邦费罗尼校正

邦费罗尼校正（英語：Bonferroni correction）是统计学中在多重比较时使用的一种校正方法，以意大利数学家卡罗·埃米利奥·邦费罗尼的名字命名。

令 $H_{1},\ldots ,H_{m}$ 为一组假设， $p_{1},\ldots ,p_{m}$ 为每一假设相对应的p值。同时， $m$ 为零假设总数， $m_{0}$ 则为实际为真的零假设总数。族错误率（familywise error rate，简称FWER）指拒绝至少一个实际为真的零假设（即出现至少一次第一类错误）的概率。此时，邦费罗尼校正是指拒绝所有 $p_{i}\leq {\frac {\alpha }{m}}$ 的零假设。在应用邦费罗尼校正后，FWER满足 ${\text{FWER}}\leq \alpha$ 。这一结论可以由布尔不等式证明：

{\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq m_{0}{\frac {\alpha }{m}}\leq m{\frac {\alpha }{m}}=\alpha .

邦费罗尼校正是一种相对保守的FWER控制方法，会增加出现第二类错误的概率。

参考文献

Dunnett, C. W. A multiple comparisons procedure for comparing several treatments with a control. Journal of the American Statistical Association. 1955, 50 (272): 1096–1121. doi:10.1080/01621459.1955.10501294.
Dunnett, C. W. New tables for multiple comparisons with a control. Biometrics. 1964, 20 (3): 482–491. JSTOR 2528490. doi:10.2307/2528490.
Shaffer, J. P. Multiple Hypothesis Testing. Annual Review of Psychology. 1995, 46: 561–584. doi:10.1146/annurev.ps.46.020195.003021.
Strassburger, K.; Bretz, Frank. Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni-based closed tests. Statistics in Medicine. 2008, 27 (24): 4914–4927. doi:10.1002/sim.3338.
Šidák, Z. Rectangular confidence regions for the means of multivariate normal distributions. Journal of the American Statistical Association. 1967, 62 (318): 626–633. doi:10.1080/01621459.1967.10482935.
Hochberg, Yosef. A Sharper Bonferroni Procedure for Multiple Tests of Significance (PDF). Biometrika. 1988, 75 (4): 800–802 [2017-02-01]. doi:10.1093/biomet/75.4.800. （原始内容存档 (PDF)于2016-10-12）.

外部連結

Bonferroni, Sidak online calculator （页面存档备份，存于互联网档案馆）
Multiple Testing Corrections in GeneSpring and Gene Expression data （页面存档备份，存于互联网档案馆）

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