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

Invited Article: Akaike Memorial Lecture


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.


Conditional main effects Fractional factorial designs Nonregular designs Orthogonal arrays 



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


  1. Addelman, S. (1961). Irregular fractions of the \(2^n\) factorial experiments. Technometrics, 3(4), 479–496.MathSciNetMATHGoogle Scholar
  2. Addelman, S. (1962). Augmenting factorial plans to accommodate additional two-level factors. Biometrics, 18(3), 308–322.MathSciNetCrossRefMATHGoogle Scholar
  3. Barker, T. B., Milivojevich, A. (2016). Quality by experimental design (4th ed.). New York: CRC Press.Google Scholar
  4. Box, G. E. P., Hunter, J. S. (1961). The \(2^{k-p}\) fractional factorial designs Part I. Technometrics, 3(3), 311–351.Google Scholar
  5. Box, G. E. P., Tyssedal, J. (1996). Projective properties of certain orthogonal arrays. Biometrika, 83(4), 950–955.Google Scholar
  6. 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.Google Scholar
  7. 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.Google Scholar
  8. Box, G. E. P., Hunter, J. S., Hunter, W. G. (2005). Statistics for experimenters: Design, innovation, and discovery (2nd ed.). New York: Wiley.Google Scholar
  9. Cheng, C.-S. (2014). Theory of factorial design: Single- and multi-stratum experiments. New York: CRC Press.MATHGoogle Scholar
  10. 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.Google Scholar
  11. Cox, D. R. (1984). Interaction. International Statistical Review, 52(1), 1–24.MathSciNetCrossRefMATHGoogle Scholar
  12. Daniel, C. (1976). Applications of statistics to industrial experiments. New York: Wiley.CrossRefGoogle Scholar
  13. 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.Google Scholar
  14. Finney, D. (1945). The fractional replication of factorial arrangements. Annals of Eugenics, 12, 291–303.MathSciNetCrossRefMATHGoogle Scholar
  15. Fisher, R. A. (1971). The design of experiments (7th ed.). London: Oliver and Boyd.Google Scholar
  16. Hamada, M., Wu, C. F. J. (1992). Analysis of designed experiments with complex aliasing. Journal of Quality Technology, 24(3), 130–137.Google Scholar
  17. Hastie, T., Tibshirani, R., Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction (2nd ed.). New York: Springer.Google Scholar
  18. Hedayat, A. S., Sloane, N. J. A., Stufken, J. (1999). Orthogonal arrays: Theory and applications. New York: Springer.Google Scholar
  19. 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.Google Scholar
  20. John, P. W. M. (1971). Statistical design and analysis of experiments. New York: Macmillan.MATHGoogle Scholar
  21. Lin, D. K. J., Draper, N. R. (1992). Projection properties of Plackett and Burman designs. Technometrics, 34(4), 423–428.Google Scholar
  22. 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. (submitted).
  23. 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
  24. Mazumder, R., Friedman, J. H., Hastie, T. (2011). SparseNet: Coordinate descent with nonconvex penalties. Journal of the American Statistical Association, 106(495), 1125–1138.Google Scholar
  25. Mukerjee, R., Wu, C. F. J. (2006). A modern theory of factorial design. New York: Springer.Google Scholar
  26. 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.Google Scholar
  27. Plackett, R. L., Burman, J. P. (1946). The design of optimum multifactorial experiments. Biometrika, 33(4), 305–325.Google Scholar
  28. Raktoe, B., Hedayat, A., Federer, W. T. (1981). Factorial designs. New York: Wiley.Google Scholar
  29. Sabbaghi, A., Dasgupta, T., Wu, C. F. J. (2014). Indicator functions and the algebra of the linear-quadratic parametrization. Biometrika, 101(2), 351–363.Google Scholar
  30. SAS Institute Inc. (2015). JMP®12 design of experiments guide. Cary, NC: SAS Institute Inc.Google Scholar
  31. Seiden, E. (1954). On the problem of construction of orthogonal arrays. The Annals of Mathematical Statistics, 25(1), 151–156.MathSciNetCrossRefMATHGoogle Scholar
  32. 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.Google Scholar
  33. 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.Google Scholar
  34. Taguchi, G. (1987). System of experimental design. Whilt Plains, NY: Unipub/Kraus International Publications.MATHGoogle Scholar
  35. Tang, B., Deng, L.-Y. (1999). Minimum \(g_2\)-aberration for nonregular fractional factorial designs. Annals of Statistics, 27(6), 1914–1926.Google Scholar
  36. Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58(1), 267–288.MathSciNetMATHGoogle Scholar
  37. Wang, J.-C., Wu, C. F. J. (1992). Nearly orthogonal arrays with mixed levels and small runs. Technometrics, 34(4), 409–422.Google Scholar
  38. Wang, J. C., Wu, C. F. J. (1995). A hidden projection property of Plackett–Burman and related designs. Statistica Sinica, 5(1), 235–250.Google Scholar
  39. Wu, C. F. J. (2015). Post-Fisherian experimentation: From physical to virtual. Journal of the American Statistical Association, 110(510), 612–620.MathSciNetCrossRefMATHGoogle Scholar
  40. Wu, C. F. J., Hamada, M. S. (2000). Experiments: Planning, analysis and parameter design optimization. New York: Wiley.Google Scholar
  41. Wu, C. F. J., Hamada, M. S. (2009). Experiments: Planning, analysis, and optimization (2nd ed.). New York: Wiley.Google Scholar
  42. Xu, H. (2003). Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica, 13(3), 691–708.MathSciNetMATHGoogle Scholar
  43. Xu, H., Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Annals of Statistics, 29(2), 549–560.Google Scholar
  44. Ye, K. Q. (2004). A note on regular fractional factorial designs. Statistica Sinica, 14(4), 1069–1074.MathSciNetMATHGoogle Scholar
  45. Yuan, M., Joseph, V. R., Zou, H. (2009). Structured variable selection and estimation. The Annals of Applied Statistics, 3(4), 1738–1757.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  1. 1.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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