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研究生:廖邦幸
研究生(外文):Bang-Xing Liao
論文名稱:變異數不等時與平均之多重比較程序之模擬研究
論文名稱(外文):Simulation Study of Multiple Comparisons with the Average under Heteroscedasticity
指導教授:吳淑妃
指導教授(外文):Shu-Fei Wu
學位類別:碩士
校院名稱:淡江大學
系所名稱:統計學系
學門:商業及管理學門
學類:會計學類
論文種類:學術論文
論文出版年:2001
畢業學年度:89
語文別:中文
論文頁數:42
中文關鍵詞:雙階段程序單階段程序蒙地卡羅模擬法平均數分析
外文關鍵詞:two-stage proceduresingle-stage procedureMonte-Carlo techniquesanalysis of means(ANOM)
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在許多實驗中我們常常將所有實驗對象的平均值當作一個評量的標準,希望能夠從中找出比整體平均好,比整體平均差,或是與整體平均無明顯差異的子集合。例如在農業上可用來篩選出比平均優良的小麥品種,使種植時的效益更高﹔或是在醫學臨床上,可針對具有相同療效的藥品中,找出比平均效果更好的。
本文中,當變異數未知且可能不相等時,我們採用了三種不同的方法來求取常態分配平均數與平均之聯立信賴區間,分別是傳統法、單階段和雙階段與平均之多重比較程序。並使用蒙地卡羅模擬法來模擬出傳統法,單階段和雙階段與平均之多重比較程序的信賴區間長度及信心水準。由模擬結果發現發現單階段和雙階段與平均之多重比較程序之模擬信心水準皆可達到名目信心水準。並舉一個生統的例子以示範單階段,雙階段與平均之多重比較程序。同時利用此同時信賴區間來做平均數分析(Analysis of Means(ANOM))以和陳順益(1998)之變異數分析做顯著水準和檢定力之差異比較。更廣義的加權平均也會在本研究中予以推廣。

In many experimental situations, the average treatment performance within its own group is used as a benchmark to be compared with each individual treatment. Our study propose is to identify better than the average, worse than the average and not much difference from the average subsets based on the simultaneous two-sided confidence interval of each normal mean away from its average. One can use this procedure to screen a great number of wheat varieties better than the average in an agricultural experiment, or to screen drugs with longer hours of pain-relief than the average in a clinical trial.
In this article, a simulation study of traditional, single-stage and two-stage multiple comparison procedures with the average for normal distribution under heteroscedasticity is investigated by the Monte-Carlo techniques. The length of simultaneous confidence interval and confidence coefficient for three procedures are simulated and it’s found that the simulated confidence coefficients can reach its nominal confidence coefficients for single-stage and two-stage multiple comparison procedures with the average while the traditional procedure fails to under heteroscedasticity. A biometrical example is given to illustrate the single-stage and two-stage procedures. These simultaneous confidence intervals are also used as an Analysis of Means(ANOM) to compared with the analysis of Variance proposed by 陳順益(1998) by their level of significance and their power. A generalized arithmetic mean as a benchmark is also considered.

目錄
摘要……………………………………………………………1
第一章 緒論…………………………………………………2
第二章 文獻探討……………………………………………4
第三章 傳統法,單階段,雙階段與平均的多重比較程序
…………………………………………………………………6
第一節 傳統法與平均的多重比較程序……………………6
第二節 雙階段與平均的多重比較程序……………………10
第三節 單階段與平均的多重比較程序……………………16
第四章 生統例子……………………………………………19
第五章 模擬比較……………………………………………23
第六章 加權平均數…………………………………………35
第七章 結論…………………………………………………40
參考文獻………………………………………………………41

中文部份
陳順益(1998). 「變異數不相等時的單階段變異數分析法」,中國統計學報}, 36(4), 321-338.
英文部份
Bishop, T. A. and Dudewicz, E. J. (1978). Exact analysis of variance with unequal variances: test procedures and tables. Technometrics, 20, 419-430.
Chen, H. J. and Dudewicz, E. L.(1976).
Procedures for fixed-width interval estimation of the largest normal mean. Journal of American Statistical Association, 71, 752-756.
Chen, H. J. and Lam, K.(1989).
Single-stage interval estimation of the largest normal mean under
heteroscedasticity. Communications in Statistics - Theory and Methods., 18(10), 3703-3718.
Dudewicz,E. J. and Dalal, S. R.(1975).
Allocation of observations in ranking and selection with unequal variances. Sankhya, Ser. B., 37, 28-78.
Halperin,M.,Greenhouse,S.W.,Cornfield,J.and Zalokar,J.(1955).
Tables of percentage points for the studentized maximum absolute deviate in normal samples. Journal of American Statistical Association, 50, 185-195.
Nelson, L. S.(1983).
Exact critical values for use with the analysis of means.
Journal of Quality Technology, 15, 40-44.
Nelson, P. R.(1993).
Additional uses for the analysis of means and extended tables of critical values. Technometrics, 35, 61-71.
Nelson, P. R.(1982).
Exact critical points for the analysis of means.
Communications in Statistics-Theory and Methods, 11, 699-709.
Stein. C. M. (1945). A two-sample test for a linear hypothesis whose power is independent of variance. Annals of Mathematical Statistics., 16, 243-258.
Tong. Y. L. (1980).
Probability inequalities in multivariate distribution.
Academic Press, New York.
Wen, M. J. and Chen, H. J.(1994).
Single-stage multiple comparison procedure under heteroscedasticity.
American Journal of Mathematical and Management Sciences, 14, Nos 1 and 2, 1-48.
Wu, S. F. and Chen, H. J.(2000).
Two-stage multiple comparisons with the average for normal distributions under heteroscedasticity.
Computational statistics & data analysis, 33, 201-213.
Wu, S. F. and Chen, H. J.(1998).
Multiple comparisons with the average for normal distributions.
American Journal of Mathematical and Management Sciences, 18, 193-218.

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