在變異數已知的線性模型下辨識協同與拮抗交互作用的交集聯集檢定法

在各式各樣的資料中，常會出現只有兩種水準 (或稱為設定值)的解釋變數。而在許多情況下，這兩個水準常分別代表“有”和“無”某種性質。考慮一筆包含兩個此類解釋變數$A$和$B$以及反應變數$Y$的資料。當無性質$A$且無性質$B$、僅有性質$A$、或僅有性質$B$時，$Y$都有相同的表現，但當有性質$A$且有性質$B$時，$Y$卻會有異常變大(或變小)的表現，則稱$A$和$B$對$Y$有協同(或拮抗)交互作用。若要辨識一組資料是否有協同或拮抗交互作用(或只辨識是否有協同交互作用，或只辨識拮抗交互作用)，Lin (2015)建議可使用Helmert coding之線性模型來做檢定。本論文亦採用此線性模型，且假設$Y$之變異數為已知，但將此檢定問題透過等效性檢定法和交集聯集檢定法來重新定義。對交集聯集檢定法，Berger (1982)曾証明，在某些條件下，利用個別拒絕域之交集所形成的拒絕域，可建構出一個size-$alpha$的檢定。在本論文中，我們將指出Berger (1982)所提出的條件，在我們的檢定問題下並不成立，故無法直接套用其方法來產生一個size-$alpha$之檢定。本論文將提出改良Berger (1982)的方法，以使檢定之最大型一錯誤機率能等於$alpha$，並在對立假設下能有較高的檢定力。對此新檢定法，我們亦討論與其相關的一些課題，包括等效性邊界之設定，與其在變異數未知的情況下所會遭遇的問題。最後我們用電腦模擬來驗證此新檢定法之效力，並與Berger (1982)之方法相比較。

關鍵字

Helmert編碼；多重檢定；檢定力； size-$alpha$檢定；等效性檢定

並列摘要

In the thesis, we consider a linear model with a continuous response $Y$ and two 2-level predictors $A$ and $B$, where the two levels are labeled by 0 and 1. Denote by $Y_{00}$, $Y_{01}$, $Y_{10}$, and $Y_{11}$ respectively, the responses obtained on the four level combinations of $A$ and $B$ (i.e., (0, 0), (0, 1), (1, 0), and (1, 1)), and assume that they have equal variance. The two predictors $A$ and $B$ are said to have a synergistic (or antagonistic) interaction on the response $Y$ if $Y_{00},$ $Y_{01},$ and $Y_{10}$ are normally distributed with almost identical means, but $Y_{11}$ follows a normal distribution with a significantly larger (or smaller) mean. To identify whether a synergistic or antagonistic interaction (or a synergistic interaction, or an antagonistic interaction) exists, Lin (2015) suggested a linear model based on Helmert coding. In the thesis, we adopt the model to develop tests for the presence of the interactions under the assumption that the variance of $Y$ is known. We formulate the test problem as an intersection-union test (IUT), which is composed of three individual tests: two equivalence tests and one two-sided (or one-sided) test. The rejection region of an IUT is the intersection of the rejection regions of its individual tests. Berger (1982) gave a condition, under which an IUT is a size-$alpha$ test if its individual tests are of size $alpha$. However, the condition does not hold in our case so that the IUT for the interactions based on Berger's method is not a level-$alpha$ test. Furthermore, Berger's method does not consider the correlations between the test statistics of the individuals tests. To address the issues, we study the maximum probability of type I error of the IUT for the interactions and obtain some theorems. Based on the theorems, we propose a new method to construct a size-$alpha$ IUT for the interactions. The new IUT has higher power than the one based on Berger's method. We also discuss some questions about the new IUT, including the choice of equivalence margins, and why our method cannot be directly generalized to the case of unknown variance. In the end, we use a computer simulation to study the performance of the new IUT, and compare it with the one based on Berger's method.

並列關鍵字

Helmert coding ； multiple testing ； power ； size-$alpha$ test ； test of equivalence

參考文獻

Bailey, D. G., Dresser, G.,and Arnold, J. M. O. (2013). “Grapefruit–medication interactions: Forbidden fruit or avoidable consequences?”Canadian Medical Association Journal, 185(4), 309-316.

Berger, R. L. (1982). “Multiparameter hypothesis testing and acceptance sampling.” Technometrics, 24(4), 295-300.

Tiong, K. L., Chang, K. C., Yeh, K. T., Liu, T. Y., Wu, J. H., Hsieh, P. H.,Lin, S. H., Lai, W. Y., Hsu, Y. C., Chen, J. Y., Chang, J. G., and Shieh, G. S. (2014). “CSNK1E/CTNNB1 are synthetic lethal to TP53 in colorectal cancer and are markers for prognosis.”Neoplasia, 16(5), 441-450.

Wellek, S. (2010). Testing statistical hypotheses of equivalence and noninferiority. CRC Press.

Lin ,W.T. (2015).“Identification of synergistic and antagonistic interactions,”Master thesis, National Tsing Hua University, Hsinchu, Taiwan.

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在變異數已知的線性模型下辨識協同與拮抗交互作用的交集聯集檢定法

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