Locally Optimal Designs for Estimating One or More Functions of Shared Parameters Between Two Groups in Biomedical Studies

Liu, Xin; Yue, Rong-Xian; Wong, Weng Kee

doi:10.5705/ss.202023.0284

Abstract

Models with shared parameters arise quite naturally in the biological

sciences and we use optimal design theory to construct c-optimal approximate

designs for estimating one or more functions of the model parameters in two

regression models with shared parameters. We assume sample sizes for the two

groups are fixed and establish equivalence theorems to confirm the optimality of

the design. As applications, we consider the parallel dose response model, the

EMAX model and the Exponential model, each with shared parameters. The

methodology is general and can be applied to other models or design problems.

For example, we show the theoretical framework can be directly extended to

the case when we are interested to find a c-optimal design to estimate the mean

difference between the expected responses at an extrapolated dose for a nonlinear

model, or when the total sample size for the whole study is fixed, and we wish to

determine the optimal proportions of observations to allocate to the two groups,

or we have multivariate responses.

Information

Preprint No.	SS-2023-0284
Manuscript ID	SS-2023-0284
Complete Authors	Xin Liu, Rong-Xian Yue, Weng Kee Wong
Corresponding Authors	Weng Kee Wong
Emails	wkwong@ucla.edu

References

Al-Sharadqah, A. and Nguyen, P. (2022). Statistical analysis of concentric objects estimation problem under the heteroscedastic Berman model.Journal of Statistical Planning and Inference 218, 9–24.
Berman, M. (1983). Estimating the parameters of a circle when angular differences are known. J. Roy. Statist. Soc. Ser. C 32, 1–6.
Butler, G.J. and Wolkowicz, G.S.K. (1985). A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM Journal on Applied Mathematics 45, 138–151.
Chakraborty, Bibhas, Ghosh, Palash, Moode, Erica E. M. and Rush, J. A. (2016). Estimating optimal shared-parameter dynamic regimens with application to a multistage depression clinical trial. Biometrics 73, 865–876.
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. PlosOne 15(10):e0239864.
Chen, P. Y., Chen, R. B. and Wong, W. K. (2022). Particle swarm optimization for searching efficient experimental designs – a review. Wires Journal: Computational Statistics 14, e1578. doi:abs/10.1002/wics.1578.
Chen, P. Y., Chen, R. B. and Wong, W. K. (2023). Particle swarm optimization for finding efficient longitudinal exact designs for nonlinear models. The New England Journal of Statistics in Data Science 1, 299–313.
Cook, R. D. and Wong, W. K. (1994). On the equivalence of constrained and compound optimal designs. Journal of the American Statistician Association 89, 687–692.
Dette, H., Melas, V. B. and Wong, W. K. (2005). Optimal designs for goodness of fit of the Michaelis-Menten enzyme kinetic function. Journal of American Statistical Association 100, 1370–1381.
Dette, H., Kiss, C., Bevanda, M. and Bretz, F. (2010). Optimal designs for the EMAX, log-linear and exponential models. Biometrika 97, 513–518.
Dette, H., Pepelyshev, A. and Wong, W. K. (2011). Experimental design strategies for detecting hormesis. Risk Analysis 31, 1949–1960.
Duarte, B. P. M., Jos´e F. O. Granjo abd Wong, W. K. (2020). Optimal exact designs of experiments via Mixed Integer Nonlinear Programming. statistics & Computing 30, 93–112.
Faya P., Rauk A. P., Griffiths K. L. and Parekh B. (2020). A curve similarity approach to parallelism testing in bioassay. J.Biopharm. Stat. 30, 721–733.
Feller, C., Schorning, K., Dette, H., Bermann, G. and Bornkamp, B. (2017). Optimal designs for dose response curves with common parameters. Annals of Statistics 45, 2102–2132.
Fleetwood K., Bursa F. and Yellowlees A. (2015). Parallelism in practice: approaches to parallelism in bioassays. PDA. J. Pharm. Sci. Technol. 69, 248–263.
Huang, M. N. L., Chen, R. B., Lin, C. S. and Wong, W. K. (2006). Optimal designs for parallel models with correlated responses. Statist. Sinica 16, 121–133.
Liu, X., Yue, R. X., Zhang, Z. and Wong, W. K. (2021). G-optimal designs for hierarchical linear models: an equivalence theorem and a nature-inspired meta-heuristic algorithm. Soft Computing 5, 13549–13565.
Lopez-Fidalgo, J. and Wong, W. K. (2002). Optimal designs for the Michaelis-Menten model. Journal of Theoretical Biology 215, 1–11.
Love, S. B., Brown, S., Weir, C. J., Harbron, C., Yap, C., Gascher-Markefski, B., Matcham,
J., Caffrey, L., McKevitt, C., Clive, S., Craddock, C., Spicer, J. and Cornelius, V. (2017). Embracing model-based designs for dosefinding trials. British Journal of Cancer 117, 332– 339.
Masoudi, E., Holling, H., Duarte, B. P. M. and Wong, W. K. (2019). Metaheuristic adaptive cubature based algorithm to find Bayesian optimal designs for nonlinear models. Journal of Computational and Graphical Statistics 28, 861–876.
Novick, S. and Harry Yang, H. (2019). A fast and reliable test for parallelism in bioassay. J. Biopharm. Stat. 29, 1011–1023.
Pinheiro, J. Bretz, F. and Branson, M. (2006). Analysis of dose response studies: Modeling approaches. Pp. 146-171 in Ting, N. (ed.) Dose finding in drug development. Springer, New York.
Pukelsheim, F. and Rieder, S. (1992). Efficient rounding of approximate designs. Biometrika 79, 763–770.
Vaquez, A., Wong, W. K. and Goos, P. (2022). Constructing two-level-optimal screening designs using mixed-integer programming and heuristic algorithms. Statistics & Computing 33, DOI https://doi.org/10.1007/s11222-022-10168-.
Yu, X. Z. and Gu, J. D. (2007). Differences in Michaelis-Menten kinetics for different cultivars of maize during cyanide removal. Ecotoxicology and Environmental Safety 67, 254–259.

Acknowledgments

Dr. Liu and Dr. Yue were partially supported by the National Natural Science Foundation of China (No.11971318) and Natural Science Foundation of

Shanghai (No.24ZR401500). Wong was partially supported by the Yushan

Fellowship Award from the Ministry of Education (MOE), Taiwan and he

is grateful for the additional support and hospitality from The National

Cheng Kung University in Tainan, Taiwan.

[1] Al-Sharadqah, A. and Nguyen, P. (2022). Statistical analysis of concentric objects estimation problem under the heteroscedastic Berman model.Journal of Statistical Planning and Inference 218, 9–24.

[2] Berman, M. (1983). Estimating the parameters of a circle when angular differences are known. J. Roy. Statist. Soc. Ser. C 32, 1–6.

[3] Butler, G.J. and Wolkowicz, G.S.K. (1985). A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM Journal on Applied Mathematics 45, 138–151.

[4] Chakraborty, Bibhas, Ghosh, Palash, Moode, Erica E. M. and Rush, J. A. (2016). Estimating optimal shared-parameter dynamic regimens with application to a multistage depression clinical trial. Biometrics 73, 865–876.

[5] 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. PlosOne 15(10):e0239864.

[6] Chen, P. Y., Chen, R. B. and Wong, W. K. (2022). Particle swarm optimization for searching efficient experimental designs – a review. Wires Journal: Computational Statistics 14, e1578. doi:abs/10.1002/wics.1578.

[7] Chen, P. Y., Chen, R. B. and Wong, W. K. (2023). Particle swarm optimization for finding efficient longitudinal exact designs for nonlinear models. The New England Journal of Statistics in Data Science 1, 299–313.

[8] Cook, R. D. and Wong, W. K. (1994). On the equivalence of constrained and compound optimal designs. Journal of the American Statistician Association 89, 687–692.

[9] Dette, H., Melas, V. B. and Wong, W. K. (2005). Optimal designs for goodness of fit of the Michaelis-Menten enzyme kinetic function. Journal of American Statistical Association 100, 1370–1381.

[10] Dette, H., Kiss, C., Bevanda, M. and Bretz, F. (2010). Optimal designs for the EMAX, log-linear and exponential models. Biometrika 97, 513–518.

[11] Dette, H., Pepelyshev, A. and Wong, W. K. (2011). Experimental design strategies for detecting hormesis. Risk Analysis 31, 1949–1960.

[12] Duarte, B. P. M., Jos´e F. O. Granjo abd Wong, W. K. (2020). Optimal exact designs of experiments via Mixed Integer Nonlinear Programming. statistics & Computing 30, 93–112.

[13] Faya P., Rauk A. P., Griffiths K. L. and Parekh B. (2020). A curve similarity approach to parallelism testing in bioassay. J.Biopharm. Stat. 30, 721–733.

[14] Feller, C., Schorning, K., Dette, H., Bermann, G. and Bornkamp, B. (2017). Optimal designs for dose response curves with common parameters. Annals of Statistics 45, 2102–2132.

[15] Fleetwood K., Bursa F. and Yellowlees A. (2015). Parallelism in practice: approaches to parallelism in bioassays. PDA. J. Pharm. Sci. Technol. 69, 248–263.

[16] Huang, M. N. L., Chen, R. B., Lin, C. S. and Wong, W. K. (2006). Optimal designs for parallel models with correlated responses. Statist. Sinica 16, 121–133.

[17] Liu, X., Yue, R. X., Zhang, Z. and Wong, W. K. (2021). G-optimal designs for hierarchical linear models: an equivalence theorem and a nature-inspired meta-heuristic algorithm. Soft Computing 5, 13549–13565.

[18] Lopez-Fidalgo, J. and Wong, W. K. (2002). Optimal designs for the Michaelis-Menten model. Journal of Theoretical Biology 215, 1–11.

[19] Love, S. B., Brown, S., Weir, C. J., Harbron, C., Yap, C., Gascher-Markefski, B., Matcham,

[20] J., Caffrey, L., McKevitt, C., Clive, S., Craddock, C., Spicer, J. and Cornelius, V. (2017). Embracing model-based designs for dosefinding trials. British Journal of Cancer 117, 332– 339.

[21] Masoudi, E., Holling, H., Duarte, B. P. M. and Wong, W. K. (2019). Metaheuristic adaptive cubature based algorithm to find Bayesian optimal designs for nonlinear models. Journal of Computational and Graphical Statistics 28, 861–876.

[22] Novick, S. and Harry Yang, H. (2019). A fast and reliable test for parallelism in bioassay. J. Biopharm. Stat. 29, 1011–1023.

[23] Pinheiro, J. Bretz, F. and Branson, M. (2006). Analysis of dose response studies: Modeling approaches. Pp. 146-171 in Ting, N. (ed.) Dose finding in drug development. Springer, New York.

[24] Pukelsheim, F. and Rieder, S. (1992). Efficient rounding of approximate designs. Biometrika 79, 763–770.

[25] Vaquez, A., Wong, W. K. and Goos, P. (2022). Constructing two-level-optimal screening designs using mixed-integer programming and heuristic algorithms. Statistics & Computing 33, DOI https://doi.org/10.1007/s11222-022-10168-.

[26] Yu, X. Z. and Gu, J. D. (2007). Differences in Michaelis-Menten kinetics for different cultivars of maize during cyanide removal. Ecotoxicology and Environmental Safety 67, 254–259.