在高維度資料研究中，控制偽發現率（FDR）已快速地被使用
於解決多重性問題。當同時執行大量的假設檢定時，FDR 已經成為
控制型I 誤差率膨脹的重要議題。在多重比較檢定中，傳統上常使
用整體錯誤率（FWER）來控制整體的型I 誤差率。然而，當很多
虛無假設是錯誤的情況下，FWER 會變的太保守以至於降低檢定
力。為了改善FWER 的缺點，Benjamini and Hochberg（1995）提出
較簡易且可提高檢定力的FDR 方法。FDR 有逐步向上與逐步向下
之檢定程序，在本篇論文中，主要的目的在於比較逐步向上與逐步
向下程序的表現，並且指出各個檢定程序的優缺點。模擬的結果指
出，當檢定個數很少和大部分假設都是錯誤時，Benjamini and Liu
（1999a、1999b）所提出之方法比其他程序更具有檢定力；而在檢
定個數很多時，Benjamini and Hochberg（1995）程序有較高之檢定
力。

Controlling false discovery rate (FDR) has been increasingly utilized in
high dimensional screening studies where the multiplicity is a problem.
It becomes an important issue to control the inflating type I error rate
when tons of tested hypotheses are simultaneously conducted.
Traditionally, familywise error rate (FWER) is used to control the overall
type I error in the area of multiple comparison. However, when many null
hypotheses are false, FWER tends to be more conservative and has less
power. To improve the drawbacks of FWER, a simple approach based on FDR
can be used. Two types of FDR procedures for multiple comparison are
step-up and step-down procedures. The objective of this article is to
compare the performance of current step-up and step-down procedures, and
detect the pros and cons of these procedures. The simulated results
indicate that Benjamini-Liu (1999a,1999b) procedures are more powerful
if the number of tested hypotheses is small and many of the hypotheses
are far from true, whereas Benjamini- Hochberg (1995) procedure has large
power if the number of tested hypotheses is large.

Contents
1 Introduction 1
2 Description of Methodology 4
2.1 Step-up Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Step-down Procedures . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Simulation Study 10
3.1 Independent Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Dependent Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Discussion and Conclusion 34

List of Tables
1 The possible outcomes of testing m hypotheses. . . . . . . . . . . . 5
2 The estimated FDR at   = 0.01 for independent case. . . . . . . . . 15
3 The estimated FDR at   = 0.05 for independent case. . . . . . . . . 16
4 The estimated FDR at   = 0.10 for independent case. . . . . . . . . 17
5 The estimated power at   = 0.01 for independent case. . . . . . . . 18
6 The estimated power at   = 0.05 for independent case. . . . . . . . 19
7 The estimated power at   = 0.10 for independent case. . . . . . . . 20
8 The estimated FDR at   = 0.05 and   = 0.5 for dependent case. . 22
9 The estimated FDR at   = 0.05 and   = 0.9 for dependent case. . 23
10 The estimated FDR at   = 0.05 and   = −0.5 for dependent case. 24
11 The estimated FDR at   = 0.05 and   = −0.9 for dependent case. 25
12 The estimated power at   = 0.05 and   = 0.5 for dependent case. . 26
13 The estimated power at   = 0.05 and   = 0.9 for dependent case. . 27
14 The estimated power at   = 0.05 and   = −0.5 for dependent case. 28
15 The estimated power at   = 0.05 and   = −0.9 for dependent case. 29
16 The ordered p-values of t test and corresponding critical values of
current procedures for gene expression data. . . . . . . . . . . . . . 31
17 Scheff´e Test at   = 0.05 for tensile strength data, compare to critical
value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
18 The ordered p-values of pair comparison and each of procedures
critical value at   = 0.05. . . . . . . . . . . . . . . . . . . . . . . . 33

List of Figures
1 The estimated FDR of step-up and step-down procedures at   =
0.05 for independent case. . . . . . . . . . . . . . . . . . . . . . . . 12
2 The estimated power of step-up and step-down procedures at   =
0.05 for independent case. . . . . . . . . . . . . . . . . . . . . . . . 13

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