Summary
We present a brief review of classical experimental design in the spirit of algebraic statistics. Notion of identifiability, aliasing and estimability of linear parametric functions, confounding are expressed in relation to a set of polynomials identified by the design, called the design ideal. An effort has been made to indicate the classical linear algebra counterpart of the objects of interest in the polynomial space, and to indicate how the algebraic statistics approach generalizes the classical theory. In the second part of this chapter we address new questions: a seemingly limitation of the algebraic approach is discussed and resolved in the ideas of minimal and maximal fan designs, again generalizing classical notions; an algorithm is provided to switch between two major representations of a design, one of which uses Gröbner bases and the other one uses indicator functions. Finally, all the theory in the chapter is applied and extended to the class of mixture designs which present a challenging structure and questions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abbott, J., Bigatti, A., Kreuzer, M., Robbiano, L. (2000). Computing ideals of points. Journal of Symbolic Computation, 30, (4)341–356.
Babson, E., Onn, S., Thomas, R. (2003). The Hilbert zonotope and a polynomial time algorithm for universal Gröbner bases. Advances in Applied Mathematics, 30, (3)529–544.
Cox, D., Little, J., O'Shea, D. (1997). Ideals, Varieties, and Algorithms. second edition.Springer-Verlag, New York,
Cox, D., Little, J., O'Shea, D. (2005). Using Algebraic Geometry. second edition.Springer, New York,
Fontana, R., Pistone, G., Rogantin, M.P. (2000). Classification of two-level factorial fractions. Journal of Statistical Planning and Inference, 87, (1)149–172.
Galetto, F., Pistone, G., Rogantin, M.P. (2003). Confounding revisited with commutative computational algebra. Journal of Statistical Planning and Inference, 117, (2)345–363.
Hedayat, A.S., Sloane, N.J.A., Stufken, J. (1999). Orthogonal Arrays. Springer-Verlag, New York.
Hinkelmann, K. Kempthorne, O. (2005). Design and Analysis of Experiments. Vol. 2. Advanced Experimental Design. John Wiley & Sons, Hoboken, NJ.
Holliday, T., Pistone, G., Riccomagno, E., Wynn, H.P. (1999). The application of computational algebraic geometry to the analysis of designed experiments: a case study. Computational Statistics, 14, (2)213–231.
Kobilinsky, A. (1997). Les Plans Factoriels, chapter 3, pages 879–883. ASU–SSdF, Éditions Technip.
Kreuzer, M. Robbiano, L. (2000). Computational Commutative Algebra 1. Springer, Berlin-Heidelberg.
Kreuzer, M. Robbiano, L. (2005). Computational Commutative Algebra 2. Springer-Verlag, Berlin.
Maruri-Aguilar, H. (2007). Algebraic Statistics in Experimental Design. Ph.D. thesis, University of Warwick, Statistics (March 2007).
Maruri-Aguilar, H., Notari, R., and Riccomagno, E. (2007). On the description and identifiability analysis of mixture designs. Statistica Sinica 17(4), 1417–1440.
McCullagh, P. Nelder, J.A. (1989). Generalized Linear Models. second edition.Chapman & Hall, London,
Onn, S. Sturmfels, B. (1999). Cutting corners. Advances in Applied Mathematics, 23, (1)29–48.
Peixoto, J.L. (1990). A property of well-formulated polynomial regression models. The American Statistician, 44, (1)26–30.
Pistone, G. and Rogantin, M. (2008). Algebraic statistics of codings for fractional factorial designs. Journal of Statistical Planning and Inference 138, 234–244.
Pistone, G. Wynn, H.P. (1996). Generalised confounding with Gröbner bases. Biometrika, 83, (3)653–666.
Pistone, G., Riccomagno, E., Wynn, H.P. (2000). Gröbner basis methods for structuring and analyzing complex industrial experiments. International Journal of Reliability, Quality, and Safety Engineering, 7, (4)285–300.
Pistone, G., Riccomagno, E., Wynn, H.P. (2001). Algebraic Statistics: Computational Commutative Algebra in Statistics. Chapman & Hall, London.
Raktoe, B.L., Hedayat, A., Federer, W.T. (1981). Factorial Designs. John Wiley & Sons Inc., New York.
Scheffé, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society B, 20, 344–360.
Wu, C.F.J. Hamada, M. (2000). Experiments. Planning, Analysis, and Parameter Design Optimization. John Wiley & Sons Inc., New York.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media LLC
About this chapter
Cite this chapter
Pistone, G., Riccomagno, E., Rogantin, M.P. (2009). Methods in Algebraic Statistics for the Design of Experiments. In: Pronzato, L., Zhigljavsky, A. (eds) Optimal Design and Related Areas in Optimization and Statistics. Springer Optimization and Its Applications, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-0-387-79936-0_5
Download citation
DOI: https://doi.org/10.1007/978-0-387-79936-0_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-79935-3
Online ISBN: 978-0-387-79936-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)