An overview of non-parametric estimation methods used in population analysis
Robert H. Leary
Certara, Cary, NC. USA.
Objectives: To present an overview of non-parametric (NP) estimation methods used to analyze PK/PD population models. Emphasis will be placed on current generation NP maximum likelihood (ML) methods, although semiparametric ML and Bayesian NP methods based on Dirichlet priors will be briefly described. Differences with parametric methods will be discussed, as will advantages and limitations of the NP approach. We will also discuss the range of mainstream current implementations now in use, and their various approaches to dealing with some of the limitations of earlier NP methods.
Overview: In 1986 Mallet wrote a seminal paper [1] which described an NP approach (Non-Parametric Maximum Likelihood, or NPML) for the PK/PD population estimation problem in which the assumption that the random effects (ETAs) are based on a multivariate normal distribution is discarded in favor of an arbitrary distribution. The paper described the key mathematical property of ML-based NP estimators, namely that the form of the optimal solution is a discrete distribution with support on a limited number (generally not greater than Nsub, the number of subjects) of support points. Mallet's algorithm allowed the support points to move (but only one per iteration) based on some ideas from optimal design (OD) theory, but suffered from a relatively slow algorithm for computing probabilities. In 1991, Schumitzky proposed a new NPEM (Non-Parametric Expectation Maximization) algorithm in which the support points were fixed and the probabilities computed via an EM algorithm [2]. Both NPML and NPEM were viable but slow and had a fundamental limitation in that the residual error model had to be input a priori. However, both algorithms were demonstrably successful in identifying, for example, mixtures of fast and slow metabolizers, even with relatively sparse data, in cases where standard parametric methods failed.
NPEM was ultimately succeeded by Leary's 2001 NPAG (Non-Parametric Adaptive Grid) algorithm [3], which allowed much smaller initial grids, movable support points (several per iteration), and had a far faster primal-dual probability optimization algorithm which took advantage of the convexity (in probabilities) of the negative NP log likelihood function. NPAG resolved some of the NP performance issues and is still the primary NP method in use by the Laboratory of Applied Pharmacokinetics in its Pmetrics and BestDose software as well as the method current used in Certara’s Phoenix NLME software.
NP algorithms gained a great deal of exposure when a rather simple NP variant NONP appeared in NONMEM 6 in 2007. NONP fixed the support points at the EBE posthoc values of a prior parametric run, and that parametric run was also used to import covariate fixed effect and residual error model parameters. The initial NONP version was usually quite fast due to its limited grid size, but suffered to some extent from that limited grid and shrinkage effects in the sparse data case. Later work in Karlsson's laboratory at Uppsala to some extent resolved the NONP grid problem by augmenting the starting grid with several strategies [4]. These modifications have now made their way into PsN and hence are readily available to the community at large. The Karlsson laboratory also did some pioneering work on bootstrap and SSE estimation of standard errors, which appear to be the only viable approaches for the NP case [5].
NP methods are still an active area of research with an increasing user base and there are now a variety of quite effective methods available to the PK/PD population community. There are still limitations and some performance issues, but these are gradually being overcome. We expect this trend to continue in the future and NP methods to take a more visible and effective role in the armamentarium of population methods.
References:
[1] A. Mallet. A maximum likelihood estimation method for random coefficient regression models. Biometrika, 73(3):645–656, 1986.
[2] A. Schumitzky. Nonparametric EM algorithms for estimating prior distributions. Applied Mathematics and Computation, 45(2, part II):143–157, 1991.
[3] R. Leary et al. An adaptive grid non-parametric approach to pharmacokinetic and pharmacodynamic population modeling. Proceedings 14th IEEE Symposium on Computer- Based Medical Systems. 26-27 July 2001.
[4] R. Savic et al. Evaluation of an extended grid method for estimation using nonparametric distributions. AAPS J. 11(3):615-27, 2009.
[5] P. Baverel et al. Two bootstrapping routines for obtaining imprecision estimates for nonparametric parameter distributions in nonlinear mixed effects models. J Pharmacokinet Pharmacodyn. 38(1):63-82, 2011.
