The Annals of Statistics

A code arithmetic approach for quaternary code designs and its application to (1/64)th-fractions

Frederick K. H. Phoa

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Abstract

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper aims at exploring the fundamental structure and developing a theory to characterize the wordlengths and aliasing indexes for a general $(1/4)^{p}$th-fraction QC design. Then the theory is applied to $(1/64)$th-fraction QC designs. Examples are given, indicating that there exist some QC designs that have better design properties, and are thus more cost-efficient, than the regular fractional factorial designs of the same size. In addition, a result about the periodic structure of $(1/64)$th-fraction QC designs regarding resolution is stated.

Article information

Source
Ann. Statist., Volume 40, Number 6 (2012), 3161-3175.

Dates
First available in Project Euclid: 22 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1361542078

Digital Object Identifier
doi:10.1214/12-AOS1069

Mathematical Reviews number (MathSciNet)
MR3097973

Zentralblatt MATH identifier
1296.62154

Subjects
Primary: 62K15: Factorial designs

Keywords
Quaternary-code design generalized minimum aberration generalized resolution generalized wordlength pattern aliasing index structure periodicity

Citation

Phoa, Frederick K. H. A code arithmetic approach for quaternary code designs and its application to (1/64)th-fractions. Ann. Statist. 40 (2012), no. 6, 3161--3175. doi:10.1214/12-AOS1069. https://projecteuclid.org/euclid.aos/1361542078


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References

  • Cheng, S.-W., Li, W. and Ye, K. Q. (2004). Blocked nonregular two-level factorial designs. Technometrics 46 269–279.
  • 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.
  • Mukerjee, R. and Wu, C. F. J. (2006). A Modern Theory of Factorial Designs. Springer, New York.
  • Phoa, F. K. H., Mukerjee, R. and Xu, H. (2012). One-eighth- and one-sixteenth-fraction quaternary code designs with high resolution. J. Statist. Plann. Inference 142 1073–1080.
  • Phoa, F. K. H. and Xu, H. (2009). Quarter-fraction factorial designs constructed via quaternary codes. Ann. Statist. 37 2561–2581.
  • Tang, B. (2001). Theory of $J$-characteristics for fractional factorial designs and projection justification of minimum $G_2$-aberration. Biometrika 88 401–407.
  • Tang, B. and Deng, L.-Y. (1999). Minimum $G_2$-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926.
  • Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.
  • Xu, H., Phoa, F. K. H. and Wong, W. K. (2009). Recent developments in nonregular fractional factorial designs. Stat. Surv. 3 18–46.
  • Xu, H. and Wong, A. (2007). Two-level nonregular designs from quaternary linear codes. Statist. Sinica 17 1191–1213.
  • Zhang, R., Phoa, F. K. H., Mukerjee, R. and Xu, H. (2011). A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions. Ann. Statist. 39 931–955.