The Annals of Statistics

A classification criterion for definitive screening designs

Eric D. Schoen, Pieter T. Eendebak, and Peter Goos

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A conference design is a rectangular matrix with orthogonal columns, one zero in each column, at most one zero in each row and $-1$’s and $+1$’s elsewhere. A definitive screening design can be constructed by folding over a conference design and adding a row vector of zeroes. We prove that, for a given even number of rows, there is just one isomorphism class for conference designs with two or three columns. Next, we derive all isomorphism classes for conference designs with four columns. Based on our results, we propose a classification criterion for definitive screening designs founded on projections into four factors. We illustrate the potential of the criterion by studying designs with 24 and 82 factors.

Article information

Ann. Statist., Volume 47, Number 2 (2019), 1179-1202.

Received: January 2018
Revised: May 2018
First available in Project Euclid: 11 January 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K20: Response surface designs
Secondary: 05B20: Matrices (incidence, Hadamard, etc.) 94C30: Applications of design theory [See also 05Bxx]

Conference matrix isomorphism class $J_{4}$-characteristic projection


Schoen, Eric D.; Eendebak, Pieter T.; Goos, Peter. A classification criterion for definitive screening designs. Ann. Statist. 47 (2019), no. 2, 1179--1202. doi:10.1214/18-AOS1723.

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  • Delsarte, P., Goethals, J. M. and Seidel, J. J. (1971). Orthogonal matrices with zero diagonal. II. Canad. J. Math. 23 816–832.
  • 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.
  • Deng, L.-Y. and Tang, B. (2002). Design selection and classification for Hadamard matrices using generalized minimum aberration criteria. Technometrics 44 173–184.
  • Dougherty, S., Simpson, J. R., Hill, R. R., Pignatiello, J. J. and White, E. D. (2015). Effect of heredity and sparsity on second-order screening design performance. Qual. Reliab. Eng. Int. 31 355–368.
  • Fidaleo, M., Lavecchia, R., Petrucci, E. and Zuorro, A. (2016). Application of a novel definitive screening design to decolorization of an azo dye on boron-doped diamond electrodes. Int. J. Environ. Sci. Technol. 13 835–842.
  • Georgiou, S. D., Stylianou, S. and Aggarwal, M. (2014). Efficient three-level screening designs using weighing matrices. Statistics 48 815–833.
  • Greig, M., Haanpää, H. and Kaski, P. (2006). On the coexistence of conference matrices and near resolvable $2$-$(2k+1,k,k-1)$ designs. J. Combin. Theory Ser. A 113 703–711.
  • Hurkens, C. A. J. and Seidel, J. J. (1985). Conference matrices from projective planes of order $9$. European J. Combin. 6 49–57.
  • Jones, B. and Nachtsheim, C. J. (2011). A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 43 1–15.
  • Jones, B. and Nachtsheim, C. J. (2013). Definitive screening designs with added two-level categorical factors. J. Qual. Technol. 45 121–129.
  • Jones, B. and Nachtsheim, C. J. (2017). Effective design-based model selection for definitive screening designs. Technometrics 59 319–329.
  • Mee, R. W., Schoen, E. D. and Edwards, D. J. (2017). Selecting an orthogonal or nonorthogonal two-level design for screening. Technometrics 59 305–318.
  • Nachtsheim, A. C., Shen, W. and Lin, D. K. J. (2017). Two-level augmented definitve screening designs. J. Qual. Technol. 49 93–107.
  • Nguyen, N. K. and Pham, T. D. (2016). Small mixed-level screening designs with orthogonal quadratic effects. J. Qual. Technol. 48 405–414.
  • Patil, M. V. (2017). Multi response simulation and optimization of gas tungsten arc welding. Appl. Math. Model. 42 540–553.
  • Schoen, E. D., Eendebak, P. T. and Goos, P. (2019). Supplement to “A classification criterion for definitive screening designs”. DOI:10.1214/18-AOS1723SUPP.
  • Schoen, E. D., Vo-Thanh, N. and Goos, P. (2017). Two-level orthogonal screening designs with 24, 28, 32, and 36 runs. J. Amer. Statist. Assoc. 112 1354–1369.
  • Xiao, L., Lin, D. K. and Bai, F. (2012). Constructing definitive screening designs using conference matrices. J. Qual. Technol. 44 1–7.

Supplemental materials

  • Conference matrices of order 82. We provide Matlab code to construct the 26 nonisomorphic conference matrices of order 82 from Hurkens and Seidel (1985) and to evaluate the $F_{4}$ vector of the definitive screening designs with 82 factors based on these matrices.