Bonferroni校正

如果在同一数据集上同时检验n个独立的假设，那么用于每一假设的统计显著水平，应为仅检验一个假设时的显著水平的1/n。举个例子：如要在同一数据集上检验两个独立的假设，显著水平设为常见的0.05。此时用于检验该两个假设应使用更严格的0.025。即0.05* (1/2)。该方法是由Carlo Emilio Bonferroni发展的，因此称Bonferroni校正。

这样做的理由是基于这样一个事实：在同一数据集上进行多个假设的检验，每20个假设中就有一个可能纯粹由于概率，而达到0.05的显著水平。

维基百科原文：

Bonferroni correction

Bonferroni correction states that if an experimenter is testing n independent hypotheses on a set of data, then the statistical significance level that should be used for each hypothesis separately is 1/n times what it would be if only one hypothesis were tested.

For example, to test two independent hypotheses on the same data at 0.05 significance level, instead of using a p value threshold of 0.05, one would use a stricter threshold of 0.025.

The Bonferroni correction is a safeguard against multiple tests of statistical significance on the same data, where 1 out of every 20 hypothesis-tests will appear to be significant at the α = 0.05 level purely due to chance. It was developed by Carlo Emilio Bonferroni.

A less restrictive criterion is the rough false discovery rate giving (3/4)0.05 = 0.0375 for n = 2 and (21/40)0.05 = 0.02625 for n = 20.

数据分析中常碰见多重检验问题(multiple testing).Benjamini于1995年提出一种方法,通过控制FDR(False Discovery Rate)来决定P值的域值. 假设你挑选了R个差异表达的基因，其中有S个是真正有差异表达的，另外有V个其实是没有差异表达的，是假阳性的.实践中希望错误比例Q=V/R平均而言不能超过某个预先设定的值（比如0.05），在统计学上，这也就等价于控制FDR不能超过5%.

根据Benjamini在他的文章中所证明的定理，控制fdr的步骤实际上非常简单。

设总共有m个候选基因，每个基因对应的p值从小到大排列分别是p(1),p(2),...,p(m),则若想控制fdr不能超过q，则只需找到最大的正整数i，使得 p(i)<= (i*q)/m.然后，挑选对应p(1),p(2),...,p(i)的基因做为差异表达基因，这样就能从统计学上保证fdr不超过q。

The False Discovery Rate (FDR) of a set of predictions is the expected percent of false predictions in the set of predictions. For example if the algorithm returns 100 genes with a false discovery rate of .3 then we should expect 70 of them to be correct.

The FDR is very different from a p-value, and as such a much higher FDR can be tolerated than with a p-value. In the example above a set of 100 predictions of which 70 are correct might be very useful, especially if there are thousands of genes on the array most of which are not differentially expressed. In contrast p-value of .3 is generally unacceptabe in any circumstance. Meanwhile an FDR of as high as .5 or even higher might be quite meaningful.

FDR错误控制法是Benjamini于1995年提出一种方法,通过控制FDR(False Discovery Rate)来决定P值的域值. 假设你挑选了R个差异表达的基因，其中有S个是真正有差异表达的，另外有V个其实是没有差异表达的，是假阳性的。实践中希望错误比例Q=V/R平均而言不能超过某个预先设定的值（比如0.05），在统计学上，这也就等价于控制FDR不能超过5%.

对所有候选基因的p值进行从小到大排序，则若想控制fdr不能超过q，则只需找到最大的正整数i，使得 p(i)<= (i*q)/m.然后，挑选对应p(1),p(2),...,p(i)的基因做为差异表达基因，这样就能从统计学上保证fdr不超过q。因此，FDR的计算公式如下：

q-value(i)=p(i)*length(p)/rank(p)

参考文献：

1.Audic, S. and J. M. Claverie (1997). The significance of digital gene expression profiles. Genome Res 7(10): 986-95.

2.Benjamini, Y. and D. Yekutieli (2001). The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics. 29: 1165-1188.

计算方法 请参考 R统计软件的p.adjust函数：

> p<-c(0.0003,0.0001,0.02)

> p

[1] 3e-04 1e-04 2e-02

>

> p.adjust(p,method="fdr",length(p))

[1] 0.00045 0.00030 0.02000

>

> p*length(p)/rank(p)

[1] 0.00045 0.00030 0.02000

> length(p)

[1] 3

> rank(p)

[1] 2 1 3

sort(p)

[1] 1e-04 3e-04 2e-02