Some Properties Of β-wordlength Pattern For Four-level Designs

Fractional factorial designs have played a prominent role in the theory and prac-tice of experimental design, thus it becomes a key issue for statisticians to provide an appropriate criterion to compare different fractional factorial designs. For designs with qualitative factors under an ANOVA model, the minimum aberration criterion has been frequently used; however, for designs with quantitative factors, a polynomial regression model is often established, then the β-wordlength pattern can be employed to compare different fractional factorial designs. Introduced in2004by Cheng and Ye, the β-wordlength pattern can be regarded as a modification of wordlength pattern. It is originally defined using characteristic polynomials, which makes it difficult to calculate. To simply the computation complexity, some properties of β-wordlength pattern for three-level designs have not been investigated. However, due to the different construc-tion, the properties of β-wordlength pattern for four-level designs will be more compli-cated. In this paper, we show that if a unified transformation is used for a minimum aberration design4n-1, it is impossible to obtain a design with βn=0. However, for some designs, there exist appropriate maps, which make the designs mirror-symmetric. These properties can help find better designs with four-level quantitative factors.