分散性效應存在下位置效應之 D-最適部分複因子設計之研究

Abstract

在使用變方分析處理試驗資料時，同質變方是基本前提，若有某個因子在不同變級時，具有不同的變方，我們稱此因子為具有分散效應的因子。在過往的研究中，多為分析資料是否具有分散效應的方法，對於如何選擇設計則較少著墨，然而若能有效的找出合適的設計，不但可以大幅減低試驗的成本，也可以提高試驗的效率。 本研究主要針對 2^{n-p} 部分複因子設計，分析同質變方 (沒有分散效應存在) 與異質變方 (有分散效應存在) 的情形，並由 eta (位置效應) 的 BLUE (Best Linear Unbiased Estimator) 觀察當分散效應存在時 eta 的訊息矩陣的型態。接著建立一套方法來描述訊息矩陣中各元素的位置及大小，發現若分散效應存在時，不可能會有正交設計 (orthogonal designs) 產生。 另為了要計算試驗效率，我們整理出一簡單的公式，試驗者僅需知道設計的別名關係便可得到訊息矩陣的行列式值，以便比較不同設計間的差異，從中發現若分散效應越顯著，則試驗效率降低的越快。 在挑選最適設計方面，在給定一組欲估計的位置效應以及具有分散效應的因子後，使用 Franklin and Bailey (1977)的演算法找出可用 (eligible) 的設計，再配合之前建立的 D-最適準則 (計算每個設計的 D-最適分數) 以及最小偏差法準則 (minimum aberration criterion) ，撰寫成 R-電腦程式以供使用，並提供一些常用的列表於文中。 此外，因分散效應存在時會造成異質變方的情形，我們使用 Box and Meyer (1986) 提出的方法，利用常態機率圖找出較顯著的位置效應及分散效應，再使用 MLE (Maximum Likelihood Estimation)來估計各參數的估值，使研究人員能夠在分散效應存在時檢定有興趣的位置效應。

Although homogeneity of variance is a basic assumption in most ANOVA analyses, it is not uncommon to encounter the situations that the variance of the response variable changes from one experimental setting to another. In factorial designs, the factors responsible for such change are called dispersion factors. Recently, several articles study on how to identify the dispersion fac- tors from experimental data. Clearly, it is still important to address the design issue concerning the estimation of location effects when there exist dispersion factors. This study focuses on regular 2n−p fractional factorial designs (FFDs). We simply consider the situation that there is exact one dispersion factor in the experiment. The task is to estimate a set of specified location effects in this situation. The BLUE (Best Linear Unbiased Estimator) of using GLSE (Generalized Least Square Estimation) is applied and its information matrix is shown to have a special pattern when using 2n−p FFDs. Namely, we estab- lish a connection between the D-efficiency for with the alias relations of the used 2n−p FFDs. Specifically, we show that there is no orthogonal design for provided that the general mean and all location main effects are included in it. An algorithm modified from that of Franklin and Baily (1977) is given to search for D-optimal designs for any specified within the class of 2n−p FFDs. Moreover, the minimum aberration criterion is used to determine the final de- sign if there are multiple equally D-optimal designs for a . The algorithm is implemented in R language. Some classes of designs generated from the algorithm are also reported. The existence of dispersion factor results in heterogeneity of variance when analyzing experimental data. According to the method proposed by Box and Meyer (1986), we first use normal plotting to identify unusually large location effects and dispersion effects, simultaneously. Then we apply MLE (Maximum likelihood Estimation) for the identified location and dispersion effects. Con- sequently, the Wald’s test is used for the significance test of location effects. Some data set is given to illustrate this analysis approach.