R-Optimal Designs for Linear Regression Model in Two Explanatory Variables
Abstract
Locally R-optimal design is derived for the linear regression model with two explanatory variables using the Identity link function. The R-optimality criterion has been proposed in the literature as an alternative to the most frequently used D-optimality criterion when the experimenter wishes to minimize the volume of the confidence region for unknown parameters based on Bonferroni t-intervals. The necessary and sufficient conditions of this optimality criterion are confirmed through the equivalence theorem. Also, the robustness issue of the proposed optimal design is examined through a simulation study. All of the numerical calculations are handled in Mathemaica software.
References
1. A.C. Atkinson and A.N. Doney; Optimum Experimental Design. In: KhuriA.I.(ed)(2006) Response Surface Methodology and Related Topics, World Scientific, New Jersey (1992).
2. A.C. Atkinson and L.M. Haines; Designs for Nonlinear and Generalized Linear Models. In: Ghosh.S, Rao C.R.(ed) Handbook of Statistics, Vol. 13, Elsevier, Amsterdam (1996).
3. H. Chernoff, Locally Optimal Designs for Estimating Parameters. The Annals of Mathematical Statistics,
24(4), 586-602 (1953).
4. H. Dette and L.M. Haines; E-Optimal Designs for Linear and Nonlinear Models with Two Parameters. Biometrika, 81(4), 739-754 (1994).
5. H. Dette; Designing Experiments with Respect to Some 'Standardized' Optimality Criteria. Journal of Royal Statistical Society, 59(B), 97-110 (1997).
6. H. Hao; X. Zhu; X. Zhang and C. Zhang; R-Optimal Design of the Second-Order Scheffé Mixture Model. Statistics & Probability Letters, 173, p. 109069 (2021).
7. I. Ford; B. Torsney and C.J. Wu, The Use of a Canonical Form in the Construction of Locally Optimal Designs for Non-Linear Problems. Journal of the Royal Statistical Society: Series B
(Methodological), 54(2), 569-583 (1992).
8. J. Kiefer; Optimum Experimental Designs. Journal of the Royal Statistical Society: Series B (Methodological), 21 (2), 272-304 (1959).
9. J. Kiefer and J. Wolfowitz; Optimal Designs in Regression Problems. Annals of Mathematical Statistics,
30, 271-294 (1959).
10. K.M. Abde basit and R.L. Plackett; Experimental Design for Binary Data. Journal of the American Statistical Association, 78(381), 90-98 (1983)
11. L.M. Haines; G. Kabera; P. Ndlovu and T.E. O' Brien (2007); D-Optimal Designs for Logistic Regression in Two Variables. In mODa 8-Advances in Model-Oriented Design and Analysis: Proceedings of the 8th International Workshop in Model-Oriented Design and Analysis, Almagro, Spain, June 4-8, 91-98.
12. L. He and R.X. Yue; R-Optimal Designs for Multi-Factor Models with Heteroscedastic Errors. Metrika, 80
717-732 (2017).
13. L. He and R.X. Yue; R-Optimality Criterion for Regression Models with Asymmetric Errors. Journal of Statistical Planning and Inference, 199, 318-326 (2019).
14. P. Liu; L. Gao and J. Zhou; R-Optimal Designs for Multi-Response Regression Models with Multi-Factors. Communications in Statistics-Theory and Methods, 51(2), 340-355 (2022).
15. P. McCullough and J.A. Nelder; Generalized Linear Models. Chapman and Hall, New York (1989).
16. R.R. Sitter and C.F.J. Wu; Optimal Designs for Binary Response Experiments: Fieller, D, and A Critena. Scandinavian Journal of Statistics, 20(4), 329-341 (1993).
17. S.D. Silvey; Optimal Design. Chapman and Hall, London (1980).
18. S. Lall; S. Jaggi; E. Varghese; C. Varghese and A. Bhowmik; D-Optimal Designs for Exponential and Poisson Regression Models. Journal of the Indian Society of Agricultural Statistics, 72, 27-32 (2018).
19.T. Mathew and B.K. Sinha; Optimal Designs for Binary Data under Logistic Regression. Journal of Statistical Planning and Inference, 93(1-2), 295-307 (2001).
20. T.K. Biswal; R-Optimal Designs for Poisson Regression Model in Two Parameters. Pakistan Journal of Statistics, 40(3), 285-300 (2024).
21. Y. Wang; R.H. Myers; E.P. Smith and K. Ye; D-Optimal Designs for Poisson Regression Models. Journal of Statistical Planning and Inference, 136(8), 2831-2845 (2006).
22. Y. Wang; E.P. Smith and K. Ye; Sequential Designs for a Poisson Regression Model. Journal of Statistical Planning and Inference, 136(9), 3187-3202 (2006).
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