Minimum Contamination and $β$-Aberration Criteria for Screening Quantitative Factors

Tang and Xu [Biometrika 101 (2014) 333-350] applied the minimum $\beta$-aberration criterion to selecting optimal designs for screening quantitative factors. They provided a statistical justification showing that minimum $\beta$-aberration criterion minimizes contamination of nonnegligible $k$th-order effects on the estimation of linear effects for $k=2,\cdots,r$, where $r$ is the strength of a design. Unfortunately, this result does not hold for $k>r$. In this paper, we provide a complete mathematical connection between $\beta$-wordlength patterns and contaminations (on the estimation of linear effects) and reveal that the minimum $\beta$-aberration criterion is not necessarily equivalent to the minimum contamination criterion for ranking designs. We prove that they are equivalent only when the number of factors of a design equals the strength plus one. We emphasize that the minimum $\beta$-aberration criterion, in fact, sequentially minimizes the contamination of nonnegligible $k$th-order effects on the estimation of the general mean, not on the estimation of linear effects. Therefore, the minimum contamination criterion should be more appropriate than the minimum $\beta$-aberration criterion for selecting optimal designs for screening quantitative factors.