分散因子存在下之最小偏差2-變級部份複因子設計

Abstract

在實驗初期通常實驗資料的變因有很多,因此2-變級複因子設計被廣泛用於估計實驗中重要的位置效應。而同質變方是變方分析中最基本的假設,在此假設成立下,最適2n-p部分複因子設計的研究已相當完善。但是當實驗的反應變數之變異程度,會因某些因子在不同變級而有顯著的差異時,我們稱這些因子為分散因子。然而在分散效應存在下,如何決定最適設計是一個在文獻上少見的研究議題,若能找出合適的設計,不但可以減低實驗的成本,也可以提高實驗的效率,因此這是個值得深入探討的研究議題。

本研究是針對在已知具有分散因子存在,而欲估計的位置效應未知的情況下,找尋最適的2n-p部分複因子設計。所以利用最小偏差準則的概念,建構出最適設計。首先從最簡單的模式:僅有一個實驗因子會影響分散效應,開始探討。由於最適設計的決定會受分散效應的影響,所以字元長度需要作調整以建立新的最小偏差準則。然後藉由R-電腦程式做完整的搜尋,找出最適的2n-p部分複因子設計以及在此設計哪些因子適合被指定為分散因子,並將一些實用的設計列表以供查詢使用。進一步我們探討當有兩個實驗因子會影響分散效應的模式,並將一些實用的設計列表以供查詢使用。

During the initial stages of experimentation, two-level regular fractional factorial designs (FFDs) are commonly used to identify important factors which may significantly affect the response(s) of the experiment. The homogeneity of variance is a basic assumption in the ANOVA for location effects. The design issue of optimal 2n-p regular FFDs based on the homogenous variance assumption has been studied extensively. However, when the variance of the response variable changes as some specific factors change from one setting to another, these factors affecting the variation of the response are called dispersion factors in this study. Interestingly, to the best of our knowledge, the issue addressing the minimum aberration designs for location effects in the presence of dispersion factors has not been found in the literature.

In this study, we shall investigate the minimum aberration 2n-p regular FFDs under the assumption that there are some specific factors responsible for the dispersion of the response. The dispersion effects may violate the usual assumption of variance homogeneity in ANOVA. Therefore, the aberration criterion needs to be modified in order to discuss this issue. It is anticipated that the choice of minimum aberration designs may depend upon the prior information on the dispersion effects. Specific attention will first be given to the simplest situation that there is exactly one factor responsible for the dispersion effects. After a thorough investigation on this, we extend the results to the situation that two factors involve the dispersion effects.