Skip to main content
SpringerLink
Log in

A fresh look at effect aliasing and interactions: some new wine in old bottles

  • Invited Article: Akaike Memorial Lecture
  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Interactions and effect aliasing are among the fundamental concepts in experimental design. In this paper, some new insights and approaches are provided on these subjects. In the literature, the “de-aliasing” of aliased effects is deemed to be impossible. We argue that this “impossibility” can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality. This approach is successfully applied to three classes of designs: regular and nonregular two-level fractional factorial designs, and three-level fractional factorial designs. For reparametrization, the notion of conditional main effects (cme’s) is employed for two-level regular designs, while the linear-quadratic system is used for three-level designs. For nonregular two-level designs, reparametrization is not needed because the partial aliasing of their effects already induces non-orthogonality. The approach can be extended to general observational data by using a new bi-level variable selection technique based on the cme’s. A historical recollection is given on how these ideas were discovered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Addelman, S. (1961). Irregular fractions of the \(2^n\) factorial experiments. Technometrics, 3(4), 479–496.

    MathSciNet  MATH  Google Scholar 

  • Addelman, S. (1962). Augmenting factorial plans to accommodate additional two-level factors. Biometrics, 18(3), 308–322.

    Article  MathSciNet  MATH  Google Scholar 

  • Barker, T. B., Milivojevich, A. (2016). Quality by experimental design (4th ed.). New York: CRC Press.

  • Box, G. E. P., Hunter, J. S. (1961). The \(2^{k-p}\) fractional factorial designs Part I. Technometrics, 3(3), 311–351.

  • Box, G. E. P., Tyssedal, J. (1996). Projective properties of certain orthogonal arrays. Biometrika, 83(4), 950–955.

  • Box, G. E. P., Hunter, W. G., Hunter, J. S. (1978). Statistics for experimenters: An introduction to design, data analysis, and model building. New York: Wiley.

  • Box, G. E. P., Kackar, R., Nair, V., Phadke, M., Shoemaker, A. C., Wu, C. F. J. (1988). Quality practices in Japan. Quality Progress, 21(3), 37–41.

  • Box, G. E. P., Hunter, J. S., Hunter, W. G. (2005). Statistics for experimenters: Design, innovation, and discovery (2nd ed.). New York: Wiley.

  • Cheng, C.-S. (2014). Theory of factorial design: Single- and multi-stratum experiments. New York: CRC Press.

    MATH  Google Scholar 

  • Chipman, H., Hamada, M., Wu, C. F. J. (1997). A Bayesian variable-selection approach for analyzing designed experiments with complex aliasing. Technometrics, 39(4), 372–381.

  • Cox, D. R. (1984). Interaction. International Statistical Review, 52(1), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  • Daniel, C. (1976). Applications of statistics to industrial experiments. New York: Wiley.

    Book  Google Scholar 

  • Deng, L.-Y., Tang, B. (1999). Generalized resolution and minimum aberration criteria for Plackett–Burman and other nonregular factorial designs. Statistica Sinica, 9(4), 1071–1082.

  • Finney, D. (1945). The fractional replication of factorial arrangements. Annals of Eugenics, 12, 291–303.

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher, R. A. (1971). The design of experiments (7th ed.). London: Oliver and Boyd.

    Google Scholar 

  • Hamada, M., Wu, C. F. J. (1992). Analysis of designed experiments with complex aliasing. Journal of Quality Technology, 24(3), 130–137.

  • Hastie, T., Tibshirani, R., Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction (2nd ed.). New York: Springer.

  • Hedayat, A. S., Sloane, N. J. A., Stufken, J. (1999). Orthogonal arrays: Theory and applications. New York: Springer.

  • Hunter, G. B., Hodi, F. S., Eagar, T. W. (1982). High-cycle fatigue of weld repaired cast Ti–6AI–4V. Metallurgical Transactions A, 13(9), 1589–1594.

  • John, P. W. M. (1971). Statistical design and analysis of experiments. New York: Macmillan.

    MATH  Google Scholar 

  • Lin, D. K. J., Draper, N. R. (1992). Projection properties of Plackett and Burman designs. Technometrics, 34(4), 423–428.

  • Mak, S., Wu, C. F. J. (2017). cmenet: A new method for bi-level variable selection of conditional main effects. Journal of the American Statistical Association. https://arxiv.org/abs/1701.05547 (submitted).

  • Masuyama, M. (1957). On difference sets for constructing orthogonal arrays of index two and of strength two. Report on Statistical Application and Research, JUSE, 5, 27–34.

    Google Scholar 

  • Mazumder, R., Friedman, J. H., Hastie, T. (2011). SparseNet: Coordinate descent with nonconvex penalties. Journal of the American Statistical Association, 106(495), 1125–1138.

  • Mukerjee, R., Wu, C. F. J. (2006). A modern theory of factorial design. New York: Springer.

  • Mukerjee, R., Wu, C. F. J., Chang, M.-C. (2017). Two-level minimum aberration designs under a conditional model with a pair of conditional and conditioned factors. Statistica Sinica, 27(3), 997--1016.

  • Plackett, R. L., Burman, J. P. (1946). The design of optimum multifactorial experiments. Biometrika, 33(4), 305–325.

  • Raktoe, B., Hedayat, A., Federer, W. T. (1981). Factorial designs. New York: Wiley.

  • Sabbaghi, A., Dasgupta, T., Wu, C. F. J. (2014). Indicator functions and the algebra of the linear-quadratic parametrization. Biometrika, 101(2), 351–363.

  • SAS Institute Inc. (2015). JMP®12 design of experiments guide. Cary, NC: SAS Institute Inc.

    Google Scholar 

  • Seiden, E. (1954). On the problem of construction of orthogonal arrays. The Annals of Mathematical Statistics, 25(1), 151–156.

    Article  MathSciNet  MATH  Google Scholar 

  • Su, H., Wu, C. F. J. (2017). CME analysis: A new method for unraveling aliased effects in two-level fractional factorial experiments. Journal of Quality Technology, 49(1), 1–10.

  • Sun, D. X., Wu, C. F. J. (1993). Statistical properties of hadamard matrices of order 16. In W. Kuo (Ed.), Quality through engineering design (pp. 169–179). New York: Elsevier.

  • Taguchi, G. (1987). System of experimental design. Whilt Plains, NY: Unipub/Kraus International Publications.

    MATH  Google Scholar 

  • Tang, B., Deng, L.-Y. (1999). Minimum \(g_2\)-aberration for nonregular fractional factorial designs. Annals of Statistics, 27(6), 1914–1926.

  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58(1), 267–288.

    MathSciNet  MATH  Google Scholar 

  • Wang, J.-C., Wu, C. F. J. (1992). Nearly orthogonal arrays with mixed levels and small runs. Technometrics, 34(4), 409–422.

  • Wang, J. C., Wu, C. F. J. (1995). A hidden projection property of Plackett–Burman and related designs. Statistica Sinica, 5(1), 235–250.

  • Wu, C. F. J. (2015). Post-Fisherian experimentation: From physical to virtual. Journal of the American Statistical Association, 110(510), 612–620.

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, C. F. J., Hamada, M. S. (2000). Experiments: Planning, analysis and parameter design optimization. New York: Wiley.

  • Wu, C. F. J., Hamada, M. S. (2009). Experiments: Planning, analysis, and optimization (2nd ed.). New York: Wiley.

  • Xu, H. (2003). Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica, 13(3), 691–708.

    MathSciNet  MATH  Google Scholar 

  • Xu, H., Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Annals of Statistics, 29(2), 549–560.

  • Ye, K. Q. (2004). A note on regular fractional factorial designs. Statistica Sinica, 14(4), 1069–1074.

    MathSciNet  MATH  Google Scholar 

  • Yuan, M., Joseph, V. R., Zou, H. (2009). Structured variable selection and estimation. The Annals of Applied Statistics, 3(4), 1738–1757.

Download references

Acknowledgements

Research supported by ARO W911NF-17-1-0007. The author is grateful to the referees for helpful comments.

Author information

Authors and Affiliations

  1. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Drive, NW, Atlanta, GA, 30332, USA

    C. F. Jeff Wu

Authors
  1. C. F. Jeff Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. F. Jeff Wu.

Additional information

The Related Articles are https://doi.org/10.1007/s10463-017-0639-4; https://doi.org/10.1007/s10463-017-0640-y; https://doi.org/10.1007/s10463-017-0641-x.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, C.F.J. A fresh look at effect aliasing and interactions: some new wine in old bottles. Ann Inst Stat Math 70, 249–268 (2018). https://doi.org/10.1007/s10463-018-0646-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-018-0646-0

Keywords

Search

Navigation