Four Component Optimal and Nearly Optimal Orthogonally Blocked Designs for Husain and Parveen’s Model
Abstract
Four component mixture experiments involving process variables are considered. When two different blocks are orthogonal to each other, then the least square estimation of the parameters are independent of estimation of the parameters of the process variables. Optimal mixture designs involve only binary blends but many practical situations may require the minimum proportion of all the ingredients to be present in the mixture such as manufacturing of any drug requires all the needed components to be present in the mixture, in making pizza all the ingredients should be present to enhance the taste. So, in order to meet these requirements, Prescott (1998) suggested nearly optimal designs. In this paper, we have obtained Latin square based optimal and nearly optimal designs in four components for Husain and Parveen’s (2016) quadratic mixture model.
References
- A. K. Nigam; Block Designs for Mixture Experiments. Annals of Mathematical Statistics, 41, 1861-1869 (1970).
- A.K.Nigam;CorrectionstoBlockingConditionsforMixtureExperiments.AnnalsofStatistics,47,1294- 1295 (1976).
- B.HusainandS.Parveen;F-squareBasedFourComponentsD-,A-,andE-OptimalOrthogonalDesigns for an Additive Quadratic Mixture Model. Journal of the Indian Society for Probability and Statistics, 17, 95-109 (2016).
- B.HusainandS.Parveen;FourComponentF-squaresBasedNearlyD-andA-OptimalOrthogonalBlock Designs for Additive Quadratic Mixture Model and Reduced Cubic Canonical Model. Aligarh Journal of Statistics, 38, 149-161(2018).
- B. Husain and S. Sharma; Optimal Orthogonal Designs in Two Blocks Based on F-squares for Mixture Inverse Model in Four Components. International Journal of Experimental Design and Process Optimization, 4(3/4), 206-215 (2015).
- B.HusainandS.Sharma;F-squaresBasedOptimalDesignsforReducedCubicCanonicalModelsinFour Components. International Journal of Experimental Design and Process Optimization, 5(3), 206-221 (2017).
- G.E.P.BoxandJ.S.Hunter;MultifactorExperimentalDesignsforExploringResponseSurfaces.Annals of Mathematical Statistics, 28, 195-242 (1957).
- H. Scheffé; Experiments with Mixtures. Journal of the Royal Statistical Society, Series B, 20, 344-360 (1958).
- H. Scheffé; Simplex-Centroid Designs for Experiments with Mixtures. Journal of the Royal Statistical Society, Series B, 25, 235-263 (1963).
- J. N. Darroch and J. Waller; Additivity and Interactions in Three Component Experiments with Mixtures. Biometrika, 72, 153-163 (1985).
- L. Y. Chan and Y. N. Guan; A- and D- Optimal Designs for a Log Contrast Model for Experiments with Mixtures. Journal of Applied Statistics, 28, 537-546 (2001).
- L. Y. Chan and M. K. Sandhu; Optimal Orthogonal Block Designs for a Quadratic Mixture Model for Three Components. Journal of Applied Statistics, 26(1), 19-34 (1999).
- M. L. Aggarwal; V. Sarin and P. Singh; Orthogonal Block Designs in Two Blocks for Becker’s Mixture Model in Three and Four Components. Statistics and Probability Letters, 59, 385-396 (2002).
- N. G. Becker; Models for the Response of a Mixture. Journal of the Royal Statistical Society, Series B 30, 349-358 (1968).
- N. R. Draper; P. Prescott; S. M. Lewis; A. M. Dean; P. W. M. John and M. G. Tuck; Mixture Designs for Four Components in Orthogonal Blocks, Technometrics, 35, 268-276 (1993).
- P. W. M. John; Experiments with Mixture Involving Process Variables. Technical report 8, Centre for Statistical Sciences, University of Texas, Austin, TX, 1-17 (1984).
- P. Prescott; Nearly Optimal Orthogonally Blocked Designs for a Quadratic Mixture Model with q Component. Communication in Statistics- Theory and Methods, 27(10), 2559-2580 (1998).
- P. Prescott; N. R. Draper; A. M. Dean and S. M. Lewis; Mixture Designs for Five Components in Orthogonal Blocks. Journal of Applied Statistics, 20(1), 268-276 (1993).
- P. Singh; Optimal Orthogonal Block Design in Two Blocks for Darroch and Waller’s Quadratic Mixture Model in Three and Four Components. METRON- International Journal of Statistics, LXI 3, 419-430 (2003).
- S. Ghosh and T. Liu; Optimal Mixture Designs for Four Components in Two Orthogonal Block. Journal of Statistical Planning and Inference, 78, 219-228 (1999).
- V. Czitrom; Mixture Experiments with Process Variables, D-Optimal Orthogonal Experimental Designs, Communication in Statistics-Theory and Methods, 17, 105-121 (1988).
- V. Czitrom; Experimental Designs for Four Mixture Components with Process Variables. Communication in Statistics-Theory and Methods, 18, 4561-4581 (1989).