Abstract

Given a set of parameters, several non-isomorphic order-of-addition

orthogonal arrays can be generated to design an order-of-addition experiment.

Under resource constraints, selecting the best from these candidate designs for the

experiment can be practical to extract as much information as possible from the

observed data. Based on some theoretical results developed for two-level orthogonal arrays, a series of numerical indices called centralized generalized wordlength

pattern is proposed in this paper to characterize and compare order-of-addition

orthogonal arrays. Specifically, the J-characteristics are first justified for pairwise

order matrices when the transitive property of pairwise order factors is taken into

account. The centralized generalized wordlength pattern is then defined based

on the sums of squared differences between the normalized J-characteristics of

the pairwise order matrices determined by the fractional and full designs. Essentially, it can be viewed as a natural extension of the generalized wordlength

pattern used for two-level orthogonal arrays.

Their functional relationship is

further simplified such that the computational cost can be reduced significantly.

Some optimal order-of-addition orthogonal arrays with economical run sizes are

identified from existing catalogues for future work.

Information

Preprint No.SS-2024-0191
Manuscript IDSS-2024-0191
Complete AuthorsShin-Fu Tsai
Corresponding AuthorsShin-Fu Tsai
Emailsshinfu@ntu.edu.tw

References

  1. B´ona, M. (2022). Combinatorics of permutations. Boca Raton, FL: Chapman and Hall/CRC.
  2. Chen, J., Mukerjee, R. and Lin, D. K.-J. (2020). Construction of optimal fractional order-ofaddition designs via block designs. Statistics and Probability Letters, 161:108728.
  3. Cheng, C.-S., Deng, L.-Y. and Tang, B. (2002). Generalized minimum aberration and design efficiency for nonregular fractional factorial designs. Statistica Sinica, 12, 991-1000.
  4. Cheng, C.-S., Mee, R. W. and Yee, O. (2008). Second order saturated orthogonal arrays of strength three. Statistica Sinica, 18, 105-119.
  5. Deng, L.-Y. and Tang, B. (1999). Generalized resolution and minimum aberration criteria for Plackett-Burman and other nonregular factorial design. Statistica Sinica, 9, 1071-1082.
  6. Deng, L.-Y. and Tang, B. (2002). Design selection and classification for Hadamard matrices using generalized minimum aberration criteria. Technometrics, 44, 173-184.
  7. Fries, A. and Hunter, W. G. (1980). Minimum aberration 2k−p designs. Technometrics, 22, 602-608.
  8. Huang, Y. and Yang, J.-F. (2025). Robust design for order-of-addition experiments. Technometrics, 67, 168-176.
  9. Ma, C.-X. and Fang, K.-T. (2001). A note on generalized aberration in factorial designs. Metrika, 53, 85-93.
  10. Mee, R. W. (2020). Order-of-addition modeling. Statistica Sinica, 30, 1543-1559.
  11. Peng, J., Mukerjee, R. and Lin, D. K.-J. (2019). Design of order-of-addition experiments. Biometrika, 106, 683-694.
  12. Schoen, E. D. and Mee, R. W. (2023). Order-of-addition orthogonal arrays to study the effect of treatment ordering. The Annals of Statistics, 51, 1877-1894.
  13. Seiden, E. and Zemach, R. (1966). On orthogonal arrays. The Annals of Mathematical Statistics, 37, 1355-1370.
  14. Stokes, Z. and Xu, H. (2022). A position-based approach for design and analysis of order-ofaddition experiments. Statistica Sinica, 32, 1467-1488.
  15. Tang, B. (2001). Theory of J-characteristics for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika, 88, 401-407.
  16. Tang, B. and Deng, L.-Y. (1999). Minimum G2-aberration for nonregular fractional factorial designs. The Annals of Statistics, 27, 1914-1926.
  17. Tsai, S.-F. (2022). Generating optimal order-of-addition designs with flexible run sizes. Journal of Statistical Planning and Inference, 218, 147-163.
  18. Tsai, S.-F. (2023a). Analyzing dispersion effects from replicated order-of-addition experiments.
  19. Journal of Quality Technology, 55, 271-288.
  20. Tsai, S.-F. (2023b). Dual-orthogonal arrays for order-of-addition two-level factorial experiments. Technometrics, 65, 388-395.
  21. Van Nostrand, R. C. (1995). Design of experiments where the order of addition is important. In ASA Proceedings of the Section on Physical and Engineering Sciences, 155-160. Alexandria, VA: American Statistical Association.
  22. Voelkel, J. G. (2019). The design of order-of-addition experiments. Journal of Quality Technology, 51, 230-241.
  23. Voelkel, J. G. and Gallagher, K. P. (2019). The design and analysis of order-of-addition experiments: An introduction and case study. Quality Engineering, 31, 627-638.
  24. Wang, A., Xu, H. and Ding, X. (2020). Simultaneous optimization of drug combination dose-ratio sequence with innovative design and active learning. Advanced Therapeutics, 3:1900135.
  25. Wang, C. and Lin, D. K.-J. (2023). Interaction effects in pairwise ordering model. Journal of Quality Technology, 55, 463-468.
  26. Wang, C. and Mee, R. W. (2022). Saturated and supersaturated order-of-addition designs. Journal of Statistical Planning and Inference, 219, 204-215.
  27. Xu, H. (2002). An algorithm for constructing orthogonal and nearly-orthogonal arrays with mixed levels and small runs. Technometrics, 44, 356-368.
  28. Xu, H. (2003). Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica, 13, 691-708.
  29. Xu, H. and Wu, C.-F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. The Annals of Statistics, 29, 1066-1077.
  30. Yamada, S. and Lin, D. K.-J. (1999). Three-level supersaturated designs. Statistics and Probability Letters, 45, 31-39.
  31. Zhao, S., Dong, Z. and Zhao, Y. (2022). Order-of-addition orthogonal arrays with high strength. Mathematics, 10:1187.
  32. Zhao, Y., Lin, D. K-J. and Liu, M.-Q. (2022). Optimal designs for order-of-addition experiments. Computational Statistics and Data Analysis, 165:107320. Shin-Fu Tsai

Acknowledgments

I would like to thank the editor, associate editor and all anonymous referees for their valuable comments and constructive suggestions. I am also

grateful to Computer and Information Networking Center, National Taiwan University for the support of high-performance computing facilities.

Shin-Fu Tsai’s research was supported by National Science and Technology

Council of Taiwan (Grant Number NSTC 113-2118-M-002-010-MY2).

Supplementary Materials

of this paper include the following sections.

(S1) Some values of Aa(U)

(S2) Code

(S3) Design matrix of an OofA-OA(48, 9, 2)

Supplementary materials are available for download.