The Annals of Statistics

A general theory of minimum aberration and its applications

Ching-Shui Cheng and Boxin Tang

Full-text: Open access


Minimum aberration is an increasingly popular criterion for comparing and assessing fractional factorial designs, and few would question its importance and usefulness nowadays. In the past decade or so, a great deal of work has been done on minimum aberration and its various extensions. This paper develops a general theory of minimum aberration based on a sound statistical principle. Our theory provides a unified framework for minimum aberration and further extends the existing work in the area. More importantly, the theory offers a systematic method that enables experimenters to derive their own aberration criteria. Our general theory also brings together two seemingly separate research areas: one on minimum aberration designs and the other on designs with requirement sets. To facilitate the design construction, we develop a complementary design theory for quite a general class of aberration criteria. As an immediate application, we present some construction results on a weak version of this class of criteria.

Article information

Ann. Statist., Volume 33, Number 2 (2005), 944-958.

First available in Project Euclid: 26 May 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K15: Factorial designs

Blocking design resolution fractional factorial design linear graph orthogonal array requirement set robust parameter design split plot design


Cheng, Ching-Shui; Tang, Boxin. A general theory of minimum aberration and its applications. Ann. Statist. 33 (2005), no. 2, 944--958. doi:10.1214/009053604000001228.

Export citation


  • Bingham, D. and Sitter, R. R. (1999). Minimum aberration two-level fractional factorial split-plot designs. Technometrics 41 62–70.
  • Chen, H. and Cheng, C.-S. (1999). Theory of optimal blocking of $2^{n-m}$ designs. Ann. Statist. 27 1948–1973.
  • Chen, H. and Hedayat, A. S. (1996). $2^{n-m}$ designs with weak minimum aberration. Ann. Statist. 24 2536–2548.
  • Chen, J. (1992). Some results on $2^{n-k}$ fractional factorial designs and search for minimum aberration designs. Ann. Statist. 20 2124–2141.
  • Chen, J. and Wu, C. F. J. (1991). Some results on $s^{n-k}$ fractional factorial designs with minimum aberration or optimal moments. Ann. Statist. 19 1028–1041.
  • Cheng, C.-S. and Mukerjee, R. (2001). Blocked regular fractional factorial designs with maximum estimation capacity. Ann. Statist. 29 530–548.
  • Cheng, C.-S. and Mukerjee, R. (2003). On regular-fractional factorial experiments in row-column designs. J. Statist. Plann. Inference 114 3–20.
  • Cheng, C.-S., Steinberg, D. M. and Sun, D. X. (1999). Minimum aberration and model robustness for two-level fractional factorial designs. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 85–93.
  • Cheng, S.-W. and Wu, C. F. J. (2002). Choice of optimal blocking schemes in two-level and three-level designs. Technometrics 44 269–277.
  • Franklin, M. F. (1984). Constructing tables of minimum aberration $p^{n-m}$ designs. Technometrics 26 225–232.
  • Fries, A. and Hunter, W. G. (1980). Minimum aberration $2^{k-p}$ designs. Technometrics 22 601–608.
  • Greenfield, A. A. (1976). Selection of defining contrasts in two-level experiments. Appl. Statist. 25 64–67.
  • Hedayat, A. S. and Stufken, J. (1999). Compound orthogonal arrays. Technometrics 41 57–61.
  • Huang, P., Chen, D. and Voelkel, J. O. (1998). Minimum-aberration two-level split-plot designs. Technometrics 40 314–326.
  • Ke, W. and Tang, B. (2003). Selecting $2^{m-p}$ designs using a minimum aberration criterion when some two-factor interactions are important. Technometrics 45 352–360.
  • Mukerjee, R. and Wu, C. F. J. (1999). Blocking in regular fractional factorials: A projective geometric approach. Ann. Statist. 27 1256–1271.
  • Sitter, R. R., Chen, J. and Feder, M. (1997). Fractional resolution and minimum aberration in blocked $2^{n-k}$ designs. Technometrics 39 382–390.
  • Suen, C.-Y., Chen, H. and Wu, C. F. J. (1997). Some identities on $q^{n-m}$ designs with application to minimum aberration designs. Ann. Statist. 25 1176–1188.
  • Tang, B. and Deng, L.-Y. (1999). Minimum $G_2$-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926.
  • Tang, B. and Wu, C. F. J. (1996). Characterization of minimum aberration $2^{n-k}$ designs in terms of their complementary designs. Ann. Statist. 24 2549–2559.
  • Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.
  • Wu, C. F. J. and Zhu, Y. (2003). Optimal selection of single arrays for parameter design experiments. Statist. Sinica 13 1179–1199.
  • Zhu, Y. (2003). Structure function for aliasing patterns in $2^{l-n}$ designs with multiple groups of factors. Ann. Statist. 31 995–1011.