Using Hamiltonian Monte-Carlo to design longitudinal count studies accounting for parameter and model uncertainties
Florence Loingeville (1), Thu Thuy Nguyen (1), Marie-Karelle Rivière (1,2), and France Mentré (1)
(1) INSERM, IAME, UMR 1137, F-75018 Paris, France; (2) Statistical Methodology Group, Biostatistics & Programming Department, Sanofi-Aventis R&D, Chilly-Mazarin, France
Objectives: Nonlinear mixed effect models (NLMEM) are widely used for the analysis of longitudinal data. To design these studies, optimal design based on the expected Fisher information matrix (FIM) can be used instead of performing clinical trial simulations. A method evaluating the FIM, without any linearization, based on Monte-Carlo Hamiltonian Monte-Carlo (MC/HMC) has been proposed and implemented in the R package MIXFIM [1] using Stan for HMC sampling [2]. This approach however requires a priori knowledge on models and parameters, which lead to designs that are locally optimal. The objective of this work was to extend this MC/HMC-based method to evaluate the FIM in NLMEM accounting for uncertainty in parameters and in models. We showed an illustration of this approach to optimize robust designs for repeated count data.
Methods: When introducing uncertainty on the population parameters, we evaluated the robust FIM as the expectation of the FIM computed by MC/HMC on the population parameters. Then, the compound D-optimality criterion [3, 4] was used to find a common CD-optimal design for several candidate models. Finally, a compound DE-criterion combining the determinants of the robust FIMs was calculated to find the CDE-optimal design which was robust with respect to both model and parameters. These methods were applied in a longitudinal Poisson count model where the event rate parameter (𝛌) is function of the dose level. We assumed a log-normal a priori distribution characterizing the uncertainty on the population parameter values as well as several candidate models describing the relationship between log(𝛌) and the dose level (linear, log-linear, Imax, full Imax, or quadratic functions). Assuming the first dose fixed to 0, we performed combinatorial optimization of 2 among 10 doses between 0.1 and 1, corresponding to 45 possible elementary designs.
Results: We found that accounting or not for uncertainty on parameters does not have a striking impact on the allocation of optimal doses in this study. However misspecification of models could lead to low D-efficiencies of only 30%. The CD- or CDE-optimal designs provided then a good compromise for different candidate models, with D-efficiencies of at least 80% for each model.
Conclusions: MC/HMC is a relevant approach allowing for the first time optimization of design for repeated discrete data accounting for uncertainty in parameters and in candidate models.
References:
[1] Riviere, M.K., Ueckert, S., and Mentré, F. (2016). An MCMC method for the evaluation of Fisher information matrix for non-linear mixed effect models. Biostatistics.
[2] Stan Development Team (2016). RStan: the R interface to stan, version 2.12.0. http://mc-stan.org/
[3] Atkinson, A. (2008). DT-optimum designs for model discrimination and parameter estimation. Journal of Statistical Planning and Inference.
[4] Nguyen, T. T., Benech, H., Delaforge, M., and Lenuzza, N. (2016). Design optimisation for pharmacokinetic modeling of a cocktail of phenotyping drugs. Pharmaceutical Statistics.
