Constructing D-Efficient Mixed-Level Foldover Designs Using Hadamard Matrices

This paper introduces a new class of Hadamard matrix-based mixed-level foldover designs (MLFODs) and an algorithm which facilitates the construction of MLFODs. Our new MLFODs were constructed by converting some 2-level columns of a Hadamard matrix to 3-level ones. Like the 2-level foldover designs (FODs), each new MLFOD was constructed by a half fraction and its foldover. Our Hadamard-matrix based MLFODs are compared with the conference matrix-based FODs of Jones &amp; Nachtsheim (2013) in terms of the D-efficiencies and the maximum of the absolute values of the correlation coefficients among the columns of the model matrix. Like the latter, our designs are also <i>definitive</i> in the sense that the estimates of all main effects are unbiased with respect to any active second order effects. In addition, they require fewer runs and can be used to study the presence of the second-order effects more efficiently. Examples illustrating the use of our new MLFODs are given.

本文提出了一类新型的基于哈达玛矩阵（Hadamard matrix）的混合水平折叠设计（mixed-level foldover designs，MLFODs），以及一种可辅助构建MLFODs的算法。我们通过将哈达玛矩阵的部分二水平列转换为三水平列，搭建了该类新型MLFODs。与二水平折叠设计（foldover designs，FODs）一致，每个新型MLFOD均通过半分式设计及其折叠版本构建得到。本文将基于哈达玛矩阵的MLFODs与Jones & Nachtsheim（2013）提出的基于会议矩阵（conference matrix）的FODs展开对比，对比维度涵盖D效率（D-efficiencies）以及模型矩阵各列间相关系数绝对值的最大值。正如上述设计，我们的方案同样具备确证性：即所有主效应的估计值对于任意活跃二阶效应均无偏。此外，该类设计所需试验次数更少，且可更高效地探究二阶效应的存在性。文末给出了若干示例，以说明新型MLFODs的应用方法。