Abstract
摘要:單一重複(unreplicated)的完全(full factorial)或部份(fractional factorial)複因子設計經常被用來搜尋實驗中顯著效應(active effects)。然而此類型的實驗每一個處理組合(treatment combination)至多僅被重複一次,所以實驗中機差變異數(error variance)無法以真正重複(pure replicates)來估計。如果設計中所有的處理組合均執行重複,則實驗的規模將會急遽增加。所以一個折衷的方式便是在單一重複的複因子實驗中,重複執行某部份的處理組合,再利用真正重複的處理組合來估計機差變異數。最近Liao and Chai (2004)及Liao and Chai (2008)等研究有系統地研究如何建構包含一個重複板(twice-replicated flat)的平行板設計(parallel-flats designs)。Liao and Chai (2008) 提出一個此類型設計滿足D-最適設計的充分條件,並且發展一個演譯法(algorithm)有系統地產生這些D-最適設計。然而這些研究仍然存在著某些限制。舉例來說,部分重複的處理組合數必須等於一個板(flat)所包含的處理組合數,亦即部分重複的處理組合數的大小必定是變級數(levels)的冪次方(power);並且可能顯著效應(possibly active effects)也必須在建構D-最適設計時就清楚地被指定(specified)。
除了搜尋顯著位置效應(location effects),分散效應(dispersion effects)的尋找及檢定是複因子實驗中另一個被關切的議題。在單一重複的複因子實驗中,分散效應無法避免會與位置效應混雜(confounded),使得這兩種效應經常無法被正確地搜尋出來。而部份重複的複因子設計,可以利用真正重複來估計分散效應以避免和位置效應混雜,這將有助於增進對分散效應的檢定力。然而目前現有檢定分散效應的統計方法大部分都建構於單一重複且為直交(orthogonal)的設計上。由於部份重複的複因子設計多為非直交的設計,因此需要更進一步發展新的檢定方法。
本計畫的第一年計畫我們將專注於如何在直交表(orthogonal arrays)上選擇最適的部份重複;並且放寬先前研究(Liao and Chai, 2008)的限制,考慮使部分重複的處理組合數為小於原直交表大小的任意整數,而且必須指定的可能顯著效應希望能放寬至僅須確定其數量即可。第二年計畫則專注於部分重複的複因子設計用於搜尋顯著分散效應的研究,我們將嘗試建構最適部分重複的設計,用於同時搜尋顯著位置效應及分散效應;並且發展適當的統計方法來檢定這兩種效應。
Abstract: An unreplicated full or fractional factorial design is commonly used to identify which of the experimental factors have an impact on the responses. For such kind of designs, the pure error variance cannot be estimated since every treatment combination is run at most once in the experiments. The cost of experimentation could increase excessively if all treatment combinations are fully replicated. Therefore, an acceptable compromise is to add partial replication to the original unreplicated design, and to obtain a replication-based estimate of the error variance. Most recently, Liao and Chai (2004) and Liao and Chai (2008) systematically investigated the construction of parallel-flats type designs with a twice-duplicated flat. A set of sufficient conditions and an algorithm for searching D-optimal ones over such type of designs are proposed in Liao and Chai (2008). However, there are still some limitations in their study. For example, the amount of twice-duplicated runs must be equal to the size of a flat, which is a power of two; and the candidates of possibly active location effects must be precisely specified in construction of their designs.
The identification of active dispersion effects is another interesting issue in factorial experiments. Dispersion effects are unavoidably confounded with location effects in the unreplicated factorial designs. As a result, these two kinds of factorial effects cannot be identified correctly. On the other hand, partially replicated designs provide replication-based estimates for dispersion effects, and this would help disentangle the confounding problem between dispersion effects and location effects. Therefore, the power in correctly identifying truly active dispersion effects can be improved. Most of the existing methods for identifying active dispersion effects are developed for unreplicated and orthogonal designs. However, partially replicated designs are usually not orthogonal. Hence, an appropriate analysis method for identifying dispersion effects, as well as an optimal design, need further investigation for the class of partially replicated designs.
During the first year of this research proposal, we will focus on extension of the results presented in Liao and Chai (2008). We will consider the optimal selection of twice-replicated runs on an orthogonal array. In particular, the size of twice-duplicated runs will be set to be any possible number less than the run size of the original design; and the specification of possibly active location effects will be modified to be only the number of them. In the second year, we may further investigate construction of optimal designs with partial replication for simultaneously identifying truly active location effects and dispersion effects. In addition, we hope to propose an appropriate analysis for partially replicated designs in identification of both effects.
Keyword(s)
複因子設計
D-最適設計
直交設計
直交表
位置效應
分散效應 表
factorial design
D-optimal design
orthogonal design
orthogonal array
location effect
dispersion effect.