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研究生:徐君岳
研究生(外文):Chun-yueh Hsu
論文名稱:兩條迴歸線交叉點X座標之信賴區間的比較
論文名稱(外文):Comparison of Confidence Intervals on the Abscissa of the Point of Intersection for Two Least Squares Fitted Lines
指導教授:吳榮彬吳榮彬引用關係林文欽林文欽引用關係
指導教授(外文):Jungpin WuWin-Chin Lin
學位類別:碩士
校院名稱:逢甲大學
系所名稱:統計與精算所
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2009
畢業學年度:97
語文別:中文
論文頁數:71
中文關鍵詞:拔靴法交叉點顯著區域詹森-內曼技巧共變數共變異數分析
外文關鍵詞:bootstrappingcovariateANCOVAJohnson-Neyman techniquepoint of intersectionsignificant region
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考慮兩種處理加上一種共變數的共變異數分析模型,當反應變數(y)受到共變數(x)的影響,且反應變數與共變數的組內母體迴歸直( )不平行的時候,可使用詹森-內曼技巧(Johnson-Neyman technique)求得共變數的顯著區域;在此顯著區域內處理效果是有差異的。本研究針對兩條組內母體迴歸直線在有限區間內相交的情況,當共變數在這交叉點的 x 座標時,統計上兩處理效果是沒有差異的。考慮 )以及由詹森-內曼技巧所衍生之非顯著區域的中點之 x 座標估計該交叉點的 x座標,由於這兩種估計式的抽樣分配之理論結果不易得知,因此我們考慮利用拔靴法(bootstrap)估計它們的抽樣分配,並進而計算交叉點 x 座標的信賴區間。本研究利用模擬檢視一些情況下,上述兩種估計式的表現,並且與理論信賴區間或是蒙地卡羅區間比較。
For an ANCOVA model with two treatments and one covariate,when the regression lines of the response variable on the covariate in two groups are not parallel,the Johnson-Neyman (JN) technique can be used to find the significant region.In this region, the treatment effect is significantly different. In this study,we focus on the situation that the two regression lines have an intersection in the finite interval. Statistically, the treatment effect is the same as the covariate takes value at the x-component of this intersection point.For estimating the x-component of this point, we consider an intuitive geometry solution and the center abscissa of the point of the JN non-significant region (called JN Evolved Estimator).Because the distributions of these two estimators are difficult to obtain theoretically,we use the bootstrap technique to estimate the sampling distribution of the estimators aforementioned,
and use the resulting bootstrap distribution to derive the confidence interval of the x-component of this intersection point.A set of simulations are used to examine the performance of the resulting confidence intervals by comparing then intervals with theoretical confidence intervals or Monte Carlo confidence intervals .
第1章 緒論 1
1.1 研究背景與說明 1
1.2 研究動機與目的 3
1.3 論文架構 4
第2章 兩條迴歸線交叉點 X 座標的區間估計 5
2.1 交叉點 X 座標的估計式 6
2.1.1 最大概似估計式 6
2.1.2 JN Evolved Estimator 8
2.2 拔靴法的信賴區間 9
2.2.1 Bootstrap normal 10
2.2.2 Bootstrap percentile 10
2.2.3 Basic bootstrap 11
2.2.4 BCa (bias-correction acceleration) percentile 11
第3 章模擬研究 13
3.1 模擬例子的設計 13
3.2 結果分析 15
第4 章實例分析 55
4.1 例子一: 林清山 (1992) 55
4.1.1 數據分析 55
4.1.2 模擬分析 58
4.2 例子二: Huitema (1980) 59
第5章 結語 64
參考文獻 65
附錄A R 程式碼 67
附錄B Fortran 程式碼 67
附錄C 公式推導 67
附錄D 信賴區間之模擬資料 67
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[4] Efron, B. (1979). Bootstrap Methods: Another Look at the Jackknife. Annals of
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[5] Efron, B. (1987). Better Bootstrap Con dence Intervals. J. Amer. Statist. Assoc,
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[6] Efron, B., Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman &
Hall, New York.
[7] Huitema, B. E. (1980). The analysis of covariance and alternatives. John wiley &
Sons, New York.
[8] Hunka, S. (1995). Identifying region of signi cance in ANCOVA problems having
non-homogeneous regressions. British Journal of Mathematical and Statistical Psy-
chology, 48 : 161-188.
[9] Johnson, P. O., Fay, L. C. (1950). The Johnson-Neyman technique, its theory and
application. Psychometrika, 15 : 349-367.
[10] Johnson, P. O., Hoyt, C. (1947). On determining three-dimensional regions of sig-
ni cance. Journal of Experimental Education, 15 : 342-353.
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[12] Jennings, E. (1988). Analysis of covariance with nonparallel regression lines. Jour-
nal of Experimental Education, 56 : 129-134.
[13] Kastenbaum, M. A. (1959). A con dence interval on the abscissa of the point of
intersection of two tted linear regressions. Biometrics, 15 : 323-324.
[14] Larholt, K. M., Sampson, A. R. (1995). E ects of heteroscedasticity upon certain
analyses when regression lines are not parallel. Biometrics, 51 : 731-737.
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thereof. Psychometrika, 29 : 241-256.
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[17] Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer-Verlag, New
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[18] 林清山 (1992)。心理與教育統計學。東華書局。
[19] 周心怡 (2004)。拔靴法(Bootstrap)之探討及其應用,中央大學統計研究所碩士論文。
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