最小污染和$β$ -筛选定量因子的畸变标准
Minimum Contamination and $β$-Aberration Criteria for Screening Quantitative Factors
摘要：
Tang和Xu [Biometrika 101（2014）333-350]应用最小$β$ -畸变标准来选择用于筛选定量因子的最优设计。他们提供了统计学依据，表明最小$β$ -畸变标准可最大限度地减少不可忽略的$k$第三级效应对$k=2,\cdots,r$线性效应估计的污染，其中$r$是设计的强度。不幸的是，这个结果并不适用于$k>r$。在本文中，我们提供了一个完整的数学连接之间的$β$ -字长模式和污染（对线性效应的估计），并揭示了最小的$β$ -畸变准则不一定是等价的最小污染准则的排名设计。我们证明，他们是等价的，只有当设计的因素的数量等于强度加1。我们强调，最小$β$ -畸变准则，事实上，顺序最小化污染的不可忽略的$k$第三阶效应的估计的一般平均值，而不是估计的线性效应。因此，在筛选定量因子的最优设计时，最小污染准则应比最小$β$ -畸变准则更合适。
英文摘要：
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 $k$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 $β$-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 $k$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 $β$-aberration criterion for selecting optimal designs for screening quantitative factors.
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