Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions

Abstract

Finding a model-based optimal design that can optimally discriminate among a class of plausible models is a difficult task because the design criterion is non-differentiable and requires 2 or more layers of nested optimization. We propose hybrid algorithms based on particle swarm optimization (PSO) to solve such optimization problems, including cases when the optimal design is singular, the mean response of some models are not fully specified and problems that involve 4 layers of nested optimization. Using several classical examples, we show that the proposed PSO-based algorithms are not models or criteria specific, and with a few repeated runs, can produce either an optimal design or a highly efficient design. They are also generally faster than the current algorithms, which are generally slow and work for only specific models or discriminating criteria. As an application, we apply our techniques to find optimal discriminating designs for a dose-response study in toxicology with 5 possible models and compare their performances with traditional and a recently proposed algorithm. In the supplementary material, we provide a R package to generate different types of discriminating designs and evaluate efficiencies of competing designs so that the user can implement an informed design.

Particle swarm optimization for searching efficient experimental designs: A review

“…To facilitate use of such designs in more complicated setups, such as the one just described, the first author has created an interactive website with downloadable PSO and enhanced PSO codes for finding standardized maximin optimal designs and for finding optimal designs for discriminating among several nonlinear models useful in toxicology (R.-B. Chen et al, 2020). The codes for finding standard maximin optimal designs are at https://github.com/PingYangChen/ stdmmOptDesignInhibition and https://pingyangchen.shinyapps.io/stdmmoptdesigninhibition/.…”

“…Extensions to more complicated cases, such as, when errors are not normally distributed or are correlated or the model is hierarchical, are possible and can be found in some of the references cited herein; see for example, R.‐B. Chen et al (2020); Zhang et al (2020), and X. Liu et al (2021).…”

“…R.‐B. Chen et al (2020) wanted to find a standardized maximin T ‐optimal design for discriminating among multiple nonlinear models and showed that the design problem can be reformulated as a four‐layer optimization problem. Let the probability distribution function of the continuous variable y be f ( y ∣ x , θ , σ 2 ) and let its mean response be η ( x , θ ), where x is the independent variable and θ is the model parameter vector.…”

“…R.‐B. Chen et al (2020) adopted a numerical approach and wanted to modify PSO to find a standardized maximin T ‐optimal design problem for a broader setup. To tackle the four‐layer optimization problem, R.‐B.…”

Metaheuristics for pharmacometrics

“…For such problems, metaheuristics has good potential to give satisfactory solutions where traditional methods fail. For example, hybrid algorithms were proposed for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions, 29 and PSO was hybridized with random forest to predict disease progression in patients with idiopathic pulmonary fibrosis. 30 …”