在全因子設計下，Huang (2020) 提出方法來定義和估計每個因子的本質編碼以及本質效應。但在其所得的配適模型中，仍常會有過多顯著(係數值非零或是偏大)的本質效應。為了進一步簡化配適模型，減少顯著本質效應的個數，我們考慮對本質編碼做旋轉，並稱旋轉後的本質編碼為簡化本質編碼。在本文中，我們提出兩種旋轉方法來定義簡化本質編碼。第一種方法對同一因子，在其與不同因子的各階效應中允許使用不同的旋轉矩陣。第二種方法則是每個因子皆只使用一個旋轉矩陣，用其來同時簡化牽涉此因子的各種效應。一般而言，第一種方法在旋轉後能得到比第二種方法更簡化的配適模型，但其所使用的編碼數則較多。我們並使用一組模擬數據，來比較兩種旋轉方法的簡化結果。

Under full factorial design, Huang (2020) defined essence codings, and proposed a method to estimate them for each factors and also the essence effects given the estimates of essence codings. However, the use of essence codings might still cause too many significant essence effects in the fitted model. To simplify it, i.e., reduce the number of significant effects, we consider orthogonally rotating the essence codings. The new codings obtained after rotation for this purpose are referred to as simplified essence codings. In this work, we propose two rotation methods for identifying simplified essence codings. For a factor x, the first one allows using a different rotation matrix for each group of effects involving x and a set of the other factors. The second one restricts using only one rotation matrix for x to simultaneously simplify all the effects involving x. In general, the first one can obtain a simpler fitted model after rotation while the second one use less simplified essence codings. We also illustrate and compare the performance of the two methods through a simulation study.

目錄
1 緒論 1
2 文獻回顧 5
2.1 本質編碼的估計準則與個數挑選. . . . . . . . . . . . . . . . . . . 5
2.2 三維直交旋轉準則 . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 梯度投影法 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 研究方法 11
3.1 多重旋轉. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 單一旋轉. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 數據分析 23
4.1 真實模型介紹 . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 全因子設計下之本質編碼. . . . . . . . . . . . . . . . . . . . . . 24
4.3 簡化結果 . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.1 多重旋轉 . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.2 單一旋轉. . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 結論與討論 36

1. Huang, J.-M. (2020). Identifying essence codings and effects for multiple factors under full factorial designs. Master’s thesis, National Tsing Hua University, Hsinchu, Taiwan.

2. Jennrich, R. (2001). A simple general procedure for orthogonal rotation. Psychometrika, 66(2):289–306.

3. Kiers, H. (1997). Three-mode orthomax rotation. Psychometrika, 62(4):579–598.

4. Kiers, H. (1998). Joint orthomax rotation of the core and component matrices resulting from three-mode principal components analysis. Journal of Classification, 15(2):245–263.