Evaluation of Non Parametric Methods for population PK/PD
Antic, Julie (1,2) ; Chafaï, Djalil (1) ; Laffont, Céline (2) ; Concordet, Didier (1)
(1) UMR181 Physiopathologie et Toxicologie Expérimentales, INRA, ENVT, Toulouse, France ; (2) Servier Research Group, Courbevoie, France
Introduction: In population pharmacokinetics/pharmacodynamics (PK/PD), the normality of the population distribution is often assumed by standard parametric estimation methods (e.g. FO, FOCEI). This assumption is usually checked a posteriori using the Empirical Bayes Estimates (EBE). In that context, NONMEM VI proposes a non parametric option (NP-NONMEM) for estimation of the population distribution. The distribution estimate is discrete and uses EBE as support points. Savic  revealed that, when the distribution is not normal, NP-NONMEM estimate is closer to the true population distribution compared to FO or FOCEI. However, the statistical properties of this estimator need to be studied. In contrast, nonparametric maximum likelihood methods (NP-MLE) have well-known statistical properties . Besides, when no constraint is imposed on the distribution, it has been shown  that likelihood reaches a maximum on a discrete distribution with at most as many support points as individuals in the sample. Since the 80's, several algorithms have been proposed to compute this distribution: nonparametric maximum likelihood (NPML) , nonparametric EM (NPEM) . Their use is still limited mainly because of computational cost, while their practical performances are not well documented. Thus, the potential benefit of NP methods needs to be investigated.
Objectives: To compare different NP methods (NPML, NPEM, NP-NONMEM) with their usual parametric counterparts (NONMEM FO or FOCEI EBE histogram) for detecting departures from normality. We especially focused on the ability of these methods to detect bimodality.
Methods: We performed simulations using a one compartment iv bolus model parameterised in terms of clearance and volume of distribution. Model was heteroscedastic. Different distributions of volume of distribution and clearance were simulated : log-normal, multimodal or heavy-tailed. For each population distribution, we considered several : (i)samples sizes from small (50 individuals) to larger (300), (ii)measurement design from sparse (1 observation per individual) to richer (3 observations per individual). Hundred different independent samples were simulated for each configuration. Each data set was analysed by standard parametric (FO, FOCEI) and nonparametric (NP-NONMEM, NP-MLE) methods. Different criteria were used to evaluate the performances of the tested methods : Kolmogorov Smirnov distance, Kurtosis and skewness indexes, Relative Estimation Error ...
Results and discussion: As expected, estimation accuracy rises with the sample size or the number of observations per individual for all methods investigated. With sparse data (one single observation per individual), NP-MLE methods are closer than EBE ones to the true distribution with respect to Kolmogorov Smirnov distance. However, the benefit of NP-MLE methods is not showed by all criteria. Our preliminaries studies do not establish the supremacy of one of the methods in their ability to detect departures form normality.
 Djalil Chafaï and Didier Concordet, On the strong consistency of asymptotic M-estimators, (2006), To appear in JSPI
 Lindsay B. G., The geometry of mixture likelihoods: a general theory. The annals of statistics, 11(1):86-91, 1983.
 Mallet A., A maximum likelihood estimation method for random coefficient regression models. Biometrika, 73(3) : 645-656,1986.
 Schumitzky A., Nonparametric EM algorithms for estimating prior distributions. Applied Mathematics and Computation, 45(2, partII) : 143-157, 1991.
 Davidian M. and Gallant A.R., The nonlinear mixed effect model with a smooth random density. Biometrika, 1993, 80, 475-488.
 Savic R., Kjellson M., Karslsson M., Evaluation of the nonparametric estimation method in NONMEM VI beta. PAGE meeting, 2006.