Electronic Journal of Statistics

Construction of minimum generalized aberration two-level orthogonal arrays

Haralambos Evangelaras

Full-text: Open access


In this paper we explore the problem of constructing two-level Minimum Generalized Aberration (MGA) orthogonal arrays with strength $t$, $n$ runs and $q>t$ columns, using a method that employs the $J$-characteristics of a two-level design. General results for the construction of MGA orthogonal arrays with $t+1$, $t+2$ and $t+3$ columns are given, while all MGA designs with strength $t\ge 2$, $n \equiv$ 0 mod 4 runs and $q\le 6$ are constructed. Results are also given for two-level orthogonal arrays with $q=7$ factors, but with strength greater than two. Projection properties of the MGA designs that have been identified, are also discussed.

Article information

Electron. J. Statist., Volume 9, Number 2 (2015), 2689-2705.

Received: July 2015
First available in Project Euclid: 8 December 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K15: Factorial designs
Secondary: 05B20: Matrices (incidence, Hadamard, etc.)

Orthogonal arrays $J$-characteristics minimum generalized aberration


Evangelaras, Haralambos. Construction of minimum generalized aberration two-level orthogonal arrays. Electron. J. Statist. 9 (2015), no. 2, 2689--2705. doi:10.1214/15-EJS1091. https://projecteuclid.org/euclid.ejs/1449582160

Export citation


  • [1] Bulutoglu, D. A. and Kaziska, D. M. (2010). Improved WLP and GWP lower bounds based on exact integer programming, Journal of Statistical Planning and Inference, 140 1154–1161.
  • [2] Bulutoglu, D. A. and Ryan, K. J. (2015). Algorithms for finding generalized minimum aberration designs, Journal of Complexity, 31 577–589.
  • [3] Butler, N. A. (2003). Minimum aberration construction results for nonregular two-level fractional factorial designs, Biometrika, 90 891–898.
  • [4] Butler, N. A. (2004). Minimum $G_2$-aberration properties of two-level foldover designs, Statistics and Probability Letters, 67 121–132.
  • [5] Deng, L.-Y. and Tang, B. (1999). Generalized resolution and minimum aberration criteria for Plackett-Burman and other nonregular factorial designs, Statistica Sinica, 9 1071–1082.
  • [6] Deng, L.-Y. and Tang, B. (2002). Design selection and classification for Hadamard matrices using generalized minimum aberration criteria, Technometrics, 44 173–184.
  • [7] Dey A. and Mukerjee, R. (1999)., Fractional Factorial Plans, Wiley, New York.
  • [8] Evangelaras, H., Koukouvinos, C. and Lappas, E. (2007). Further contributions to nonisomorphic two level orthogonal arrays, Journal of Statistical Planning and Inference, 137 2080–2086.
  • [9] Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999)., Orthogonal Arrays: Theory and Applications, Springer-Verlag, New York.
  • [10] Lin, C. D., Sitter, R. R. and Tang, B. (2012). Creating catalogues of two-level nonregular fractional factorial designs based on the criteria of generalized aberration, Journal of Statistical Planning and Inference, 142 445–456.
  • [11] Ma, C. X. and Fang, K. T. (2001). A note on generalized aberration in factorial designs, Metrika, 53 85–93.
  • [12] Schoen, E. D., Eendebak, P. T. and Nguyen, M. V. M. (2010). Complete Enumeration of Pure-Level and Mixed-Level Orthogonal Arrays, Journal of Combinatorial Designs, 18 123–140.
  • [13] Seiden, E. and Zemach, R. (1966). On orthogonal arrays, The Annals of Mathematical Statistics, 37 1355–1370.
  • [14] Stufken, J. and Tang, B. (2007). Complete enumeration of two-level orthogonal arrays of strength $d$ with $d+2$ constraints, The Annals of Statistics, 35 793–814.
  • [15] Tang, B. (2001). Theory of $J$-characteristics for fractional factorial designs and projection justification of minimum $G_2$-aberration., Biometrika, 88 401–407.
  • [16] Tang, B. and Deng, L.-Y. (1999). Minimum $G_2$-aberration for Nonregular Fractional Factorial designs, The Annals of Statistics, 27 1914–1926.
  • [17] Tang, B. and Deng, L.-Y. (2003). Construction of generalized minimum aberration designs of 3, 4 and 5 factors, Journal of Statistical Planning and Inference 113 335–340.
  • [18] Xu, H., Phoa, F. K. H. and Wong, W. K. (2009). Recent developments in nonregular fractional factorial designs, Statistic Surveys, 3 18–46.
  • [19] Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs, The Annals of Statistics, 29 1066–1077.