Literature on Nonlinear Mixed Models

Davidian and Giltinan (1995) and Vonesh and Chinchilli (1997) provide good overviews as well as general theoretical developments and examples of nonlinear mixed models. Pinheiro and Bates (1995) is a primary reference for the theory and computational techniques of PROC NLMIXED. They describe and compare several different integrated likelihood approximations and provide evidence that adaptive Gaussian quadrature is one of the best methods. Davidian and Gallant (1993) also use Gaussian quadrature for nonlinear mixed models, although the smooth nonparametric density they advocate for the random effects is currently not available in PROC NLMIXED.

Traditional approaches to fitting nonlinear mixed models involve Taylor series expansions, expanding around either zero or the empirical best linear unbiased predictions of the random effects. The former is the basis for the well-known first-order method of Beal and Sheiner (1982, 1988) and Sheiner and Beal (1985), and it is optionally available in PROC NLMIXED. The latter is the basis for the estimation method of Lindstrom and Bates (1990), and it is not available in PROC NLMIXED. However, the closely related Laplacian approximation is an option; it is equivalent to adaptive Gaussian quadrature with only one quadrature point. The Laplacian approximation and its relationship to the Lindstrom-Bates method are discussed by Beal and Sheiner (1992), Wolfinger (1993), Vonesh (1992, 1996), Vonesh and Chinchilli (1997), and Wolfinger and Lin (1997).

A parallel literature exists in the area of generalized linear mixed models, in which random effects appear as a part of the linear predictor inside a link function. Taylor-series methods similar to those just described are discussed in articles such as Harville and Mee (1984), Stiratelli, Laird, and Ware (1984), Gilmour, Anderson, and Rae (1985), Goldstein (1991), Schall (1991), Engel and Keen (1992), Breslow and Clayton (1993), Wolfinger and O’Connell (1993), and McGilchrist (1994), but such methods have not been implemented in PROC NLMIXED because they can produce biased results in certain binary data situations (Rodriguez and Goldman 1995, Lin and Breslow 1996). Instead, a numerical quadrature approach is available in PROC NLMIXED, as discussed in Pierce and Sands (1975), Anderson and Aitkin (1985), Crouch and Spiegelman (1990), Hedeker and Gibbons (1994), Longford (1994), McCulloch (1994), Liu and Pierce (1994), and Diggle, Liang, and Zeger (1994).

Nonlinear mixed models have important applications in pharmacokinetics, and Roe (1997) provides a wide-ranging comparison of many popular techniques. Yuh et al. (1994) provide an extensive bibliography on nonlinear mixed models and their use in pharmacokinetics.