Further Study On Properties Of β-wordlength Pattern For Three-level Designs

Minimum aberration criterion is an important concept in Experimental Designand is often adopted to compare diferent fractional factorial designs with qualitativefactors. However, when a design contains quantitative factors, a polynomial modelis often established to conduct statistical inference. Two designs that are combina-torially equivalent to each other may have diferent statistical inference ability underthe polynomial model. However, these two designs cannot be distinguished by mini-mum aberration criterion. Under this consideration, Cheng and Ye (2004) proposedthe concept of β-wordlength pattern, which can be regarded as a generalized versionof the minimum aberration criterion. They pointed out that a good design shouldsequentially minimize its β1, β2,...,βK. The β-wordlength pattern is defned usingcharacteristic polynomial, which makes its calculation much more complicated. Tangand Xu (2014) provided some interesting properties of the β-wordlength pattern forthree-level regular designs, but their method and result cannot be directly applied fornon-regular designs. In this article, we will introduce a parameter m, and provide asimple condition, which can be used to determine whether the β3of the design is0. So,we can compare all permuted designs having β3=0, and fnally select the best one.