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Biostatistics Advance Access originally published online on June 25, 2008
Biostatistics 2009 10(1):121-135; doi:10.1093/biostatistics/kxn020
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects

Xavière Panhard*

INSERM U738 and UFR de Médecine, University Paris 7, 16 rue Huchard, 75018 Paris, France xaviere.panhard{at}bichat.inserm.fr

Adeline Samson

Laboratoire MAP5, UMR CNRS 8145, University Paris 5, France

* To whom correspondence should be addressed.

This article focuses on parameter estimation of multilevel nonlinear mixed-effects models (MNLMEMs). These models are used to analyze data presenting multiple hierarchical levels of grouping (cluster data, clinical trials with several observation periods, ...). The variability of the individual parameters of the regression function is thus decomposed as a between-subject variability and higher levels of variability (e.g. within-subject variability). We propose maximum likelihood estimates of parameters of those MNLMEMs with 2 levels of random effects, using an extension of the stochastic approximation version of expectation–maximization (SAEM)–Monte Carlo Markov chain algorithm. The extended SAEM algorithm is split into an explicit direct expectation–maximization (EM) algorithm and a stochastic EM part. Compared to the original algorithm, additional sufficient statistics have to be approximated by relying on the conditional distribution of the second level of random effects. This estimation method is evaluated on pharmacokinetic crossover simulated trials, mimicking theophylline concentration data. Results obtained on those data sets with either the SAEM algorithm or the first-order conditional estimates (FOCE) algorithm (implemented in the nlme function of R software) are compared: biases and root mean square errors of almost all the SAEM estimates are smaller than the FOCE ones. Finally, we apply the extended SAEM algorithm to analyze the pharmacokinetic interaction of tenofovir on atazanavir, a novel protease inhibitor, from the Agence Nationale de Recherche sur le Sida 107-Puzzle 2 study. A significant decrease of the area under the curve of atazanavir is found in patients receiving both treatments.

Keywords: Bioequivalence trials; Crossover trial; Multilevel nonlinear mixed-effects models; Multiple periods; SAEM algorithm

Received December 21, 2007; revised February 19, 2008; revised April 2, 2008; accepted for publication May 12, 2008.


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