Evaluation of Fisher information matrix using Gaussian quadrature in nonlinear mixed effects models: application to dose-response trials
Thu Thuy Nguyen, France Mentré
UMR738 INSERM and University Paris Diderot, Paris, France
Objectives: Nonlinear mixed effects models (NLMEM) can be used to analyse dose-response trials where each patient received several doses. Design in NLMEM can be evaluated/optimised using the population Fisher information matrix (MF), with first order approximation of the model [1,2]. This approach was implemented in the R function PFIM [3,4] and in other software. Adequacy of this approximation is however influenced by model nonlinearity. We aim to: i) propose a new approach to evaluate MF in NLMEM without linearisation, based on Gaussian quadrature [5]; ii) evaluate this method by simulations and compare it to first order approximation for dose-response sigmoid Emax models with various nonlinearity levels.
Methods: In NLMEM, the likelihood expressed as an integral has no analytical form. We approximate it by quadrature rule [6] using Gauss-Hermite nodes and weights [7]. We then can derive the expression of MF without linearising the model. We evaluate the relevance of this method and compare it to the linearisation approach for dose-response sigmoid Emax models, inspired from a previous example [8]. Various nonlinearity levels of the model are studied with different sigmoidicity factors γ(the larger γ is, the more nonlinear is the model). Dose-response trial simulations are performed for a rich design (7 doses) as well as for a sparser design (4 doses). For each scenario, we simulate 1 trial of 10000 subjects then 1000 trials of 100 subjects. We compare the standard errors (SE) obtained from different approaches: simulations analysed in MONOLIX 3.2 [9] with SAEM algorithm [10] vs. linearisation approach in PFIM vs. Gaussian quadrature.
Results: We have implemented this new method using the function gauss.quad of the R package statmod to calculate Gauss-Hermite nodes and weights in a working version of PFIM. For γ = 3, with the rich design, the SE predicted by linearisation approach are very close to those obtained from simulations. However, with the sparser design, we observe large differences in SE of fixed effects between linearisation approach and simulations.
Conclusions: The linearisation approach seems to work well for a great nonlinearity level with a very rich design but not with a sparse design. We propose an approach without linearisation to evaluate MF in NLMEM for designing dose-response trials. This approach, if relevant, can be applied to more complex models with great nonlinearity level and will be implemented in a future version of PFIM.
References:
[1] Mentré, F., Mallet, A., Baccar, D. (1997) Optimal design in random effect regression models, Biometrika, 84(2), 429-442.
[2] Bazzoli, C., Retout, S., Mentré, F. (2009) Fisher information matrix for nonlinear mixed effects multiple response models: Evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model, Statistics in Medicine, 28, 1940-1956.
[3] Bazzoli, C., Retout, S., Mentré, F. (2010) Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Computer Methods and Programs in Biomedicine, 98, 55-65.
[4] http://www.pfim.biostat.fr/.
[5] Guedj, J., Thiébaut, R., Commenges, D. (2007) Maximum likelihood estimation in dynamical models of HIV, Biometrics, 63, 1198-206.
[6] Lange, K. (1999), Numerical Analysis for Statisticians, New York: Springer-Verlag.
[7] Press, W.H, Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. (1992), Numerical Recipes in C - The Art of Scientific Computing, 2nd ed., Cambridge Press.
[8] Plan, E.L., Maloney, A., Mentré, F., Karlsson, M.O., Bertrand, J. (2010) Nonlinear mixed effects estimation algorithms: a performance comparison for continuous pharmacodynamic population Models, 19th Meeting of the Population Approach Group in Europe (PAGE) Conference, Berlin, Germany, Abstract 1880 [www.page-meeting.org/?abstract=1880].
[9] http://www.monolix.org/
[10] Kuhn, E., Lavielle, M. (2005) Maximum likelihood estimation in nonlinear mixed effects model, Computational Statistics and Data Analysis, 49, 1020-1038.
