Nonparametric Approach using Gaussian Kernels Estimates Multivariate Probability Densities in Population Pharmacokinetics
H. Marouani (1), L. Claret (2), C. Faivre (1), A. Iliadis (1)
(1) Dpt. of Pharmacokinetics, UMR-MD3, Univ. of Méditerranée, Marseilles, France
Objectives: Describe the interindividual variability (interIV) by means of a multivariate probability density function (pdf) using a two-stage nonparametric procedure. Our proposal enables incorporating intraindividual variability (intraIV) and out-performs the traditional two-stage method, which ignores intraIV and overestimates random effects.
Methods: In the first stage of this procedure, covariates are recorded and individual parameters xi are estimated by using the maximum likelihood principle. Because the maximum likelihood is asymptotically Gaussian, precisions of estimates Pi are computable and quantify the intraIV. In second stage, the nonparametric kernel approach compiles the n available training data xi and Pi to reliably estimate the underlying pdf:
f(x,h)= (nh)-1 Σ K[(x-xi)/h]
with K the Gaussian kernel and h the bandwidth. We propose to individualize h by incorporating the intraIV so as hi = sPi½. The choice of the new bandwidth s in the above relationship is typically critical in terms of performance when implementing kernel smoothers .
Results: The feasibility of the proposed procedure is illustrated by a simulation study. The raw data were "prewhitened" to scale equally in all directions in the parameter space. Several approaches for selecting the "optimal bandwidth" are studied . Performances of this procedure were evaluated by establishing consistency of estimates and rate of convergence for bandwidth selection. This procedure is used to obtain nonparametric conditional pdf of the kinetic parameters given the individual covariates. We show how to use this conditional pdf as prior information in Bayesian estimation.
Conclusion: Both intraIV and interIV are incorporated in the proposed nonparametric procedure. The optimal bandwidth in the kernel pdf estimator is computed. This procedure does not require the structural model between covariates and kinetic parameters but it establishes conditional distribution between them. The method is easy to implement and quick for data processing.
 M.P. Wand, M.C. Jones. Kernel smoothing. Chapman and Hall, London, 1995.
 Martin L, Hazelton, Berwin A, Turlach. Nonparametric density deconvolution by weighted kernel estimators. Stat Comput 2009;19:217-228.