Volume 3, Issue 4, October 2017, Page: 86-91
Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments
Wambua Alex Mwaniki, Department of Planning and Statistics, Ministry of Agriculture, Livestock and Fisheries, Nairobi, Kenya
Njoroge Elizabeth, Department of Business Administration, Chuka University, Chuka, Kenya
Koske Joseph, Department of Mathematics and Computer Science, Moi University, Eldoret, Kenya
John Mutiso, Department of Mathematics and Computer Science, Moi University, Eldoret, Kenya
Kuria Joseph Gikonyo, Department of Mathematics, Statistics and Actuarial Sciences, Karatina University, Karatina, Kenya
Muriungi Robert Gitunga, Department of Mathematics, Meru University of Science and Technology, Meru, Kenya
Cheruiyot Kipkoech, Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya
Received: Mar. 31, 2017;       Accepted: Apr. 14, 2017;       Published: Oct. 26, 2017
DOI: 10.11648/j.ijamtp.20170304.12      View  2253      Downloads  111
The aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker model for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.
Slope Mixture designs Kronecker product, Optimal Designs, Weighted Centroid Designs, A-, D-, E-Optimality and H- invariant Symmetric Matrices
To cite this article
Wambua Alex Mwaniki, Njoroge Elizabeth, Koske Joseph, John Mutiso, Kuria Joseph Gikonyo, Muriungi Robert Gitunga, Cheruiyot Kipkoech, Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments, International Journal of Applied Mathematics and Theoretical Physics. Vol. 3, No. 4, 2017, pp. 86-91. doi: 10.11648/j.ijamtp.20170304.12
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Box, G. E. P and Hunter, J. S (1957). Multifactorial experimental designs for exploring response surfaces. Ann. math. statist.28 195-241.
Cornell, J. A. (1981). Experiments with Mixtures designs: Models and analysis of mixture data. John Willy & Sons, New York.
Cornell, J. A. (1990). Experiments with Mixtures. Wiley, New York.
Draper N. R., and Pukelsheim, F., (1998), Kiefer ordering of simplex designs for first-and second degree mixture models, Journal of statistical planning and inference, 79:325-348.
Draper, N. R. and Pukelsheim, F. (1998a). Polynomial representations for response surface modelling. In New Developments and Applications in Experimental Design (N. Flournoy, W. F. Rosenberger and W. K. Wong, eds.), 34 199–212. IMS, Hayward, CA.
Draper, N. R. and Pukelsheim, F. (1998b). Mixture models based on homogeneous polynomials. J. Statist. Plann. Inference 71 303–311.
Draper, N. R. and Pukelsheim, F. (1999). Kiefer ordering of simplex designs for first- and second degree mixture models. J. Statist. Plann. Inference 79 325–348.
Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiefer ordering of second-degree mixture designs for four ingredients. In Proceedings of the American Statistical Association, Annual Meeting, Baltimore MD, August 1999. Amer. Statist. Assoc., Alexandria, VA.
Friedrich Pukelsheim, (1993) “Optimal designs of experiment”.
Hader, R. J and Park, S. H slope rotatable central composite designs, Technometrics, 1978; 20:413-417.
John A. Cornell, (1990) Experiments with Mixtures, second edition.
Kinyanjui J. K., (2007), Some Optimal Designs for Second-Degree Kronecker Model Mixture Experiments, PhD, Thesis, Moi University, Eldoret.
Koech Eliud, Koech Milton, Koske Joseph, Kerich Gregory, Argwings Otieno (2014) E-Optimal designs for maximal parameter subsystem second degree Kronecker model mixture experiments.
Norman R. Draper, Berthold Heiligers and Fredrich Pukelshiem (2000) Kiefer ordering of Simplex designs for second degree mixture models with four or more ingredients.
Park S. H and Kim H. J, A measure of slope rotatability for second order response surface experimental designs, Journal of Applied Statistics, 1992; 19: 391-404
Peter Goos, Poradley Jones and Utain Syafitri (2013) I-Optimal mixture designs.
Scheffé, H., 1958. Experiments with Mixtures, Journal of Royal Statistical Society. Series B, 20, 344-360.
Scheffé, H., 1963. Simplex-centroid designs for experiments with Mixtures, Journal of Royal Statistical Society. Series B, 25, 235-263.
Thomas Klein (2001) Invariant Symmetric block matrices for design of mixture of mixture experiments.
Browse journals by subject