Methods in Algebraic Statistics for the Design of Experiments

  • G. Pistone
  • E. Riccomagno
  • M. P. Rogantin
Part of the Springer Optimization and Its Applications book series (SOIA, volume 28)


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.


Indicator Function Orthogonal Array Full Factorial Design Fractional Factorial Design Counting Function 
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  1. Abbott, J., Bigatti, A., Kreuzer, M., Robbiano, L. (2000). Computing ideals of points. Journal of Symbolic Computation, 30, (4)341–356.MATHCrossRefMathSciNetGoogle Scholar
  2. 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.MATHCrossRefMathSciNetGoogle Scholar
  3. Cox, D., Little, J., O'Shea, D. (1997). Ideals, Varieties, and Algorithms. second edition.Springer-Verlag, New York,Google Scholar
  4. Cox, D., Little, J., O'Shea, D. (2005). Using Algebraic Geometry. second edition.Springer, New York,MATHGoogle Scholar
  5. Fontana, R., Pistone, G., Rogantin, M.P. (2000). Classification of two-level factorial fractions. Journal of Statistical Planning and Inference, 87, (1)149–172.MATHCrossRefMathSciNetGoogle Scholar
  6. Galetto, F., Pistone, G., Rogantin, M.P. (2003). Confounding revisited with commutative computational algebra. Journal of Statistical Planning and Inference, 117, (2)345–363.MATHCrossRefMathSciNetGoogle Scholar
  7. Hedayat, A.S., Sloane, N.J.A., Stufken, J. (1999). Orthogonal Arrays. Springer-Verlag, New York.MATHGoogle Scholar
  8. Hinkelmann, K. Kempthorne, O. (2005). Design and Analysis of Experiments. Vol. 2. Advanced Experimental Design. John Wiley & Sons, Hoboken, NJ.MATHGoogle Scholar
  9. 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.MATHCrossRefMathSciNetGoogle Scholar
  10. Kobilinsky, A. (1997). Les Plans Factoriels, chapter 3, pages 879–883. ASU–SSdF, Éditions Technip.Google Scholar
  11. Kreuzer, M. Robbiano, L. (2000). Computational Commutative Algebra 1. Springer, Berlin-Heidelberg.CrossRefGoogle Scholar
  12. Kreuzer, M. Robbiano, L. (2005). Computational Commutative Algebra 2. Springer-Verlag, Berlin.Google Scholar
  13. Maruri-Aguilar, H. (2007). Algebraic Statistics in Experimental Design. Ph.D. thesis, University of Warwick, Statistics (March 2007).Google Scholar
  14. Maruri-Aguilar, H., Notari, R., and Riccomagno, E. (2007). On the description and identifiability analysis of mixture designs. Statistica Sinica 17(4), 1417–1440.MATHMathSciNetGoogle Scholar
  15. McCullagh, P. Nelder, J.A. (1989). Generalized Linear Models. second edition.Chapman & Hall, London,MATHGoogle Scholar
  16. Onn, S. Sturmfels, B. (1999). Cutting corners. Advances in Applied Mathematics, 23, (1)29–48.MATHCrossRefMathSciNetGoogle Scholar
  17. Peixoto, J.L. (1990). A property of well-formulated polynomial regression models. The American Statistician, 44, (1)26–30.CrossRefMathSciNetGoogle Scholar
  18. Pistone, G. and Rogantin, M. (2008). Algebraic statistics of codings for fractional factorial designs. Journal of Statistical Planning and Inference 138, 234–244.MATHCrossRefMathSciNetGoogle Scholar
  19. Pistone, G. Wynn, H.P. (1996). Generalised confounding with Gröbner bases. Biometrika, 83, (3)653–666.MATHCrossRefMathSciNetGoogle Scholar
  20. 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.CrossRefGoogle Scholar
  21. Pistone, G., Riccomagno, E., Wynn, H.P. (2001). Algebraic Statistics: Computational Commutative Algebra in Statistics. Chapman & Hall, London.MATHGoogle Scholar
  22. Raktoe, B.L., Hedayat, A., Federer, W.T. (1981). Factorial Designs. John Wiley & Sons Inc., New York.MATHGoogle Scholar
  23. Scheffé, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society B, 20, 344–360.MATHGoogle Scholar
  24. Wu, C.F.J. Hamada, M. (2000). Experiments. Planning, Analysis, and Parameter Design Optimization. John Wiley & Sons Inc., New York.MATHGoogle Scholar

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© Springer Science+Business Media LLC 2009

Authors and Affiliations

  1. 1. Department of Mathematics, Politecnico di Torino Torino Italy
  2. 2. Dipartimento di Matematica, Universit`a degli Studi di Genova Genova Italia
  3. 3.Cardiff Dipartimento di Matematica, Universit`a degli Studi di Genova GenovaItalia

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