Hierarchical-likelihood approach for nonlinear mixed-effects models
Introduction
NLMMs have been widely used for the analysis of repeated measurements, particularly in pharmacokinetic studies (Sheiner and Beal, 1980). Unlike LMMs, in which the normality assumption on the random effects yields closed-form expressions for the marginal likelihood, the normality assumption in NLMMs leads to an analytically intractable and computationally intensive likelihood that involves multi-dimensional integration. Numerical integration such as Gauss–Hermite quadrature (GHQ) is often not feasible when high-dimensional integrals are required. A commonly used approach, as adopted in the software package Nonmen (Sheiner and Beal, 1980) and the nlme procedure in S-plus due to Lindstrom and Bates (1990), is to develop iterative schemes by using the restricted maximum likelihood (REML) procedure for LMMs to estimate parameters in NLMMs.
Wolfinger and Lin (1997) pointed out that these methods for NLMMs are based on a Taylor-series expansion around specific values for the random effects and are closely related to the those for generalized linear mixed models (GLMMs); the approaches of Sheiner and Beal (1980) and Lindstrom and Bates (1990) for NLMMs correspond respectively to Breslow and Clayton’s (1993) marginal quasi-likelihood (MQL) and their penalized quasi-likelihood (PQL) for GLMMs. Wolfinger and Lin (1997) found that the PQL estimator works better than the MQL estimator in NLMMs. However, the improvement is slight and their numerical studies show that the PQL estimator still suffers from severe biases. These results agree with the previous work of Breslow and Lin (1995) in GLMMs. Breslow and Lin (1995) and Lin and Breslow (1996) studied bias-correction methods for PQL, but their methods fail seriously when values of the variance components are large or the number of observations for each subject is small. Vonesh (1996) and Shun (1997) proposed to use the Laplace (LAPL) approximation to the marginal likelihood in NLMMs and GLMMs, respectively. Lai and Shih, 2003a, Lai and Shih, 2003b proposed a hybrid method that combines the Laplace approximation and Monte Carlo simulations to evaluate the integrals in the marginal likelihood. However, these methods are computationally intensive and still give biased estimation.
Lee and Nelder (1996) extended GLMMs to hierarchical generalized linear models (HGLMs) by allowing non-normal distributions for random effects. They proposed the use of the hierarchical likelihood (h-likelihood) to avoid the often intractable integration and extended the LAPL approximation to allow the REML procedure, which is important for reducing biases in finite samples. Recently, Noh and Lee (2007) investigated performances of various methods numerically for GLMMs and found that h-likelihood methods, if properly implemented, give the best estimates among methods they considered: for more discussion see Lee and Nelder (2006). In this paper we propose an h-likelihood procedure for the analysis of NLMMs and show that it gives statistically and computationally efficient estimates. Table 1 lists methods related to the h-likelihood method in NLMMs and GLMMs, and this shows a parallel development of methods in the two classes.
In Section 2 we review existing methods in NLMMs related to LAPL approximations and propose the h-likelihood method. In Section 3 we discuss how they differ and study how the current REML procedure for LMMs can be modified to compute h-likelihood estimators for NLMMs. Using the one-compartment model, in Section 4, we compare performances of these estimators numerically and show that the h-likelihood estimators work best. In Section 5, a real data example is provided using guinea pig data of Johansen (1984). The proofs are in the Appendix.
Section snippets
Model and method
A widely used model in population pharmacokinetics and pharmacodynamics is the NLMM of the form where is the model matrix for fixed effects , is the model matrix for random effects , and with diagonal matrix .
Golub and Pereyra (1973) showed how the estimating procedure in regression models can be modified to fit nonlinear regression models. In this paper we study how the REML procedure in LMMs can be modified to obtain the h-likelihood
Implementation of the h-likelihood procedure
In this section, we discuss how to modify the current REML procedure (3), (4) of LMMs (2) to obtain the h-likelihood estimators for . We take and which result in . The proofs are in the Appendix.
Lemma Given , let . Then we havewhere , , is th diagonal element of , is th element of ,
Simulation study
Following Lai and Shih (2003b) consider a one-compartment model: for , , where Here , , denote the th subject’s volume of distribution, absorption rate and elimination rate constants, and Assume that is normal with standard deviation and that . Random effects are assumed to follow a
Example: Guinea pig data
We investigate the guinea pig data of Johansen (1984). For eight guinea pigs, five tissue samples from each guinea pig’s intestine were assigned randomly to each of ten different concentrations of B-methyl-glucoside. The proposed mechanistic model of Johansen (1984) for the mean of the logarithms of the uptake volumes of the five samples from th guinea pig at concentration ( and ) is where and is independent normal with
Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00061).
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