Design and Analysis of Partially Replicated Factorial Experiments

因子設計為研究試驗反應變數與多個因子之關係的重要統計工具。目前已被廣泛地應用於農業田間試驗、醫學臨床試驗、社會科學、品質管制與工業產品製程改進等相關領域。試驗初期, 單一重複的部分二變級或三變級因子設計常被用於搜尋對試驗結果有重要影響的因子效應。在重要因子效應稀少的原則之下, 許多分析單一重複試驗資料的統計分析方法已陸續被提出。然而實際應用上, 單一重複試驗的分析結果有時令人不甚滿意。主要原因在於試驗缺乏重複, 導致試驗機差變方無法有效估計。完全重複所有處理組合是一個執行重複試驗並改善上述問題的方法, 但試驗成本將因此而倍增。 本論文提出一個適用於搜尋重要因子效應的折衷試驗方法: 部分重複的因子設計。首先藉由幾組實際試驗的資料分析, 說明部分重複的因子設計在應用上的優勢。針對研究者在實際試驗上的需要, 我們討論以下幾個重要的試驗設計議題。(1) 部分重複二變級主效應設計之最適性。(2) 當不可忽略的交感效應存在時, 部分重複二變級主效應設計之穩健性。(3) 當研究者指定需求集時, 二變級與三變級混合部分重複因子設計之最適性。我們提出建構最適設計的充分條件及建構方法, 並將一些常用的設計製成表格提供研究者進行試驗時參考使用。Factorial design is a useful statistical tool for simultaneously investigating the relationship between the experimental response and multiple factors of interest. It has been successfully used in a wide range of applications, such as agricultural field experiments, biomedical trials, social sciences, quality controls and industrial product/process improvements, etc. At the preliminary stage of a scientific investigation, an unreplicated two-level or three-level fractional factorial design is commonly used to identify important factorial effects, which have an impact on the experimental response. Under the effect sparsity principle, several statistical analysis methods are developed for analyzing the outcomes of unreplicated experiments. However, there are many experimental scenarios in which unreplicated experiments typically lead to unsatisfactory findings. This is due mainly to the lack of a replication-based estimate of the experimental error variance. Fully replicating all the treatment combinations is a straightforward method to obtain the pure replicates. However, the experimental cost is rapidly outgrown. In this dissertation, partially replicated factorials, which serve as a compromise between unreplicated and fully replicated experiments, are proposed for effectively and economically screening important effects. First, several practical examples are provided to demonstrate that the partially replicated factorials are competitive in practical applications. To meet the requirements of experimental designs raised by investigators, the following practical design issues regarding the selection of partial replication on fractional factorial designs are explored, respectively. (1) The optimality of partially replicated two-level main-effect plans. (2) The robustness of partially replicated two-level main-effect plans against non-negligible two-factor and higher order interactions. (3) The optimality of partially replicated two-level and three-level mixed factorials for a user pre-specified requirement set. For these important design issues, sufficient conditions and construction approaches to derive the desired designs are investigated. In addition, a series of designs are provided for practical uses