Niklas Hartung (1), Martin Wahl (2), Abhishake Rastogi (1), Gilles Blanchard (1,3), Markus Reiss (2), Wilhelm Huisinga (1)
(1) Institute of Mathematics, University of Potsdam, Germany (2) Institute of Mathematics, Humboldt University Berlin, Germany (3) Departement de Mathematiques d’Orsay, University of Paris-Saclay, France
OBJECTIVES:
Covariates allow to identify patient subpopulations at risk or to individualize drug treatment. While covariate selection criteria have been studied extensively, the choice of the functional relationship between covariates and parameters has received less attention. Indeed, except for particular cases such as allometric scaling, where mechanistic covariate models have been derived [1,2], the choice for a particular class of covariate-to-parameter relationships (linear, exponential, etc.) is largely empirical [3]. When the appropriateness of a covariate model is at stake, goodness-of-fit tests provide a means for its evaluation. In this work, we therefore derived and evaluated a goodness-of-fit test for parametric covariate models, based on a nonparametric covariate modelling framework.
METHODS:
Based on the theory of vector-valued reproducing kernel Hilbert spaces [4,5,6], we propose a nonparametric covariate modelling approach in which the functional relationship between covariates and model parameters is given as the sum of a parametric and a kernel-based part. To estimate this combined covariate-to-parameter relationship, a regularization is applied to the kernel-based part (penalized maximum likelihood estimation) [7]. To evaluate a particular parametric covariate model, the normalized discrepancy in goodness-of-fit between the purely parametric vs. the combined covariate model is used as a test statistic; critical values are determined by Monte Carlo simulations. In a simulation study, we explore the suggested approach in the context of estimating the effect of enzyme maturation on clearance [8] by evaluating the appropriateness of different possible parametric models for an age- and weight-dependent maturation function: (i) a saturable age-dependency (true model used for simulation), and (ii) an affine linear age-dependency (misspecified model). We investigated two scenarios, data-rich (rich sampling, large population) and data-poor (sparse sampling, small population).
RESULTS:
A tailored numerical scheme was developed and implemented in R (version 3.5.1) [9], extending the backfitting algorithm [10], to solve the resulting high-dimensional and non-convex optimization problem robustly by iterating between the parametric part (low-dimensional, non-convex) and the linearized kernel-based part (high-dimensional, closed-form solution). As expected from asymptotic theory, the implementation of the proposed goodness-of-fit test maintained Type I error bounds (5%) for the data-rich situation and approximately (7%) in the data-poor case. Furthermore, a misspecified parametric model (affine linear instead of saturable) could be detected with power 100% (data-rich) and 88% (data-poor).
CONCLUSIONS:
Kernel methods are classically used for machine learning applications such as classification and smoothing [4]. Integrating this methodology into covariate modelling, the presented proof-of-concept study demonstrates of the feasibility of goodness-of-fit testing for parametric covariate models through a combined parametric/kernel-based approach, both from a conceptual and an implementational point of view. As a next step, we will integrate random effects into the proposed framework.
ACKNOWLEDGEMENTS:
The presented work was funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1294 Data Assimilation.
References:
[1] Anderson BJ, Holford NH Drug Metab Pharmacokinet (2009) 24(1):25-36
[2] Huisinga W et al. CPT:PSP (2012) 1:e4
[3] Joerger M. AAPS J (2012) 14(1):119-132
[4] Alvarez M et al. Found Trends Mach Learn (2012) 4(3):195-266
[5] De Vito E et al. Analysis and Applications (2006) 4(1):81-99
[6] Hein M, Bousquet O, Kernels, Associated Structures and Generalizations (2004).
[7] Poggio T, Girosi F Proceedings of the IEEE (1990) 78(9):1481-1497
[8] Robbie GJ et al Antimicrob. Agents Chemother (2012) 56(9):4927-4936
[9] R Core Team (2018) https://www.R-project.org/.
[10] Hastie T, Tibshirani R, Generalized additive models. Chapman & Hall/CRC, 1990
Reference: PAGE () Abstr 9385 [www.page-meeting.org/?abstract=9385]
Poster: Methodology - Covariate/Variability Models