Mélanie Guhl (1), Julie Bertrand (1), Emmanuelle Comets (1)
(1) University of Paris, INSERM, IAME, UMR 1137, F-75018 Paris, France
Introduction: Longitudinal data are often collected in clinical trials, and non linear mixed effects models (NLMEM) are powerful to model them. In the frequentist paradigm, the population parameter vector theta is considered as a point estimate, and the standard error of the maximum likelihood estimator (MLE) can be computed asymptotically based on the Fisher Information Matrix (FIM). However, when working at finite distance (i.e. far from the asymptotic), using the FIM underestimates the SE of NLMEM. It has been proposed to use bootstrap [1] or SIR [2] methods.
Objectives: In this work, we want to borrow from the Bayesian paradigm, to compute SEs of NLMEM at finite distance.
Methods: Here, the population parameter vector theta is considered as a random variable with a prior. Ueckert et. al. [3] proposed to use the standard deviation of the posterior distribution of theta as a proxy for the standard error. Indeed, under some regularity conditions on the prior, the limit distributions of the MLE and the maximum a posteriori (MAP) estimator are equivalent (Bernstein-von Mises theorem). This method has been implemented via HMC algorithm in Stan (Post) [4].
We propose to integrate a similar method in the SAEM algorithm. To do so, we draw in the conditional distribution of the estimated population parameter vector, at finite distance, and evaluate its standard deviation. We implemented this solution using the saemix R package. We do this using a Metropolis Hastings (MH) type of algorithm included into the SAEM algorithm in saemix, after the initial exploration phase, in parallel of the convergence phase. The parameters that have to be calibrated in this algorithm are the length of the chain, the prior distribution and the kernel distribution. For now, The Bayesian step uses the frequentist estimations as parameters of the proposal kernel, but does not influence the frequentist estimation. We end up with a frequentist and a Bayesian estimation of the standard error, respectively obtained from the FIM and the standard deviation of the chain sampled.
Our set of simulations is inspired by the settings of theophylline data: a one-compartment structural model with linear absorption and elimination, and a combined error model. We simulate two equal treatment groups where one was assumed to receive placebo and the other a treatment of interest a treatment effect on Cl and V at log(1.25).The number of patients varies from N=150 with n=10 sampling points per patient, to N=12 and n=3.
The evaluation of the method is based on the acceptation rates, the histograms of posterior distributions, the boxplots of SEs and the 90% coverage rates. We compare the results obtained with our method to those obtained with Asympt (FIM) and Post (HMC).
Results: On N=150 and n=10, the SE obtained from the FIM are close to the target, the empirical SE. The SE from our method and Post are also well estimated. Coverage rates (CR) are appropriate. These results are satisfactory but not surprising, because we are in an asymptotic setting.
On N=12 and n=3, Asympt underestimates the SE. The coverage rates are below the prediction interval. We find the same results with our method. The acceptance rates are high (about 90%), compared to the 40% acceptance rates found in the first scenario.
In that case, inflating the kernel variance by a factor 2 lowers the acceptance rate (about 60%) and increases the effective sample size. The SE become higher, well estimated, and the CR are within the prediction interval. In this scenario, the SE are overestimated by Post, and further work is needed to investigate suitable priors.
Conclusions: These first results show that, as expected, the FIM-based method of computation of SEs degrades at finite distance. In this case, MH seems promising but we need to further investigate the calibration of the prior and kernel distributions.
References:
[1] Thai HT, Mentré F, Holford N, Veyrat-Follet C, Comets E, Evaluation of bootstrap methods for estimating uncertainty of parameters in nonlinear mixed-effects models: a simulation study in population pharmacokinetics, J Pharmacokin Pharmacodyn 2013.
[2] Dosne AG, Bergstrand M, Harling K, Karlsson MO, Improving the estimation of parameter uncertainty distributions in nonlinear mixed effects models using sampling importance
resampling, J Pharmacokin Pharmacodyn 2016.
[3] Ueckert S, Riviere MK, Mentré F, Alternative to resampling methods in maximum likelihood Estimation for NLMEMs by borrowing from Bayesian methodology, PAGE 2015 (Abstr 3632).
[4] Loingeville F, Bertrand J, Nguyen T, Sharan S, Feng K, Sun W, Han J, Grosser S, Zhao L, Fang L, Möllenhoff K, Dette H, Mentré F, New model-based bioequivalence statistical approaches for pharmacokinetic studies with sparse sampling, AAPS J 2020.
Reference: PAGE 30 (2022) Abstr 10152 [www.page-meeting.org/?abstract=10152]
Poster: Methodology - Other topics