Improved numerical stability for the bounded integer model

Sebastian Ueckert,Mats O Karlsson

doi:10.1007/s10928-020-09727-8

Sebastian Ueckert, Mats O Karlsson

Open Access

https://doi.org/10.1007/s10928-020-09727-8

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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 within-subject 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.

Highlights

Composite scores are an outcome type of importance in many clinical trials
In scenario S2, the improved implementation showed significantly lower bias and imprecision when used with the Laplace
Our results show that the improved implementation of the bounded integer (BI) model has a considerably higher numerical stability and that this improvement can translate to meaningful differences in estimation performance

Summary

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. Pharmacometric models often ignored these constraints and assumed a normal distribution for the residual error. 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]. The BI model assumes a latent grid defined by quantiles of the normal distribution.

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