Theory of optimal blocking of $2^{n-m}$ designs

Abstract

In this paper, we define the blocking wordlength pattern of a blocked fractional factorial design by combining the wordlength patterns of treatment-defining words and block-defining words. The concept of minimum aberration can be defined in terms of the blocking wordlength pattern and provides a good measure of the estimation capacity of a blocked fractional factorial design. By blending techniques of coding theory and finite projective geometry, we obtain combinatorial identities that govern the relationship between the blocking wordlength pattern of a blocked 2 ny m design and the split wordlength pattern of its blocked residual design. Based on these identities, we establish general rules for identifying minimum aberration blocked 2 ny m designs in terms of their blocked residual designs. Using these rules, we study the structures of some blocked 2 ny m designs with minimum aberration.

Reverse Foldovers for Blocked 2m−kFractional Factorial Designs

“…Then one design in this class which might potentially be used is D having treatment group G t D generators 1235 and 1246 and block group G b D generators 134b 1 and 234b 2 . We note that D is optimal among all 2 6+2 − 2+2 under the criteria of Sun et al (1997) and Chen and Cheng (1999). Then and W t D = 0 3 0 0 and W b D ∪ W bt D = 3 8 0 0 1 .…”

“…This inherent difference between added factors and blocking factors must be properly taken into account when assigning interactions in the basic factors to treatment and blocking factors in a 2 m+p − p+k designs. Bisgaard (1994), Sun et al (1997), Sitter et al (1997), and Chen and Cheng (1999) have all proposed different methods for obtaining optimal blocked regular 2 m−k FF designs. These different methods of selecting 2 m+p − p+k designs yields in some cases the same and in other cases different optimal designs.…”

Optimal foldover plans for regular -level fractional factorial designs

“…This includes Sitter et al (1997), Chen and Cheng (1999), Zhang and Park (2000), Cheng and Wu (2002) and Ai and Zhang (2004a). For detailed comparisons and reviews on the prevailing criteria refer to Ai and Zhang (2004b).…”

Assessing the bias in blocked two level fractional factorial designs with respect to confounding pattern

“…Sitter, et al [9] provided some collections of minimum aberration blocked designs with all 8 and 16 runs, 32 runs up to 15 factors, 64 runs up to 9 factors, and 128 runs up to 9 factors. Chen and Cheng [2] developed a theory to characterize minimum aberration blocked designs in terms of their blocked residual designs and gave collection of minimum aberration blocked designs with all 8 and 16 runs, and 32 runs up to 20 factors. Cheng and Wu [4] compared minimum aberration blocked designs with respect to different combined word length patterns for 8, 16 ,32 ,64 and128 runs up to 9 factors; they also provide collections of minimum aberration blocked designs with all 27 runs, and 81 runs up to 10 factors.…”