Evaluation of the expected Fisher information matrix without linearization, in nonlinear mixed effect models for discrete and continuous outcomes
Marie-Karelle Riviere, Sebastian Ueckert, and France Mentré
IAME, UMR 1137, INSERM, F-75018 Paris, France University Paris Diderot, Sorbonne Paris Cité, F-75018 Paris, France
Objectives: In recent years, estimation algorithms for NLMEMs have transitioned from linearization-based approaches towards more exact higher-order methods. Optimal design, on the other hand, has mainly relied on first-order linearization (FO) to calculate the expected Fisher information matrix (FIM) [1]. Although efficient in general, FO precludes the application of optimal design with complex non-linear models and in studies with discrete endpoints [2,3]. The objective of this work was to apply integration algorithms, which have proven to be efficient for estimation, to evaluate the asymptotically exact FIM in NLMEM for both discrete and continuous outcomes.
Methods: In NLMEMs, the FIM has no analytical form as its calculation involves multiple integrations. We used either Adaptive Gaussian Quadrature (AGQ) [4] or Markov Chain Monte Carlo (MCMC) to integrate the derivatives of the log-likelihood over the random effects, and Monte Carlo (MC) approximation to evaluate its expectation w.r.t. the observations. The proposed methods were implemented in R and used the probabilistic programming language STAN for MCMC sampling [5]. Evaluation was performed with models for continuous, binary, count and repeated time-to-event outcomes by comparing the FIM based relative standard errors (RSE) to the relative root mean square errors (RRMSE) from clinical trials simulation. The RRMSEs were obtained by simulating 1000 data sets in R and subsequently analyzing them with MONOLIX [6].
Results: Both AGQ and MCMC-based approaches showed good performance on scenarios for continuous and discrete outcomes with RSEs close to the RRMSEs obtained by simulations. In general, RSE predicted by linearization gave close results for rich designs, but showed larger deviations for sparse designs and very non-linear models. We compared the pros and cons of the proposed methods: especially computation of the FIM with AGQ took only seconds for models with few random effects (as commonly encountered in discrete outcome models), but models with more than 4 random effects became infeasible. The MCMC approach on the other hand was notably slower than AGQ for simple models, but can be applied to complex ones with similar time calculation.
Conclusions: Two complementing methods for calculating the exact FIM were proposed and evaluated: AGQ as fast algorithm for simple discrete models, and MCMC which suited even for large complex models where FO fails to correctly evaluate the FIM.
References:
[1] Nyberg, J. et al. Methods and software tools for design evaluation in population pharmacokinetics-pharmacodynamics studies. Br J Clin Pharmacol 79, 6–17 (2015).
[2] Nyberg J, Karlsson MO, Hooker AC. Population optimal experimental design for discrete type data. PAGE (Population Approach Group Europe), St. Petersburgh, Russia, 2009.
[3] Ogungbenro K, Aarons L. Population fisher information matrix and optimal design of discrete data responses in population pharmacodynamic experiments. Journal of Pharmacokinetics and Pharmacodynamics Aug 2011; 38(4):449–469, doi:10.1007/s10928-011-9203-7.studies. Journal of Statistical Planning and Inference, 78(1-2):307 { 316, 1999.
[4] Lange, K. (1999), Numerical Analysis for Statisticians, New York: Springer-Verlag. [5] Stan Development Team. Rstan: the r interface to stan, version 2.6.0, 2014. [6] http://www.monolix.org/ This work was supported by DDMoRe and IDEAL.
