A comparison of bootstrap approaches for estimating standard errors of parameters in linear mixed effects models
Hoai-Thu Thai (1), Christine Veyrat-Follet (2), France Mentrť (1), Emmanuelle Comets (1)
(1) INSERM UMR738, University Paris Diderot-Paris 7, Paris, France; (2) Global Metabolism and Pharmacokinetics Department, Sanofi-aventis, Paris, France
Objectives: Paired bootstrap which resamples the individuals from original data, has been used in PK/PD for estimating the standard errors (SE) of parameters in mixed-effects models (MEM) [1,2]. Alternative bootstrap methods, such as residual bootstrap and parametric bootstrap have been proposed to better take into account the hierarchical structure of multi-level data [3-6]. Our objectives are to study and propose appropriate bootstrap methods in MEM and to evaluate their performance by simulation. We present the results obtained with linear mixed-effect models, using an example of disease progression model in Parkinsonís disease .
Methods: We implemented the paired, residual and parametric bootstraps in R, fitting the simulated data with the R function lme. We investigated corrections to residuals using the ratio between empirical and estimated variance-covariance matrix  to account for shrinkage. The bootstrap approaches were compared in term of bias, delta SE (difference between bootstrap and empirical SE), root mean squared error (RMSE), and coverage rate of the 95% confidence interval of all parameter estimates. A subset of a study describing the natural evolution of Parkinson's disease over a 2-year period was used to motivate the simulations and illustrate the results. The rich design (100 subjects and 7 samples per subject) and the sparse design (30 subjects and 3 samples per subject) were investigated by simulation using 1000 replicates and 1000 bootstrap samples per replicate for each bootstrap scenario.
Results: The paired and parametric bootstraps proved good approaches for both evaluated designs with small bias and RMSE of parameter estimates, SE close to empirical values and a good coverage rate. The residual bootstrap also performed well when resampling with corrected residuals. For these three approaches, the bias of all parameters were lower than 2.1% and 8%, and their SE remains within 6.4 and 10.4% of the true values, respectively for the rich and sparse design. The coverage rates obtained by bootstrap were better than the asymptotic coverage rates for random effects and residual error.
Conclusions: The paired bootstrap works as well as the residual bootstrap and the parametric bootstraps for both designs, although only the interindividual variability is resampled, possibly due to the large interindividual variability in the data. A correction was necessary for the residual bootstrap to account for the variance underestimation.
 Ette EI et Williams PJ. Pharmacometrics†: The science of quantitative pharmacology. 2007, John Wiley & Son, Inc. Chapter Resampling techniques and their application to pharmacometrics.
 Parke J, Holford NHG, Charles BG. A procedure for generating bootstrap samples for the validation of nonlinear mixed-effects population models. Computer Methods and Programs in Biomedicine. 1999; 59:19-29.
 Van der Leeden R, Busing F. M. T. A., and Meijer E. Bootstrap methods for two-level models. Technical Report PRM 97-04, Leiden University, Department of Psychology, Leiden, 1997.
 Ocana J, Halimi RE , Ruiz de Villa MC, and Sanchez JA. Bootstrapping repeated measures data in a nonlinear mixed-models context. Mathematics Prepint Series. 2005.
 Das S, Krishen A. Some bootstrap methods in nonlinear mixed-effects models. Journal of Statistical Planning and Inference. 1999; 75: 237-245.
 Wang J, Carpenter JR, Kepler MA. Using SAS to conduct nonparametric residual bootstrap multilevel modeling with a small number of groups. Computer Methods and Programs in Biomedicine. 2006; 82: 130-143.
 Holford NHG, Chan PLS, Nutt JG, Kieburtz K, Shoulson I and Parkinson Study Group. Disease progression and pharmacodynamics in Parkinson disease-evidence for functional protection with Levodopa and other treatments. Journal of Pharmacokinetics and Pharmacodynamics. 2006; 33: 281-311.