Bonferroni Correction¶

Control the family-wise probability of any false rejection across m tests by comparing each p-value with alpha/m, or equivalently multiplying each p-value by m, without requiring independence.

Version: v3 · 2026-09-19 · History
Domain-specific #: 117
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics
Subdomain: Multiple Testing → Experimental Design & Statistics

Core Idea¶

Bonferroni correction controls the probability of one or more false rejections across a fixed family of \(m\) hypothesis tests by testing each hypothesis at \(\alpha/m\), or equivalently by multiplying each raw p-value by \(m\) and truncating at one. Its guarantee does not require test independence. The procedure requires a declared test family, a family-wise error budget, a finite count \(m\), and a uniform division of that budget across family members. Its conservatism is the price of a simple, dependence-robust union-bound guarantee.

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Stricter Rules for Many Tries

If you check lots of guesses, a few might look right just by luck, like flipping many coins and seeing a streak. The Bonferroni correction says: if you check many guesses, make each one pass a much harder test. That way, the chance of being fooled even once stays small.

Splitting the False-Alarm Budget

Scientists often run a test to decide if a result is real or just luck, and they accept a small chance of being fooled, like 5 percent. But if you run many tests, the chance of being fooled at least once goes way up. The Bonferroni correction fixes this by dividing the allowed chance by the number of tests: with 10 tests, each one must pass at 0.5 percent instead of 5 percent. This keeps the chance of any false alarm across the whole group at or below the original limit. It works even if the tests are connected to each other, but it's quite strict, so it may miss some real effects.

Family-Wise Error Control by Division

The Bonferroni correction controls the family-wise error rate, the probability of making at least one false rejection (false positive) across a set of m hypothesis tests. It tests each hypothesis at level alpha/m instead of alpha, or equivalently multiplies each p-value by m, capping it at 1, and compares it to alpha. The guarantee comes from a simple probability bound (the union bound), so it holds whether or not the tests are independent. To use it properly you must define the family of tests in advance, choose the overall error budget alpha, know the number m, and split the budget evenly. Its downside is that it is conservative: making each test so strict reduces the power to detect real effects, especially when m is large.

The Bonferroni correction controls the family-wise error rate (FWER), the probability of at least one false rejection, across a declared, fixed family of m hypothesis tests. Each hypothesis is tested at level alpha/m, or equivalently each raw p-value is multiplied by m and truncated at one before comparison with alpha. The guarantee follows from the union bound: the probability of any false rejection is at most the sum of the individual false-rejection probabilities, which is at most m times alpha/m = alpha. Because the union bound needs no assumptions about joint distribution, the guarantee holds under arbitrary dependence among tests. The cost is conservatism, especially with many or positively correlated tests, which reduces power. The canonical procedure requires a declared test family, a family-wise error budget, a finite m, and a uniform split of the budget; everyday use of 'Bonferroni' for any multiplicity adjustment is looser than this identity.

Scope of Application¶

The recipe runs unchanged in genomics, physics, medicine, and online experiments, but each use imports the same statistical-testing apparatus. Its reach is broad application of a statistical method, not substrate removal.

Clarity¶

It is one procedure within Multiple Comparisons Correction, not an alias for that family. False Discovery Rate is a criterion about the proportion of false discoveries; Benjamini–Hochberg is a different procedure targeting that criterion. Look-Elsewhere problems can require an effective rather than literal count of opportunities.

Manages Complexity¶

The abstraction turns a scattered family of cases into one role-based test: identify the required elements, test their relation, and reject the classification when a constitutive commitment is missing.

Abstract Reasoning¶

Use the structural signature rather than the label, then challenge the nearest counterexample. Dividing alpha by the number of observations rather than the number of tested claims is not Bonferroni correction.

Knowledge Transfer¶

The recipe runs unchanged in genomics, physics, medicine, and online experiments, but each use imports the same statistical-testing apparatus. Its reach is broad application of a statistical method, not substrate removal.

Example¶

A case qualifies only when the definition and every load-bearing role remain present. Dividing alpha by the number of observations rather than the number of tested claims is not Bonferroni correction.

Relationships to Other Abstractions¶

Current abstraction Bonferroni Correction Domain-specific

Parents (1) — outgoing typed relations

Bonferroni Correction is a kind of Multiple Comparisons Correction Prime

Bonferroni Correction is a strict specialization of Multiple Comparisons Correction.

Hierarchy paths (5) — routes to 5 parentless roots

Bonferroni Correction → is a kind of Multiple Comparisons Correction → presupposes Type I & Type II Errors → presupposes Hypothesis Testing (Null vs. Alternative) → is a kind of Statistical Inference → is a kind of Inductive Reasoning

Show alternative paths (4)

Neighborhood in Abstraction Space¶

Bonferroni Correction sits in a sparse region of the domain-specific corpus (100^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Combinatorial Set Systems & Counting (9 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With¶

It is one procedure within Multiple Comparisons Correction, not an alias for that family. False Discovery Rate is a criterion about the proportion of false discoveries; Benjamini–Hochberg is a different procedure targeting that criterion. Look-Elsewhere problems can require an effective rather than literal count of opportunities.

Notes¶

(Canonical first draft; queued for Claude house-style re-authoring and citation review.)