Minimum Contamination and β-Aberration Criteria for Screening Quantitative Factors

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Tang and Xu [Biometrika 101 (2014) 333-350] applied the minimum β-aberration criterion to selecting optimal designs for screening quantitative factors. They provided a statistical justification showing that minimum β-aberration criterion minimizes contamination of nonnegligible kth-order effects on the estimation of linear effects for k=2,,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 β-wordlength patterns and contaminations (on the estimation of linear effects) and reveal that the minimum β-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 β-aberration criterion, in fact, sequentially minimizes the contamination of nonnegligible kth-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 β-aberration criterion for selecting optimal designs for screening quantitative factors.