Benjamini–Hochberg Procedure¶
A step-up rule that sorts m p-values and rejects through the largest rank k where p(k) ≤ (k/m)·α, bounding the false discovery rate — the expected proportion of false rejections among discoveries — rather than the probability of any false positive.
Core Idea¶
The Benjamini–Hochberg (BH) procedure controls the false discovery rate (FDR) — the expected proportion of rejected nulls that are false rejections — when testing many hypotheses at once. Given m tests with p-values sorted ascending, it finds the largest k with p(k) ≤ (k/m)·α and rejects ranks 1 through k. Under independence or positive regression dependence the FDR is bounded by α. Its structural commitment is a shift in what error to control: rather than the familywise error rate (any false positive), which Bonferroni's α/m enforces and which eliminates discovery at scale, BH bounds the proportion of false discoveries among discoveries.
Scope of Application¶
Because BH is a named statistical algorithm mapping sorted p-values to a rejection set, it applies wherever its precondition holds — many tests evaluated together, statistics independent or positively dependent (PRDS).
- Genomics and bioinformatics — the enabling use; differential expression "significant at FDR 0.05."
- Neuroimaging — voxel- and cluster-level fMRI inference at hundred-thousand-test multiplicity.
- Mass-screening clinical/epidemiological studies — prioritising among many candidate biomarkers.
- Astronomy and physics — source, transit, and signal-candidate detection.
- At-scale A/B testing — thousands of parallel experiments in the reporting layer.
Clarity¶
BH makes legible the distinction single-test thinking hides: familywise error rate (the probability of any false discovery) versus false discovery rate (the expected proportion among discoveries). That dissolves the false dilemma in which rigor and discovery look incompatible at scale — FWER was the wrong target, not screening hopeless. It also makes the rank-dependent threshold inspectable and reconstructable, with the q-value as a per-test summary.
Manages Complexity¶
The multiplicity problem otherwise presents as open-ended per-study bargaining over how harshly to penalize. BH fixes the controlled quantity: commit in advance to one rate α, and the procedure deterministically returns the rejection set whose false-discovery proportion is bounded by it. The analyst tracks just three things — the error notion, α, and the sorted list with length m.
Abstract Reasoning¶
BH supports a boundary-drawing move on the error notion (stakes select FWER versus FDR before any computation), a deterministic construction move (walk the ranks to the step-up index with an attached guarantee), a diagnostic move on the dependence structure (switch to Benjamini–Yekutieli under arbitrary dependence), and a predictive-yield move forecasting how much discovery the linear relaxation buys.
Knowledge Transfer¶
BH is an instrument, transferring literally wherever calibrated p-values and the PRDS assumption hold — the substrate enters only as a list of p-values and a count. The boundary is instrument-reach versus over-reading: the bound fails under arbitrary dependence (use Benjamini–Yekutieli), and FDR controls a proportion, not a count or a set of confirmed truths. A genuinely cross-domain idea belongs to the parents false_discovery_rate, multiple_comparisons_correction, hypothesis_testing_null_vs_alternative, and type_i_type_ii_errors.
Relationships to Other Abstractions¶
Current abstraction Benjamini–Hochberg Procedure Domain-specific
Parents (2) — more general patterns this builds on
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Benjamini–Hochberg Procedure is a kind of Multiple Comparisons Correction Prime
Benjamini-Hochberg is multiple-comparisons correction specialized to a rank-based step-up rule that controls false discovery rate rather than familywise error rate.
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Benjamini–Hochberg Procedure presupposes False Discovery Rate Domain-specific
Benjamini Hochberg Procedure presupposes False Discovery Rate.
Hierarchy paths (12) — routes to 6 parentless roots
- Benjamini–Hochberg Procedure → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Inductive Reasoning
- Benjamini–Hochberg Procedure → False Discovery Rate → Type I & Type II Errors → Trade-offs → Constraint
- Benjamini–Hochberg Procedure → Multiple Comparisons Correction → Type I & Type II Errors → Trade-offs → Constraint
- Benjamini–Hochberg Procedure → False Discovery Rate → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Inductive Reasoning
- Benjamini–Hochberg Procedure → False Discovery Rate → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Uncertainty
- Benjamini–Hochberg Procedure → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Uncertainty
- Benjamini–Hochberg Procedure → False Discovery Rate → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Verification → Evaluation → Comparison → Self Checking
- Benjamini–Hochberg Procedure → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Verification → Evaluation → Comparison → Self Checking
- Benjamini–Hochberg Procedure → False Discovery Rate → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Probability → Measure → Set and Membership
- Benjamini–Hochberg Procedure → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Probability → Measure → Set and Membership
- Benjamini–Hochberg Procedure → False Discovery Rate → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Probability → Measure → Aggregation → Micro Macro Linkage
- Benjamini–Hochberg Procedure → Multiple Comparisons Correction → Type I & Type II Errors → Hypothesis Testing (Null vs. Alternative) → Statistical Inference → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Benjamini–Hochberg Procedure sits in a moderately populated region (54th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Statistical Inference & Model Failure Modes (16 abstractions)
Nearest neighbors
- Congruence Bias — 0.86
- Jeffreys-Lindley Paradox — 0.85
- Cherry Picking — 0.85
- Null Ritual — 0.83
- Gambler's Fallacy — 0.83
Computed from structural-signature embeddings · 2026-07-12