Minimum $$\theta $$ -aberration criterion for designs with qualitative and quantitative factors

The minimum aberration criterion is popular for selecting good designs with qualitative factors under an ANOVA model, and the minimum \(\beta \)-aberration criterion is more suitable for selecting designs with quantitative factors under a polynomial model. However, numerous computer experiments involve both qualitative and quantitative factors, while there is a lack of a reasonable criterion to assess the effectiveness of such designs. This paper proposes some important properties of the \(\beta \)-wordlength pattern for mixed-level designs, and introduces the minimum \(\theta \)-aberration criterion for comparing and selecting designs with qualitative and quantitative factors based on a full model involving all interactions of the factors. The computation of the \(\theta \)-wordlength pattern is optimized by the generalized wordlength enumerator. Then we provide some construction methods for designs with less \(\theta \)-aberration, and apply this criterion to screen the marginally coupled designs and the doubly coupled designs.