The performances of Custom A-, D-, and I-optimal designs on non-standard second-order models are examined using the alphabetic A-, D-, and G-optimality efficiencies, as well as the Average Variance of Prediction. Designs of varying sizes are constructed with the help of JMP Pro 14 software and are customized for specified non-standard models, optimality criteria, prespecified experimental runs, and a specified range of input variables. The results reveal that Custom-A optimal designs perform generally better in terms of G-efficiency. They show high superiority to A-efficiency as the worst G-efficiency value of the created Custom-A optimal designs exceeds the best A-efficiency value of the designs, and also does well in terms of D-efficiency. Custom-D optimal designs perform generally best in terms of G-efficiency, as the worst G-efficiency value exceeds all A- and D-efficiency values. Custom-I optimal designs perform generally best in terms of G-efficiency as the worst G-efficiency value is better than the best A-efficiency value and performs generally better than the corresponding D-efficiency values. For the Average Variance of Prediction, Custom A- and I-optimal designs perform competitively well, with relatively low Average Variances of Prediction. On the contrary, the Average Variance of Prediction is generally larger for Custom-D optimal designs. Hence when seeking designs that minimize the variance of the predicted response, it suffices to construct Custom A-, D-, or I-optimal designs, with a preference for Custom-D optimal designs.
| Published in | American Journal of Theoretical and Applied Statistics (Volume 13, Issue 5) |
| DOI | 10.11648/j.ajtas.20241305.11 |
| Page(s) | 92-114 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Custom A-, D-, and I-optimal Designs, Non-Standard Second-Order Model, Average Variance of Prediction, D-, G-, A-efficiency
| [1] | Akinlana, D. M. (2022). New Developments in Statistical Optimal Designs for Physical and Computer Experiments. University of South Florida. |
| [2] | Gaifman, H. (2004). Non-standard models in a broader perspective. In Nonstandard Models of Arithmetic and Set Theory (pp. 1–22). American Mathematical Society. |
| [3] | Goos, P., & Jones, B. (2011). Optimal Designs of Experiments: A Case Study Approach, 1st Edition, Wiley, New York. ISBN: 978-0-470-74461-1 |
| [4] | Goos, P., Jones, B., & Syafitri, U. (2016). I-optimal design of mixture experiments. Journal of the American Statistical Association, 111(514), 899–911. |
| [5] | Iwundu M. P. & Otaru O. A. P. (2019). Construction of Hat-Matrix Composite Designs for Second-Order Models. American Journal of Computational and Applied Mathematics. |
| [6] | Iwundu, M. (2017). Missing observations: The loss in relative A-, D- and G-efficiency. International Journal of Advanced Mathematical Sciences, 5(2), 43. |
| [7] | Iwundu, M. P. (2018). Construction of Modified Central Composite Designs for Non-standard Models. Canadian Center of Science and Education. |
| [8] | Iwundu, Mary Paschal, & Cosmos, J. (2022). The efficiency of seven-variable box-Behnken experimental design with varying center runs on full and reduced model types. Journal of Mathematics and Statistics, 18(1), 196–207. |
| [9] | Johnson, R. T., Montgomery, D. C., & Jones, B. A. (2011). An expository paper on optimal design. Quality Engineering, 23(3), 287–301. |
| [10] | Jones, B., & Goos, P. (2012). I-optimal versus D-optimal split-plot response surface designs. Journal of Quality Technology, 44(2), 85–101. |
| [11] | Kiefer, J., & Wolfowitz, J. (1959). Optimum Designs in Regression Problems. The Annals of Mathematical Statistics, 30(2), 271–294. |
| [12] | Kiefer, J., & Wolfowitz, J. (1960). The equivalence of two extremum problems. Canadian Journal of Mathematics. Journal Canadien de Mathematiques, 12(0), 363–366. |
| [13] | Montgomery, D. C. (2001). Design and Analysis of Experiment 5th ed. John Wiley & Sons Inc. |
| [14] | Myers R. H, M. D. C. &. A.-C. C. M. (2009). Response surface methodology: Process and product using designed experiments (3rd ed.). Wiley-Blackwell. |
| [15] | Ossia C. V. and Big-Alabo, A. (2022). Performance Evaluation of Apricot Kernel Oil-Based Cutting-Fluid Using L27 Orthogonal Arrays Design. Journal of the Egyptian Society of Tribology. |
| [16] | Rady E. A., M. M. E. Abd El-Monsef, and M. M. Seyam. (2009). Relationships among Several Optimality Criteria. Cairo University. |
| [17] | Smucker, B., Krzywinski, M., & Altman, N. (2018). Optimal experimental design. Nature Methods, 15(8), 559–560. |
| [18] | Walsh, S. J., Lu, L., & Anderson-Cook, C. M. (2024). I -optimal or G -optimal: Do we have to choose? Quality Engineering, 36(2), 227–248. |
| [19] | Warren F. Kuhfeld, SAS Institute, Inc. (1997). Efficient Experimental Designs Using Computerized Searches. Sawtooth Software, Inc. |
| [20] | Wong, W. K. (1994). Comparing robust properties of A, D, E, and G-optimal designs. Computational Statistics & Data Analysis, 18(4), 441–448. |
| [21] | Yeh Mei-Fen, Anthony Cece, Mark Presser. (2010). Custom Designs Using JMP® Design of Experiments and SAS® PROC OPTEX. Unilever, Trumbull, CT. |
| [22] | Zhou, Y. D., & Xu, H. (2017). Composite designs based on orthogonal arrays and definitive screening designs. Journal of the American Statistical Association, 112(520), 1675–1683. |
APA Style
Paschal, I. M., Fortune, I. C. (2024). Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models. American Journal of Theoretical and Applied Statistics, 13(5), 92-114. https://doi.org/10.11648/j.ajtas.20241305.11
ACS Style
Paschal, I. M.; Fortune, I. C. Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models. Am. J. Theor. Appl. Stat. 2024, 13(5), 92-114. doi: 10.11648/j.ajtas.20241305.11
AMA Style
Paschal IM, Fortune IC. Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models. Am J Theor Appl Stat. 2024;13(5):92-114. doi: 10.11648/j.ajtas.20241305.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajtas.20241305.11,
author = {Iwundu Mary Paschal and Israel Chinomso Fortune},
title = {Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models
},
journal = {American Journal of Theoretical and Applied Statistics},
volume = {13},
number = {5},
pages = {92-114},
doi = {10.11648/j.ajtas.20241305.11},
url = {https://doi.org/10.11648/j.ajtas.20241305.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20241305.11},
abstract = {The performances of Custom A-, D-, and I-optimal designs on non-standard second-order models are examined using the alphabetic A-, D-, and G-optimality efficiencies, as well as the Average Variance of Prediction. Designs of varying sizes are constructed with the help of JMP Pro 14 software and are customized for specified non-standard models, optimality criteria, prespecified experimental runs, and a specified range of input variables. The results reveal that Custom-A optimal designs perform generally better in terms of G-efficiency. They show high superiority to A-efficiency as the worst G-efficiency value of the created Custom-A optimal designs exceeds the best A-efficiency value of the designs, and also does well in terms of D-efficiency. Custom-D optimal designs perform generally best in terms of G-efficiency, as the worst G-efficiency value exceeds all A- and D-efficiency values. Custom-I optimal designs perform generally best in terms of G-efficiency as the worst G-efficiency value is better than the best A-efficiency value and performs generally better than the corresponding D-efficiency values. For the Average Variance of Prediction, Custom A- and I-optimal designs perform competitively well, with relatively low Average Variances of Prediction. On the contrary, the Average Variance of Prediction is generally larger for Custom-D optimal designs. Hence when seeking designs that minimize the variance of the predicted response, it suffices to construct Custom A-, D-, or I-optimal designs, with a preference for Custom-D optimal designs.
},
year = {2024}
}
TY - JOUR T1 - Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models AU - Iwundu Mary Paschal AU - Israel Chinomso Fortune Y1 - 2024/09/26 PY - 2024 N1 - https://doi.org/10.11648/j.ajtas.20241305.11 DO - 10.11648/j.ajtas.20241305.11 T2 - American Journal of Theoretical and Applied Statistics JF - American Journal of Theoretical and Applied Statistics JO - American Journal of Theoretical and Applied Statistics SP - 92 EP - 114 PB - Science Publishing Group SN - 2326-9006 UR - https://doi.org/10.11648/j.ajtas.20241305.11 AB - The performances of Custom A-, D-, and I-optimal designs on non-standard second-order models are examined using the alphabetic A-, D-, and G-optimality efficiencies, as well as the Average Variance of Prediction. Designs of varying sizes are constructed with the help of JMP Pro 14 software and are customized for specified non-standard models, optimality criteria, prespecified experimental runs, and a specified range of input variables. The results reveal that Custom-A optimal designs perform generally better in terms of G-efficiency. They show high superiority to A-efficiency as the worst G-efficiency value of the created Custom-A optimal designs exceeds the best A-efficiency value of the designs, and also does well in terms of D-efficiency. Custom-D optimal designs perform generally best in terms of G-efficiency, as the worst G-efficiency value exceeds all A- and D-efficiency values. Custom-I optimal designs perform generally best in terms of G-efficiency as the worst G-efficiency value is better than the best A-efficiency value and performs generally better than the corresponding D-efficiency values. For the Average Variance of Prediction, Custom A- and I-optimal designs perform competitively well, with relatively low Average Variances of Prediction. On the contrary, the Average Variance of Prediction is generally larger for Custom-D optimal designs. Hence when seeking designs that minimize the variance of the predicted response, it suffices to construct Custom A-, D-, or I-optimal designs, with a preference for Custom-D optimal designs. VL - 13 IS - 5 ER -
Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria
Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria
Information