多階層因子實驗在各領域都很常被使用，諸如農業實驗以及工業實驗。當執行實驗時，若實驗單位的隨機化有所限制，則此實驗即為多階層實驗，反之則稱為一單階層實驗。多階層實驗的特色之一為反應變數的總變異量可歸因於數個來源，此時實驗單位也稱擁有一區集結構 (如區集設計與裂區設計)。已有許多文獻探討最佳多階層實驗設計的議題，然而它們都有各自的限制。部分文獻的結果只能使用在兩水準的實驗因子上；有些文獻只探討很特定的區集結構，像是區集設計與裂區設計；有些則只能處理直交正規的部分因子設計。 本計畫致力於在很一般的實驗設定下發展多階層實驗設計的統一理論，包含它們的數理結構刻化以及建構方法與可能的應用。本計畫所探討的多階層設計除了直交正規設計外，也涵蓋非正規或超飽和設計。同時，實驗因子可為質性或量性，且水準數不受限。此外我們允許實驗因子對反應變數的影響程度可以不一致並區分成數組，使得每一組的實驗因子對反應變數有一致的影響，且不同組造成的影響不一致。除此之外，我們也將探討具備「等價估計」性質的設計，其中該種設計可使得最「小平方法估計式」與「一般最小平方法估計式」有相同的參數估計值。此計畫亦將深入探討多階層等價估計設計的理論以及建構方法。 ;Multi-stratum factorial experiments, commonly conducted in diverse fields such as agriculture and industrial investigations, refer to those with the experimental units that cannot be completely randomized during experimental processes. Multiple error terms with different variances arise then. When complete randomizations are available, the experimental units are said to be unstructured and the resulting experiments can be viewed as single-stratum or unstructured experiments. If, instead, there are restrictions on the randomizations of experimental units, then these units are said to have a block structure (e.g., block designs and split-plot designs). Many works in the literature have already paid attention to the construction of optimal multi-stratum designs. However, some are restricted to two-level treatment factors; others considered specific block structures such as block designs or split-plot designs, or they can deal with orthogonal regular fractional factorial designs only. This project aims at developing a general and unified approach to the selection and construction of optimal and efficient multi-stratum designs, including theoretical structure of optimal designs and their efficient construction algorithms as well as useful applications. The multi-stratum designs considered in this project involve not only orthogonal regular designs but also nonregular/supersaturated ones. Meanwhile, they can have very complex block structures as well as multi-level treatment factors with multiple groups, where the treatment factors in the same group have equal importance different from the other groups. In addition to the above features, we also incorporate the equivalent-estimation (EE) property into design structure. A design is said to possess the EE property if under the design, the ordinary least square estimator gives the same estimates as the generalized least square estimator. This property ensures the accuracy of the estimates since the estimates can be obtained without estimating unknown variance components. The theoretical construction and design tables of optimal multi-stratum EE designs are investigated and provided.
