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.
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 canonical identity is narrower than the phrase’s everyday use. 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.
Structural Signature¶
- 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.
What It Is Not¶
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.
- 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.
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.
A shared label or downstream consequence is insufficient; the load-bearing roles must survive.
Clarity¶
Bonferroni Correction separates a specific relation from neighboring ideas that can produce similar observations. 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 compresses recurring cases into one inspectable model. An analyst can track the structural roles, compare mechanisms, and locate exactly which missing commitment invalidates an analogy.
Abstract Reasoning¶
Identify the candidate roles, test the defining relation, then challenge the nearest boundary case. 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. Transfer is warranted only when the same causal, formal, or relational work survives.
Examples¶
Qualifying pattern. 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.
Boundary case. Dividing alpha by the number of observations rather than the number of tested claims is not Bonferroni correction.
Structural Tensions¶
T1 — Reach versus identity inflation. Broad use is valuable only while every defining role survives.
T2 — Observation versus mechanism. Similar outcomes can arise from neighboring mechanisms, so classification follows the relation and its counterfactual rather than appearance.
Structural Core vs. Domain Accent¶
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. The transferable residue is thinner than the named mechanism, whose domain vocabulary and causal apparatus remain constitutive.
Relationships to Other Abstractions¶
Current abstraction Bonferroni Correction Domain-specific
Parents (1) — more general patterns this builds on
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Bonferroni Correction is a kind of Multiple Comparisons Correction Prime
Bonferroni Correction is a strict specialization of Multiple Comparisons Correction.The parent supplies the genus and can occur without the child; the child preserves that structure while adding commitments the parent does not require.
Hierarchy paths (6) — routes to 6 parentless roots
- Bonferroni Correction → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Inductive Reasoning
- Bonferroni Correction → Multiple Comparisons Correction → Type I & Type II Errors → Trade-offs → Constraint
- Bonferroni Correction → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Uncertainty
- Bonferroni Correction → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Verification → Evaluation → Comparison → Self Checking
- Bonferroni Correction → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Probability → Measure → Set and Membership
- Bonferroni Correction → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Probability → Measure → Aggregation → Micro Macro Linkage
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.
- 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 from the adjudicated missing-node gate. Queued for Claude house-style re-authoring and independent citation review; no citations have been fabricated.)