Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model

TLDR

Nonlinear mixed‑effects models are widely used for unbalanced repeated‑measure data in fields such as pharmacokinetics and economics, but estimating their parameters—especially the log‑likelihood via multiple integrals—poses significant numerical challenges. The study focuses on maximum likelihood and restricted maximum likelihood estimation, evaluating four approximations to the log‑likelihood for their computational and statistical properties. The authors compare four log‑likelihood approximations—LME, Laplacian, Gaussian quadrature at conditional modes, and importance sampling—within a nonlinear mixed‑effects framework. The LME, Laplacian, and Gaussian‑quadrature‑conditional‑mode approximations are accurate and efficient, whereas Gaussian quadrature at the expected value is inaccurate or inefficient depending on abscissas, and importance sampling, while accurate, is computationally inefficient.

Abstract

Abstract 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 the 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 in nonlinear mixed-effects models is the evaluation of the log-likelihood function of the data, because 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 log-likelihood, comparing their computational and statistical properties. We conclude that the linear mixed-effects (LME) approximation suggested by Lindstrom and Bates, 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.