邦費羅尼校正

邦費羅尼校正（英語：Bonferroni correction）是統計學中在多重比較時使用的一種校正方法，以意大利數學家卡羅·埃米利奧·邦費羅尼的名字命名。

令${\displaystyle H_{1},\ldots ,H_{m}}$為一組假設，${\displaystyle p_{1},\ldots ,p_{m}}$為每一假設相對應的p值。同時，${\displaystyle m}$為零假設總數，${\displaystyle m_{0}}$則為實際為真的零假設總數。族錯誤率（familywise error rate，簡稱FWER）指拒絕至少一個實際為真的零假設（即出現至少一次第一類錯誤）的概率。此時，邦費羅尼校正是指拒絕所有${\displaystyle p_{i}\leq {\frac {\alpha }{m}}}$的零假設。在應用邦費羅尼校正後，FWER滿足${\displaystyle {\text{FWER}}\leq \alpha }$。這一結論可以由布爾不等式證明：

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

邦費羅尼校正是一種相對保守的FWER控制方法，會增加出現第二類錯誤的概率。

參考文獻

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• 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. （原始內容存檔 (PDF)於2016-10-12）.

外部連結

• Bonferroni, Sidak online calculator （頁面存檔備份，存於網際網路檔案館）
• Multiple Testing Corrections in GeneSpring and Gene Expression data （頁面存檔備份，存於網際網路檔案館）