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

An evaluation of various pairwise multiple comparison procedures under several distributions by Monte Carlo simulation

黃琬庭(Wan-Ting Huang)

指導教授 ： 陳順益

淡江大學/理學院/數學學系碩士班/碩士(2010年)

https://doi.org/10.6846/TKU.2010.00215

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。

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

並列摘要

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.

並列關鍵字

Multiple comparison procedures ； Monte Carlo simulation ； Analysis of variance ； Stepwise multiple comparison ； Type I error rate

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,

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,

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.