D-Optimality for Gamma Regression Model
Abstract
D-Optimality is a concept in experimental design. Certain treatments combinations in Experimental design may be too costly or difficult to measure in some circumstances. D-optimal designs are model-specific designs that address Conventional design constraints. In any field, the tests are frequently related to binary or count data, such as failure/success or the number of defects. Generalized Linear Models (GLM) and optimality are related in the context of experimental design and statistical analysis. To develop a D-optimal design criterion for GLM, we utilize the asymptotic variance-covariance matrix, which is a weighted form of the covariance matrix for the linear situation. We have used a Gamma GLM with any number of predictor variables and a log link modelled linear predictor to find the best optimal design. To prove local D-optimal design for a class of bisection designs, we employ a standard version of the problem and a generic equivalence theorem. Combining the generic equivalence theorem with clustering strategies yields an immediate method for identifying optimal designs points that are robust to a broad range of model parameter increment values.
References
1. A.J. Dobson and A.G. Barnett; An Introduction to Generalized Linear Models. 4th Edition, Chapman and Hall, London (2018).
2. A. Rahman; D-Optimal Experimental Design (2019). URL: https://cran.r-project.org/web/packages /sdpt3r/vignettes/doptimal.pdf.
3. B. Duraković; Design of Experiments Application, Concepts, Examples: State of the Art. Periodicals of Engineering and Natural Sciences, 5(3), 421-439 (2017).
4. E.M. Omshi and S. Shemehsavar.; Optimal Design for Accelerated Degradation Test Based on D- Optimality. Iranian Journal of Science and Technology, Transactions A: Science, 43, 1811-1818 (2018).
5. F. Kurtoğlu and M.R. Özkale; Liu Estimation in Generalized Linear Models: Application on Gamma
Distributed Response Variable. Statistical Papers, 57(4), 911–928 (2016).
6. G. Gea-Izquierdo and I. Cañellas; Analysis of Holm Oak Intraspecific competition using Gamma
Regression. Forest Science 55(4), 310–322 (2009).
7. G. Grover; A. Sabharwal and J. Mittal; An Application of Gamma Generalized Linear Model for
Estimation of Survival Function of Diabetic Nephropathy Patients. International Journal of Statistics in
Medical Research, 2(3), 209–219 (2013).
8. I. Osama; Locally Optimal Designs for Generalized Linear Models with Applications to Gamma Models.
Dissertation, Otto Von Guericke University Magdeburg (2019).
9. J.BarberandS.Thompson;MultipleRegressionofCostData:UseofGeneralizedLinearModels.Journal
of Health Services Research & Policy, 9(4), 197-204 (2004).
10. J. Fellman; Gustav Elfving‟s Contribution to the Emergence of the Optimal Experimental Design Theory.
Statistical Science, 14(2), 197-200 (1999).
11. J.L. Moran; P.J. Solomon; A.R. Peisach and J. Martin; New Models for Old Questions: Generalized Linear
Models for Cost Prediction. Journal of Evaluation in Clinical Practice, 13(3), 381–389 (2007).
12. L. Pronzato and A. Pázman; Design of Experiments in Nonlinear Models. Springer-Verlag, New York,
(2013).
13. M. Pal; N.K. Mandal and H.K. Maity; D-Optimal Design for Estimation of Optimum Mixture in a Three- Component Mixture Experiment with Two Responses. Communications in Statistics - Simulation and Computation, 52(5), 2012-2021 (2023).
14. M. Singh and W. Xie; Approximation Algorithms for D-optimal Design. arXiv:1802.08372 (2019).
15. N.G. Emenogu and M.O.Adenomon; Design and Analysis of Experiments with Examples in R. Jube-Evans
Books and Publishers (2018).
16. P. Goos and B. Jones; Optimal Design of Experiments: A Case Study Approach. John Wiley & Sons, Inc.,
New York (2011).
17.P. McCullagh and J.A. Nelder; Generalized Linear Models. 2nd Edition, Chapman and Hall, London
(1989).
18. R.H. Myers and D.C. Montgomery; Response Surface Methodology: Process and Product Optimization
using Designed Experiments. John Wiley and Sons, New York (1995).
19. S.W. Wanyonyi; A. Okango and J. Koech; Exploration of D-, A-, I-and G-Optimality Criteria in Mixture
Modelling. Asian Journal of Mathematics & Statistics 12(4), 15-28 (2021).
20. T. Holland-Letz and A. Kopp-Schneider; An R-Shiny Application to Calculate Optimal Designs for Single
Substance and Interaction Trials in Dose Response Experiments. Toxicology Letters, 337, 18–27 (2021).
21. V. Ng and R.A. Cribbie; Using the Gamma Generalized Linear Model for Modelling Continuous, Skewed
and Heteroscedastic Outcomes in Psychology. Current Psychology, 36(2), 225-235 (2017).
22. V.V. Fedorov and S.L. Leonov; Optimal Design for Nonlinear Response Models. CRC Press, Boca Raton
(2013).
23. W.G. Manning and J. Mullahy; Estimating Log Models: To Transform or Not To Transform?. Journal of
Health Economics, 20(4), 461-494 (2001).
24. X. Chen; A.Y. Aravkin and R.D. Martin; Generalized Linear Model for Gamma Distributed Variables via
Elastic Net Regularization. arXiv:1804.07780 (2018).
Trion Studio
