臺灣博碩士論文加值系統

本文針對具有常態分布誤差的數個迴歸模式，於其共變數可能範圍內，建立多條迴歸直線與一條對照迴歸直線差異之聯合單尾信賴域。針對具有單一共變數的兩條迴歸直線差異，本文求出確實的雙尾信賴帶或單尾信賴域之後應用Bonferroni不等式調整族誤差率以建立相關的聯合雙尾信賴帶或聯合單尾信賴域。針對具有多個共變數的數條迴歸直線比較，則是利用模擬的方法求得建立此一聯合單尾信賴域所需之臨界值。本文並進一步利用模擬方法，研究所提聯合單尾信賴域的覆蓋機率。最後，以實例說明所提方法之應用。

The problem of interest in this article is to construct simultaneous one-sided confidence regions for the difference between one controlled regression line with other several regression lines when the random errors are normally distributed. We propose an exact two-sided confidence band or one-sided confidence region for the difference of two simple linear regression lines and then suggest to construct such a simultaneous two-sided confidence band or one-sided confidence region by applying Bonferroni’s inequality for controlling the experiment error rate. For the comparison of several regression lines with one regression line when two or more covariates are involved, we consider to use a simulation-based method for finding the required critical value in the simultaneous one-sided confidence region. A simulation study is then conducted to investigate the coverage probability of the proposed confidence region. Finally, the application of the proposed procedures are demonstrated by illustrating a real data set.

第一章 研究動機及目的……………………………… 1
第二章 文獻回顧……………………………………… 3
2.1單一共變數之下平均反應的單尾聯合信賴域…… 3
2.2 數條迴歸直線的多重比較………………………… 7
第三章 統計方法……………………………………… 11
3.1具單一共變數數條迴歸直線之多重比較………… 11
3.2具多個共變數數條迴歸直線之多重比較………… 14
3.3 封閉性多重比較 ……………………………………18
第四章 模擬研究……………………………………… 19
第五章 實例分析……………………………………… 22
第六章 結論及未來研究……………………………… 25
參考文獻 ……………………………………………… 26
附錄一 定理證明……………………………………… 44
附錄二 實例原始資料………………………………… 51

1.Bradstreet, T.E. (1991). Some favorite data set from early phase of drug research. Proceeding of the Section on Statistical Education of the American Statistical Association, pp. 190-195.

2.Hochberg, Y., Tamhane, A. C. and Dunnett, C. W. (1996). Multiple test procedures for dose finding. Biometrics, 52, 21-37.

3.Hochberg, Y. and Tamhane, A. C. (1987). Multiple Comparison Procedures. New York: Wiley.

4.Hotelling, H. and Working, H. (1929). Applications of the theory of error to the interpretation of trends. Journal of the American Statistical Association, 24, 73-85.

5.Hsu, J. C. (1996). Multiple Comparisons: Theory and Methods. New York: Chapman & Hall.

6.Hsu, J. C. and Berger, R. L. (1999). Stepwise confidence intervals without multiplicity adjustment for dose response and toxicity studies. Journal of the American Statistical Association, 94, 468-482.

7.Kodell, R. L. and West, R. W. (1993). Upper confidence intervals on excess risk for quantitative responses. Risk Analysis, 13, 177-182.

8.Liu, W. and Zhang, Y. (2004). Multiple comparison of several linear regression models. Journal of the American Statistical Association, 99, 395-403.

9.Pan, W., Piegorsch, W. W. and West, R. W. (2003). Exact one-sided simultaneous confidence bands via Uusipaikka’s method. Annals of the Institute of Statistic at Mathematics, 55, 243-250.
10.Piegorsch, W. W., West, R. W., Pan, W., Kodell, R. L. (2005). Low dose risk estimation via simultaneous statistical inferences. Journal of the Royal Statistical Society, Series C, 54, 245-258.

11.Robertson, J.D. and Armitage, P. (1959). Comparison of two hypotensive agents. Anaesthesia, 14, 53-64.

12.Ruberg, S. J. (1989). Contrasts for identifying the minimun effective dose. Journal of the American Statistical Association, 84, 816-822.

13.Scheffe, H. (1953). A method for judging all contrasts in the analysis of variance. Biometrika. 40, 87-110

14.Serfling, R. J. (1980). Approximation Theorem of Mathematical Statistic. New York: Wiley.

15.Spurrier, J. D. (1999). Exact confidence bounds for all contrasts of three or more regression lines. Journal of the American Statistical Association, 94, 483-488.

16.Uusipaikka, E. (1983). Exact confidence bands for linear regression over intervals. Journal of the American Statistical Association, 78, 638-644.