Approximations to the Loglikelihood Function in the Nonlinear Mixed Effects Model

Nonlinear mixed effects models have received a great deal of attention in the statistical literature in recent years because of the flexibility they offer in handling unbalanced repeated measures data that arise in different areas of investigation, such as pharmacokinetics and economics. Several different methods for estimating the parameters in nonlinear mixed effects model have been proposed. We concentrate here on two of them: maximum likelihood and restricted maximum likelihood. A rather complex numerical issue for (restricted) maximum likelihood estimation is the evaluation of the loglikelihood function of the data, since it involves the evaluation of a multiple integral that in most cases does not have a closed form expression. We consider here four different approximations to the loglikelihood, comparing their computational and statistical properties. We conclude that the alternating approximation suggested by Lindstrom and Bates (1990), the Laplacian approximation, and Gaussian quadrature centered at the conditional modes of the random effects are quite accurate and computationally efficient. Gaussian quadrature centered at the expected value of the random effects is quite inaccurate for a smaller number of abscissas and computationally inefficient for a larger number of abscissas. Importance sampling is accurate but quite inefficient computationally.