An approximate likelihood approach to nonlinear mixed effects models via spline approximation

In dealing with parametric nonlinear mixed effects models, intensive numerical integration often makes exact maximum likelihood estimation impractical given the current computing capacity. Algorithms based on linearization, such as the first order method and the conditional first order method, have the potential of producing highly inconsistent estimates, although numerically they are more efficient. We propose an approximate likelihood approach via spline approximation, which significantly reduces the numerical difficulty associated with the exact maximum likelihood estimation and can give estimates asymptotically equivalent to MLE or up to a controllable asymptotic bias. Theoretical properties of the new algorithm are established for parametric nonlinear mixed effects models with normal additive measurement error. We apply our algorithm to the population pharmacokinetics of phenobarbital and compare results to those obtained with nlme() in S-PLUS. Simulation studies show that our algorithm works equally well as the nlme() for small variability of random effects and outperforms the nlme() for large variability of random effects.