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Two-level screening designs derived from binary nonlinear codes

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Abstract

Nonregular fractional factorial designs can provide economical designs in screening experiments. In this paper, two criteria are proposed for evaluating the projectivity and uniformity properties of projections onto active factors in two-level nonregular fractional factorial designs. Moreover, two-level nonregular fractional factorial designs derived from binary nonlinear codes with 12, 24, 32 and 40 codewords and various lengths are evaluated using the new criteria. Such designs are also evaluated under the known E(s2) criterion for optimal designs in screening experiments, and are compared to Plackett-Burman designs or to projections of Plackett-Burman designs. Results show that some binary nonlinear codes can provide useful two-level nonregular fractional factorial designs in screening experiments. A search method is proposed for finding good designs with a large number of factors, starting from a good design with the same number of runs but with a smaller number of factors.

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References

  • Assmus, E. F., Jr., & Key, J. D. (1992). Hadamard matrices and their designs: a coding theoretic approach. Transactions of the American Mathematical Society, 330, 269–293.

    Article  MathSciNet  Google Scholar 

  • Best, M. R. (1980). Binary codes with minimum distance of four. IEEE Transactions on Information Theory, 26, 738–742.

    Article  MathSciNet  Google Scholar 

  • Booth, K. H. V., & Cox, D. R. (1962). Some systematic supersaturated designs. Technometrics, 4, 489–495.

    Article  MathSciNet  Google Scholar 

  • Bose, R. C. (1961). On some connections between the design of experiments and information theory. Bulletin of the International Statistical Institute, 38, 257–271.

    MathSciNet  MATH  Google Scholar 

  • Box, G. E. P., & Hunter, J. S. (1961). The 2kp fractional factorial designs. Technometrics, 3, 311–351.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Bulutoglu, D. A., & Cheng, C. S. (2003). Hidden projection properties of some nonregular fractional factorial designs and their applications. The Annals of Statistics, 31, 1012–1026.

    Article  MathSciNet  Google Scholar 

  • Cheng, C. S. (1995). Some projection properties of orthogonal arrays. Annals of Statistics, 23, 1223–1233.

    Article  MathSciNet  Google Scholar 

  • Cheng, C. S., Deng, L. Y., & Tang, B. (2002). Generalized minimum aberration and design efficiency for nonregular fractional factorial designs. Statistica Sinica, 12, 991–1000.

    MathSciNet  MATH  Google Scholar 

  • Cohen, G., Lobstein, A. C., & Sloane, N. (1986). Further results on the covering radius of codes. IEEE Transactions on Information Theory, 32, 680–694.

    Article  MathSciNet  Google Scholar 

  • Dean, A., & Lewis, S. (2006). Screening: Methods for experimentation in industry, drug discovery, and genetics. New York: Springer Science + Business Media.

    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, 1071–1082.

    MathSciNet  MATH  Google Scholar 

  • Eccleston, J. A., & Hedayat, A. (1974). On the theory of connected designs: Characterization and optimality. Annals of Statistics, 2, 1238–1255.

    Article  MathSciNet  Google Scholar 

  • Fang, K. T. (1980). The uniform design: application of number theoretic methods in experimental design. Acta Mathematicae Applicatae Sinica, 3, 363–372.

    MathSciNet  Google Scholar 

  • Fang, K. T., Lin, D. K. J., Winker, P., & Zhang, Y. (2000). Uniform design: Theory and application. Technometrics, 42, 237–248.

    Article  MathSciNet  Google Scholar 

  • Fang, K. T., & Ma, C. X. (2001). Wrap-around L2-discrepancy of random sampling, Latin hypercube and uniform designs. Journal of Complexity, 17, 608–624.

    Article  MathSciNet  Google Scholar 

  • Grassl, M., Shor, P., Smith, G., Smolin, J., & Zeng, B. (2012). New constructions of codes for asymmetric channels via concatenation. In International symposium on information theory proceedings (pp. 751–755). Cambridge, MA: IEEE.

    Google Scholar 

  • Hickernell, F. J. (1998). A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67, 299–322.

    Article  MathSciNet  Google Scholar 

  • Hickernell, F. J. (1999). Goodness-of-fit statistics, discrepancies and robust designs. Statistics & Probability Letters, 44, 73–78.

    Article  MathSciNet  Google Scholar 

  • Jaffe, D. B. (2000). Optimal binary linear codes of length ≤ 30. Discrete Mathematics, 223, 135–155.

    Article  MathSciNet  Google Scholar 

  • Katsaounis, T. I., & Dean, A. M. (2008). A survey and evaluation of methods for determination of combinatorial equivalence of factorial designs. Journal of Statistical Planning and Inference, 138, 245–258.

    Article  MathSciNet  Google Scholar 

  • Kole, B., Gangwani, J., Gupta, V. K., & Parsad, R. (2010). Two-level supersaturated designs, a review. Journal of Statistical Theory and Practice, 4, 598–608.

    Article  MathSciNet  Google Scholar 

  • Liu, M. Q., & Hickernell, F. J. (2002). E(s2)-optimality and minimum discrepancy in 2-level supersaturated designs. Statistica Sinica, 12, 931–939.

    MathSciNet  MATH  Google Scholar 

  • Ma, C. X., Fang, K. T., & Lin, D. K. J. (2001). On the isomorphism of fractional factorial designs. Journal of Complexity, 17, 86–97.

    Article  MathSciNet  Google Scholar 

  • Ma, C. X., Fang, K. T., & Lin, D. K. J. (2003). A note on uniformity and orthogonality. Journal of Statistical Planning and Inference, 113, 323–334.

    Article  MathSciNet  Google Scholar 

  • McWilliams, F. J., & Sloane, N. J. A. (1977). The theory of error correcting codes. New York: North Holland Mathematical Library.

    Google Scholar 

  • Nordstrom, A. W., & Robinson, J. P. (1967). An optimum nonlinear code. Information and Control, 11, 613–616.

    Article  Google Scholar 

  • Phoa, F. K. H., Xu, H., & Wong, W. K. (2009). The use of nonregular fractional factorial designs in combination toxicity studies. Food and Chemical Toxicology, 47, 2183–2188.

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Pless, V. (1989). Wiley-Interscience Series in Discrete Mathematics and Optimization, Introduction to the Theory of Error-Correcting Codes (2nd ed.). New York: Wiley & Sons, inc.

    Google Scholar 

  • Plotkin, M. (1960). Binary codes with specified minimum distance. IRE Transactions on Information Theory, 6, 445–450.

    Article  MathSciNet  Google Scholar 

  • Qin, H., & Ai, M. (2007). A note on connection between uniformity and generalized minimum aberration. Statistical Papers, 48, 491–502.

    Article  MathSciNet  Google Scholar 

  • Satterthwaite, F. E. (1959). Random balance experimentation (with discussions). Technometrics, 1, 111–137.

    Article  MathSciNet  Google Scholar 

  • Sloane, N. J. A., & Whitehead, D. (1970). New family of single-error correcting codes. IEEE Transactions on Information Theory, 16, 717–719.

    Article  MathSciNet  Google Scholar 

  • Sloane, N. J. A., & Wiley, J. (1993). Covering arrays and intersecting codes. Journal of Combinatorial Designs, 1, 51–63.

    Article  MathSciNet  Google Scholar 

  • Tang, B. (2001). Theory of J-characteristic for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika, 88, 401–407.

    Article  MathSciNet  Google Scholar 

  • Tang, B., & Deng, L. Y. (1999). Minimum G2-aberration for nonregular fractional factorial designs. Annals of Statistics, 27, 1914–1926.

    Article  MathSciNet  Google Scholar 

  • Xu, H., & Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Annals of Statistics, 29, 1066–1077.

    Article  MathSciNet  Google Scholar 

  • Ye, K. Q. (2003). Indicator function and its application in two-level factorial designs. Annals of Statistics, 31, 984–994.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Mansfield, OH, 44906, USA

    T. I. Katsaounis

  2. Department of Mathematics, University of Memphis, Memphis, TN, 38152, USA

    Manohar L. Aggarwal

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  1. T. I. Katsaounis
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  2. Manohar L. Aggarwal
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Correspondence to T. I. Katsaounis.

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Katsaounis, T.I., Aggarwal, M.L. Two-level screening designs derived from binary nonlinear codes. J. Korean Stat. Soc. 45, 210–220 (2016). https://doi.org/10.1016/j.jkss.2015.10.001

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  • DOI: https://doi.org/10.1016/j.jkss.2015.10.001

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