Two-Level Designs to Estimate All Main Effects and Two-Factor Interactions

We study the design of two-level experiments with N runs and n factors large enough to estimate the interaction model, which contains all the main effects and all the two-factor interactions. Yet, an effect hierarchy assumption suggests that main effect estimation should be given more prominence than the estimation of two-factor interactions. Orthogonal arrays (OAs) favor main effect estimation. However, complete enumeration becomes infeasible for cases relevant for practitioners. We develop a partial enumeration procedure for these cases and we establish upper bounds on the D-efficiency for the interaction model based on arrays that have not been generated by the partial enumeration. We also propose an optimal design procedure that favors main effect estimation. Designs created with this procedure have smaller D-efficiencies for the interaction model than D-optimal designs, but standard errors for the main effects in this model are improved. Generated OAs for 7–10 factors and 32–72 runs are smaller or have a higher D-efficiency than the smallest OAs from the literature. Designs obtained with the new optimal design procedure or strength-3 OAs (which have main effects that are not correlated with two-factor interactions) are recommended if main effects unbiased by possible two-factor interactions are of primary interest. D-optimal designs are recommended if interactions are of primary interest. Supplementary materials for this article are available online.本研究针对含N次试验、n个因子且规模足以拟合包含所有主效应与所有两因子交互效应的交互效应模型的二水平试验设计展开探讨。然而，效应层级假设主张主效应估计的优先级应高于两因子交互效应估计。正交阵列（Orthogonal Arrays，OAs）更适配主效应估计，但针对从业者常用的试验场景，完全枚举法已不再可行。针对此类场景，本文提出一种部分枚举流程，并基于未通过该部分枚举生成的阵列，推导得到交互效应模型下D-效率的上界。此外，本文还提出一种优先保障主效应估计的最优设计流程。采用该流程生成的设计方案，其针对交互效应模型的D-效率低于D最优设计，但模型中主效应的标准误得到了优化。针对7至10个因子、32至72次试验的场景，本文生成的正交阵列尺寸更小，或D-效率高于现有文献中最小的正交阵列。若研究核心关注不受两因子交互效应偏倚的主效应，则推荐采用本文提出的新型最优设计流程生成的方案，或强度为3的正交阵列（其主效应与两因子交互效应互不相关）。若研究核心关注交互效应，则推荐采用D最优设计。本文配套补充材料可在线获取。