Nature-inspired Metaheuristics for finding Optimal Designs for the Continuation-Ratio Models

Qiu, Jiaheng; Wong, Weng Kee

doi:10.51387/23-NEJSDS44

The New England Journal of Statistics in Data Science

Nature-inspired Metaheuristics for finding Optimal Designs for the Continuation-Ratio Models

Volume 2, Issue 1 (2024), pp. 15–29

Jiaheng Qiu Weng Kee Wong

https://doi.org/10.51387/23-NEJSDS44

Pub. online: 7 August 2023 Type: Biomedical Research

Open Access

Accepted
24 June 2023

Published
7 August 2023

Abstract

The continuation-ratio (CR) model is frequently used in dose response studies to model a three-category outcome as the dose levels vary. Design issues for a CR model defined on an unrestricted dose interval have been discussed for estimating model parameters or a selected function of the model parameters. This paper uses metaheuristics to address design issues for a CR model defined on any compact dose interval when there are one or more objectives in the study and some are more important than others. Specifically, we use an exemplary nature-inspired metaheuristic algorithm called particle swarm optimization (PSO) to find locally optimal designs for estimating a few interesting functions of the model parameters, such as the most effective dose ($MED$), the maximum tolerated dose ($MTD$) and for estimating all parameters in a CR model. We demonstrate that PSO can efficiently find locally multiple-objective optimal designs for a CR model on various dose intervals and a small simulation study shows it tends to outperform the popular deterministic cocktail algorithm (CA) and another competitive metaheuristic algorithm called differential evolutionary (DE). We also discuss hybrid algorithms and their flexible applications to design early Phase 2 trials or tackle biomedical problems, such as different strategies for handling the recent pandemic.

References

[1]

AbdelAziz, A. M., Alarabi, L., Basalamah, S. and Hendawi, A. (2021). A Multi-objective optimization method for hospital admission problem – a case study on COVID-19 patients. Algorithms 14(2) 38. https://doi.org/10.3390/a14020038.

[2]

Agresti, A. (1990) Categorical Data Analysis. Wiley. MR1044993

[3]

Atashpaz-Gargari, C. E. ; Lucas (2007). Imperialist Competitive Algorithm: An algorithm for optimization inspired by imperialistic competition. IEEE Congress on Evolutionary Computation 7 4661–4666.

[4]

Atkinson, A. C. (1992). A segmented algorithm for simulated annealing. Statistics & Computing 2 221–230.

[5]

Atkinson, A. C. and Donev, A. N. (1992) Optimum Experimental Designs. OXFORD University Press.

[6]

Atkinson, A. C. and Haines, L. M. (1996). Designs for nonlinear and generalized linear models. In Handbook of Statistics 13 437–475. https://doi.org/10.1016/S0169-7161(96)13016-9. MR1492576

[7]

Blum, C., Puchinger, J., Raidl, G. R. and Roli, A. (2011). Hybrid metaheuristics in combinatorial optimization: A survey. Applied Soft Computing 11 4135–4151. https://doi.org/10.1007/978-1-4419-1644-0_9. MR2766839

[8]

Bonyadi, M. R. and Michalewicz, Z. (2014). A locally convergent rotationally invariant particle swarm optimization algorithm. Swarm Intelligence 8(3) 159–198.

[9]

Chen, R. -B., Chen, P. -Y., Hsu, C. -L. and Wong, W. K. (2020). Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions. PLOS ONE 15(10). https://doi.org/10.1371/journal.pone.0239864.

[10]

Chernoff, H. (1953). Locally Optimal Designs for Estimating Parameters. Annals of Mathematical Statistics 24(4) 586–602. https://doi.org/10.1214/aoms/1177728915. MR0058932

[11]

Clyde, M. and Chaloner, K. (1996). The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems. Journal of the American Statistician Association 91(43) 1236–1244. https://doi.org/10.2307/2291742. MR1424621

[12]

Collins, M. D., Cui, E., H., S. W. Hyun and Wong, W. K. (2021). A Model-based Approach to Designing Developmental Toxicology Experiments using Sea Urchin Embryos. Archives of Toxicology 96 919–932.

[13]

Cook, R. D. and Wong, W. K. (1994). On the Equivalence of Constrained and Compound Optimal Designs. Journal of the American Statistician Association 89(426) 687–692. MR1294092

[14]

Eberhart, R. and Kennedy, J. (1995). A new optimizer using particle swarm theory. In Micro Machine and Human Science, 1995. MHS’95., Proceedings of the Sixth International Symposium on 39–43. IEEE.

[15]

Falco, I. D., Cioppa, A. D., Scafuri, U. and Tarantino, E. (2020). Coronavirus COVID-19 spreading in Italy: optimizing an epidemiological model with dynamic social distancing through Differential Evolution. arXiv preprint arXiv:2004.00553v3.

[16]

Fan, S. K. and Chaloner, K. (2004). Optimal designs and limiting optimal designs for a trinomial response. Journal of Statistical Planning and Inference 126 347–360. https://doi.org/10.1016/j.jspi.2003.08.004. MR2090863

[17]

Fan, S. q. K. S., Liang, Y. -c. and Zahara, E. (2004). Hybrid simplex search and particle swarm optimization for the global optimization of multimodal functions. Engineering Optimization 36(4) 401–418.

[18]

Fedorov, V. V. and Wu, Y. (2007). Dose finding designs for continuous responses and binary utility. Journal of Biopharmaceutical Statistics 17 1085–1096. https://doi.org/10.1080/10543400701645132. MR2414563

[19]

Fedorov, V. (1972). Theory of optimal design. Academic, New York. MR0403103

[20]

Fumoleau, P., Trigo, J. M., Isambert, N., Simiond, D., Gupta, S. and Campone, M. (2013). Phase I dose-finding study of cabazitaxel administered weekly in patients with advanced solid tumours. BMC Cancer 13 1471–2407.

[21]

Gomes, D. C. D. S. and Serra, G. L. d. O. (2021). Machine learning model for computational tracking forecasting the COVID-19 dynamic propagation. IEEE Journal of Biomedical and Health Informatics 25(3) 515–622.

[22]

Haines, L. M. (1987). The application of the annealing algorithm to the construction of exact optimal designs for linear regression models. Technometrics 29(4) 439–447. MR2637962

[23]

He, S., Peng, Y. and Sun, K. (2020). SEIR modeling of the COVID-19 and its dynamics. Nonlinear Dynamics 101 10–100071107102005743.

[24]

Heise, M. A. and Myers, R. H. (1996). Optimal designs for bivariate logistic regression. Biometrics 52 613–624.

[25]

Hosseini, E., Ghafoor, K. Z., Sadiq, A. S., Guizani, M. and Emrouznejad, A. (2020). COVID-19 Optimizer algorithm, modeling and controlling of coronavirus distribution process. IEEE Journal of Biomedical and Health Informatics 24(10) 2765–2775.

[26]

Kiefer, J. (1974). General Equivalence Theory for optimum designs (Approximate Theory). The Annals of Statistics 2 849–879. MR0356386

[27]

Kim, S. and Wong, W. K. (2018). Extended two-stage adaptive designs with three target responses for phase II clinical trials. Statistical methods in Medical Research 27(12) 3628–3642. https://doi.org/10.1177/0962280217709817. MR3878648

[28]

Lai, T. L., Choi, K. P., Tong, T. X. and Wong, W. K. (2021). A Statistical Approach to Adaptive Parameter Tuning in Nature-inspired Optimization and Optimal Sequential Design of Dose-Finding Trials. Statistica Sinica. In press. https://doi.org/10.5705/ss.20. MR4338089

[29]

Li, G. and Majumdar, D. (2009). Some results on D-optimal designs for nonlinear models with applications. Biometrika 96(2) 487–493. https://doi.org/10.1093/biomet/asp004. MR2507158

[30]

Lin, Y. and Shih, W. J. (2004). Adaptive two-stage designs for single-arm phase IIA cancer clinical trials. Biometrics 60 482–490. https://doi.org/10.1111/j.0006-341X.2004.00193.x. MR2067004

[31]

Makade, R. G., Chakrabarti, S. and Jamil, B. (2020). Real time estimation and prediction of the mortality caused due to COVID-19 using particle swarm optimization and finding the most influential parameter. Infectious Disease Modelling 5 772–782.

[32]

Mohammadi, M., Nastaran, M. and Sahebgharani, A. (2016). Development, application, and comparison of hybrid meta-heuristics for urban land-use allocation optimization: Tabu search, genetic, GRASP, and simulated annealing algorithms. Computers, Environment and Urban Systems 60 23–36.

[33]

Montepiedra, G., Myers, D. and B., Y. A. (1998). Application of genetic algorithms to the construction of exact D-optimal designs. Journal of Applied Statistics 25 817–826.

[34]

Pinter, G., Felde, I., Mosavi, A., Ghamisi, P. and Gloaguen, R. (2020). COVID-19 Pandemic prediction for Hungary; a hybrid machine learning approach. MDPI Mathematics 8 890.

[35]

Simon, R. (1989). Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10 1–10.

[36]

Singh, D., Kumar, V., Vaishali and Kaur, M. (2020). Classification of COVID-19 patients from chest CT images using multi-objective Differential Evolution-based convolutional neural networks. European Journal of Clinical Microbiology and Infectious Diseases 39 1379–21389.

[37]

Slob, W. (2002). Dose-response modeling of continuous endpoints. Toxicological Sciences 66 298–312.

[38]

Storn, R. and Price, K. (1997). Differential Evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11 341–359. https://doi.org/10.1023/A:1008202821328. MR1479553

[39]

Sun, J., Xu, W. and Feng, B. (2004). A global search strategy of quantum-behaved particle swarm optimization 1 111–116.

[40]

Thall, P. F. and Cook, D. J. (2004). Dose-finding based on efficacy-toxicity trade-offs. Biometrics 60(3) 684–693. https://doi.org/10.1111/j.0006-341X.2004.00218.x. MR2089444

[41]

Thall, P. F. and Russell, K. E. (1998). A Strategy for Dose-Finding and Safety Monitoring Based on Efficacy and Adverse Outcomes in Phase I/II Clinical Trials. Biometrics 54 251–264.

[42]

Thall, P. F., Cook, D. J. and Estey, E. H. (2006). Adaptive dose selection using efficacy-toxicity trade-offs: Illustrations and practical considerations. Journal of Biopharmaceutical Statistics 16(5) 623–638. https://doi.org/10.1080/10543400600860394. MR2252311

[43]

Tian, Y., Yang, S. and Zhang, X. (2020). An Evolutionary Multiobjective Optimization Based Fuzzy Method for Overlapping Community Detection. IEEE Transactions on Fuzzy Systems 28(11) 2841–2855.

[44]

Tian, Y., Zhang, X., Wang, C. and Jin, Y. (2018). An Evolutionary Algorithm for Large-Scale Sparse Multiobjective Optimization Problems. IEEE Transactions on Evolutionary Computation 24(2) 380–393.

[45]

Tian, Y., Su, X., Su, Y. and Zhang, X. (2020). EMODMI: A Multi-Objective Optimization Based Method to Identify Disease Modules. IEEE Transactions on Emerging Topics in Computational Intelligence.

[46]

Tian, Y., Liu, R., Zhang, X., Ma, H., Tan, K. C. and Jin, Y. (2020). A Multi-Population Evolutionary Algorithm for Solving Large-Scale Multi-Modal Multi-Objective Optimization Problems. IEEE Transactions on Evolutionary Computation. MR4361281

[47]

Tong, T. X., Choi, K. P., Lai, T. L. and Wong, W. K. (2021). Stability Bounds and Almost Sure Convergence of Improved Particle Swarm Optimization Methods. Research in Mathematical Sciences. https://doi.org/10.1007/s40687-020-00241-4. MR4257863

[48]

van den Bergh, F. and Engelbrecht, A. P. (2002). A new locally convergent particle swarm optimiser. In Systems, Man and Cybernetics, 2002 IEEE International Conference on 3 6. IEEE.

[49]

Whitacre, J. M. (2011). Recent trends indicate rapid growth of nature-inspired optimization in academia and industry. Computing 93 121–133. https://doi.org/10.1007/s00607-011-0154-z. MR2860177

[50]

Whitacre, J. M. (2011). Survival of the flexible: explaining the recent popularity of nature-inspired optimization within a rapidly evolving world. Computing 93 135–146. https://doi.org/10.1007/s00607-011-0156-x. MR2860178

[51]

Wu, M., Hu, W., Hu, J. and Yen, G. G. (2021). Adaptive Multiobjective Particle Swarm Optimization Based on Evolutionary State Estimation. IEEE Transactions on Cybernetics 51(7) 3738–3751.

[52]

Xu, W. W., K., W., Tan, K. C. and Xu, J. X. (2019). Finding High-Dimensional D-Optimal Designs for Logistic Models via Differential Evolution. IEEE Access 7 7133–7146.

[53]

Yang, X. S. (2010). Engineering Optimization: An Introduction with Metaheuristic Applications. In Particle Swarm Optimization John Wiley and Sons.

[54]

Yeap, B. Y., Catalano, P. J., Ryan, L. M. and Davidian, M. (2003). Robust two-stage approach to repeated measurements analysis of chronic ozone exposure in rats. Journal of Agricultural, Biological, and Environmental Statistics 8(4) 438–454.

[55]

Yu, Y. (2011). D-optimal designs via a cocktail algorithm. Stat Compute 21 475–481. https://doi.org/10.1007/s11222-010-9183-2. MR2826686

[56]

Zhang, W., Sargent, D. J. and Mandrekar, S. (2006). An adaptive dose-finding design incorporating both toxicity and efficacy. Statistics in Medicine 25 2365–2383. https://doi.org/10.1002/sim.2325. MR2240943

[57]

Zhu, W. and Wong, W. K. (2000). Multi-objective Designs in A Dose-response Experiment. Journal of Biopharmaceutical Statistics, 10:1 10 1–14.

[58]

Zocchi, S. S. and Atkinson, A. C. (1999). Optimum Experimental Designs for Multinomial Logistic Models. Biometrics 55 437–444. https://doi.org/10.1111/j.0006-341X.1999.00437.x. MR1705110

Full article

Open access article under the CC BY license.

Keywords

Dose response Models Implicit function theorem Maximin tolerated dose Most effective dose Multiple-objective optimal design

Funding

The research of Wong reported in this publication was partially supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award Number R01GM107639.

Metrics

since December 2021

237 Article info views	122 Full article views
114 PDF downloads	36 XML downloads

RSS

Authors

Abstract

References

Export citation

Copy and paste formatted citation

Download citation in file