Improved numerical stability for the bounded integer model
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ORIGINAL PAPER
Improved numerical stability for the bounded integer model Sebastian Ueckert1
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Mats O. Karlsson1
Received: 7 September 2020 / Accepted: 5 November 2020 Ó The Author(s) 2020
Abstract This article highlights some numerical challenges when implementing the bounded integer model for composite score modeling and suggests an improved implementation. The improvement is based on an approximation of the logarithm of the error function. After presenting the derivation of the improved implementation, the article compares the performance of the algorithm to a naive implementation of the log-likelihood using both simulations and a real data example. In the simulation setting, the improved algorithm yielded more precise and less biased parameter estimates when the withinsubject variability was small and estimation was performed using the Laplace algorithm. The estimation results did not differ between implementations when the SAEM algorithm was used. For the real data example, bootstrap results differed between implementations with the improved implementation producing identical or better objective function values. Based on the findings in this article, the improved implementation is suggested as the new default log-likelihood implementation for the bounded integer model. Keywords NONMEM Composite score Numeric stability
Introduction Composite scores are an outcome type of importance in many clinical trials. The observations are discrete numbers between a minimum and a maximum value. Early pharmacometric models often ignored these constraints and assumed a normal distribution for the residual error. In recent years, however, more sophisticated approaches have been proposed that respect the discrete, bounded nature of the data [1, 2]. An example of such a model is the bounded integer (BI) model that we recently proposed [3]. Conceptually, the BI model assumes a latent grid defined by quantiles of the normal distribution. Each subject’s location in that latent grid is described both in terms of its mean and
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10928-020-09727-8) contains supplementary material, which is available to authorized users. & Sebastian Ueckert [email protected] Mats O. Karlsson [email protected] 1
Department of Pharmacy, Uppsala University, Uppsala, Sweden
variance over time. The BI model is conceptually straight forward and flexible. Previously, we showed the advantages in data-description of the BI model over a continuous variable model. In this work, the numeric stability of the implementation of the BI log-likelihood function will be the focus. Numerical analysis is the field in applied mathematics that studies algorithms for solving the problems of continuous mathematics [4]. It is a common misconception to think that a solution for a mathematical model can be obtained by providing a computer with a formula and some numbers and that the machine will simply use those numbers to compute every term in the form
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