Benjaminihochberg ProcedureEdit
The Benjamini-Hochberg procedure is a statistical method designed to control the false discovery rate (FDR) when many hypotheses are tested at once. Introduced in 1995 by Yoav Benjamini and Yosef Hochberg, it offers a practical alternative to the hard guardrails of family-wise error rate control, such as the Bonferroni correction. In data-rich fields like genomics Genomics, neuroimaging neuroimaging, and proteomics, researchers prize a method that preserves power—the ability to detect true effects—without letting false claims proliferate. The procedure has become a staple in both exploratory and confirmatory research, where large numbers of simultaneous tests are the norm.
At a high level, the Benjamini-Hochberg procedure uses the p-values from individual tests, orders them from smallest to largest, and compares them to a growing threshold. Let m be the number of hypotheses and p_(1) ≤ p_(2) ≤ ... ≤ p_(m) the ordered p-values. For a chosen target FDR level q, the method finds the largest k such that p_(k) ≤ (k/m) q, and rejects the null hypotheses for all i ≤ k. The parameter q represents the acceptable average proportion of false discoveries among the rejected hypotheses—commonly set at 0.05. The approach is simple, scalable, and has robust performance under the right conditions. Extensions and refinements address practical issues that show up in real data, such as dependence among tests and the proportion of true null hypotheses in the dataset.
How the Benjamini-Hochberg Procedure Works
- Input: a set of m hypotheses, each with an associated p-value.
- Step 1: sort p-values in ascending order: p_(1), p_(2), ..., p_(m).
- Step 2: choose a desired FDR level q (often 0.05).
- Step 3: find the largest k with p_(k) ≤ (k/m) q.
- Step 4: reject the null hypotheses for all tests with indices i ≤ k; keep the rest as non-rejected.
This step-up approach is what gives BH its power relative to more conservative approaches, while still offering a formal guarantee about the expected rate of false discoveries across the set of rejected hypotheses.
Assumptions and Extensions
- Assumptions: The classic BH procedure provides FDR control under independence of test statistics or under certain kinds of positive dependence among tests (a condition sometimes summarized as PRDS, positive regression dependence on a subset). In practical terms, many real-world datasets approximate these conditions, making BH a good default choice.
- Extensions for dependence: When tests exhibit arbitrary dependence, the Benjamini-Yekutieli (BY) procedure offers a way to preserve FDR control, though at the cost of reduced power. This trade-off between rigor and discovery is a recurring theme in multiple testing.Benjamini-Yekutieli procedure
- Adaptive and variant methods: Researchers have developed adaptive versions of BH that try to estimate the proportion of true null hypotheses (pi0) to improve power. The concept of q-values, popularized in later work, provides a local measure of significance for each test. These refinements keep the core BH logic intact while tailoring it to the data at hand.
Variants and Related Methods
- Adaptive BH: Uses an estimate of pi0 to adjust the threshold, potentially allowing more discoveries when a large fraction of hypotheses are non-null.
- BY procedure: A more conservative alternative that remains valid under arbitrary dependence among tests.
- q-values: A way of reporting the smallest FDR at which a given test would be deemed significant, extending the BH framework to a continuous significance landscape.
- Weighted BH and other extensions: Allow incorporating prior information or hierarchical structure among hypotheses to improve power without sacrificing control.
In practice, BH and its variants are widely used in fields such as genomics, GWAS, neuroimaging, and bioinformatics. Prevalent software and packages in R and other languages implement these procedures, often under the function name p.adjust(method = "BH") or equivalent.
Practical Considerations and Debates
- Power vs. error control: The appeal of BH is that it maintains a reasonable level of power in large-scale testing while keeping false discoveries in check. Critics who favor stricter control (FWER) argue that even a small fraction of false positives can be costly in clinical or regulatory contexts; supporters contend that in exploratory science, missing true signals (false negatives) is a greater systemic risk than a modest number of false leads.
- Dependence structure: Real-world data often contain complex correlations. BH performs well under independence or PRDS, but critics point out that certain dependence patterns can undermine FDR guarantees. Proponents respond with empirical evidence that BH remains robust in many practical settings, especially when the goal is discovery rather than definitive proof.
- Interpretation and misuse: Some commentators caution that FDR is an average property across repeated experiments, which can be abstract for individual studies. In practice, researchers should pre-specify hypotheses when possible, report effect sizes alongside p-values, and consider replication efforts to bolster credibility.
- Woke criticism and p-values: In some circles, critiques of p-values and multiple testing are couched in broader debates about scientific methodology and incentives. Proponents of BH emphasize that, when used properly, BH provides a transparent, decision-theoretic framework for balancing the risk of false discoveries with the desire to learn from data. They argue that dismissing p-values or treating them as inherently flawed misses the point of a well-calibrated error-control procedure; the remedy is better practice, not wholesale rejection of the method.
From a practical, results-focused standpoint, BH is valued for enabling researchers to make credible claims in the face of abundant data. It aligns with a prudent allocation of research resources: it reduces the chance that a large number of reported findings are false, while preserving enough power to uncover genuine effects that warrant further study, replication, or translation.