線性模型下之協同與拮抗交互作用檢定法

Tests for Identifying Synergistic and Antagonistic Interactions under Linear Model

曹立綸(Li-Lun Thao)

指導教授 ： 鄭少為

國立清華大學/理學院/統計學研究所/碩士(2017年)

摘要

在各式各樣的資料中，常會出現僅有兩種水準的解釋變數，而這兩種水準常分別代表“有”和“無”某種性質。現考慮一筆包含兩個此類解釋變數A和B以及連續型反應變數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。為了辨識協同或拮抗交互作用是否存在，我們利用統計上稱為“等效性檢定”(equivalence test)與“交聯集檢定”(intersection-union test)的兩種方法來發展檢定法。Chen (2016)曾在假設Y服從變異數已知的常態分配下探討此問題的檢定法，本論文則在Y之變異數未知的條件下探討此問題，故參數空間及拒絕域之維度將與Chen (2016)不同。對交聯集檢定法，Berger and Hsu (1996)曾證明，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-α的拒絕域。在本論文中，我們證明當Y變異數未知時，我們使用交聯集法來檢定協同與拮抗交互作用時所建構的拒絕域，將滿足Berger and Hsu (1996)所提出的條件，亦即透過交聯集檢定所形成之拒絕域將是一個size-α的拒絕域。我們並探討此拒絕域在哪些參數點下將發生最大的型一錯誤機率值。利用此結果，我們推導出在建構此拒絕域時，無需考慮檢定統計量之間的相關性之結論。最後我們將利用電腦模擬來驗證此檢定法之效力。

關鍵字

交聯集檢定 ； 等效性檢定 ； size-α檢定

並列摘要

In this thesis, we consider a linear model with two 2-level predictors A and B and a continuous response Y. The two levels of A and B are labeled by 0 and 1. The responses obtained on the four level combinations of A and B, i.e., (0, 0), (0, 1), (1, 0), and (1, 1), are denoted by Y_{00}, Y_{01}, Y_{10}, and Y_{11} respectively. They are
assumed to be normally distributed with means µ_{00}, µ_{01}, µ_{10}, and µ_{11} respectively, and an unknown variance σ^{2}. The two predictors A and B are said to have a synergistic (or antagonistic) interaction on the response Y if µ_{00}=µ_{01}=µ_{10}=µ, and µ_{11} is larger (or smaller) than µ. In this thesis, we utilize the methods of intersection-union test (IUT) and equivalence test to resolve the test problem of identifying such intersections. Chen (2016) also studied this problem but under the stronger assumption that σ is known. When σ is unknown, the parameter space and the rejection region we develop are quite different from what presented in Chen (2016). Berger and Hsu (1996) gave some conditions under which the rejection region of an IUT is of size α. We prove that our rejection region satisfies these conditions, and therefore it is a size-α rejection region. We also identify the parameter values on which the probability of this rejection region reaches α, and use this result to conclude that it is unnecessary to consider the correlation between the test statistics in the construction of this rejection region. A computer
simulation is performed to study and verify the power of our method.

並列關鍵字

intersection-union test ； equivalence test ； size-α test

參考文獻

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

Berger, R. L. and Hsu, J. C. (1996). “Bioequivalence trials, intersection-union tests and equivalence confidence sets.” Statistical Science, 11(4):283–319.

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., and Chen, J.-Y. (2014). “CSNK1E/CTNNB1 are synthetic lethal to TP53 in colorectal cancer and are markers for prognosis.” Neoplasia, 16(5):441–450.

Chen, T.-C. (2016). “Intersection-union test for identifying synergistic and antagonistic interactions under linear model with known variance.” Master’s thesis, National Tsing Hua University, Hsinchu, Taiwan.

Genz, A. and Bretz, F. (2009). Computation of multivariate normal and t probabilities. Berlin-Heidelberg: Springer-Verlag.