An Optimal Split-Plot Design for Performing a Mixture-Process Experiment
Science Journal of Applied Mathematics and Statistics
Volume 5, Issue 1, February 2017, Pages: 15-23
Received: Oct. 30, 2016; Accepted: Nov. 21, 2016; Published: Jan. 18, 2017
Views 3358      Downloads 155
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
Article Tools
Follow on us
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.
Optimality, Split-Plot Design, Efficiency, Mixture Components, Process Variables
To cite this article
Gladys Gakenia Njoroge, Jemimah Ayuma Simbauni, Joseph Arap Koske, An Optimal Split-Plot Design for Performing a Mixture-Process Experiment, Science Journal of Applied Mathematics and Statistics. Vol. 5, No. 1, 2017, pp. 15-23. doi: 10.11648/j.sjams.20170501.13
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Box, G. E. P. & Hunter, J., “Multifactor experimental designs for exploring response surfaces,” The Annals of Mathematical Statistics, 28: 195-241, 1957.
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.
Cornell, J. A., Experiments with Mixtures: designs, models and the analysis of mixture data, 3rd Edition; New York, NY. John Wiley & Sons, Inc, 2002.
Design-Expert® Version 10 Software. Minneapolis, MN. State-Ease Inc, 2016.
Giovannitti-Jensen, A. & Myers, R. H., “Graphical assessment of the prediction capability of response surface designs,” Technometrics, 31: 159-171, 1989.
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.
Goldfarb, H. B., Borror, C. M. & Montgomery, D. C., “Mixture-process variable experiments with noise variables,” Journal of Quality Technology, 35:393-405, 2003.
Goos, P., The optimal design of blocked and split-plot experiments, New York: Springer –Verlang, 2002.
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.
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.
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.
Montgomery, D. C., Design and Analysis of Experiments, New York, John Wiley & Sons, Inc.2009.
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.
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.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186