查看“︁邦费罗尼校正”︁的源代码

{{NoteTA|G1=Math}}

'''邦费罗尼校正'''（{{lang-en|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<math>H_1,\ldots,H_m</math>为一组假设，<math>p_1,\ldots,p_m</math>为每一假设相对应的[[p值]]。同时，<math>m</math>为[[零假设]]总数，<math>m_0</math>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<math>p_i\leq\frac \alpha m</math>的零假设。在应用邦费罗尼校正后，FWER满足<math>\text{FWER} \leq \alpha</math>。这一结论可以由[[布尔不等式]]证明：
: <math> \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.</math>

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 ==
{{Refbegin|2}}
*{{cite journal |last=Dunnett |first=C. W. |year=1955 |title=A multiple comparisons procedure for comparing several treatments with a control |url=https://archive.org/details/sim_journal-of-the-american-statistical-association_1955-12_50_272/page/1096 |journal=Journal of the American Statistical Association |volume=50 |issue=272 |pages=1096–1121 |doi=10.1080/01621459.1955.10501294 }}
*{{cite journal |last=Dunnett |first=C. W. |year=1964 |title=New tables for multiple comparisons with a control |url=https://archive.org/details/sim_biometrics_1964-09_20_3/page/482 |journal=[[Biometrics (journal)|Biometrics]] |volume=20 |issue=3 |pages=482–491 |jstor=2528490 |doi=10.2307/2528490}}
*{{cite journal |last=Shaffer |first=J. P. |year=1995 |title=Multiple Hypothesis Testing |url=https://archive.org/details/sim_annual-review-of-psychology_1995_46/page/561 |journal=[[Annual Review of Psychology]] |volume=46 |issue= |pages=561–584 |doi=10.1146/annurev.ps.46.020195.003021 }}
*{{cite journal |last=Strassburger |first=K. |last2=Bretz |first2=Frank |year=2008 |title=Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni-based closed tests |journal=[[Statistics in Medicine (journal)|Statistics in Medicine]] |volume=27 |issue=24 |pages=4914–4927 |doi=10.1002/sim.3338 }}
*{{cite journal |last=Šidák |first=Z. |year=1967 |title=Rectangular confidence regions for the means of multivariate normal distributions |url=https://archive.org/details/sim_journal-of-the-american-statistical-association_1967-06_62_318/page/626 |journal=Journal of the American Statistical Association |volume=62 |issue=318 |pages=626–633 |doi=10.1080/01621459.1967.10482935 }}
*{{cite journal | last1 = Hochberg | first1 = Yosef | year = 1988 | title = A Sharper Bonferroni Procedure for Multiple Tests of Significance | journal = [[Biometrika]] | volume = 75 | issue = 4 | pages = 800–802 | url = http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/MultipleComparision/Hochberg88.pdf | doi = 10.1093/biomet/75.4.800 | access-date = 2017-02-01 | archive-date = 2016-10-12 | archive-url = https://web.archive.org/web/20161012012123/http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/MultipleComparision/Hochberg88.pdf | dead-url = no }}
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==外部連結==
*[http://www.quantitativeskills.com/sisa/calculations/bonfer.htm Bonferroni, Sidak online calculator] {{Wayback|url=http://www.quantitativeskills.com/sisa/calculations/bonfer.htm |date=20181229123418 }}
*[http://nebc.nerc.ac.uk/courses/GeneSpring/GS_Mar2006/Multiple%20testing%20corrections.pdf Multiple Testing Corrections in GeneSpring and Gene Expression data] {{Wayback|url=http://nebc.nerc.ac.uk/courses/GeneSpring/GS_Mar2006/Multiple%20testing%20corrections.pdf |date=20160112121600 }}

{{DEFAULTSORT:Bonferroni Correction}}
[[Category:多重比較]]
[[Category:假設檢定]]