Christelle Rodrigues (1), Vincent Jullien (1), Emmanuelle Comets (2,3)* (* Presenting author)
(1) UMR1129, INSERM, Paris, France (2) INSERM, IAME, UMR 1137, F-75018 Paris, France; Univ Paris Diderot, Sorbonne Paris Cité, F-75018 Paris, France (3) INSERM, CIC 1414, 35700 Rennes, France; Univ Rennes-1, 35700 Rennes, France
Introduction: Standard errors of estimation (SE) measure the precision of estimation for the parameters in a statistical model. A commonly used approach in non-linear mixed effect models (NLMEM) is to use the inverse of the Fisher information matrix to derive the SE. How well this asymptotic approximation holds in practice is a matter of debate, and alternative approaches have been proposed such as log-likelihood profiling and bootstrap methods.
Bootstrap approaches have been extended to NLMEM by considering the different levels of variability involved at the individual and population level. They include the case bootstrap, resampling individuals, the non-parametric bootstrap, which resamples residuals within and across individuals and the parametric bootstrap, resampling from a distribution. Thai et al. have investigated their properties in NLMEM for different numbers of subjects in sparse or rich designs, and found contrasted results [1]. A question arising was whether it would be possible to improve the correction of the residuals, performed before resampling to compound for shrinkage, to benefit from the non-parametric bootstrap’s ability to maintain the structure of the original dataset, while being less dependent on model assumptions than the parametric bootstrap.
In the present work, we propose an alternative non-parametric bootstrap, resampling from the conditional distribution of the individual parameters, and evaluate it in a simple framework.
Methods: A general bootstrap algorithm consists in creating bootstrapped datasets through resampling, fitting a model to the data, and storing the parameter estimates from each replicate to form a bootstrap distribution of the parameters. We implemented four bootstrap approaches: case, parametric, non-parametric [1], and conditional non-parametric bootstrap. In this new approach, instead of correcting the estimated random effects for shrinkage, we sample random effects within a set of 100 samples from the conditional distribution [2].
The bootstrap approaches were evaluated in a simulation study, using a one-compartment model, simplifying the model developed to describe the pharmacokinetics of valproic acid in children in [3]. Both single dose and SS profiles were simulated, and the design included 100 subjects with 6 sampling times, which were obtained by optimal design using PFIM [4].
The bootstraps and the asymptotic method were compared in terms of bias, standard errors (SE) and coverage rate of the 95% confidence interval for all parameter estimates. We used the SAEM algorithm implemented in R in the saemix package [5,6], as well as the MlxConnectors library, interfacing with Monolix [7], for comparison.
Results: In the single dose scenario, all methods provided unbiased estimates of the fixed parameters and of the variabilities, with the exception of the conditional non-parametric bootstrap which underestimated the variability of clearance and volume of distribution. All methods yielded biased estimations of the SE for at least some of the parameters, although the absolute difference was actually small, and again, the conditional non-parametric bootstrap showed more bias than the non-parametric bootstrap. In the steady-state scenario, biases appeared also for the bootstrap estimates of the parameters themselves, and were more pronounced for the SEs. The single-dose simulation was run in MlxConnectors and showed qualitatively similar results.
Conclusions: Using samples from the conditional distribution in the non-parametric bootstrap increased the bias on bootstrap estimates of both parameters and SE, compared to the standard non-parametric bootstrap. We chose a simple scenario with linear pharmacokinetics for this first evaluation, as a method needs to perform well in simple settings before we use it in more complex conditions. Unfortunately none of the bootstraps, including our new proposal, clearly improves over the asymptotic method, and in particular they are unable to correct for estimation bias in the asymptotic method.
References:
[1] Thai H, Mentré F, Holford NH, Veyrat-Follet C, Comets E. (2014). Evaluation of bootstrap methods for estimating uncertainty of parameters in nonlinear mixed-effects models: a simulation study in population pharmacokinetics. Journal of Pharmacokinetics and Pharmacodynamics, 41:15–33.
[2] Lavielle M and Ribba B (2016). Enhanced method for diagnosing pharmacometric models: random sampling from conditional distributions. Pharmaceutical Research, 33:2979–88.
[3] Rodrigues C, Chhun S, Chiron C, Dulac O, Rey E, Pons G, and Jullien V (2018). A population pharmacokinetic model taking into account protein binding for the sustained-release granule formulation of valproic acid in children with epilepsy. Journal of Clinical Pharmacology, 74:793–803.
[4] Dumont C, Lestini G, Le Nagard H, Mentré F, Comets E, Nguyen T, the PFIM Group (2018). PFIM 4.0, an extended R program for design evaluation and optimization in nonlinear mixed-effect models. Computer Methods and Programs Biomedicine, 156 : 217–29.
[5] Comets E, Lavenu A, Lavielle M (2017). Parameter estimation in nonlinear mixed effect models using saemix, an R implementation of the SAEM algorithm. Journal of Statistical Software, 801–41.
[6] https://CRAN.R-project.org/package=saemix
[7] Lavielle M (2014). Mixed Effects Models for the Population Approach – Models, Tasks, Methods and Tools. Chapman & Hall/CRC, Boca Raton.
Reference: PAGE 28 (2019) Abstr 8871 [www.page-meeting.org/?abstract=8871]
Poster: Methodology - Estimation Methods