Abstract
Designing their experiments is the significant problem that experimenters face. Maximin distance designs, supersaturated designs, minimum aberration designs, uniform designs, minimum moment designs and orthogonal arrays are arguably the most exceedingly used designs for many real-life experiments. From different perspectives, several criteria have been proposed for constructing these designs for investigating quantitative or qualitative factors. Each of those criteria has its pros and cons and thus an optimal criterion does not exist, which may confuse investigators searching for a suitable criterion for their experiment. Some logical questions are now arising, such as are these designs consistent, can an optimal design via a specific criterion perform well based on another criterion and can an optimal design for screening quantitative factors be optimal for qualitative factors? Through theoretical justifications, this paper tries to answer these interesting questions by building some bridges among these designs. Some conditions under which these designs agree with each other are discussed. These bridges can be used to select a suitable criterion for studying some hard problems effectively, such as detection of (combinatorial/geometrical) non-isomorphism among designs and construction of optimal designs. Benchmarks for reducing the computational complexity are given.
Similar content being viewed by others
References
Booth, K. H. V., & Cox, D. R. (1962). Some systematic supersaturated designs. Technometrics, 4, 489–495.
Chen, W., Qi, Z. F., & Zhou, Y. D. (2015). Constructing uniform designs under mixture discrepancy. Statistics and Probability Letters, 97, 76–82.
Cheng, C. S. (1997). \(E(s^2)\)-optimal supersaturated designs. Statistica Sinica, 7, 929–939.
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.
Cheng, C. S., Steinberg, D. M., & Sun, D. X. (1999). Minimum aberration and model robustness for two-level fractional factorial designs. Journal of the Royal Statistical Society: Series, 61, 85–93.
Cheng, S.-W., & Ye, K. Q. (2004). Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. The Annals of Statistics, 32(5), 2168–2185.
Dey, A., & Mukerjee, R. (1999). Fractional factorial plans. New York: Wiley.
Elsawah, A. M. (2019). Constructing optimal router bit life sequential experimental designs: New results with a case study. Communications in Statistics-Simulation and Computation, 48(3), 723–752.
Elsawah, A. M. (2018). Choice of optimal second stage designs in two-stage experiments. Computational Statistic, 33(2), 933–965.
Elsawah, A. M. (2017a). A closer look at de-aliasing effects using an efficient foldover technique. Statistics, 51(3), 532–557.
Elsawah, A. M. (2017b). A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs. Australian & New Zealand Journal of Statistics, 59(1), 17–41.
Elsawah, A. M., Fang, K. T., & Ke, X. (2019). New recommended designs for screening either qualitative or quantitative factors. Statistical Papers,. https://doi.org/10.1007/s00362-019-01089-9.
Elsawah, A. M., & Qin, H. (2015a). A new strategy for optimal foldover two-level designs. Statistics & Probability Letters, 103, 116–126.
Elsawah, A. M., & Qin, H. (2015b). Mixture discrepancy on symmetric balanced designs. Statistics & Probability Letters, 104, 123–132.
Fang, K. T. (1980). The uniform design: Application of number-theoretic methods in experimental design. Acta Mathematicae Applicatae Sinica, 3, 363–372.
Fang, K. T., Ke, X., & Elsawah, A. M. (2017). Construction of uniform designs via an adjusted threshold accepting algorithm. Journal of Complexity, 43, 28–37.
Fang, K. T., Lu, X., & Winker, P. (2003). Lower bounds for centered and wrap-around \(L_2\)-discrepancies and construction of uniform designs by threshold accepting. Journal of Complexity, 19, 692–711.
Fang, K. T., Li, R., & Sudjianto, A. (2006). Design and modeling for computer experiments. New York: CRC Press.
Fang, K. T., Lin, D. K. J., Winker, P., & Zhang, Y. (2000). Uniform design: Theory and application. Technometrics, 42, 237–248.
Fang, K. T., Ma, C. X., & Mukerjee, R. (2002). Uniformity in fractional factorials. In K. T. Fang, F. J. Hickernell, & H. Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo methods in scientific computing. Berlin: Springer.
Fang, K. T., & Mukerjee, R. (2000). A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika, 87, 93–198.
Fang, K. T., & Wang, Y. (1994). Number-theoretic methods in statistics. London: Chapman and Hall.
Franek, L., & Jiang, X. (2013). Orthogonal design of experiments for parameter learning in imagesegmentation. Signal Processing, 93, 1694–1704.
Fries, A., & Hunter, W. G. (1980). Minimum aberration \(2^{k-p}\) designs. Technometrics, 22, 601–608.
Hedayat, A. S., Sloane, N. J., & Stufken, J. (1999). Orthogonal arrays: Theory and application. Berlin: Springer.
Hickernell, F. J. (1998a). A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67, 299–322.
Hickernell, F. J. (1998b). Lattice rules: How well do they measure up? In P. Hellekalek & G. Larcher (Eds.), Random and Quasi-random point sets. In: Lecture notes in statistics (Vol. 138, pp. 109–166). New York: Springer.
Hou, H., Yue, S., Huang, X., & Wang, H. (2015). Application of orthogonal design to optimize flow pattern transition conditions. Sensor Review, 35(4), 425–431.
Johnson, M. E., Moore, L. M., & Ylvisaker, D. (1990). Minimax and maximin distance design. Journal of Statistical Planning and Inference, 26, 131–148.
Lin, D. K. J. (1993). A new class of supersaturated designs. Technometrics, 35, 28–31.
Liang, Y. Z., Fang, K. T., & Xu, Q. S. (2001). Uniform design and its applications in chemistry and chemical engineering. Chemometrics and Intelligent Laboratory System, 58, 43–57.
Liang, J., Dong, L., Wang, M. D., Zhang, X. Y., & Hou, J. Z. (2016). Application of orthogonal design to the extraction and HPLC analysis of sedimentary pigments from lakes of the Tibetan Plateau. Science China Earth Sciences, 59(6), 1195–1205.
Liu, M. Q., Fang, K. T., & Hickernell, F. J. (2006). Connection among different criteria for asymmetrical fractional factorial designs. Statistica Sinica, 16, 1285–297.
Lu, X., Fang, K. T., Xu, Q. F. & Yin, J. X. (2002). Balance pattern and BP-optimal factorial designs. Technical Report-324, Hong Kong Baptist University.
Luss, H. (1999). On equitable resource allocation problems: A lexicographic minimax approach. Operations Research, 47(3), 361–378.
Ma, C. X., & Fang, K. T. (2001). A note on generalized aberration in factorial designs. Metrika, 53, 85–93.
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.
Morris, M. D., & Mitchell, T. J. (1995). Exploratory designs for computational experiments. Journal of Statistical Planning and Inference, 43, 381–402.
Owen, B. (1992). Orthogonal arrays for computer experiments, integration and visualization. Statistica Sinica, 2(2), 439–452.
Pang, F., & Liu, M. Q. (2012). A note on connections among criteria for asymmetrical factorials. Metrika, 75, 23–32.
Qin, H., & Ai, M. (2007). A note on the connection between uniformity and generalized minimum aberration. Statistical Papers, 48, 491–502.
Qin, H., & Chen, Y. B. (2004). Some results on generalized minimum aberration for symmetrical fractional factorial designs. Statistics & Probability Letters, 66, 51–57.
Qin, H., & Fang, K. T. (2004). Discrete discrepancy infactorial designs. Metrika, 60, 59–72.
Qin, H., & Li, D. (2006). Connection between uniformity and orthogonality for symmetrical factorial designs. Journal of Statistical Planning and Inference, 136, 2770–2782.
Qin, H., Zou, N., & Chatterjee, K. (2009). Connection between uniformity and minimummoment aberration. Metrika, 70, 79–88.
Sun, F., Chen, J., & Liu, M. Q. (2011). Connections between uniformity and aberration in general multi-level factorials. Metrika, 73, 305–315.
Tang, B. (2001). Theory of J-characteristics for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika, 88, 401–407.
Tang, B., & Deng, L. Y. (1999). Minimum \(G_2\)-aberration for non-regular fractional factorial designs. Annals of Statistics, 27, 1914–1926.
Tang, Y., & Xu, H. (2013). An effective construction method for multi-level uniform designs. Journal of Statistical Planning and Inference, 143, 1583–1589.
Tang, Y., Xu, H., & Lin, D. K. J. (2012). Uniform fractional factorial designs. Annals of Statistics, 40, 891–907.
Vanli, O. A., Zhang, C., Nguyen, A., & Wang, B. (2012). A minimax sensor placement approach for damage detection in composite structures. Journal of Intelligent Material Systems and Structures, 23, 919–932.
Wang, Y., & Fang, K. T. (1981). A note on uniform distribution and experimental design. Chinese Science Bulletin, 26, 485–489.
Wu, H. (2013). Application of orthogonal experimental design for the automatic software testing. Proceedings of the 2nd ICCSEE.
Xu, H. (2003). Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica, 13, 691–708.
Xu, H., & Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. The Annals of Statistics, 29, 549–560.
Xu, G., Zhang, J., & Tang, Y. (2014). Level permutation method for constructing uniform designs under the wrap-around \(L_2\)-discrepancy. Journal of Complexity, 30, 46–53.
Xu, Q. S., Xu, Y. D., Li, L., & Fang, K. T. (2018). Uniform experimental design in chemometrics. Journal of Chemometrics,. https://doi.org/10.1002/cem.3020.
Yamada, S., & Lin, D. K. J. (1999). Three-level supersaturated designs. Statistics & Probability Letters, 45, 31–39.
Yamada, S., & Matsui, T. (2002). Optimality of mixed-level supersaturated designs. The Journal of Statistical Planning and Inference, 104, 459–468.
Zhang, A., Fang, K. T., Li, R., & Sudjianto, A. (2005). Majorization framework for balanced lattice designs. The Annals of Statistics, 33(6), 2837–2853.
Zhou, Y. D., Fang, K. F., & Ning, J.-H. (2013). Mixture discrepancy for quasi-randompoint sets. Journal of Complexity, 29, 283–301.
Zhou, Y. D., & Xu, H. (2014). Space-filling fractional factorial designs. Journal of the American Statistical Association, 109(507), 1134–1144.
Zhou, Y. D., Ning, J. H., & Song, X. B. (2008). Lee discrepancy and its applications in experimental designs. Statistics & Probability Letters, 78, 1933–1942.
Acknowledgements
The author greatly appreciate valuable comments and suggestions of the two referees and the Associate Editor that significantly improved the paper. The author greatly appreciate the kind support of Prof. Kai-Tai Fang and Prof. Hong Qin. This work was partially supported by the UIC Grants (Nos. R201810 and R201912) and the Zhuhai Premier Discipline Grant.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
There is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Elsawah, A.M. Building some bridges among various experimental designs. J. Korean Stat. Soc. 49, 55–81 (2020). https://doi.org/10.1007/s42952-019-00004-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42952-019-00004-0
Keywords
- Orthogonality
- Uniformity
- Discrepancy
- Aberration
- Moment aberration
- Hamming distance
- Combinatorial isomorphism
- Geometrical isomorphism