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On the aberrations of mixed level orthogonal arrays with removed runs

  • Roberto FontanaEmail author
  • Fabio Rapallo
Regular Article
  • 13 Downloads

Abstract

Given an orthogonal array we analyze the aberrations of the sub-fractions which are obtained by the deletion of some of its points. We provide formulae to compute the Generalized Word-Length Pattern of any sub-fraction. In the case of the deletion of one single point, we provide a simple methodology to find which the best sub-fractions are according to the Generalized Minimum Aberration criterion. We also study the effect of the deletion of 1, 2 or 3 points on some examples. The methodology does not put any restriction on the number of levels of each factor. It follows that any mixed level orthogonal array can be considered.

Keywords

Orthogonal arrays Generalized word-length pattern Generalized minimum aberration criterion Incomplete designs 

Mathematics Subject Classification

62K15 

Notes

Acknowledgements

Both authors are members of GNAMPA-INdAM. This research has a financial support from Politecnico di Torino and Università del Piemonte Orientale.

References

  1. Butler NA, Ramos VM (2007) Optimal additions to and deletions from two-level orthogonal arrays. J R Stat Soc Ser B 69(1):51–61MathSciNetCrossRefGoogle Scholar
  2. Chatzopoulos SA, Kolyva-Machera F, Chatterjee K (2011) Optimality results on orthogonal arrays plus \(p\) runs for \(s^m\) factorial experiments. Metrika 73(3):385–394MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cheng SW, Ye KQ (2004) Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. Ann Stat 32(5):2168–2185MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dey A, Mukerjee R (2009) Fractional Factorial Plans. Wiley, New YorkzbMATHGoogle Scholar
  5. Eendebak P, Schoen E (2018) Complete series of non-isomorphic orthogonal arrays. http://pietereendebak.nl/oapage/. Accessed 31 July 2018
  6. Fontana R, Rapallo F (2018) Unions of orthogonal arrays and their aberrations via Hilbert bases. Tech. Rep. arXiv:1801.00591, submitted
  7. Fontana R, Rapallo F, Rogantin MP (2016) Aberration in qualitative multilevel designs. J Stat Plan Inference 174:1–10MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fries A, Hunter WG (1980) Minimum aberration \(2^{k-p}\) designs. Technometrics 22(4):601–608MathSciNetzbMATHGoogle Scholar
  9. Grömping U, Xu H (2014) Generalized resolution for orthogonal arrays. Ann Stat 42(3):918–939MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hedayat AS, Sloane NJA, Stufken J (2012) Orthogonal arrays: theory and applications. Springer, New YorkzbMATHGoogle Scholar
  11. Mukerjee R, Wu CFJ (2007) A modern theory of factorial design. Springer, New YorkGoogle Scholar
  12. Pistone G, Rogantin MP (2008) Indicator function and complex coding for mixed fractional factorial designs. J Stat Plan Inference 138(3):787–802MathSciNetCrossRefzbMATHGoogle Scholar
  13. Street DJ, Bird EM (2018) \({D}\)-optimal orthogonal array minus \(t\) run designs. J Stat Theory Pract 12(3):575–594MathSciNetCrossRefGoogle Scholar
  14. Wang P, Jan H (1995) Designing two-level factorial experiments using orthogonal arrays when the run order is important. The Statistician 44(2):379–388CrossRefGoogle Scholar
  15. Xampeny R, Grima P, Tort-Martorell X (2018) Which runs to skip in two-level factorial designs when not all can be performed. Qual Eng.  https://doi.org/10.1080/08982112.2018.1428751
  16. Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Stat 29(4):1066–1077MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department DISMA, Dipartimento di eccellenza 2018-2022Politecnico di TorinoTurinItaly
  2. 2.Department DISITUniversità del Piemonte OrientaleAlessandriaItaly

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