An Optimal Split-Plot Design for Performing a  Mixture-Process Experiment

Gladys Gakenia Njoroge; Jemimah Ayuma Simbauni; Joseph Arap Koske

doi:doi:10.11648/j.sjams.20170501.13

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An Optimal Split-Plot Design for Performing a Mixture-Process Experiment

Gladys Gakenia Njoroge, Jemimah Ayuma Simbauni, Joseph Arap Koske

Published in Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 1)

Received: 30 October 2016 Accepted: 21 November 2016 Published: 18 January 2017

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Abstract

In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 2² factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 1)
DOI	10.11648/j.sjams.20170501.13
Page(s)	15-23
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

Optimality, Split-Plot Design, Efficiency, Mixture Components, Process Variables

References

[1]	Anderson-Cook, C. M., Borror, C. M. & Montgomery, D. C., “Response surface design evaluation and comparison,” Journal of Statistical Planning and Inferences; 139: 629-641, 2009.
[2]	Box, G. E. P. & Hunter, J., “Multifactor experimental designs for exploring response surfaces,” The Annals of Mathematical Statistics, 28: 195-241, 1957.
[3]	Cho, T-Y., Mixture-process variable design experiments with control and noise variables within a split-plot structure. A PhD Dissertation, Arizona State University, 2010.
[4]	Cornell, J. A., Experiments with Mixtures: designs, models and the analysis of mixture data, 3^rd Edition; New York, NY. John Wiley & Sons, Inc, 2002.
[5]	Design-Expert^® Version 10 Software. Minneapolis, MN. State-Ease Inc, 2016.
[6]	Giovannitti-Jensen, A. & Myers, R. H., “Graphical assessment of the prediction capability of response surface designs,” Technometrics, 31: 159-171, 1989.
[7]	Goldfarb, H. B., Anderson-Cook, C. M., Borror, C. M. & Montgomery, D. C., “Fraction of design space plots for assessing mixture-process designs,” Journal of Quality Technology, 36(2): 169-179, 2010.
[8]	Goldfarb, H. B., Borror, C. M. & Montgomery, D. C., “Mixture-process variable experiments with noise variables,” Journal of Quality Technology, 35:393-405, 2003.
[9]	Goos, P., The optimal design of blocked and split-plot experiments, New York: Springer –Verlang, 2002.
[10]	Kowalski, S. M., Cornell, J. A., & Vining, G. G., “Split-Plot Designs and estimation methods for mixture experiments with process variables,” Technometrics, 44 (1): 72-79, 2002.
[11]	Liang, J. D., Myers, R. A. & Robinson, T. J., “Fraction of Design Space Plots for Split-Plot Designs,” Quality and Reliability Engineering International, 22: 275-289, 2006.
[12]	Lin, L-Y., Robust split-plot designs for model misspecification. Department of Applied Mathematics and Institute of Statistics, National Chung Hsing University, Taichung, Taiwan, 40227, 2016.
[13]	Montgomery, D. C., Design and Analysis of Experiments, New York, John Wiley & Sons, Inc.2009.
[14]	Schoonees, P., Niel le Roux, N. J. & Coetzer, R. L. J., “Flexible Graphical Assessment of Experimental Designs in R: The vdg package,” Journal of Statistical Software, 74(3), 2016.
[15]	Zahran, A., Anderson-Cook, C. M. & Myers, R. H., “Fraction of Design Space to Assess Prediction Capability of Response Surface Designs,” Journal of Quality Technology, 35: 377-386, 2003.

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Gladys Gakenia Njoroge, Jemimah Ayuma Simbauni, Joseph Arap Koske. (2017). An Optimal Split-Plot Design for Performing a Mixture-Process Experiment. Science Journal of Applied Mathematics and Statistics, 5(1), 15-23. https://doi.org/10.11648/j.sjams.20170501.13

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ACS Style

Gladys Gakenia Njoroge; Jemimah Ayuma Simbauni; Joseph Arap Koske. An Optimal Split-Plot Design for Performing a Mixture-Process Experiment. Sci. J. Appl. Math. Stat. 2017, 5(1), 15-23. doi: 10.11648/j.sjams.20170501.13

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AMA Style

Gladys Gakenia Njoroge, Jemimah Ayuma Simbauni, Joseph Arap Koske. An Optimal Split-Plot Design for Performing a Mixture-Process Experiment. Sci J Appl Math Stat. 2017;5(1):15-23. doi: 10.11648/j.sjams.20170501.13

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@article{10.11648/j.sjams.20170501.13,
  author = {Gladys Gakenia Njoroge and Jemimah Ayuma Simbauni and Joseph Arap Koske},
  title = {An Optimal Split-Plot Design for Performing a  Mixture-Process Experiment},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {5},
  number = {1},
  pages = {15-23},
  doi = {10.11648/j.sjams.20170501.13},
  url = {https://doi.org/10.11648/j.sjams.20170501.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20170501.13},
  abstract = {In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 22 factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.},
 year = {2017}
}

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TY  - JOUR
T1  - An Optimal Split-Plot Design for Performing a  Mixture-Process Experiment
AU  - Gladys Gakenia Njoroge
AU  - Jemimah Ayuma Simbauni
AU  - Joseph Arap Koske
Y1  - 2017/01/18
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170501.13
DO  - 10.11648/j.sjams.20170501.13
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 15
EP  - 23
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170501.13
AB  - In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 22 factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.
VL  - 5
IS  - 1
ER  -

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Author Information

Gladys Gakenia Njoroge

Department of Physical Sciences, Chuka University, Chuka, Kenya
Jemimah Ayuma Simbauni

Department of Zoological Sciences, Kenyatta University, Nairobi, Kenya
Joseph Arap Koske

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

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APA Style

Gladys Gakenia Njoroge, Jemimah Ayuma Simbauni, Joseph Arap Koske. (2017). An Optimal Split-Plot Design for Performing a Mixture-Process Experiment. Science Journal of Applied Mathematics and Statistics, 5(1), 15-23. https://doi.org/10.11648/j.sjams.20170501.13

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ACS Style

Gladys Gakenia Njoroge; Jemimah Ayuma Simbauni; Joseph Arap Koske. An Optimal Split-Plot Design for Performing a Mixture-Process Experiment. Sci. J. Appl. Math. Stat. 2017, 5(1), 15-23. doi: 10.11648/j.sjams.20170501.13

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AMA Style

Gladys Gakenia Njoroge, Jemimah Ayuma Simbauni, Joseph Arap Koske. An Optimal Split-Plot Design for Performing a Mixture-Process Experiment. Sci J Appl Math Stat. 2017;5(1):15-23. doi: 10.11648/j.sjams.20170501.13

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@article{10.11648/j.sjams.20170501.13,
  author = {Gladys Gakenia Njoroge and Jemimah Ayuma Simbauni and Joseph Arap Koske},
  title = {An Optimal Split-Plot Design for Performing a  Mixture-Process Experiment},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {5},
  number = {1},
  pages = {15-23},
  doi = {10.11648/j.sjams.20170501.13},
  url = {https://doi.org/10.11648/j.sjams.20170501.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20170501.13},
  abstract = {In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 22 factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - An Optimal Split-Plot Design for Performing a  Mixture-Process Experiment
AU  - Gladys Gakenia Njoroge
AU  - Jemimah Ayuma Simbauni
AU  - Joseph Arap Koske
Y1  - 2017/01/18
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170501.13
DO  - 10.11648/j.sjams.20170501.13
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 15
EP  - 23
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170501.13
AB  - In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 22 factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.
VL  - 5
IS  - 1
ER  -

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