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

We study the design of two-level experiments with <i>N</i> runs and <i>n</i> 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.

本文针对包含N次试验、n个因子的两水平试验设计展开研究，此类设计需具备足够规模以拟合涵盖所有主效应与全部两因子交互效应的交互模型。然而，效应层级假设主张，主效应的估计应较两因子交互效应的估计获得更高优先级。正交表（Orthogonal Arrays, OAs）对主效应估计更为友好，但针对从业者实际关注的应用场景，完全枚举法已不再具备可行性。针对此类场景，本文提出一种部分枚举方法，并基于未通过该部分枚举生成的试验数组，推导得到交互模型的D效率（D-efficiency）上界。此外，本文提出一种偏向主效应估计的最优设计流程。采用该流程生成的试验设计，其针对交互模型的D效率相较于D最优设计更低，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，本文生成的正交表相较于现有文献中的最小正交表，要么规模更小，要么D效率更高。若研究核心为不受潜在两因子交互效应干扰的主效应，则推荐采用本文提出的新型最优设计流程生成的试验设计，或强度为3的正交表（此类正交表的主效应与两因子交互效应互不相关）；若研究核心为交互效应，则推荐采用D最优设计。本文的补充材料可在线获取。