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

PLOS ONE RESEARCH ARTICLE Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions Ray-Bing Chen1,2, Ping-Yang Chen1, Cheng-Lin Hsu1, Weng Kee Wong ID3* a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 1 Department of Statistics, National Cheng Kung University, Tainan, Taiwan, 2 Institute of Data Science, National Cheng Kung University, Tainan, Taiwan, 3 Department of Biostatistics, University of California, Los Angeles, California, United States of America * Abstract OPEN ACCESS Citation: Chen R-B, Chen P-Y, Hsu C-L, Wong WK (2020) Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions. PLoS ONE 15(10): e0239864. https://doi.org/10.1371/journal.pone.0239864 Editor: Ping He, Jinan University, China, HONG KONG Received: April 15, 2020 Accepted: September 14, 2020 Published: October 5, 2020 Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https://doi.org/10.1371/journal.pone.0239864 Copyright: © 2020 Chen et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: They are no real data involved because this work describes how to collect data to identify the right statistical models in the most efficient manner under various criteria 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. Introduction Much of the work in optimal design of experiments assumes a known parametric model, apart from the unknown model parameters and the objective is to develop a plan to collect data judiciously for accurate statistical inference. For example, one may wish to design a study to estimate parameters in a nonlinear regression model. In practice, the model is rarely known with certainty and it is likely that there are a few plausible models. Optimal design problems concern identifying the best design, i.e. how to collect data to judiciously select the right model among the plausible models. When there are 2 models and errors are normally distributed and one of the 2 models is fully known, [1] introduced T-optimality as a design discrimination PLOS ONE | https://doi.org/10.1371/journal.pone.0239864 October 5, 2020 1 / 30 PLOS ONE and assumptions. However codes are available to reproduce the results in the paper. Funding: Authors P-YC, R-BC, and WKW received partial support for this study in the form of a grant from the National Institute of General Medical Sciences of the National Institutes of Health under Award Number R01GM107639. R-BC is also partially supported by the Mathematics Division of the National Center for Theoretical Sciences in Taiwan. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. Hybrid algorithms for discriminating multiple nonlinear models using various criteria criterion based on the squared difference between the 2 mean predictions. [2] reviewed optimal discriminating design problems and since then, locally T-optimal designs have been applied and studied in various setups, see for example, [3–8] and [9]. When the outcomes are binary [10] or model errors are not normally distributed, [11] proposed KL-optimality criterion based on the Kullback-Leibler (KL) divergence as the distance measure between the 2 competing models. Analytical descriptions of optimal discriminating designs rarely exist unless there are simple settings, such as when we want to find an optimal design to discriminate between a constant model and a quadratic model, and both models have homoscedastic errors [1]. When there are multiple models to discriminate, [12] proposed a Fedorov-Wynn type algorithm to find a T-optimal design and the convergence of such an algorithm to the optimal discriminating design was established recently under some restrictive conditions [13]. Over time, there were several modifications of the algorithm to find various optimal designs, including [11], who amended it to find KL-optimal designs. Algorithms are a practical way to find optimal discriminating designs. Recently, natureinspired metaheuristic algorithms have been repeatedly shown to be fast, flexible and efficient for solving hard and high dimensional optimization problems in engineering and computer science. 2 such algorithms are differential evolutionary (DE) algorithm proposed by [14] and particle swarm optimization (PSO) proposed by [15]. [16] was the first to show that PSO outperformed traditional algorithms in statistics for finding a variety of optimal designs. Maximin design problems are much harder problems to solve because the design criterion is non-differentiable and require multiple nested optimization. [17] developed hybridized PSO-based algorithms to solve more complicated optimal design problems such as the standardized maximin optimal criteria, which includes the simpler minimax design problems. Most recently, [18] applied DE to find optimal approximate designs for logistic models with up to 5 factors with all pairwise interaction terms. The number of variables to optimize for such a model is at least 95 if the optimal design is minimally supported; otherwise, there will be many more vari (...truncated)