Bonferroni correction

The Bonferroni correction or Bonferroni method (after Carlo Emilio Bonferroni ) is a method of mathematical statistics that neutralizes the accumulation of alpha errors in multiple comparisons.

It says: If you test independent hypotheses on a data set, the statistical significance that should be used for each hypothesis separately is times the significance that would result if only one hypothesis were tested. ${\displaystyle n}$ ${\displaystyle {\tfrac {1}{n}}}$

example

Let us assume that there is an experiment in which gene expression differences between healthy cells and cancer cells are looked for. 10,000 genes were tested. The number of tests is denoted by n in this example . If one uses the usual level of significance (p <0.05), one obtains 587 significantly different genes. However, due to errors caused by multiple testing, very many of these genes will not be different in reality.

Instead, the Bonferroni correction results in an adjusted significance level . In the above example (corresponds to ). As an alternative procedure, one can also adjust the p-values ​​with the number of tests n : adjusted . ${\displaystyle {p}^{*}<{\frac {\alpha }{n}}}$ ${\displaystyle {p}^{*}<{\frac {0,05}{10\,000}}}$ ${\displaystyle {p}^{*}<0{,}000005}$ ${\displaystyle p_{i}^{*}=p_{i}*n}$

The adjusted significance value (or the adjusted p-values ) are compared with its p-values ​​and obtain e.g. B. only 6 significant genes. This rules out a great many false-positive results. The remaining results are therefore more reliable, but at the same time many genes that really differ were also excluded. The Bonferroni correction is therefore more conservative than, for example, the correction of the False Discovery Rate (FDR) according to Benjamini-Hochberg. ${\displaystyle {p}^{*}}$ ${\displaystyle p_{i}^{*}}$

background

If one examines a hypothesis family with pairwise comparisons and tests each associated individual hypothesis for the level of significance , then the following inequality exists between the risk of the individual test and the multiple overall risk (also referred to as ): ${\displaystyle m\geq 2}$ ${\displaystyle \mathbf {\alpha } '}$ ${\displaystyle \mathbf {\alpha } '}$ ${\displaystyle \mathbf {\alpha } }$ ${\displaystyle \mathbf {\alpha } _{exp}}$

${\displaystyle \mathbf {\alpha } '\leq \mathbf {\alpha } \leq m\cdot \mathbf {\alpha } '.}$

This relationship follows from the Bonferroni inequality (Boolean inequality) and states that the multiple total risk has an upper limit. If one chooses as the significance level for each individual test , then the multiple total risk cannot be greater than . ${\displaystyle \mathbf {\alpha } '=\mathbf {\alpha } /m}$ ${\displaystyle \mathbf {\alpha } }$

So in order to comply with the multiple overall risk , the significance level must be adjusted accordingly in each individual test . The Bonferroni method is a very rough approximation and very conservative. Therefore, more precise methods have been developed that control the error less conservatively and further exploit the significance level of the multiple test procedure (see alpha error accumulation ). ${\displaystyle \mathbf {\alpha } }$ ${\displaystyle \mathbf {\alpha } '}$ ${\displaystyle \alpha }$

Bonferroni in signal processing

There is a voxel map with many statistical values, some of which are independent, others in turn are dependent on one another. The Bonferroni correction can be used to find out special features of this distribution. However, this only applies to independent tests and is too strict for only partially dependent tests. Therefore, when a significance limit ( p-value or level of significance ) is found in such a statistical map, the values ​​of which are only partially dependent or independent, it is often mixed with the Gaussfeld method. The lower p-value of the two correction methods is specified for a voxel and the limit is thus determined.

literature

• Abdi, H: Bonferroni and Sidak corrections for multiple comparisons . In: NJ Salkind (ed.) (Ed.): Encyclopedia of Measurement and Statistics . Sage, Thousand Oaks, CA 2007.

Web links

• Manitoba Center for Health Policy (MCHP) 2003, Concept: Multiple Comparisons.
• Perneger, Thomas V, What's wrong with Bonferroni adjustments, BMJ 1998; 316: 1236-1238 (April 18).
• School of Psychology, University of New England, New South Wales, Australia, 2000.
• Eric W. Weisstein : Bonferroni Correction . In: MathWorld (English).