Bonferroni Correction
Bonferroni Family-Wise Error Rate Correction · Also known as: Bonferroni adjustment, Bonferroni method, Bonferroni procedure, FWER correction, alpha correction for multiple comparisons
The Bonferroni correction is a conservative, universally applicable method for controlling the family-wise error rate (FWER) when conducting multiple simultaneous hypothesis tests. Grounded in Bonferroni's 1936 probability inequality and formalized for multiple comparisons by Olive Jean Dunn in 1961, the procedure divides the target significance level α by the number of tests m, ensuring that the probability of making even one false rejection across the entire family of tests does not exceed α.
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When to use it
Apply the Bonferroni correction whenever you conduct m ≥ 2 hypothesis tests and wish to maintain the family-wise error rate at a prespecified level α (commonly 0.05). It is the natural baseline for post-hoc pairwise comparisons after a significant ANOVA, for secondary endpoint analyses in clinical trials, and for any exploratory analysis where the number of planned comparisons is small to moderate. The correction is valid regardless of the dependence structure among the tests and requires no distributional assumption beyond those of the individual tests themselves. It becomes overly conservative when m is large (e.g., genomics with thousands of tests) — in that regime, false discovery rate (FDR) control via the Benjamini-Hochberg procedure is usually preferred. Also consider the less conservative Holm step-down procedure, which uniformly controls the FWER at the same α level while rejecting at least as many hypotheses as Bonferroni.
Strengths & limitations
- Simplicity and transparency: the adjusted threshold α/m can be calculated mentally and communicated to any audience.
- Universal validity: the union bound holds under arbitrary dependence among tests — no knowledge of the correlation structure is required.
- Conservative guarantee: the FWER is guaranteed not to exceed α, making it suitable for confirmatory settings where any false discovery is costly.
- Widely accepted as the standard baseline for multiple comparison corrections in virtually every scientific discipline.
- Overly conservative when m is large or when tests are positively correlated, leading to reduced statistical power and increased Type II errors.
- Assumes a fixed, pre-specified set of m tests; the correction is less principled when the number of tests is chosen post-hoc or adaptively.
- Uniformly reduces power across all tests, whereas stepwise procedures (Holm, Hochberg) achieve the same FWER control with higher power.
- Not designed for large-scale simultaneous testing (e.g., genome-wide association studies) where FWER control itself may be an inappropriate criterion.
Frequently asked
Why is the Bonferroni correction called conservative?
The union bound it uses — P(A₁ ∪ … ∪ Aₘ) ≤ ΣP(Aᵢ) — is an inequality, not an equality. When the tests are positively correlated, the true FWER is lower than α, meaning the correction is more stringent than it needs to be. The actual FWER may be well below α, so the method rejects fewer true nulls than a procedure that accounts for the dependence structure would allow.
When should I use Holm's procedure instead of Bonferroni?
Holm's step-down procedure always rejects at least as many hypotheses as Bonferroni at the same FWER level, so it uniformly dominates Bonferroni in terms of power. In practice, Holm should be preferred whenever a slightly more complex calculation is acceptable. Bonferroni retains an advantage in simplicity and when a single adjusted threshold must be communicated in advance — for instance, in pre-registered protocols.
Should I use Bonferroni or Benjamini-Hochberg for many tests?
It depends on the scientific goal. Bonferroni controls the FWER — the probability of even one false positive in the entire family. Benjamini-Hochberg controls the FDR — the expected proportion of false positives among rejected hypotheses. For exploratory large-scale testing (genomics, metabolomics, survey data mining), FWER control via Bonferroni is often too stringent; FDR control via Benjamini-Hochberg is typically preferred. For confirmatory settings with a small number of primary hypotheses, Bonferroni or Holm remains appropriate.
Does Bonferroni assume the tests are independent?
No. The Bonferroni inequality (Boole's inequality) holds under any dependence structure — tests may be independent, positively correlated, or negatively correlated. This generality is one of its main strengths. However, it also means the correction does not exploit positive correlation to gain power, which is why it is conservative when tests are strongly correlated.
Sources
- Bonferroni, C. E. (1936). Teoria statistica delle classi e calcolo delle probabilità. Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze, 8, 3–62. link ↗
- Dunn, O. J. (1961). Multiple comparisons among means. Journal of the American Statistical Association, 56(293), 52–64. DOI: 10.1080/01621459.1961.10482090 ↗
- Miller, R. G. (1981). Simultaneous Statistical Inference (2nd ed.). Springer-Verlag. ISBN: 978-1-4613-8124-2
How to cite this page
ScholarGate. (2026, June 3). Bonferroni Family-Wise Error Rate Correction. ScholarGate. https://scholargate.app/en/statistics/bonferroni-correction
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Benjamini-Hochberg ProcedureStatistics↔ compare
- Holm CorrectionStatistics↔ compare
- Independent t-testStatistics↔ compare
- One-way ANOVAStatistics↔ compare
- Scheffé TestStatistics↔ compare