令D1與D2分別表示兩個試驗次數相同的二水準因子設計矩陣，而D1(r)為D1透過某種列互換r後所得到的設計矩陣。令D(r)為將D1(r)與D2列與列合併後所得的設計矩陣，稱D(r)為逐列合併設計。本研究中，我們將設計矩陣D1, D2，透過類似耶茲序列(Yates order)的轉換，把每個不同的設計點轉換成實數線上不同的數值，將D1,D2轉換成維度等同於試驗次數的序向量V1,V2，並以V1(r)代表D1(r)轉換過後所得的序向量。藉由V1(r)與V2之間的樣本相關係數，我們可探討逐列合併設計D(r)的性質。我們發現D(r)的強度為2時，V1(r),V2的樣本相關係數必為0，我們並將此性質推廣到D(r)的強度大於2的情況。另外若考慮對D1(r),D2執行行互換，我們可將不同行互換下所產生的序向量之樣本相關係數皆等於0發展成為D(r)為強度2之設計的充要條件。我們亦提出一個進位係數挑選法，其可提供一個較佳的序向量來更快速地檢驗D(r)之強度是否為2。而透過V1與V2之間的組合，我們提出一列互換演算法，來搜尋能使D(r)成為強度2設計的列互換。

Let D1 and D2 be the same run size 2-level factorial design matrix, respectively, and D1(r) is a design matrix which obtained by D1 through some row permutation r.We merge D1(r) and D2 row-wise and then get a design matrix D(r), called the row-wise merged design. In this work, we transform design matrix D1 and D2 through a similar Yates order conversion. Each design point with a value of the real number line to replace, respectively. Through this transformation, D1 and D2 converted into two vectors V1 and V2, respectively, which dimension equal to the run size of D1 and D2. And V1(r) represents D1(r) conversion after the resulting vector. We can investigate the property of the row-wise merged design D(r) by the sample correlation coefficient between V1(r) and V2. We found that when the strength of D(r) is 2, the sample correlation coefficient of V1(r) and V2 must be 0, and we also generalized this nature such that strength greater than 2. In addition, if D1(r) and D2 more consider the transformation of column permutation, we can develope conditions that the sample correlation coefficient of V1(r) and V2 is equal to 0 if and only if D(r) is a design of strength 2. Finally, we propose a positional notation coefficient selection method that perform a faster way to detect whether the strength of D(r) is 2 than column permutation. Furthermore, we through a combination between V1 and V2 to present a row permutation algorithm that perform a row permutation such that D(r) is strength 2 design.

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