The Annals of Statistics

Uniform fractional factorial designs

Yu Tang, Hongquan Xu, and Dennis K. J. Lin

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The minimum aberration criterion has been frequently used in the selection of fractional factorial designs with nominal factors. For designs with quantitative factors, however, level permutation of factors could alter their geometrical structures and statistical properties. In this paper uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion. We show that minimum aberration designs have low discrepancies on average. An efficient method for constructing uniform minimum aberration designs is proposed and optimal designs with 27 and 81 runs are obtained for practical use. These designs have good uniformity and are effective for studying quantitative factors.

Article information

Ann. Statist., Volume 40, Number 2 (2012), 891-907.

First available in Project Euclid: 1 June 2012

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Zentralblatt MATH identifier

Primary: 62K15: Factorial designs

Discrepancy generalized minimum aberration generalized word-length pattern geometrical isomorphism uniform minimum aberration design


Tang, Yu; Xu, Hongquan; Lin, Dennis K. J. Uniform fractional factorial designs. Ann. Statist. 40 (2012), no. 2, 891--907. doi:10.1214/12-AOS987.

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  • Cheng, S.-W. and Wu, C. F. J. (2001). Factor screening and response surface exploration. Statist. Sinica 11 553–604.
  • Cheng, S.-W. and Ye, K. Q. (2004). Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. Ann. Statist. 32 2168–2185.
  • Deng, L.-Y. and Tang, B. (1999). Generalized resolution and minimum aberration criteria for Plackett–Burman and other nonregular factorial designs. Statist. Sinica 9 1071–1082.
  • Fang, K.-T., Li, R. and Sudjianto, A. (2006). Design and Modeling for Computer Experiments. Chapman & Hall/CRC, Boca Raton, FL.
  • Fang, K. T. and Ma, C. X. (2001). Uniform and Orthogonal Designs (in Chinese). Science Press, Beijing.
  • Fang, K.-T. and Mukerjee, R. (2000). A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika 87 193–198.
  • Fang, K.-T., Lin, D. K. J., Winker, P. and Zhang, Y. (2000). Uniform design: Theory and application. Technometrics 42 237–248.
  • Fries, A. and Hunter, W. G. (1980). Minimum aberration $2^k-p$ designs. Technometrics 22 601–608.
  • Hickernell, F. J. (1998). A generalized discrepancy and quadrature error bound. Math. Comp. 67 299–322.
  • Hickernell, F. J. and Liu, M.-Q. (2002). Uniform designs limit aliasing. Biometrika 89 893–904.
  • Ma, C.-X. and Fang, K.-T. (2001). A note on generalized aberration in factorial designs. Metrika 53 85–93.
  • MacWilliams, F. J. and Sloane, N. J. A. (1977). The Theory of Error-Correcting Codes. North-Holland, Amsterdam.
  • Mukerjee, R. and Wu, C. F. J. (2006). A Modern Theory of Factorial Designs. Springer, New York.
  • Tang, B. and Deng, L.-Y. (1999). Minimum $G_2$-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926.
  • Wu, C. F. J. and Hamada, M. (2009). Experiments: Planning, Analysis and Parameter Design Optimization, 2nd ed. Wiley, New York.
  • Xu, H. (2005). A catalogue of three-level regular fractional factorial designs. Metrika 62 259–281.
  • Xu, H., Cheng, S.-W. and Wu, C. F. J. (2004). Optimal projective three-level designs for factor screening and interaction detection. Technometrics 46 280–292.
  • Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29 1066–1077.