statistical correction · BEGINNER LESSON
Bonferroni
Bonferroni correction is a simple multiple-testing adjustment that divides a chosen significance level by the number of hypotheses being tested. It is used to control the family-wise error rate, the chance of making at least one false rejection among the tests.
What it is and what it measures
Bonferroni is a method for multiple testing, which means it is used when several statistical tests are performed at once. The key quantity it protects is the family-wise error rate, or FWER, meaning the probability of incorrectly rejecting at least one true null hypothesis. A null hypothesis is the default claim that there is no effect or no difference. The Bonferroni correction is the simplest conventional FWER method: take the chosen significance level alpha (the cutoff for calling a result statistically significant) and divide it by the number of null hypotheses being tested. [0]
[1] Extending effective multiplicity for independent designs in ...
How it works
If many tests are done, the chance of at least one false positive rises. Bonferroni lowers the per-test cutoff so that only very small p-values count as significant. A p-value is a measure of how surprising the data are if the null hypothesis were true. This makes the method conservative: it strongly reduces false positives, but it can also miss real effects and produce false negatives. In the cited fNIRS example, no false positives were detected with Bonferroni, but the number of detected significant channels could fall below the number that were truly significant. [0]
[1] Extending effective multiplicity for independent designs in ...
Why researchers use it
Researchers often use Bonferroni when they want a straightforward way to control false positives across a set of tests. It is commonly used in settings such as functional near-infrared spectroscopy (fNIRS), especially when independent statistical designs are analyzed and a predefined region of interest, or ROI, is tested. The sources also note that Bonferroni can combine p-values under arbitrary dependence in partial-conjunction settings, where one asks how many studies show a shared directional effect. [0,1]
[1] Extending effective multiplicity for independent designs in ... [2] Directional replicability: When can the factor of two be omitted
Limits and interpretation
Bonferroni assumes the tests are independent when used in the standard way, and that assumption can make it overly strict when tests are correlated. Because it aims to protect against any false positive in the family, it may reduce statistical power, meaning the ability to detect real effects. A later source emphasizes that whether multiple testing should be adjusted depends on how much emphasis is placed on small p-values in reporting and interpretation, but that guidance is about when to adjust, not how Bonferroni itself works. [0,2]
[1] Extending effective multiplicity for independent designs in ... [3] pmc.ncbi.nlm.nih.gov · articles · PMC13324398When to Adjust for Multiple Testing: A Unifying Guiding ...
SEE THE IDEA
How Bonferroni changes significance testing
Several hypotheses are checked at the same time.
Pick a significance cutoff such as 0.05.
Use alpha divided by the number of tests.
Only p-values below the smaller cutoff are called significant.
Each step makes the decision rule stricter to reduce the chance of any false positive across the full set of tests.
Bonferroni lowers the per-test threshold so the whole family of tests is harder to call significant.
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.