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題名 複迴歸係數排列檢定方法探討
Methods for testing significance of partial regression coefficients in regression model
作者 闕靖元
Chueh, Ching Yuan
貢獻者 江振東
Chiang, Jeng Tung
闕靖元
Chueh, Ching Yuan
關鍵詞 排列檢定
複迴歸模型
樞紐統計量
蒙地卡羅模擬
型一誤差
檢定力
Permutation test
Multiple regression model
Pivotal quantity
Monte Carlo simulation
Type I error
Power
日期 2017
上傳時間 11-Jul-2017 11:25:12 (UTC+8)
摘要 在傳統的迴歸模型架構下，統計推論的進行需要假設誤差項之間相互獨立，且來自於常態分配。當理論模型假設條件無法達成的時候，排列檢定（permutation tests）這種無母數的統計方法通常會是可行的替代方法。
在以往的文獻中，應用於複迴歸模型（multiple regression）之係數排列檢定方法主要以樞紐統計量（pivotal quantity）作為檢定統計量，進而探討不同排列檢定方式的差異。本文除了採用t統計量這一個樞紐統計量作為檢定統計量的排列檢定方式外，亦納入以非樞紐統計量的迴歸係數估計量b22所建構而成的排列檢定方式，藉由蒙地卡羅模擬方法，比較以此兩類檢定方式之型一誤差（type I error）機率以及檢定力（power），並觀察其可行性以及適用時機。模擬結果顯示，在解釋變數間不相關且誤差分配較不偏斜的情形下，Freedman and Lane (1983)、Levin and Robbins (1983)、Kennedy (1995)之排列方法在樣本數大時適用b2統計量，且其檢定力較使用t2統計量高，但差異程度不大；若解釋變數間呈現高度相關，則不論誤差的偏斜狀態，Freedman and Lane (1983)、Kennedy (1995) 之排列方法於樣本數大時適用b2統計量，其檢定力結果也較使用t2統計量高，而且兩者的差異程度比起解釋變數間不相關時更加明顯。整體而言，使用t2統計量適用的場合較廣；相反的，使用b2的模擬結果則常需視樣本數大小以及解釋變數間相關性而定。
In traditional linear models, error term are usually assumed to be independently, identically, normally distributed with mean zero and a constant variance. When the assumptions cannot meet, permutation tests can be an alternative method.
Several permutation tests have been proposed to test the significance of a partial regression coefficient in a multiple regression model. t=b⁄(se(b)), an asymptotically pivotal quantity, is usually preferred and suggested as the test statistic. In this study, we take not only t statistics, but also the estimates of the partial regression coefficient as our test statistics. Their performance are compared in terms of the probability of committing a type I error and the power through the use of Monte Carlo simulation method. Situations where estimates of the partial regression coefficients may outperform t statistics are discussed.
參考文獻 1.Anderson, M. J. (2001). Permutation tests for univariate or multivariate analysis of variance and regression. Canadian Journal of Fisheries and Aquatic Sciences, 58, 626-639.
2.Anderson, M. J., and Legendre, P. (1999). An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model. Journal of Statistical Computation and Simulation, 62, 271-303.
3.Anderson, M. J., and Robinson, J. (2001). Permutation tests for linear models. Australian and New Zealand Journal of Statistics, 43, 75-88.
4.Freedman, D., and Lane, D. (1983). A nonstochastic interpretation of reported significance levels. Journal of Business and Economic Statistics, 1, 292-298.
5.Kennedy, P. E. (1995). Randomization tests in econometrics. Journal of Business and Economic Statistics, 13, 85-94.
6.Kennedy, P. E., and Cade, B. S. (1996). Randomization tests for multiple regression. Communications in Statistics – Simulation and Computation, 25, 923-936.
7.Levin, B., and Robbins, H. (1983). Urn models for regression analysis, with applications to employment discrimination studies. Law and Contemporary Problems, 46, 247-267.
8.Manly, B. F. J. (1991). Randomization, Bootstrap and Monte Carlo methods in biology (First Edition). London: Chapman and Hall.
9.Manly, B. F. J. (1997). Randomization, Bootstrap and Monte Carlo methods in biology (Second Edition). London: Chapman and Hall.
10.Manly, B. F. J. (2006). Randomization, Bootstrap and Monte Carlo methods in biology (Third Edition). London: Chapman and Hall.
11.Oja, H. (1987). On permutation tests in multiple regression and analysis of covariance problems. Australian Journal of Statistics, 29, 91-100.
12.O’Gorman, T. W. (2005). The performance of randomization tests that use permutations of independent variables. Communication in Statistics – Simulation and Computation, 34, 895-908.
13.ter Braak, C. J. F. (1992). Permutation versus bootstrap significance tests in multiple regression and ANOVA. Bootstrapping and Related Techniques (K.-H. Jockel, G. Rothe and W. Sendler, Eds.), Berlin: Springer Verlag, 79-86.
14.Winkler, A. M., Ridgway, G. R., Webster, M. A., Smith, S. M., and Nichols, T. E. (2014). Permutation inference for the general linear model. NeuroImage, 92, 381-397.
描述 碩士
國立政治大學
統計學系
104354007
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0104354007
資料類型 thesis
dc.contributor.advisor 江振東zh_TW
dc.contributor.advisor Chiang, Jeng Tungen_US
dc.contributor.author (Authors) 闕靖元zh_TW
dc.contributor.author (Authors) Chueh, Ching Yuanen_US
dc.creator (作者) 闕靖元zh_TW
dc.creator (作者) Chueh, Ching Yuanen_US
dc.date (日期) 2017en_US
dc.date.accessioned 11-Jul-2017 11:25:12 (UTC+8)-
dc.date.available 11-Jul-2017 11:25:12 (UTC+8)-
dc.date.issued (上傳時間) 11-Jul-2017 11:25:12 (UTC+8)-
dc.identifier (Other Identifiers) G0104354007en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/110779-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 104354007zh_TW
dc.description.abstract (摘要) 在傳統的迴歸模型架構下，統計推論的進行需要假設誤差項之間相互獨立，且來自於常態分配。當理論模型假設條件無法達成的時候，排列檢定（permutation tests）這種無母數的統計方法通常會是可行的替代方法。
在以往的文獻中，應用於複迴歸模型（multiple regression）之係數排列檢定方法主要以樞紐統計量（pivotal quantity）作為檢定統計量，進而探討不同排列檢定方式的差異。本文除了採用t統計量這一個樞紐統計量作為檢定統計量的排列檢定方式外，亦納入以非樞紐統計量的迴歸係數估計量b22所建構而成的排列檢定方式，藉由蒙地卡羅模擬方法，比較以此兩類檢定方式之型一誤差（type I error）機率以及檢定力（power），並觀察其可行性以及適用時機。模擬結果顯示，在解釋變數間不相關且誤差分配較不偏斜的情形下，Freedman and Lane (1983)、Levin and Robbins (1983)、Kennedy (1995)之排列方法在樣本數大時適用b2統計量，且其檢定力較使用t2統計量高，但差異程度不大；若解釋變數間呈現高度相關，則不論誤差的偏斜狀態，Freedman and Lane (1983)、Kennedy (1995) 之排列方法於樣本數大時適用b2統計量，其檢定力結果也較使用t2統計量高，而且兩者的差異程度比起解釋變數間不相關時更加明顯。整體而言，使用t2統計量適用的場合較廣；相反的，使用b2的模擬結果則常需視樣本數大小以及解釋變數間相關性而定。		zh_TW
dc.description.abstract (摘要) In traditional linear models, error term are usually assumed to be independently, identically, normally distributed with mean zero and a constant variance. When the assumptions cannot meet, permutation tests can be an alternative method.
Several permutation tests have been proposed to test the significance of a partial regression coefficient in a multiple regression model. t=b⁄(se(b)), an asymptotically pivotal quantity, is usually preferred and suggested as the test statistic. In this study, we take not only t statistics, but also the estimates of the partial regression coefficient as our test statistics. Their performance are compared in terms of the probability of committing a type I error and the power through the use of Monte Carlo simulation method. Situations where estimates of the partial regression coefficients may outperform t statistics are discussed.		en_US
dc.description.tableofcontents 第一章 緒論 1
第一節 研究背景與動機 1
第二節 研究目的 2
第三節 本文架構 2
第二章 文獻探討 3
第一節 簡單線性迴歸模型下係數排列檢定方法 3
第二節 複迴歸模型下係數排列檢定方法 4
第三章 文獻相關議題探討 11
第一節 迴歸係數估計值與t值之對等（equivalent）關係 11
第二節 Kennedy(1995) 方法下，t_2值自由度使用之探討 17
第三節 Oja (1987)方法下設計變數值之探討 21
第四章 研究方法與設計 22
第一節 研究方法 22
第二節 研究設計 24
第五章 研究結果 27
第一節 型一誤差機率 27
第二節 檢定力 33
第三節 小結 36
第六章 結論與建議 38
第一節 結論 38
第二節 建議 39
參考文獻 40
附錄、 R程式碼 42		zh_TW
dc.format.extent 1093299 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0104354007en_US
dc.subject (關鍵詞) 排列檢定zh_TW
dc.subject (關鍵詞) 複迴歸模型zh_TW
dc.subject (關鍵詞) 樞紐統計量zh_TW
dc.subject (關鍵詞) 蒙地卡羅模擬zh_TW
dc.subject (關鍵詞) 型一誤差zh_TW
dc.subject (關鍵詞) 檢定力zh_TW
dc.subject (關鍵詞) Permutation testen_US
dc.subject (關鍵詞) Multiple regression modelen_US
dc.subject (關鍵詞) Pivotal quantityen_US
dc.subject (關鍵詞) Monte Carlo simulationen_US
dc.subject (關鍵詞) Type I erroren_US
dc.subject (關鍵詞) Poweren_US
dc.title (題名) 複迴歸係數排列檢定方法探討zh_TW
dc.title (題名) Methods for testing significance of partial regression coefficients in regression modelen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) 1.Anderson, M. J. (2001). Permutation tests for univariate or multivariate analysis of variance and regression. Canadian Journal of Fisheries and Aquatic Sciences, 58, 626-639.
2.Anderson, M. J., and Legendre, P. (1999). An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model. Journal of Statistical Computation and Simulation, 62, 271-303.
3.Anderson, M. J., and Robinson, J. (2001). Permutation tests for linear models. Australian and New Zealand Journal of Statistics, 43, 75-88.
4.Freedman, D., and Lane, D. (1983). A nonstochastic interpretation of reported significance levels. Journal of Business and Economic Statistics, 1, 292-298.
5.Kennedy, P. E. (1995). Randomization tests in econometrics. Journal of Business and Economic Statistics, 13, 85-94.
6.Kennedy, P. E., and Cade, B. S. (1996). Randomization tests for multiple regression. Communications in Statistics – Simulation and Computation, 25, 923-936.
7.Levin, B., and Robbins, H. (1983). Urn models for regression analysis, with applications to employment discrimination studies. Law and Contemporary Problems, 46, 247-267.
8.Manly, B. F. J. (1991). Randomization, Bootstrap and Monte Carlo methods in biology (First Edition). London: Chapman and Hall.
9.Manly, B. F. J. (1997). Randomization, Bootstrap and Monte Carlo methods in biology (Second Edition). London: Chapman and Hall.
10.Manly, B. F. J. (2006). Randomization, Bootstrap and Monte Carlo methods in biology (Third Edition). London: Chapman and Hall.
11.Oja, H. (1987). On permutation tests in multiple regression and analysis of covariance problems. Australian Journal of Statistics, 29, 91-100.
12.O’Gorman, T. W. (2005). The performance of randomization tests that use permutations of independent variables. Communication in Statistics – Simulation and Computation, 34, 895-908.
13.ter Braak, C. J. F. (1992). Permutation versus bootstrap significance tests in multiple regression and ANOVA. Bootstrapping and Related Techniques (K.-H. Jockel, G. Rothe and W. Sendler, Eds.), Berlin: Springer Verlag, 79-86.
14.Winkler, A. M., Ridgway, G. R., Webster, M. A., Smith, S. M., and Nichols, T. E. (2014). Permutation inference for the general linear model. NeuroImage, 92, 381-397.		zh_TW