Some Stochastic Algorithms For The Smooth Non Parametric (SNP) Estimator
Julie Antic (1,2), Céline M. Laffont (1), Didier Concordet (2).
(1) Servier Research Group, Courbevoie, France ; (2) UMR181 Physiopathologie et Toxicologie Expérimentales, INRA, ENVT, Toulouse, France.
Background: In population analyses, standard parametric methods assume the normality of the random effects (ETAs), even if this may be unrealistic (especially in phase II/III studies where the population may be very heterogeneous).
Non parametric (NP) estimators are of interest, since they do not assume the normality of ETAs. Several discrete NP estimators have been proposed: NPML, NPEM, and recently the NP method in NONMEM ([1]). These NP estimators have several drawbacks. In particular, they are difficult to interpret because they are discrete although the true distribution of ETAs is generally continuous.
The smooth NP (SNP) estimator proposed by Davidian and Gallant [2] is more attractive: it is continuous and has well established statistical properties. However, its use is limited by a complex computation: the likelihood is not explicit and thus difficult to maximize.
Objective: Find an efficient algorithm to make an easy computation of the SNP estimator.
Methods: Here, 4 stochastic algorithms are studied: a stochastic gradient algorithm [3], a stochastic Newton-Raphson algorithm [4], a new perturbed stochastic gradient algorithm, and a new particle algorithm. Convergence of each algorithm is investigated. Practical performances are illustrated on simulated data obtained with the phenobarbital model [5].
Results: The stochastic gradient algorithm and the stochastic Newton-Raphson algorithm are not very satisfactory because they sometimes converge to a maximum that is not global. The usual procedure to cope with this problem is the multistart procedure: several optimizations are performed with different initial estimates. This procedure ensures convergence to the global maximum if there are a lot of initial estimates, but is expensive in terms of computation time. We show that a new perturbed gradient algorithm converges to the global maximum of the likelihood. However, its convergence rate is slow and thus, it is not competitive with the multistart procedure. The particle algorithm, which makes a more clever use of initial estimates, appears promising.
Conclusion: The use of the SNP estimator is made easier thanks to the development of specific stochastic algorithms. Standard stochastic algorithms such as SAEM could not be applied given the complexity of the SNP model. Deterministic algorithms based on numerical approximation of the likelihood could have been used. However, they are computationally expensive and/or inaccurate in case of numerous random effects.
References:
[1] Antic J., Chafaï D., Laffont C.M., Concordet D. Evaluation of Non Parametric Methods for Population PK/PD. PAGE meeting 16, 2007, abstract 1181 [www.page-meeting.org/?abstract=1181].
[2] Davidian M. and Gallant A.R., The nonlinear mixed effect model with a smooth random density. Biometrika, 1993, 80, 475-488.
[3] Younes, L. Parameter estimation for imperfectly observed Gibbsian fields. Probab. Theory and Related Fields, 1989, 82, 625-645.
[4] Gu M.G., Kong F.H. A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems. Proceedings of the National Academy of Sciences of the USA, 1998, 13, 7270-7274.
[5] Grasela T.H., Donn S.M. Neonatal population pharmacokinetics of phenobarbital derived from routine clinical data. Developmental Pharmacology and Therapeutics, 1985, 8: 6, 374-383.