Conditional Second-Order Generalized Estimating Equations for Generalized Linear and Nonlinear Mixed-Effects Models

Abstract

Generalized linear and nonlinear mixed-effects models are used extensively in health care research, including applications in pharmacokinetics, clinical trials, and epidemiology. Because the underlying model may be nonlinear in the random effects, there will generally be no closed-form expression for the marginal likelihood or indeed for the marginal moments. Consequently, estimation is often carried out either using numerical integration techniques or by approximating the marginal likelihood and/or marginal moments using first-order expansion methods. An advantage of the first-order methods is that they do not necessarily require specification of a conditional like-lihood to estimate the regression parameters of interest. However, in many cases they may not take full advantage of the fact that the conditional variance depends on both fixed and random effects. In this article we propose using conditional second-order generalized estimating equations (CGEE2) to estimate both fixed- and random-effects parameters. Under mild regularity conditions, the CGEE2 estimator is shown to be consistent and asymptotically efficient with a rate of convergence depending on both the number of subjects and the number of observations per subject. We compare the CGEE2 estimator against alternative estimators using limited simulation and demonstrate its utility with a numerical example.

KEY WORDS:

• Asymptotics
• First-order linearization
• Gaussian random effects
• Laplace approximation
• Penalized extended least squares
• Quadratic exponential family