控制型一誤為多重假設檢定(multiple hypothesis testing)中極為重要的議題。最早使用的型一誤定義為至少有一個錯誤拒絕的機率，一般通稱為FWER(familywise error rate)。最早亦最簡單的檢定方法則為利用Bonferroni不等式所導出之檢定方法，但因Bonferroni法在假設個數增加時會趨於保守，故許多作者提出許多改善的方法。但過去
文獻在評估方法之好壞時，多半只針對是否有控制FWER及該法檢定力的大小。Pawian(2005)指出衡量各方法之方式還可以包括專一性(specificity)、FDR(falsediscovery rate)及FNDR(false non-discovery rate)這些指標，且各指標之表現不一定會一致。本論文的目的是利用蒙地卡羅模擬(Monte Carlo simulation)來討論在不同環境下，以FWER、敏感性、專一性、FDR及FNDR等指標來衡量現有檢定方法之好壞。

One of the most important issue in multiple hypothesis testing is to control Type I error. The earliest and easiest way is to define the Type I error as the probability of at least one false rejection, which is called the familywise error rate(FWER). Bonferroni method based on the Bonferroni inequality is the simplest and most common method.However, as the number of the hypothesis increases, the method becomes very conservative. There are many modified method to improve the conservativeness.Nevertheless,when evaluating the performance of the modified method, they are only focused on whether the modified method controls the FWER and how good the power is.Pawian(2005) indicates that in addition to FWER and power , specificity , false discovery rate(FDR) and false non-discovery rate(FNDR) can be also used to evaluate the performance of the method.The purpose of this thesis is to evaluate the existing methods for controlling FWER under various evaluating indexes based on various scenarios.

1 序 1

2 方法介紹 5
1 背景 5
2 控制FWER的調整p值法 6
3 文獻上方法之比較 9
4 評估各法指標 10

3 模擬方法 12
1 狀況描述 12
2 檢定統計量 14
3 模擬結果 14
3.1 FWER 15
3.2 敏感性 26
3.3 FDR 55
3.4 專一性 66
3.5 FNDR 77

4 結論與討論 102

參考文獻 106

1. Hochberg, Y. (1988). A sharper Bonferroni procedurefor multiple tests of significance. Biometrika 75:800-803.
2. Hochberg, Y. and Tamhance AC. (1987). Multiple Comparison Procedures. New York: Wiley.
3. Holland, B. (1991). On the application of three modified Bonferroni procedures to pairwise multiple comparisons in balanced repeated measures designs. Comput Stat. Q. 6:219-231.
4. Holm, S. (1979). A simple sequenttially rejective multiple test procedure. Scand. J. Stat. 6:65-70.
5. Holm, S. (1990). Review of "Multiple HypothesisTesting".Metrika 37:206.
6. Hommel, G.(1988). A stagewise rejective multiple test procedure based on a modified Bonferroni test.Biometrika 75:383-386.
7. Hommel, G.(1989). A comparison of two modified Bonferroni procedures. Biometrika 76:624-625.
8. Pawian, Y., Michiels, S., Koscielny, S., Gusnanto, A. and Ploner, A.(2005). Falsediscovery rate,sensitivity and sample size for microarray studies. Bioinformatics 21:3017-3024.
9. Shaffer, J. P. (1995). Multiple Hypothesis Testing. Annual Review of Psychology}} 46:561-584.
10. Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73:751-754.
11. van der Lann, M. J., Dudoit, S. and Pollard, K. S.(2004). Augmentation procedures for control of the generalized familywise error rate and tail probabilities for the proportion of false positives. Stat. Appl. Genet. Mol. Biol. 3:A15.