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.

Version

v3 · 2026-09-28 · History · 2 corrections

Domain-specific #

98

Domain group

Formal Sciences

Origin domain

Experimental Design & Statistics

Subdomain

Multiple Testing → Experimental Design & Statistics

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.

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Few Wrong Guesses on Average

Imagine checking a thousand cookie jars to find which ones have cookies. If you check quickly, sometimes you'll say "cookies!" when there aren't any. The Benjamini–Hochberg Procedure is a rule for choosing which jars to call "cookies!" so that, on average, only a small share of your "cookies!" calls are wrong, instead of being so careful you find almost nothing.

Keeping False Finds Rare

When scientists test thousands of things at once, like which genes are linked to a disease, some tests will look positive just by luck. One old fix is to be so strict that almost nothing wrong gets through, but then you also miss most of the real findings. The Benjamini–Hochberg Procedure takes a different approach: it keeps the fraction of your "discoveries" that are false small on average, say 5%, instead of trying to make zero mistakes. To do it, you line up all the test results from strongest to weakest and use a cutoff that gets a little looser as you go down the list. Everything above the last result that passes its cutoff counts as a discovery.

False Discovery Rate Control

The Benjamini-Hochberg (BH) procedure controls the false discovery rate (FDR), the expected proportion of rejected null hypotheses that are actually false rejections, when many hypotheses are tested at once. You sort the m p-values from smallest to largest, find the largest rank k where p(k) ≤ (k/m)·α, and reject hypotheses 1 through k. If the tests are independent (or positively dependent in a specific way), the FDR is at most α·m₀/m, which is at most α, where m₀ is the unknown number of true nulls. This differs from the Bonferroni correction, which controls the chance of even one false discovery by testing each at α/m; with tens of thousands of tests, that is so strict that true signals rarely survive. BH trades a controlled share of false positives for much more power, which is why it became standard in genomics and neuroimaging.

 

The Benjamini–Hochberg procedure (1995) is the canonical method for controlling the false discovery rate, defined as the expected proportion of false rejections among all rejected null hypotheses, in simultaneous testing. Given m p-values sorted as p(1) ≤ … ≤ p(m), it finds the largest k with p(k) ≤ (k/m)·α and rejects hypotheses 1 through k. Under independence or positive regression dependence of the test statistics, the FDR is bounded by α·m₀/m ≤ α, where m₀ is the unknown number of true nulls. This represents a shift in the target error rate. Classical multiple-testing corrections control the familywise error rate (FWER), the probability of any false rejection, as Bonferroni does by testing at α/m; at the scale of 20,000 genes or 100,000 voxels, that threshold is so stringent that true effects are essentially eliminated. BH instead tolerates a controlled fraction of false positives among discoveries to recover a usable set of true signals. Its step-up structure, sorting p-values and applying a threshold that relaxes linearly with rank, makes it computationally simple and transparent, which helped make it foundational in genomics, neuroimaging, and large-scale clinical research.

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) — outgoing typed relations

• 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.

• Benjamini–Hochberg Procedure presupposes False Discovery Rate Domain-specific

  Benjamini Hochberg Procedure presupposes False Discovery Rate.

Hierarchy paths (10) — routes to 5 parentless roots

• Benjamini–Hochberg Procedure → 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 (9)

Neighborhood in Abstraction Space

Benjamini–Hochberg Procedure sits in a sparse region of the domain-specific corpus (63^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Congruence Bias — 0.86
• Optimality criterion — 0.85
• Jeffreys-Lindley Paradox — 0.85
• Cherry Picking — 0.85
• Monotone Likelihood Ratio Property — 0.84

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