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| In [[statistics]], the '''Bonferroni correction''' is a method used to counteract the problem of [[multiple comparisons]]. It is considered the simplest and most conservative method to control the [[familywise error rate]].
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| It is named after [[Italian people|Italian]] [[mathematician]] [[Carlo Emilio Bonferroni]] for the use of [[Bonferroni inequalities]], but modern usage is credited to [[Olive Jean Dunn]], who first used it in a pair of articles written in 1959 and 1961.
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| ==Informal introduction==
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| Statistical inference logic is based on rejecting the [[null hypothesis|null hypotheses]] if the likelihood of the observed data under the null hypotheses is low. The problem of multiplicity arises from the fact that as we increase the number of hypotheses in a test, we also increase the likelihood of witnessing a rare event, and therefore, the chance to reject the null hypotheses when it's true (type I error). Bonferroni correction is the most naive way to address this issue. The correction is based on the idea that if an experimenter is testing ''n'' dependent or independent [[statistical hypothesis testing|hypotheses]] on a set of data, then one way of maintaining the [[familywise error rate]] (FWER) is to test each individual hypothesis at a [[statistical significance]] level of ''1/n'' times what it would be if only one hypothesis were tested. So, if it is desired that the significance level for the whole family of tests should be (at most) ''α'', then the Bonferroni correction would be to test each of the individual tests at a significance level of ''α/n''. ''Statistically significant'' simply means that a given result is unlikely to have occurred by chance assuming the null hypothesis is actually correct (i.e., no difference among groups, no effect of treatment, no relation among variables).
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| ==Definition==
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| Let <math>H_{1},...,H_{m}</math> be a family of hypotheses and <math>p_{1},...,p_{m}</math> the corresponding p-values. Let <math>I_{0}</math> be the subset of the (unknown) true null hypotheses, having <math>m_{0}</math> members.
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| The [[familywise error rate]] is the probability of rejecting at least one of the members in <math>I_{0}</math>, that is to make one or more [[type I error]]. The Bonferroni Correction states that rejecting all <math>p_{i}<\frac{\alpha}{m}</math> will control the <math>FWER\leq\alpha</math>. The proof follows from [[Boole's inequality]]:
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| <math>FWER=Pr\left\{ \bigcup_{I_{o}}\left(p_{i}\leq\frac{\alpha}{m}\right)\right\} \leq\sum_{I_{o}}\left\{Pr\left(p_{i}\leq\frac{\alpha}{m}\right)\right\}\leq m_{0}\frac{\alpha}{m}\leq m\frac{\alpha}{m}=\alpha</math>
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| This result does not require that the tests be [[Independence (probability theory)|independent]].
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| ==Modifications==
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| ===Generalization===
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| We have used the fact that <math>\sum_{i=1}^{n}\frac{\alpha}{n}=\alpha</math>, but the correction can be generalized and applied to any <math>\sum_{i=1}^{n}a_{i}=\alpha</math>, as long as the weights are defined prior to the test.
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| ===Confidence intervals===
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| Bonferroni correction can be used to adjust [[confidence intervals]]. If we are forming <math>m</math> confidence intervals, and wish to have overall confidence level of <math>1-\alpha</math>, then adjusting each individual confidence interval to the level of <math>1-\frac{\alpha}{m}</math> will be the analog confidence interval correction.
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| ==Simultaneous inference and selective inference==
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| Bonferroni correction is the basic type of [[Familywise_error_rate#Simultaneous_inference_vs._selective_inference|simultaneous inference]], that aims to control the familywise error rate. A significant statistical research was done in the field from early 60's until late 90's, and many improvements were offered. Most notably are the [[Holm-Bonferroni method]], which offers a uniformly more [[statistical power|powerful]] test procedure (i.e., more powerful regardless of the values of the unobservable parameters), and the [[Yosef Hochberg|Hochberg]] (1988) method, guaranteed to be no less powerful and is in many cases more powerful when the tests are independent (and also under some forms of positive dependence).
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| In 1995 [[Yoav Benjamini|Benjamini]] and Hochberg suggested to control the [[false discovery rate]] instead of the familywise error rate and do [[Familywise_error_rate#Simultaneous_inference_vs._selective_inference|selective inference]] corrections. This approach addresses the technological improvements that occurred at the end of the century, and provides the researcher with better tools to do large-scale inferences, which was considered one of the weak points of simultaneous inferences methodologies.
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| ==Alternatives==
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| {{main|Familywise error rate#Controlling procedures}}
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| This list only include some of the alternatives that control the familywise error rate.
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| ===Holm-Bonferroni method===
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| {{main|Holm–Bonferroni method}}
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| A uniformly more [[statistical power|powerful]] test procedure (i.e., more powerful regardless of the values of the unobservable parameters) is the [[Holm–Bonferroni method]].
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| ===Šidák correction===
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| {{main|Šidák correction}}
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| A more [[statistical power|powerful]] test procedure than the Bonferroni correction, which is suited when the individual tests are independent.
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| ===False discovery rate===
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| {{main|False discovery rate}}
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| A less restrictive criterion that does not control the familywise error rate is the [[False_discovery_rate#Dependent_tests|approximate false discovery rate]] that does not require ordering the p-values, then using different criteria for each test.
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| ==Criticisms==
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| The Bonferroni correction can be somewhat conservative if there are a large number of tests and/or the test statistics are positively correlated. Bonferroni correction controls the probability of [[Type I and type II errors#Type_I_error|false positives]] only. The correction ordinarily comes at the cost of increasing the probability of producing [[Type_I_and_type_II_errors#Type_II_error|false negatives]], and consequently reducing [[statistical power]]. When testing a large number of hypotheses, this can result in large critical values.
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| Another criticism concerns the concept of a [[Familywise_error_rate#The_concept_of_a_family|family]] of hypotheses. The statistical community has not yet reached a consensus on how to define such a family. Currently it is defined subjectively per test. As there is no standard definition, test results may change dramatically, only by modifying the way we consider the hypotheses families.
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| In addition, in certain situations where one wants to retain, not reject, the null hypothesis, then Bonferroni correction is non-conservative.
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| ==See also==
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| *[[Boole's inequality#Bonferroni inequalities|Bonferroni inequalities]]
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| *[[Familywise error rate]]
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| *[[Holm–Bonferroni method]]
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| *[[Multiple testing]]
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| ==References==
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| {{Reflist}}
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| {{Refbegin|60em}}
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| *{{cite book |last=Abdi |first=H. |chapter=Bonferroni and Šidák corrections for multiple comparisons |editor-first=N. J. |editor-last=Salkind |title=Encyclopedia of Measurement and Statistics | year=2007 | location=Thousand Oaks, CA | publisher=Sage | url=http://www.utdallas.edu/~herve/Abdi-Bonferroni2007-pretty.pdf }}
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| *{{cite web |author=Manitoba Centre for Health Policy |year=2008 |title=Concept: Multiple Comparisons |url=http://mchp-appserv.cpe.umanitoba.ca/viewConcept.php?conceptID=1049 }}
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| *{{cite journal |last=Dunn |first=O. J. |year=1961 |title=Multiple Comparisons Among Means |journal=[[Journal of the American Statistical Association]] |volume=56 |issue=293 |pages=52–64 |doi=10.1080/01621459.1961.10482090 }}
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| *{{cite journal |last=Dunnett |first=C. W. |year=1955 |title=A multiple comparisons procedure for comparing several treatments with a control |journal=Journal of the American Statistical Association |volume=50 |issue=272 |pages=1096–1121 |doi=10.1080/01621459.1955.10501294 }}
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| *{{cite journal |last=Dunnett |first=C. W. |year=1964 |title=New tables for multiple comparisons with a control |journal=[[Biometrics (journal)|Biometrics]] |volume=20 |issue=3 |pages=482–491 |jstor=2528490 }}
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| *{{cite journal |last=Perneger |first=Thomas V. |year=1998 |title=What's wrong with Bonferroni adjustments |journal=[[British Medical Journal]] |volume=316 |issue=7139 |pages=1236–1238 |doi=10.1136/bmj.316.7139.1236 }} See also the Rapid Response to this suggesting much of it is mistaken.
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| *School of Psychology, University of New England, New South Wales, Australia, 2000, http://www.une.edu.au/WebStat/unit_materials/c5_inferential_statistics/bonferroni.html
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| *{{MathWorld|BonferroniCorrection}}
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| *{{cite journal |last=Shaffer |first=J. P. |year=1995 |title=Multiple Hypothesis Testing |journal=[[Annual Review of Psychology|Annual Review of Psych]] |volume=46 |issue= |pages=561–584 |doi=10.1146/annurev.ps.46.020195.003021 }}
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| *{{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 }}
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| *{{cite journal |last=Šidák |first=Z. |year=1967 |title=Rectangular confidence regions for the means of multivariate normal distributions |journal=Journal of the American Statistical Association |volume=62 |issue=318 |pages=626–633 |doi=10.1080/01621459.1967.10482935 }}
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| *{{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}}
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| {{Refend}}
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| ==External links==
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| *[http://www.quantitativeskills.com/sisa/calculations/bonfer.htm Bonferroni, Sidak online calculator]
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| *[http://www.silicongenetics.com/Support/GeneSpring/GSnotes/analysis_guides/mtc.pdf Explanation of p-value correction methods under the context of differential gene expression analysis]
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| {{DEFAULTSORT:Bonferroni Correction}}
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| [[Category:Multiple comparisons]]
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| [[Category:Hypothesis testing]]
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