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標題: 最大卡方統計量在邏輯斯迴歸中之修正
Maximally selected chi-square statistic corrected for logistic regression
作者: 易緯霆
Yi, Wei-Ting
關鍵字: epidemiology;流行病學;logistic regression;maximally selected statistic;Brownian bridge;chi-square distribution;Gram-Schmidt orthogonalization;邏輯斯迴歸;最大化統計量;布朗橋;卡方分布;葛蘭-舒密正交化程序
出版社: 統計學研究所
引用: [1] Betensky, R. A. and Rabinowitz, D. (1999). "Maximally selected _2 statistics for k _ 2 tables." Biometrics 55, 317-320. [2] Billinsley, P. (1968). Convergence of Probability Measures. New York: Wiley. [3] Dirkse, J. P. (1975). "An absorption probability for the Ornstein-Uhlenbeck process." journal of Applied Probability 12, 595-599. [4] Hothorn, T. and Zdileis, A. (2008). "Generalized Maximally Selected Statistics." Biometrics, 64, 1263-1269. [5] Miller, R. and Siegmund, D. (1982). "Maximally Selected Chi Square Statistics." Biometrics, 38, 1011-1016. [6] Halpern, J. P. ( 1982). "Maximally Selected Chi Square Statistics." Biometrics, 38, 1017-1022. [7] Lausen,B. and Schumacher,M.(1992).” Maximally selected rank statistics.”Biometrics, 48-73. [8] Hothorn,Y. and Lausen,B.(2003).” On the exact distribution of maximally selected rank statistics.” Computational Statistics & Data Analysis, 43-121. [9] Shorack, R. and Wellner,J.(1986).” Empirical Processes with Applications to Statistics”. [10]Small,C.G and McLeish,D.L.(1994).” Hilbert space methods in probability and statistical inference”. New York: Wiley. [11]王冠翔(2011)” Modified Wald test corresponding to maximally selected chi-square statistic”.
流行病學研究裡,當所關心的結果變項可以被一組變數(包含主要暴露(exposure)或處方(treatment))解釋時,如果我們想要檢定某一特定之連續型變項的效果,此時邏輯斯迴歸是一個常被採用的分析模型。當考慮將該特定之連續型變項「二元化」(dichotomization)時,統計分析者常取的切點(cutoff point)多是使關聯性卡方檢定統計量最大化的切點。此種做法所得到的統計量,文獻上已知不是自由度為1的卡方分布,而是一種修正型之最大值布朗橋(Brownian bridge)分布。本研究將此問題推廣至邏輯斯迴歸分析。我們探討解釋變數獨立時,不同檢定方式的檢定力。當變數間有相關性時,我們提出逐步擬似「葛蘭-舒密」正交化程序(stepwise pseudo Gram-Schmidt orthogonalization),研究如何逐一控制各個參數檢定之型一誤差的問題。我們提出幾種不同方法檢定邏輯斯迴歸的係數,用模擬的方法比較不同方法的優劣。最後,利用實際例子來比較不同檢定方法的差異。

Epidemiologic studies concern the effect of an exposure, and a set of explanatory variables, on the outcome variable. When logistic regression model is used and some variables are dichotomized according to possible cutoff points to produce maximally selected test statistics, the distribution of the test statistics under null hypothesis (of ‘no effect') is proved to be not a chi-square distribution with 1 degree of freedom. It is a distribution of the supremum of the Brownian bridge with suitable correction. This study extends the problem to a logistic regression setting. We investigate the situation of independent explanatory variables towards the power of different tests. When there are correlations among variables, we propose a stepwise pseudo Gram-Schmidt orthogonalization process so that each individual regression parameters can have reasonable type I error. Several methods are proposed, and their powers are compared through simulations. We implement the proposed methods on an actual data set for illustration.
其他識別: U0005-1601201214510400
Appears in Collections:統計學研究所

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