of partial differential equation). A classical estimation method relies on coupling a stochastic version of the EM algorithm with a Monte Carlo Markov Chain algorithm. This algorithm requires many evaluations of the regression function. This is clearly prohibitive when the solution is numerically approximated with a time-consuming solver. In this paper a meta-model relying on a Gaussian process emulator is proposed to approximate the regression function, that leads to what is called a mixed meta-model. The uncertainty of the meta-model approximation can be incorporated in the model. A control on the distance between the maximum likelihood estimates of the mixed meta-model and the maximum likelihood estimates of the exact mixed model is guaranteed. Eventually, numerical simulations are performed to illustrate the efficiency of this approach.