Two-Factor Saturated Designs: Cycles, Gini Index, and State Polytopes

  • Roberto FontanaEmail author
  • Fabio Rapallo
  • Maria-Piera Rogantin


In this article, we analyze and characterize the saturated fractions of two-factor designs under the simple effect model. Using linear algebra, we define a criterion to check whether a given fraction is saturated or not. We also compute the number of saturated fractions, providing an alternative proof of the Cayley’s formula. Finally, we show how, given a list of saturated fractions, Gini indexes of their margins and the associated state polytopes could be used to classify them.


Estimability Gini index State polytope Universal Markov basis 

AMS Subject Classification

62K15 15B34 05B20 62H17 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aoki, S., and A. Takemura. 2010. Markov chain Monte Carlo tests for designed experiments. J. Stat. Plan. Inference, 140(3), 817–830.MathSciNetCrossRefGoogle Scholar
  2. Aoki, S., and A. Takemura. 2012. Design and analysis of fractional factorial experiments from the viewpoint of computational algebraic statistics. J. Stat. Theory Pract., 6(1), 147–161.MathSciNetCrossRefGoogle Scholar
  3. Arboretti Giancristofaro, R., R. Fontana, and S. Ragazzi. 2012. Construction and nonparametric testing of orthogonal arrays through algebraic strata and inequivalent permutation matrices. Comm. Statist. Theory Methods, 41, 16–17.MathSciNetCrossRefGoogle Scholar
  4. Bailey, R. A., 2008. Design of comparative experiments. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  5. Basso, D., L. Salmaso, H. Evangelaras, and C. Koukouvinos. 2004. Nonparametric testing for main effects on inequivalent designs. In mODa 7—Advances in model-oriented design and analysis, eds. A. Di Bucchianico, H. Läuter, and H. P. Wynn, 33–40. Heidelberg, Germany: Contrib. Statist. Physica.CrossRefGoogle Scholar
  6. Berstein, Y., H. Maruri-Aguilar, S. Onn, E. Riccomagno, and H. Wynn. 2010. Minimal average degree aberration and the state polytope for experimental designs. Ann. Inst. Stat. Math., 62(4), 673–698.MathSciNetCrossRefGoogle Scholar
  7. Dey, A., K. R. Shah, and A. Das. 1995. Optimal block designs with minimal and nearly minimal number of units. Stat. Sinica, 5(2), 547–558.MathSciNetzbMATHGoogle Scholar
  8. Drton, M., B. Sturmfels, and S. Sullivant. 2009. Lectures on algebraic statistics. Basel, Switzerland: Birkhauser.CrossRefGoogle Scholar
  9. Fontana, R. 2013. Algebraic generation of minimum size orthogonal fractional factorial designs: An approach based on integer linear programming. Comput. Stat., 28: 241–253. doi: 10.1007/s00180-011-0296-7.MathSciNetCrossRefGoogle Scholar
  10. Fontana, R., G. Pistone, and M. P. Rogantin. 2000. Classification of two-level factorial fractions. J. Stat. Plan. Inference, 87(1), 149–172.MathSciNetCrossRefGoogle Scholar
  11. Fontana, R., F. Rapallo, and M. P. Rogantin. 2012. Markov bases for Sudoku grids. In Advanced statistical methods for the analysis of large data-sets, ed. A. Di Ciaccio, M. Coli, and J. M. Angulo Ibanez, 305–315. Studies in Theoretical and Applied Statistics. Berlin, Germany: Springer.CrossRefGoogle Scholar
  12. Fontana, R., F. Rapallo, and M. P. Rogantin. 2013. A characterization of saturated designs for factorial experiments. arXiv:1304.7914v1.Google Scholar
  13. Fries, A., and W. G. Hunter. 1980. Minimum aberration 2k−p designs. Technometrics, 22(4), 601–608.MathSciNetzbMATHGoogle Scholar
  14. Gini, C. 1912. Variabilità e mutabilità: contributo allo studio delle distribuzioni e delle relazioni statistiche. Bologna, Italy: Tipogr. di P. Cuppini.Google Scholar
  15. Hassani, M. 2003. Derangements and applications. J. Integer Seq., 6(1), article 03.1.2, 8 pp. (electronic).Google Scholar
  16. Hedayat, A. S., N. J. A. Sloane, and J. Stufken. 1999. Orthogonal arrays. Theory and applications. Springer Series in Statistics. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
  17. Krafft, O., and M. Schaefer. 1997. A-optimal connected block designs with nearly minimal number of observations. J. Stat. Plan. Inference, 65(2), 375–386.MathSciNetCrossRefGoogle Scholar
  18. Kuhnt, S., F. Rapallo, and A. Rehage. 2013. Outlier detection in contingency tables based on minimal patterns. Stat. Comput. Online first. doi:10.1007/s11222-013-9382-8.MathSciNetCrossRefGoogle Scholar
  19. Li, W., D. K. J. Lin, and K. Q. Ye. 2003. Optimal foldover plans for two-level nonregular orthogonal designs. Technometrics, 45(4), 347–351.MathSciNetCrossRefGoogle Scholar
  20. Maruri-Aguilar, H., E. Sáenz-de Cabezón, and H. P. Wynn. 2012. Betti numbers of polynomial hierarchical models for experimental designs. Ann. Math. Artif. Intell., 64(4), 411–426.MathSciNetCrossRefGoogle Scholar
  21. Mukerjee, R., K. Chatterjee, and M. Sen. 1986. D-optimality of a class of saturated main-effect plans and allied results. Statistics, 17(3), 349–355.MathSciNetCrossRefGoogle Scholar
  22. Pistone, G., E. Riccomagno, and H. P. Wynn. 2001. Algebraic statistics: Computational commutative algebra in statistics. Boca Raton, FL: Chapman&Hall/CRC.zbMATHGoogle Scholar
  23. Raktoe, B. L., A. Hedayat, and W. T. Federer. 1981. Factorial designs. Wiley Series in Probability and Mathematical Statistics. New York, NY: John Wiley & Sons.zbMATHGoogle Scholar
  24. Rapallo, F., and M. P. Rogantin. 2007. Markov chains on the reference set of contingency tables with upper bounds. Metron, 65(1), 35–51.zbMATHGoogle Scholar
  25. Sturmfels, B. 1996. Gröbner bases and convex polytopes. Vol. 8 of University lecture series. Providence, RI: American Mathematical Society.zbMATHGoogle Scholar
  26. Xu, K. 2003. How has the literature on Gini’s index evolved in the past 80 years? Economics Working Paper, Dalhousie University, Halifax, NS, Canada.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2014

Authors and Affiliations

  • Roberto Fontana
    • 1
    Email author
  • Fabio Rapallo
    • 2
  • Maria-Piera Rogantin
    • 3
  1. 1.Department of Mathematical SciencesPolitecnico di TorinoTurinItaly
  2. 2.Department DISITUniversità del Piemonte OrientaleAlessandriaItaly
  3. 3.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

Personalised recommendations