Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments

Cheruiyot Kipkoech; Koske Joseph; Mutiso John

doi:doi:10.11648/j.ajtas.20170604.11

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Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments

Cheruiyot Kipkoech, Koske Joseph, Mutiso John

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 4)

Received: 13 October 2016 Accepted: 28 October 2016 Published: 1 June 2017

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Abstract

Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.

Published in	American Journal of Theoretical and Applied Statistics (Volume 6, Issue 4)
DOI	10.11648/j.ajtas.20170604.11
Page(s)	170-175
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), 2017. Published by Science Publishing Group

Keywords

Kronecker Model, Optimal Designs, Slope Information Matrices (SIM), Weighted Simplex Centroid Designs, A-, D-, E- and T-Optimality

References

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[2]	Draper, N. R. and Pukelsheim, F. (1998). Mixture models based on homogeneous polynomials. J. Statist. Plann. Inference 71 303–311.
[3]	Hader, R. J. and Park, S. H. (1978). Slope-rotatable central composite designs, Technometrics, 20, 413-417.
[4]	Herzberg, A. M. (1967). The behavior of the variance functions of the difference between two responses. J. Roy. Statist. Soc. B, 29, 174-179.
[5]	Huda, S. (2006). Minimax designs for the difference between estimated responses for the quadratic model over hypercubic regions. Commun. Statist.- Theory meth.
[6]	Huda, S. and Al-Shiha, A. A. (1999). On D-Optimal designs for estimating slope. The Indian Journal of Statistics. 61: B, 3, 488-495.
[7]	Huda, S. and Mukerjee, R. (1984). Minimizing the maximum variance of the diﬀerence between two estimated responses, Biometrika, 71, 381-385.
[8]	Kerich, G., Koske, J., Rutto, M., Korir, B., Ronoh, B., Kinyanjui, J. and Kungu, P. (2014). D-Optimal Designs for Third Degree Kronecker Model Mixture Experiments with An Application to Artificial Sweetener Experiment. IOSR Journal of Mathematics, 10 (6), 32-41.
[9]	Korir, B. C. (2008),” Kiefer ordering of simplex designs for third-degree mixture models.” P. hd. Thesis Moi University.
[10]	Montgomery, M. D. (2001). Design and Analysis of Experiments, 5^th ed., John Wiley and Sons, New York.
[11]	Pukelsheim, F. (1993). Optimal design of experiments. Wiley, New York.
[12]	Sung H. P., Hyang S. J. and Rabindra N. D. (2009). Slope –rotatbility of second order response surface regression models with correlated error. Quality technology & quantitative management 6, (4) 471-492.

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APA Style

Cheruiyot Kipkoech, Koske Joseph, Mutiso John. (2017). Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments. American Journal of Theoretical and Applied Statistics, 6(4), 170-175. https://doi.org/10.11648/j.ajtas.20170604.11

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ACS Style

Cheruiyot Kipkoech; Koske Joseph; Mutiso John. Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments. Am. J. Theor. Appl. Stat. 2017, 6(4), 170-175. doi: 10.11648/j.ajtas.20170604.11

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AMA Style

Cheruiyot Kipkoech, Koske Joseph, Mutiso John. Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments. Am J Theor Appl Stat. 2017;6(4):170-175. doi: 10.11648/j.ajtas.20170604.11

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@article{10.11648/j.ajtas.20170604.11,
  author = {Cheruiyot Kipkoech and Koske Joseph and Mutiso John},
  title = {Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {4},
  pages = {170-175},
  doi = {10.11648/j.ajtas.20170604.11},
  url = {https://doi.org/10.11648/j.ajtas.20170604.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170604.11},
  abstract = {Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.},
 year = {2017}
}

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TY  - JOUR
T1  - Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments
AU  - Cheruiyot Kipkoech
AU  - Koske Joseph
AU  - Mutiso John
Y1  - 2017/06/01
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.20170604.11
DO  - 10.11648/j.ajtas.20170604.11
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  - 170
EP  - 175
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170604.11
AB  - Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.
VL  - 6
IS  - 4
ER  -

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Author Information

Cheruiyot Kipkoech

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya
Koske Joseph

Department of Statistics and Computer Science, Moi University, Eldoret, Kenya
Mutiso John

Department of Statistics and Computer Science, Moi University, Eldoret, Kenya

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Plain Text BibTeX RIS

APA Style

Cheruiyot Kipkoech, Koske Joseph, Mutiso John. (2017). Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments. American Journal of Theoretical and Applied Statistics, 6(4), 170-175. https://doi.org/10.11648/j.ajtas.20170604.11

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ACS Style

Cheruiyot Kipkoech; Koske Joseph; Mutiso John. Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments. Am. J. Theor. Appl. Stat. 2017, 6(4), 170-175. doi: 10.11648/j.ajtas.20170604.11

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AMA Style

Cheruiyot Kipkoech, Koske Joseph, Mutiso John. Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments. Am J Theor Appl Stat. 2017;6(4):170-175. doi: 10.11648/j.ajtas.20170604.11

Copy | Download

@article{10.11648/j.ajtas.20170604.11,
  author = {Cheruiyot Kipkoech and Koske Joseph and Mutiso John},
  title = {Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {4},
  pages = {170-175},
  doi = {10.11648/j.ajtas.20170604.11},
  url = {https://doi.org/10.11648/j.ajtas.20170604.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170604.11},
  abstract = {Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments
AU  - Cheruiyot Kipkoech
AU  - Koske Joseph
AU  - Mutiso John
Y1  - 2017/06/01
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.20170604.11
DO  - 10.11648/j.ajtas.20170604.11
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  - 170
EP  - 175
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170604.11
AB  - Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.
VL  - 6
IS  - 4
ER  -

Copy | Download