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中文論文名稱 以蒙地卡羅模擬評估各種多重比較法 在不同分佈下的表現
英文論文名稱 An evaluation of various pairwise multiple comparison procedures under several distributions by Monte Carlo simulation
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 98
學期 2
出版年 99
研究生中文姓名 黃琬庭
研究生英文姓名 Wan-Ting Huang
學號 697190428
學位類別 碩士
語文別 中文
口試日期 2010-06-08
論文頁數 130頁
口試委員 指導教授-陳順益
委員-賴耀宗
委員-吳秀芬
中文關鍵字 多重比較法  蒙地卡羅模擬  變異數分析  逐步多重比較  型一誤差率 
英文關鍵字 Multiple comparison procedures  Monte Carlo simulation  Analysis of variance  Stepwise multiple comparison  Type I error rate 
學科別分類 學科別自然科學數學
中文摘要 對於實驗中所包含的許多處理,當研究者感興趣的是所有處理平均兩兩之間是否有差異時,即會進行多重比較法中的成對比較。但許多常用的多重比較法均建立在常態分佈假設的基礎上, 而現實情況中實驗所得的數據卻不一定來自於常態分佈。對於研究者之期望何種多重比較法較為適合? 而當實驗數據來自於非常態分佈時, 又是何種多重比較法較為適合? 此即為本文欲探討的問題。
本文使用Carmer 和 Swanson(1973)、Welsch (1977) 和 Hayter (1986) 文中出現的各種多重比較法,分析來自四種不同分佈的數據:常態分佈、均勻連續分佈、指數分佈與韋伯分佈,由蒙地卡羅法電腦模擬實驗可得各多重比較法在隨機化完全區集設計下的型一誤差率以及正確決策率。
得到的結論為各多重比較法的值在均勻連續分佈下與常態分佈較相似,指數分佈、韋伯分佈下的改變較大。但不同分佈之下各多重比較法,型一誤差率及正確決策率之間相對的大小關係並不改變。 對於不同研究目的:(1)欲控制比較型一誤差率,研究者可考慮LSD。(2)欲控制實驗型一誤差率,研究者可考慮Welsch (1977)文中之GAPA。(3)欲判別處理間的差異時,LSD能控制比較型一誤差率,而MRT (Duncan's multiple range test) 則是實驗型一誤差率較小且在處理間差異度小的時候擁有較高的正確決策率。依不同情況研究者可考慮LSD或是MRT。
英文摘要 Pairwise multiple comparison procedures are often used to detect the differences between treatment means in designed experiments. Most of procedures were derived based on normal theory. However in many application areas where experiment data are perfectly normal distributed could be rare. The aim of this article is to compare the performance of several pairwise multiple procedures under different distributions, and to propose appropriate methods to suit the purpose of experiments.
Ten pairwise multiple comparison procedures from Carmer and Swanson (1973), four stepwise multiple comparison procedures from Welsh (1977), and the modified Fisher's LSD in Hayter (1986) were employed in this study. These multiple comparison procedures were investigated under four different distributions: Normal, Uniform, Exponential, and Weibull. Type I error rates and correct decision rates for the randomized complete block design were evaluated by Monte Carlo simulation method.
Results indicate that the performances under Uniform distribution are similar to that of Normal distribution,and the performances under Exponential and Weibull distributions are more different. If the goal of the experiment is: (1) to control the comparisonwise type I error rate, researchers may use LSD; (2) to control the experimentwise type I error rate, researchers may use GAPA in Welsch (1977); (3) to detect the differences between treatment means, researchers may choose either LSD or Duncan's MRT, since LSD has a good control of comparisonwise type I error rate and MRT has lower experimentwise type I error rate. The another advantage of Duncan's MRT is that it has higher correct decision rate when the degree of treatments' hetrogeneity is low.
論文目次 目錄
1 緒論 1
1.1 動機..............................................1
1.2 問題與目的........................................1
1.3 重要名詞定義......................................2
1.4 文獻討論..........................................3
2 研究工具 4
2.1 多重比較法之過程..................................4
2.2 多重比較程序......................................6
3 研究計劃與實施 10
3.1 模型描述.........................................10
3.2 蒙地卡羅模擬實驗.................................13
4 發現與討論 16
4.1 常態分佈下各種多重比較法之表現...................16
4.1.1 所有處理平均皆相等之組合.......................16
4.1.2 一些處理平均相等(但不全相等)的組合.............18
4.1.3 正確決策率.....................................23
4.2 多重比較法在不同分佈下之表現.....................26
4.2.1 均勻連續分佈...................................26
4.2.2 指數分佈.......................................31
4.2.3 韋伯分佈.......................................38
4.3 綜合討論.........................................42
5 結論與建議 45
5.1 結論.............................................45
5.2 標準差不等時之比較與階級式逐步減少多重比較法.....47
5.2.1 標準差不相等時.................................47
5.2.2 階級式逐步減少多重比較法.......................52
參考文獻 53
附錄 55
附錄一不同分佈下所有k = 5 組合之型一誤差率...........55
附錄二電腦模擬程式...................................63
表目錄
表3.1 RCBD變異數分析.....................................10
表3.2 處理平均組合表.....................................13
表4.1 常態分佈下所有處理平均皆相等之型一誤差率...........17
表4.2 常態分佈下k = 5 時各多重比較法的型一誤差率.........19
表4.3 常態分佈下k = 10 時各多重比較法的型一誤差率........20
表4.4 常態分佈下k = 5 時以q 2分類之正確決策率............24
表4.5 常態分佈下k = 5 與k = 10之正確決策率...............25
表4.6 均勻連續分佈下所有處理平均皆相等之型一誤差率.......27
表4.7 均勻連續分佈下k = 5 時各多重比較法的型一誤差率.....28
表4.8 均勻連續分佈下k = 10 時各多重比較法的型一誤差率....29
表4.9 均勻連續分佈下k = 5 與k = 10之正確決策率...........30
表4.10指數分佈下所有處理平均皆相等之型一誤差率...........32
表4.11指數分佈下k = 5 時各多重比較法的型一誤差率.........33
表4.12指數分佈下處理結構對SNK、GAPA(B)、NKA(B)實驗型一誤
差率的影響...............................................34
表4.13指數分佈下k = 10 時各多重比較法的型一誤差率........36
表4.14指數分佈下k = 5 與k = 10之正確決策率...............37
表4.15韋伯分佈下所有處理平均皆相等之型一誤差率...........38
表4.16韋伯分佈下k = 5 時各多重比較法的型一誤差率.........39
表4.17韋伯分佈下k = 10 時各多重比較法的型一誤差率........40
表4.18韋伯分佈下k = 5 與k = 10之正確決策率...............41
表5.1 標準差不相等k = 5 時所有處理平均皆相等之型一誤差率.48
表5.2 標準差不相等k = 5 時各多重比較法的型一誤差率.......49
表5.2 標準差不相等k = 5 時各多重比較法的型一誤差率(續)...50
表5.3 處理平均皆相等時階級式逐步減少法的型一誤差率.......53
表A.1 常態分佈下k = 5 時各多重比較法的型一誤差率.........55
表A.1 常態分佈下k = 5 時各多重比較法的型一誤差率(續).....56
表A.2 均勻連續分佈下k = 5 時各多重比較法的型一誤差率.....57
表A.2 均勻連續分佈下k = 5 時各多重比較法的型一誤差率(續).58
表A.3 指數分佈下k = 5 時各多重比較法的型一誤差率.........59
表A.3 指數分佈下k = 5 時各多重比較法的型一誤差率(續).....60
表A.4 韋伯分佈下k = 5 時各多重比較法的型一誤差率.........61
表A.4 韋伯分佈下k = 5 時各多重比較法的型一誤差率(續).....62
圖目錄
圖3.1 不同分佈之機率密度圖...............................13
參考文獻 Carmer, S. G., and Swanson, M. R. (1973), An Evaluation of Ten Pairwise Multiple Comparison Procedures by Monte Carlo Methods. Journal of the American Statistical Assoclation,
Vol. 68, 66–74.

Dodge, Y., and Thomas, D. R. (1980), On the Performance of Non-Parametric and Normal Theory Multiple Comparison Procedures. Sankhy‾a: The Indian Journal of Statistics, Series B,Vol. 42, No. 1/2, 11–27.

Duncan, D. B. (1965), A Bayesian Approach to Multiple Comparisons. Technometrics, Vol. 7, No. 2, 171–222.

EMEA. (2002), Points to consider on multiplicity issues in clinical trials, The European Agency for the Evaluation of Medicinal Products, Evaluation of medicines for human use,
CPMP/EWP/908/99.

Hayter, A. J. (1986), The Maximum Familywise Error Rate of Fisher’s Least Significant Difference Test. Journal of the American Statistical Association, Vol. 81, No. 396, 1000–1004.

Waller, R. A., and Duncan, D. B. (1969), A Bayes Rule for the Symmetric Multiple Comparisons Problems. Journal of the American Statistical Assoclation, Vol. 94, No. 328, 1484–1503.

Waller, R. A., and Duncan, D. B. (1972), Corrigenda: A Bayes Rule for the Symmetric Multiple Comparisons Problem. Journal of the American Statistical Assoclation, Vol. 67, No. 337, 253–255.

Welsch, R. E. (1977a), Stepwise Multiple Comparison Procedures. Journal of the American Statistical Association, Vol. 72, No. 359, 566–575.

Welsch, R. E. (1977b), Tables for stepwise multiple comparison procedures. Massachusetts Institute of Technolog,Working paper, No. 949–77.

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