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In statistics, the Bonferroni correction is a method used to counteract the problem of multiple comparisons. It is named after Italian mathematician Carlo Emilio Bonferroni for the use of Bonferroni inequalities, but modern usage is often credited to Olive Jean Dunn, who described the procedure in a pair of articles written in 1959 and 1961.
Statistical inference logic is based on rejecting the 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 being tested, we also increase the likelihood of a rare event, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., make a Type I error).
The Bonferroni correction is based on the idea that if an experimenter is testing hypotheses, then one way of maintaining the familywise error rate (FWER) is to test each individual hypothesis at a statistical significance level of times what it would be if only one hypothesis were tested.
So, if the desired significance level for the whole family of tests is , then the Bonferroni correction would test each individual hypothesis at a significance level of . For example, if a trial is testing hypotheses with a desired , then the Bonferroni correction would test each individual hypothesis at .
Statistically significant simply means that a given result is unlikely to occur by chance if the null hypothesis is true (i.e., no difference among groups, no effect of treatment, no relation among variables).
Let be a family of hypotheses and their corresponding p-values for a given data. Let the null hypothesis be the union of hypotheses taken from this family of hypotheses,
The is true if all of the in are true. This means, is false if any one of the in is false. Let the decision strategy be such that is rejected if any one of in is rejected.
The familywise error rate is the probability of rejecting at least one of the members in based on their corresponding p-values; that is, to make one or more type I error during the null hypothesis testing of individual 's in . The Bonferroni Correction states that choosing all will control the . The proof follows from Boole's inequality:
This result does not require that the tests be independent.
We have used the fact that , but the correction can be generalized and applied to any , as long as the weights are defined prior to the test.
Bonferroni correction can be used to adjust confidence intervals. If we are forming confidence intervals, and wish to have overall confidence level of , then adjusting each individual confidence interval to the level of will be the analog confidence interval correction.
There are other alternatives to control the familywise error rate. For example, the Holm–Bonferroni method and the Šidák correction are universally more powerful procedures than the Bonferroni correction, meaning that they are always at least as powerful.
The Bonferroni correction can be somewhat conservative if there are a large number of tests and/or the test statistics are positively correlated. The correction also comes at the cost of increasing the probability of producing false negatives, and consequently reducing statistical power.
Another criticism concerns the concept of a family of hypotheses. There is not a definitive consensus on how to define a family in all cases. As there is no standard definition, test results may change dramatically, only by modifying the way we consider the hypotheses families.
All of these criticisms, however, apply to adjustments for multiple comparisons in general, and are not specific to the Bonferroni correction.
- Bonferroni, C. E., Teoria statistica delle classi e calcolo delle probabilità, Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 1936
- Dunn, Olive Jean (1959). "Estimation of the Medians for Dependent Variables". Annals of Mathematical Statistics 30 (1): 192–197. JSTOR 2237135.
- Dunn, Olive Jean (1961). "Multiple Comparisons Among Means" (PDF). Journal of the American Statistical Association 56 (293): 52–64. doi:10.1080/01621459.1961.10482090.
- Abdi, H. (2007). "Bonferroni and Šidák corrections for multiple comparisons". In Salkind, N. J. Encyclopedia of Measurement and Statistics (PDF). Thousand Oaks, CA: Sage.
- Manitoba Centre for Health Policy (2008). "Concept: Multiple Comparisons".
- Dunn, O. J. (1961). "Multiple Comparisons Among Means". Journal of the American Statistical Association 56 (293): 52–64. doi:10.1080/01621459.1961.10482090.
- Dunnett, C. W. (1955). "A multiple comparisons procedure for comparing several treatments with a control". Journal of the American Statistical Association 50 (272): 1096–1121. doi:10.1080/01621459.1955.10501294.
- Dunnett, C. W. (1964). "New tables for multiple comparisons with a control". Biometrics 20 (3): 482–491. JSTOR 2528490.
- Perneger, Thomas V. (1998). "What's wrong with Bonferroni adjustments". British Medical Journal 316 (7139): 1236–1238. doi:10.1136/bmj.316.7139.1236. See also the Rapid Response to this suggesting much of it is mistaken.
- Shaffer, J. P. (1995). "Multiple Hypothesis Testing". Annual Review of Psych 46: 561–584. doi:10.1146/annurev.ps.46.020195.003021.
- Strassburger, K.; Bretz, Frank (2008). "Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni-based closed tests". Statistics in Medicine 27 (24): 4914–4927. doi:10.1002/sim.3338.
- Šidák, Z. (1967). "Rectangular confidence regions for the means of multivariate normal distributions". Journal of the American Statistical Association 62 (318): 626–633. doi:10.1080/01621459.1967.10482935.
- Hochberg, Yosef (1988). "A Sharper Bonferroni Procedure for Multiple Tests of Significance" (PDF). Biometrika 75 (4): 800–802. doi:10.1093/biomet/75.4.800.
- 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
- Weisstein, Eric W., "Bonferroni correction", MathWorld.
- Bonferroni, Sidak online calculator
- Explanation of p-value correction methods under the context of differential gene expression analysis