Abstract

Two-level orthogonal arrays ensure the independent estimations of main effects when

linear models are considered, and thus are popularly used experimental designs. Such arrays

can be classified into regular and nonregular designs (Wu and Hamada, 2021). Regular designs

entertain specific algebraic structures and thus have been well studied in the literature. Their

run sizes, however, are limited to powers of 2. Nonregular designs have a more complicated

structure, but they are more flexible in the run sizes and allow the estimation of more effects.

The construction of nonregular designs remains a challenge. This paper introduces a new class

of nonregular designs called isomorphic foldovers design (IFD). Specifically, it is composed of

several foldovers of an initial design. The goal of our study is to investigate the general theory

of IFDs. We propose a method for obtaining all nonequivalent IFDs with f foldovers for any

initial design. Two algorithms are provided to construct optimal f-IFD in terms of G-aberration

(or G2-aberration) criterion. The IFD structure provides an efficient way to find good designs

in the sense that constructing good IFDs based on a nonregular initial design is often more

successful than doing so with a more granular single flat. Meanwhile, the IFDs have a parallel

flats structure and thus are much easier to understand and analyze than many other nonregular

designs. Moreover, we show that some existing designs can be viewed as special cases of IFDs.

Information

Preprint No.SS-2023-0401
Manuscript IDSS-2023-0401
Complete AuthorsChunyan Wang, Dennis K. J. Lin
Corresponding AuthorsChunyan Wang
Emailschunyanwang@ruc.edu.cn

References

  1. Addelman, S. (1961). Irregular fractions of the 2n factorial experiments. Technometrics 3, 479–496.
  2. Box, G. E. and Hunter, J. S. (1961). The 2k−p fractional factorial designs. Technometrics 3, 311–351.
  3. Butler, N. A. (2008). Defining equations for two-level factorial designs. J. Statist. Plann. Inference 138, 3157–3163.
  4. Chen, J. and Lin, D. K. J. (1991). On the identity relationships of 2k−p designs. J. Statist. Plann. Inference 28, 95–98.
  5. Chen, J., Sun, D. X. and Wu, C. F. J. (1993). A catalogue of two-level and three-level fractional factorial designs with small runs. Int. Stat. Rev. 61, 135–145.
  6. Cheng, C. S. (1998). Some hidden projection properties of orthogonal arrays with strength three. Biometrika 85, 491–495.
  7. Cheng, C. S. and Mee, R. W., and Yee, O. (2008). Second order saturated orthogonal arrays of strength three. Statist. Sinica 18, 105–119.
  8. Cheng, S. W., Li, W. and Ye, K. Q. (2004). Blocked nonregular two-level factorial designs. Technometrics 46, 269–279.
  9. Connor, W. S. and Young, S. (1961). Fractional Factorial Designs for Experiments with Factors at Two and Three Levels. National Bureau of Standards Applied Mathematics Series 58. National Bureau of Standards, Gaithersburg. TWO-LEVEL ISOMORPHIC FOLDOVERS DESIGNS
  10. Deng, L. Y. and Tang, B. (1999). Generalized resolution and minimum aberration criterion for Plackett-Burman and other nonregular factorial designs. Statist. Sinica 9, 1071–1082.
  11. Dokovi´c, D. ˇZ., Golubitsky, O. and Kotsireas, I. S. (2014). Some new orders of Hadamard and Skew-Hadamard matrices. J. Comb. Des. 22, 270–277.
  12. Draper, N. R. and Mitchell, T. J. (1967). The construction of saturated 2k−p R designs. Ann. Appl. Stat. 38, 1110–1126
  13. Eendebak, P. and Schoen, E. (2017). Complete series of non-isomorphic orthogonal arrays. http://pietereendebak.nl/oapage.
  14. Edwards, D. J. and Mee, R. W. (2023). Structure of two-level nonregular designs. J. Amer. Statist. Assoc. 118, 1222-1233.
  15. Elsawah, A. M. and Qin, H. (2015). A new strategy for optimal foldover two-level designs. Statist. Probab. Lett. 103, 116–126.
  16. Fang, K. T., Lin, D. K. J. and Qin, H. (2003). A note on optimal foldover design. Statist. Probab. Lett. 62, 245–250.
  17. Fontana, R., Pistone, G. and Rogantin, M. P. (2000). Classification of two-level factorial fractions. J. Statist. Plann. Inference 87, 149–172.
  18. Hansen, P. and Mladenovi´c, N. (2001). Variable neighborhood search: Principles and applications. Eur. J. Oper. Res. 130, 449–467. TWO-LEVEL ISOMORPHIC FOLDOVERS DESIGNS
  19. Hansen, P., Mladenovi´c, N. and ´Perez, J. A. M. (2008). Variable neighborhood search: Methods and applications. 4OR 6, 319–360.
  20. Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer, New York.
  21. 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.
  22. Li, H. and Mee, R. W. (2002). Better foldover fractions for resolution III 2k−p designs. Technometrics 44, 278–283.
  23. Li, W. W. and Lin, D. K. J. (2003). Optimal foldover two-level fractional factorial designs. Technometrics 45, 142–149.
  24. Li, W. W., Lin, D. K. J. and Ye, K. (2003). Optimal foldover plans for non-regular orthogonal designs. Technometrics 45, 347–351.
  25. Lin, D. K. J. and Draper, N. R. (1992). Projection Properties of Plackett and Burman designs. Technometrics 34, 423–428.
  26. Liu, Y., Yang, J. F. and Liu, M. Q. (2011). Isomorphism check in fractional factorial designs via letter interaction pattern matrix. J. Statist. Plann. Inference 141, 3055– 3062.
  27. Mee, R. W. (2009). A Comprehensive Guide to Factorial Two-Level Experimentation.
  28. Springer, New York. TWO-LEVEL ISOMORPHIC FOLDOVERS DESIGNS
  29. Meyer, R. K. and Nachtsheim, C. J. (1995). The coordinate-exchange algorithm for exact constructing optimal experimental designs. Technometrics 37, 60–69.
  30. Mladenovi´c, N. and Hansen, P. (1997). Variable neighborhood search. Comput. Oper. Res. 24, 1097–1100
  31. Montgomery, D. C. and Runger, G. C. (1996). Foldovers of 2k−p resolution IV experimental designs. J. Qual. Technol. 28, 446–450.
  32. Schoen, E. D., Eendebak, P. T. and Nguyen, M. V. M. (2010). Complete enumeration of pure-level and mixed-level orthogonal arrays. J. Comb. Des. 18, 123–140.
  33. Schoen, E. D. and Mee, R. W. (2012). Two-level designs of strength 3 and up to 48
  34. runs. (2012). J. R. Stat. Soc. C 61, 163–74.
  35. 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.
  36. Shi, C. and Tang, B. (2018). Designs from good Hadamard matrices. Bernoulli 24, 661–671.
  37. Shrivastava, A. K. and Ding, Y. (2010). Graph based isomorph-free generation of twolevel regular fractional factorial designs. J. Statist. Plann. Inference 140 169–179.
  38. Sun, D. X. (1993). Estimation capacity and related topics in experimental designs. unpublished doctoral thesis, University of Waterloo, Dept. of Statistics and Actuarial Science. TWO-LEVEL ISOMORPHIC FOLDOVERS DESIGNS
  39. Sun, D. X., Li, W. and Ye. K. Q. (2002) An algorithm for sequentially constructing nonisomorphic orthogonal designs and its applications. Technical report, Department of Applied Mathematics and Statistics, SUNY at Stony Brook.
  40. Tang, B. and Deng, L. Y. (1999). Minimum G2-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926.
  41. Vazquez, A. R. and Xu, H. (2019). Construction of two-level nonregular designs of strength three with large run sizes. Technometrics 61, 341–353.
  42. Vazquez, A. R, Goos, P. and Schoen, E. D. (2019). Constructing two-level designs by concatenation of strength-3 orthogonal arrays. Technometrics 61, 219-232.
  43. Vazquez, A. R., Schoen, E. D and Goos, P. (2022). Two-level orthogonal screening designs with 80, 96, and 112 runs, and up to 29 factors. J. Qual. Technol. 54, 338358.
  44. Wang, C. and Mee, R. W. (2021). Two-level parallel flats designs. Ann. Statist. 49, 3015–3042.
  45. Wang, C. and Mee, R. W. (2023). Fast construction of efficient two-level parallel flats designs. J. Statist. Plann. Inference, accepted.
  46. Webb, S. (1968). Non-orthogonal designs of even resolution. Technometrics 10 291–299.
  47. Wu, C. F. J. and Hamada, M. (2021). Experiments: Planning, Analysis and Optimization (3nd ed.). Wiley, New York. TWO-LEVEL ISOMORPHIC FOLDOVERS DESIGNS
  48. Xu, H. (2005). Some nonregular designs from the Nordstrom–Robinson code and their statistical properties. Biometrika 92, 385–397.
  49. Xu, H. (2009) Algorithmic construction of efficient fractional factorial designs with large run sizes. Technometrics 51 262–277.
  50. Xu, H. and Wong, A. (2007). Two-level nonregular designs from quaternary linear codes. Statist. Sinica 17, 1191–1213.
  51. Ye, K. Q. (2003). Indicators functions and its applications in two-level factorial design. Ann. Statist. 31, 984–994. Chunyan Wang

Acknowledgments

The authors would like to thank Professor Robert Mee for his valuable comments

and suggestions. This work was supported by the National Natural Science Foundation

of China (Grant Nos. 12301323 and 12131001), and the MOE Project of Key Research

Institute of Humanities and Social Sciences (22JJD110001).

Supplementary Materials

The online Supplementary Material includes S1: the proofs of Theorems 1–3 and

Corollaries 1 and 2; S2: the optimal foldover matrices for the three 6-IFDs in Table 1;

S3: the initial designs of the IFDs in Tables 1–4; S4: the initial designs of the IFDs in

Tables C.1 and C.2; and S5: The indicator function of design 10.48.

Supplementary materials are available for download.