Many practical problems are associated with the investigation of mixture of m ingredients, which are assumed to influence the response through the proportions in which they are blended together. Such problems lend their applicability to mixture experiments. Mixture experiments can be modeled using Scheffe’ or Kronecker models. For the first-, second-, and third-degree Kronecker models, D-optimal designs for the mixture experiments have been derived by various authors. This creates uncertainties to an experimenter, hence the need for robust designs. The objective of this study is to derive robust D-optimal designs for mixture experiments in the first- and the second-degree Kronecker model for mixture experiments. In order to achieve this, the D-optimal weights for the designs in the first-degree and those of the second degree Kronecker models are obtained. The model robust D-Optimality criterion is then used. The D-Optimal designs are obtained by maximizing this criterion which involves differentiating, equating to zero and solving for , r1 and r2 in order to obtain the optimal values. In conclusion the results of this study demonstrate the existence of model robust D-optimal Kronecker model mixture experiments for the first- and the second-degree Kronecker models.
| Published in | American Journal of Theoretical and Applied Statistics (Volume 14, Issue 4) |
| DOI | 10.11648/j.ajtas.20251404.12 |
| Page(s) | 126-137 |
| 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), 2025. Published by Science Publishing Group |
Mixture Experiment, Kronecker Product, Moment Matrix, D-optimal, Robust Designs
| [1] | Chan, L. Y. (2000). “Optimal Designs of Experiments with Mixtures: a survey.” Communications in statistics-theory and methods, Vol. 29(9 & 10), 2281-2312. |
| [2] | Box, G. E. P., and Draper, N. R. (1959) “A Basis for the selection of a response surface Design.” Journal of the American Statistical Association, Vol. 54, 622-654. |
| [3] | Draper, N. R. and Pukelsheim, F., (1998). Kiefer ordering of simplex designs for first-and second degree mixture models. In: Journal of statistical planning and inference, vol. 79: 325-348. |
| [4] | Draper, N. R., Heilingers, B., Pukelsheim, F. (2000). Kiefer ordering of simplex designs for mixture models with four or more ingredients. In: Annals of statistics, vol. 28: 578-590. |
| [5] | Galil, Z., Kiefer J. (1977). “Comparison of Simplex Designs for Quadratic Mixture” Technometrics, Vol 19, 445-453. |
| [6] | Hampel, Frank. (2001). “Robust statistics: A brief introduction and overview” ResearchGate. |
| [7] | Kerich, G. K., Koske, J. K., Cherutich, M. R. and Kungu, P. N. (2014) “D-Optimal Designs for Third-Degree Kronecker Model Mixture Experiments with an application to artificial sweetener experiment.” FJTS, India. |
| [8] | Kiefer, J. (1961). “Optimal Designs in regression Problems, II” Annals of Mathematical Statistics, Vol 32, Issue 1, 298-235. |
| [9] | Kinyanjui, J. K., (2007). Some optimal designs for second-degree Kronecker model mixture experiments. PhD. Thesis, Moi University, Eldoret. |
| [10] | Klein, T. (2004), Invariant Symmetric Block matrices for the design mixture experiments. In: Linear Algebra and its application. Vol. 388: 261-278. |
| [11] | Klein, T. (2001), Invariant Symmetric Block matrices for the design mixture experiments. University of Augsburg Germany. |
| [12] | Koech, E., Koech, M., Koske, J. K., Kerich, G., and Otieno, A. R. (2014) “E-Optimal Designs for Maximal Parameter Subsystem Second-Degree Kronecker Model Mixture Experiments.” Mathematical Theory and Modelling, Vol 4, No. 5. |
| [13] | Koske, J. K., Kinyanjui, J. K., Mutiso, J. M. and Cherutich, M. R. (2012). “Designs with Optimal Values in the Second-Degree Kronecker Model Mixture Experiments.” M. Sc Thesis, Moi University. |
| [14] | Mahesh K. P. (2024), “Model Robust Optimal Designs for Kronecker Model for Mixture experiments.” Journal of Statistical Sciences, Vol. 24(1), pp 31-48. |
| [15] | Mong-Na, L. H., Hsiang-Ling, H., Chao-Jin, C. and Klein, T. (2009), “Model-Robust D- and A-Optimal Designs for Mixture Experiments.” Statistica Sinica, Vol. 19, 1055-1075. |
| [16] | Ngigi, P. K.(2009). “Optimality criteria for second-degree Kronecker model mixture experiments with two, three and four ingredients.” M. Phil. Thesis, Moi University: Eldoret. |
| [17] | Pukelsheim, F (1993). Optimal Design of Experiments, New York: wiley interscience. |
| [18] | Quinoille, M. H. (1953). “The Design and Analysis of Experiments,” Charles Graffin and Company, London, England. |
| [19] | Scheff’e, H. (1958). “Experiments with mixtures.” In: J. Roy. Statist. Soc. Ser. Vol. B20: 344-360. |
| [20] | Scheff’e, H. (1963). “The Simplex-Centroid Design for experiments with mixtures.” In: J. Roy. Statist. Soc. Ser. Vol. B25: 235-257. |
APA Style
Cherutich, M., Koske, J. A., Kosgey, M. (2025). Robust D-optimal Designs for the First-degree and the Second-degree Kronecker Model for Mixture Experiments. American Journal of Theoretical and Applied Statistics, 14(4), 126-137. https://doi.org/10.11648/j.ajtas.20251404.12
ACS Style
Cherutich, M.; Koske, J. A.; Kosgey, M. Robust D-optimal Designs for the First-degree and the Second-degree Kronecker Model for Mixture Experiments. Am. J. Theor. Appl. Stat. 2025, 14(4), 126-137. doi: 10.11648/j.ajtas.20251404.12
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Download@article{10.11648/j.ajtas.20251404.12,
author = {Mike Cherutich and Jopseph Arap Koske and Mathew Kosgey},
title = {Robust D-optimal Designs for the First-degree and the Second-degree Kronecker Model for Mixture Experiments
},
journal = {American Journal of Theoretical and Applied Statistics},
volume = {14},
number = {4},
pages = {126-137},
doi = {10.11648/j.ajtas.20251404.12},
url = {https://doi.org/10.11648/j.ajtas.20251404.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20251404.12},
abstract = {Many practical problems are associated with the investigation of mixture of m ingredients, which are assumed to influence the response through the proportions in which they are blended together. Such problems lend their applicability to mixture experiments. Mixture experiments can be modeled using Scheffe’ or Kronecker models. For the first-, second-, and third-degree Kronecker models, D-optimal designs for the mixture experiments have been derived by various authors. This creates uncertainties to an experimenter, hence the need for robust designs. The objective of this study is to derive robust D-optimal designs for mixture experiments in the first- and the second-degree Kronecker model for mixture experiments. In order to achieve this, the D-optimal weights for the designs in the first-degree and those of the second degree Kronecker models are obtained. The model robust D-Optimality criterion is then used. The D-Optimal designs are obtained by maximizing this criterion which involves differentiating, equating to zero and solving for , r1 and r2 in order to obtain the optimal values. In conclusion the results of this study demonstrate the existence of model robust D-optimal Kronecker model mixture experiments for the first- and the second-degree Kronecker models.},
year = {2025}
}
TY - JOUR T1 - Robust D-optimal Designs for the First-degree and the Second-degree Kronecker Model for Mixture Experiments AU - Mike Cherutich AU - Jopseph Arap Koske AU - Mathew Kosgey Y1 - 2025/07/14 PY - 2025 N1 - https://doi.org/10.11648/j.ajtas.20251404.12 DO - 10.11648/j.ajtas.20251404.12 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 - 126 EP - 137 PB - Science Publishing Group SN - 2326-9006 UR - https://doi.org/10.11648/j.ajtas.20251404.12 AB - Many practical problems are associated with the investigation of mixture of m ingredients, which are assumed to influence the response through the proportions in which they are blended together. Such problems lend their applicability to mixture experiments. Mixture experiments can be modeled using Scheffe’ or Kronecker models. For the first-, second-, and third-degree Kronecker models, D-optimal designs for the mixture experiments have been derived by various authors. This creates uncertainties to an experimenter, hence the need for robust designs. The objective of this study is to derive robust D-optimal designs for mixture experiments in the first- and the second-degree Kronecker model for mixture experiments. In order to achieve this, the D-optimal weights for the designs in the first-degree and those of the second degree Kronecker models are obtained. The model robust D-Optimality criterion is then used. The D-Optimal designs are obtained by maximizing this criterion which involves differentiating, equating to zero and solving for , r1 and r2 in order to obtain the optimal values. In conclusion the results of this study demonstrate the existence of model robust D-optimal Kronecker model mixture experiments for the first- and the second-degree Kronecker models. VL - 14 IS - 4 ER -
Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
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