Holm Correction (Holm-Bonferroni)
Also known as: Holm-Bonferroni method, Holm step-down procedure, Holm's sequentially rejective procedure, Holm düzeltmesi
The Holm correction, introduced by Sture Holm in 1979, is a step-down multiple-comparison procedure that controls the family-wise error rate (FWER) at level α while rejecting at least as many hypotheses as the classical Bonferroni correction. It orders the observed p-values from smallest to largest and compares each against a threshold that starts strict and relaxes as testing proceeds, making it uniformly more powerful than Bonferroni at the same level of error control.
Key highlights
- Uniformly more powerful than Bonferroni: it rejects at least as many hypotheses at the same α, never fewer.
- Distribution-free FWER control — valid under any dependence structure among the tests, with no correlation assumptions.
- Simple to compute and explain: sort the p-values and walk down a shrinking-denominator threshold.
- Applies to any family of tests, since it operates only on the p-values rather than the underlying test statistics.
Intuition
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How it works
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When to use it
Reach for the Holm correction whenever you would otherwise use Bonferroni: you have m ≥ 2 simultaneous tests and need to control the probability of even one false rejection at a prespecified α. It requires no assumption about the dependence between tests, applies to any collection of p-values (post-hoc ANOVA contrasts, multiple endpoints, several planned comparisons), and uniformly dominates Bonferroni, so there is rarely a reason to prefer plain Bonferroni once Holm is available. When the goal shifts from controlling any false positive (FWER) to controlling the expected proportion of false positives among discoveries (FDR), as in large-scale genomic screens, the Benjamini-Hochberg procedure is the more appropriate tool.
Strengths & limitations
- Uniformly more powerful than Bonferroni: it rejects at least as many hypotheses at the same α, never fewer.
- Distribution-free FWER control — valid under any dependence structure among the tests, with no correlation assumptions.
- Simple to compute and explain: sort the p-values and walk down a shrinking-denominator threshold.
- Applies to any family of tests, since it operates only on the p-values rather than the underlying test statistics.
- Still controls the FWER, so it remains conservative and low-powered when the number of tests is very large.
- The step-down logic means a single early failure halts all further rejections, even if some later hypotheses are strongly significant.
- Like all FWER methods, it does not exploit positive correlation among tests to recover power.
- Reporting adjusted p-values requires an enforced monotonicity step, which can confuse users expecting a single fixed threshold.
Common pitfalls
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Applications
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Frequently asked
How is Holm different from Bonferroni?
Both control the family-wise error rate at α and both apply α/m to the smallest p-value. Bonferroni applies that same strict threshold to every test, while Holm relaxes the threshold to α/(m − k + 1) as it works down the ordered p-values. As a result Holm rejects at least as many hypotheses as Bonferroni and usually more, with no loss of error control.
Does Holm assume independent tests?
No. Like Bonferroni it relies on Boole's inequality, so the FWER guarantee holds under arbitrary dependence among the tests. This generality is why it is recommended as a safe default, though it also means it cannot exploit positive correlation to gain extra power.
Is there ever a reason to use Bonferroni instead of Holm?
Statistically Holm dominates Bonferroni, so the main reasons to keep Bonferroni are communicative: it yields one fixed, pre-announceable threshold α/m, which is convenient in pre-registered protocols and easy to explain to non-specialists.
When should I switch to Benjamini-Hochberg?
Switch when controlling the expected proportion of false discoveries (FDR) is more appropriate than controlling the chance of any false discovery (FWER) — typically in large-scale exploratory testing such as genomics, where FWER procedures including Holm become too conservative.
Sources
- 1.Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2), 65–70.
- 2.Bonferroni, C. E. (1936). Teoria statistica delle classi e calcolo delle probabilità. Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze, 8, 3–62.
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Cite this page
ScholarGate. (2026, June 2). Holm Correction. ScholarGate. https://scholargate.app/statistics/holm-correction