Structure of Nonregular Two-Level Designs

Two-level fractional factorial designs are often used in screening scenarios to identify active factors. This article investigates the block diagonal structure of the information matrix of nonregular two-level designs. This structure is appealing since estimates of parameters belonging to different diagonal submatrices are uncorrelated. As such, the covariance matrix of the least squares estimates is simplified and the number of linear dependencies is reduced. We connect the block diagonal information matrix to the parallel flats design (PFD) literature and gain insights into the structure of what is estimable and/or aliased using the concept of minimal dependent sets. We show how to determine the number of parallel flats for any given design, and how to construct a design with a specified number of parallel flats. The usefulness of our construction method is illustrated by producing designs for estimation of the two-factor interaction model with three or more parallel flats. We also provide a fuller understanding of recently proposed group orthogonal supersaturated designs. Benefits of PFDs for analysis, including bias containment, are also discussed.

二水平部分因子设计（two-level fractional factorial designs）常被用于筛选试验场景，以识别活跃因子。本文针对非规则二水平设计（nonregular two-level designs）的信息矩阵块对角结构展开研究。该结构具备良好的研究价值，因为属于不同对角子矩阵的参数估计量互不相关。如此一来，最小二乘估计（least squares estimates）的协方差矩阵得以简化，线性相依关系的数量也随之减少。我们将块对角信息矩阵与平行超平面设计（parallel flats design, PFD）的相关研究建立关联，并借助极小相依集（minimal dependent sets）的概念，对可估与/或混杂的效应结构展开深入剖析。本文阐明了如何针对任意给定设计确定其平行超平面的数量，以及如何构建具备指定数量平行超平面的设计。通过构建适用于三平行超平面及以上的双因子交互效应模型（two-factor interaction model）估计的设计方案，本文验证了所提构造方法的有效性。此外，本文还对近期提出的分组正交超饱和设计（group orthogonal supersaturated designs）进行了更为全面的阐释。本文同时探讨了平行超平面设计（PFD）用于数据分析的优势，包括偏差控制（bias containment）在内。