Bonferroni correction
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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.
It is named after Italian mathematician Carlo Emilio Bonferroni for the use of Bonferroni inequalities,[1] but modern usage is credited to Olive Jean Dunn, who first used it in a pair of articles written in 1959 and 1961.[2][3]
Informal introduction
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 should be (at most) , then the Bonferroni correction would test each individual hypothesis at a significance level of . For example, if a trial is testing eight 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 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).
Definition
Let be a family of hypotheses and the corresponding p-values. Let be the (unknown) subset of the true null hypotheses, having members.
The familywise error rate is the probability of rejecting at least one of the members in ; that is, to make one or more type I error. 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.
Modifications
Generalization
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.
Confidence intervals
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.
Simultaneous inference and selective inference
Bonferroni correction is the basic type of simultaneous inference, that aims to control the familywise error rate. Significant statistical research was done in the field from early 1960s until late '90s, and many improvements were offered. Most notably are the Holm–Bonferroni method, which offers a uniformly more powerful test procedure (i.e., more powerful regardless of the values of the unobservable parameters), and the 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).
In 1995 Benjamini and Hochberg suggested to control the false discovery rate instead of the familywise error rate and do 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.
Alternatives
There are other alternatives to control the familywise error rate. Both the Holm–Bonferroni method and the Šidák correction are said to be uniformly more powerful test procedures than the Bonferroni correction.
Criticisms
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. The statistical community has not reached a consensus on how to define such a family. As there is no standard definition, test results may change dramatically, only by modifying the way we consider the hypotheses families.
See also
References
- ^ 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.
Further reading
- Abdi, H. (2007). "Bonferroni and Šidák corrections for multiple comparisons". In Salkind, N. J. (ed.). Encyclopedia of Measurement and Statistics (PDF). Thousand Oaks, CA: Sage.
- Manitoba Centre for Health Policy (2008). "Concept: Multiple Comparisons".
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has generic name (help) - 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.
External links
- 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