Construction of optimal multi-level supersaturated designs. (English) Zbl 1084.62070

Xu, Hongquan; Wu, C. F. J.

Ann. Stat. 33, No. 6, 2811-2836 (2005).

Summary: A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by the authors [ibid. 29, No. 4, 1066–1077 (2001; Zbl 1041.62067)]. A new lower bound is derived and general construction methods are proposed for multi-level supersaturated designs.
Inspired by the S. Addelman and O. Kempthorne’s construction of orthogonal arrays [Ann. Math. Stat. 32, 1167–1176 (1961; Zbl 0107.36001)] several classes of optimal multi-level supersaturated designs are given in explicit form: Columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of the resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties.

MSC:

62K15	Factorial statistical designs
62K05	Optimal statistical designs
05B15	Orthogonal arrays, Latin squares, Room squares

Keywords:

Addelman-Kempthorne construction; additive character; Galois field; generalized minimum aberration; orthogonal array; supersaturated design

Citations:

Zbl 1041.62067; Zbl 0107.36001

References:

[1]	Addelman, S. and Kempthorne, O. (1961). Some main-effect plans and orthogonal arrays of strength two. Ann. Math. Statist. 32 1167–1176. · Zbl 0107.36001 · doi:10.1214/aoms/1177704855
[2]	Aggarwal, M. L. and Gupta, S. (2004). A new method of construction of multi-level supersaturated designs. J. Statist. Plann. Inference 121 127–134. · Zbl 1038.62068 · doi:10.1016/S0378-3758(02)00504-9
[3]	Ai, M.-Y. and Zhang, R.-C. (2004). Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60 279–285. · Zbl 1083.62071 · doi:10.1007/s001840300310
[4]	Booth, K. H. V. and Cox, D. R. (1962). Some systematic supersaturated designs. Technometrics 4 489–495. · Zbl 0109.12201 · doi:10.2307/1266285
[5]	Box, G. E. P. and Meyer, R. D. (1986). An analysis for unreplicated fractional factorials. Technometrics 28 11–18. · Zbl 0586.62168 · doi:10.2307/1269599
[6]	Bulutoglu, D. A. and Cheng, C.-S. (2004). Construction of \(E(s^2)\)-optimal supersaturated designs. Ann. Statist. 32 1662–1678. · Zbl 1105.62362 · doi:10.1214/009053604000000472
[7]	Butler, N. A., Mead, R., Eskridge, K. M. and Gilmour, S. G. (2001). A general method of constructing \(E(s^2)\)-optimal supersaturated designs. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 621–632. · Zbl 1040.62064 · doi:10.1111/1467-9868.00303
[8]	Chatterjee, K. and Gupta, S. (2003). Construction of supersaturated designs involving \(s\)-level factors. J. Statist. Plann. Inference 113 589–595. · Zbl 1014.62093 · doi:10.1016/S0378-3758(02)00109-X
[9]	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. · Zbl 0725.62068 · doi:10.1214/aos/1176348135
[10]	Cheng, C.-S. (1997). \(E (s^2)\)-optimal supersaturated designs. Statist. Sinica 7 929–939. · Zbl 1067.62560
[11]	Cheng, C.-S., Deng, L.-Y. and Tang, B. (2002). Generalized minimum aberration and design efficiency for nonregular fractional factorial designs. Statist. Sinica 12 991–1000. · Zbl 1004.62065
[12]	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. · Zbl 0942.62084
[13]	Fang, K.-T., Lin, D. K. J. and Ma, C.-X. (2000). On the construction of multi-level supersaturated designs. J. Statist. Plann. Inference 86 239–252. · Zbl 0964.62078 · doi:10.1016/S0378-3758(99)00163-9
[14]	Fries, A. and Hunter, W. G. (1980). Minimum aberration \(2^k-p\) designs. Technometrics 22 601–608. · Zbl 0453.62063 · doi:10.2307/1268198
[15]	Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays : Theory and Applications . Springer, New York. · Zbl 0935.05001
[16]	Li, W. W. and Wu, C. F. J. (1997). Columnwise–pairwise algorithms with applications to the construction of supersaturated designs. Technometrics 39 171–179. · Zbl 0889.62066 · doi:10.2307/1270905
[17]	Lidl, R. and Niederreiter, H. (1997). Finite Fields , 2nd ed. Cambridge Univ. Press. · Zbl 1139.11053
[18]	Lin, D. K. J. (1993). A new class of supersaturated designs. Technometrics 35 28–31.
[19]	Lin, D. K. J. (1995). Generating systematic supersaturated designs. Technometrics 37 213–225. · Zbl 0822.62062 · doi:10.2307/1269622
[20]	Liu, Y. and Dean, A. (2004). \(k\)-circulant supersaturated designs. Technometrics 46 32–43.
[21]	Lu, X., Hu, W. and Zheng, Y. (2003). A systematic procedure in the construction of multi-level supersaturated design. J. Statist. Plann. Inference 115 287–310. · Zbl 1127.62395 · doi:10.1016/S0378-3758(02)00116-7
[22]	Lu, X. and Sun, Y. (2001). Supersaturated design with more than two levels. Chinese Ann. Math. Ser. B 22 183–194. · Zbl 0972.62056 · doi:10.1142/S0252959901000188
[23]	Ma, C.-X. and Fang, K.-T. (2001). A note on generalized aberration in factorial designs. Metrika 53 85–93. · Zbl 0990.62067 · doi:10.1007/s001840100112
[24]	Mukerjee, R. and Wu, C. F. J. (1995). On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist. 23 2102–2115. · Zbl 0897.62083 · doi:10.1214/aos/1034713649
[25]	Nguyen, N.-K. (1996). An algorithmic approach to constructing supersaturated designs. Technometrics 38 69–73. · Zbl 0900.62416 · doi:10.2307/1268904
[26]	Satterthwaite, F. E. (1959). Random balance experimentation. Technometrics 1 111–137.
[27]	Tang, B. (2001). Theory of \(J\)-characteristics for fractional factorial designs and projection justification of minimum \(G_2\)-aberration. Biometrika 88 401–407. · Zbl 0984.62053 · doi:10.1093/biomet/88.2.401
[28]	Tang, B. and Deng, L.-Y. (1999). Minimum \(G_2\)-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926. · Zbl 0967.62055 · doi:10.1214/aos/1017939244
[29]	Tang, B. and Wu, C. F. J. (1997). A method for constructing supersaturated designs and its \(E (s^2)\) optimality. Canad. J. Statist. 25 191–201. · Zbl 0891.62054 · doi:10.2307/3315731
[30]	Wu, C. F. J. (1993). Construction of supersaturated designs through partially aliased interactions. Biometrika 80 661–669. · Zbl 0800.62483 · doi:10.1093/biomet/80.3.661
[31]	Wu, C. F. J. and Hamada, M. (2000). Experiments : Planning , Analysis and Parameter Design Optimization . Wiley, New York. · Zbl 0964.62065
[32]	Xu, H. (2002). An algorithm for constructing orthogonal and nearly-orthogonal arrays with mixed levels and small runs. Technometrics 44 356–368.
[33]	Xu, H. (2003). Minimum moment aberration for nonregular designs and supersaturated designs. Statist. Sinica 13 691–708. · Zbl 1028.62063
[34]	Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29 1066–1077. · Zbl 1041.62067 · doi:10.1214/aos/1013699993
[35]	Xu, H. and Wu, C. F. J. (2003). Construction of optimal multi-level supersaturated designs. UCLA Statistics Electronic Publications, Preprint 356. Available at preprints.stat.ucla.edu/. · Zbl 1084.62070 · doi:10.1214/009053605000000688
[36]	Yamada, S., Ikebe, Y. T., Hashiguchi, H. and Niki, N. (1999). Construction of three-level supersaturated design. J. Statist. Plann. Inference 81 183–193. · Zbl 0939.62083 · doi:10.1016/S0378-3758(99)00007-5
[37]	Yamada, S. and Lin, D. K. J. (1999). Three-level supersaturated designs. Statist. Probab. Lett. 45 31–39. · Zbl 0958.62071 · doi:10.1016/S0167-7152(99)00038-3
[38]	Yamada, S. and Matsui, T. (2002). Optimality of mixed-level supersaturated designs. J. Statist. Plann. Inference 104 459–468. · Zbl 0992.62069 · doi:10.1016/S0378-3758(01)00248-8