Holm–Bonferroni Correction Calculator
The formula
What the step-down family control tells you
Holm sorts your p-values and applies shrinking multipliers (m, m−1, …) — uniformly more powerful than Bonferroni with the same family-wise error guarantee.
Calculation: Holm-adjusted p for rank k = max over j≤k of p(j) × (m − j + 1), floored at the previous value. Enter your inputs and every substitution is shown below the result.
Reading the result
If the smallest p survives and the next is comfortably small, later tests face α/8, α/7… instead of α/10 forever.
Rules of thumb: Prefer Holm; reserve Bonferroni for a priori single planned contrasts
Limitations and best practice
Forgetting the monotonicity enforce: a raw step-down value can dip below an earlier one; the adjusted sequence must never decrease.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
When do Holm and Bonferroni agree?
Why is Holm valid?
Should I use Holm over Bonferroni?
Source & methodology
Holm (1979), A simple sequentially rejective multiple test procedure.
Holm–Bonferroni Correction explained: formula, worked example and honest limits
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. There are plenty of pages about Holm–Bonferroni Correction; almost none of them show the working. This one does: the formula, every input, a fully computed example, and a sensitivity analysis calculated live from this engine’s own defaults.
A calculation like this one is worth roughly as much as a careful sentence: it forces several vague considerations into one comparable quantity. Once Holm–Bonferroni Correction is on the table, the argument can stop being about impressions.
People rarely disagree about arithmetic; they disagree about assumptions that were never written down. Printing the substitution and the sensitivity ranking removes that category of disagreement for Holm–Bonferroni Correction.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to holm-adjusted p (smallest) = 0.072 on the default inputs, and the sensitivity section ranks the inputs that move the result — Smallest raw p-value first, at ≈10.0 % of the headline for a ten-percent error either way in that input. Read the sections in order and you will be able to reproduce every figure by hand.
The Holm–Bonferroni Correction formula, explained
One formula, published in full: Holm-adjusted p for rank k = max over j≤k of p(j) × (m − j + 1), floored at the previous value. Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Family-wise α (example value 0) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. This input feeds the formula directly; a defensible figure here is worth more than false precision anywhere else.
Number of comparisons (example value 12) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Count from the record, not the estimate; whole-number inputs are cheap to verify and expensive to get wrong.
Smallest raw p-value (example value 0) is a measured number in the unit named by the label; the formula does not convert for you. If you are unsure what belongs in this field, look at the printed steps: the substitution shows exactly how the engine uses it.
Holm–Bonferroni Correction on the example inputs: the complete substitution
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Family-wise α, Number of comparisons, Smallest raw p-value. The endpoint of that chain is 0.072 of holm-adjusted p (smallest), sitting in the “Even the smallest p fails; the whole family stops here” band.
- Sort p-values ascending; smallest gets multiplier m = 12
- Adjusted smallest p = 0.006 × 12 = 0.072
- Step down: release only while adjusted p ≤ α, enforcing monotonicity
Comparison tests left after this rank — 11. Compare it against the main result of 0.072 before drawing conclusions.
Alpha for the second-smallest p — 0.005. Compare it against the main result of 0.072 before drawing conclusions.
Same p under plain Bonferroni — 0.072. Read next to the headline, this is the figure that shows the shape of the trade-off.
Power gain vs Bonferroni at rank 2 — 1.09 ×. Compare it against the main result of 0.072 before drawing conclusions.
Overwrite any input above and every line of this example updates with it — the substitution is live, not a screenshot of somebody else’s numbers.
Scenario table: Holm–Bonferroni Correction at ±10 % and ±20 % of the key input
Single-point answers invite false confidence, so the same engine was run across a band around the input that matters most here. Each row is a full recalculation of Holm–Bonferroni Correction, not an interpolation.
| Smallest raw p-value setting | Holm-adjusted p (smallest) |
|---|---|
| 80 % of the example value | 0.058 |
| 90 % of the example value | 0.065 |
| example inputs | 0.072 |
| 110 % of the example value | 0.079 |
| 120 % of the example value | 0.086 |
The corridor runs from 0.058 to 0.086 across that band — which is the range worth planning inside rather than the single 0.072 point estimate. Reproduce it in the form above by editing Smallest raw p-value alone; the substitution steps will follow your figures exactly.
The levers behind Holm–Bonferroni Correction, ranked by effect
This list was computed for this specific engine just now: each numeric input was nudged ±10 % while every other field held its example value, and the holm-adjusted p (smallest) was recalculated from scratch. Read it as a map of where Holm–Bonferroni Correction is fragile — drawn for the example inputs, which is the map you want when deciding what to measure carefully.
- Smallest raw p-value +10 % → Holm–Bonferroni Correction = 0.079 · −10 % → 0.065 (the holm-adjusted p (smallest) shifts ≈10.0 %)
- Nudge Number of comparisons by ±10 % and Holm–Bonferroni Correction lands between 0.065 and 0.079 (sensitivity ≈10.0 %)
Smallest raw p-value is the dominant lever here: ten percent either way moves the result by about 10.0 %, from 0.065 to 0.079. That is where a careless figure does the most damage, so check it against a source, and Number of comparisons is effectively tied with it (≈10.0 % for the same nudge), so verify the pair together. Everything below that can stay approximate, which is a licence to stop chasing data that would not change the decision.
Interpreting the Holm–Bonferroni Correction verdict
A number on its own is trivia; a number against a range is intelligence. The verdict attached to the default calculation — “Even the smallest p fails; the whole family stops here” — is one band on a ladder your own figure may sit somewhere else on. The sensitivity list shows how far a modest input error moves you along that ladder: for Smallest raw p-value, ±10 % covers 0.065 to 0.079. The practical reading rules: ask what would have to change to move you one band. If the answer is “a small, plausible input difference”, the band itself is not the finding — the uncertainty is.
Where Holm–Bonferroni Correction sits next to similar measures: this tool answers “how much is the Holm–Bonferroni Correction”, while the related calculators answer “what do I do about it”. Reading the result alone tends to over-weight a single snapshot; pairing it with the companion tools above turns the number into a decision.
Common mistakes with Holm–Bonferroni Correction (and the fix)
Forgetting the monotonicity enforce: a raw step-down value can dip below an earlier one; the adjusted sequence must never decrease. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Confusing a large number with an important one. Holm–Bonferroni Correction can be dominated by one input whose absolute value looks unimpressive, which is why this page ranks influence separately from magnitude.
Attention is also misallocated between fields. Having measured this engine, we can say where it belongs: Smallest raw p-value first, everything else second.
The subtlest error is answering the wrong question precisely. Confirm that holm-adjusted p (smallest) is actually the quantity your decision needs — the arithmetic is flawless either way, which is what makes the mistake invisible.
Holm–Bonferroni Correction verification checklist
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- Recompute once with the dominant input deliberately set low, and see whether the decision changes.
- Put Smallest raw p-value under a magnifier; the sensitivity table above is the reason why.
- Compare the output against one published benchmark or a colleague’s independent estimate.
- Compare the result with one independent method or published benchmark.
- Confirm the answer against one independent method or published benchmark for Holm–Bonferroni Correction.
Five steps from raw inputs to a figure you can defend
Using a calculator well sounds trivial, but the difference between a decision-grade figure and a throwaway one lives entirely in the routine. The routine for Holm–Bonferroni Correction takes a minute to learn and applies to every input above.
Step 1 — start from evidence. The inputs behind Holm–Bonferroni Correction are family-wise α, number of comparisons, smallest raw p-value. Write them down with their source next to each one; if you cannot name where a figure came from, that is the input most likely to be wrong. If the answer changes your behaviour, it deserves a second run with different assumptions.
Step 2 — enter the conservative case first. Resist the flattering scenario. Type the inputs you would defend in front of a sceptical reader, read the Holm-adjusted p (smallest) (0.072 for the example values), and note the verdict band. If a result feels obviously right, that is the moment to check the inputs hardest.
Step 3 — measure what deserves measuring. The ranked levers above tell you where a careful figure pays off. Everything else can stay approximate, which is a licence to stop hunting for data you do not need. Half of a good calculation is knowing which input you would hate to be wrong about.
Step 4 — compute a corridor, not a point. Three runs — worst, likely, best — turn Holm–Bonferroni Correction from a snapshot into a range. Plan against the pessimistic end and treat the optimistic end as the upside story. Compare the calculation to the rough guess you arrived with, and ask why they differ.
Step 5 — record the context, not just the number. Note today’s date and the inputs alongside the 0.072 output. Six months from now the context is what turns this figure into a trend. Say the answer out loud before you save it; if it sounds wrong, the model did not change, your understanding did.
Who actually uses Holm–Bonferroni Correction — and when
In Statistics & Data Science the figure computed here is usually the first one somebody asks to see. Having it — with the formula behind it — changes your position in the conversation from participant to reference. Precision is a promise you make about your data; make sure your inputs can keep it.
Professionals in Statistics & Data Science use this style of calculation as a screening filter before spending real money or effort: a quick Holm–Bonferroni Correction check eliminates weak options in minutes, and the survivors earn the deeper, slower analysis. A calculation you cannot explain to somebody else is a calculation you cannot defend.
Re-running this page costs nothing and catches the slow drifts that never announce themselves. In Science & Research the expensive mistakes are rarely sudden; they are small changes in Family-wise α that nobody recalculated for a year. When in doubt, prefer the boring input: the interesting one is usually the one you misread.
The questions that come up most about Holm–Bonferroni Correction
What affects Holm–Bonferroni Correction the most?
Measured on this engine, Smallest raw p-value is the dominant lever: moving it ±10 % shifts the holm-adjusted p (smallest) across 0.065 to 0.079 — about 10.0 % of the headline, with Number of comparisons next at ≈10.0 %. The ranked list in the sensitivity section is computed live, not asserted from a template.
Do number of comparisons and smallest raw p-value matter equally?
Effectively. Measured on this engine they land at ≈10.0 % and ≈10.0 % for the same ten-percent nudge, so the honest advice is to get both right; chasing precision in only one of them buys almost nothing.
What is a typical Holm–Bonferroni Correction?
On the worked example above Holm–Bonferroni Correction comes out at 0.072. Treat that as one point on a distribution rather than a norm: the bands, the companion outputs and the sensitivity ranking tell you how wide the plausible range really is.
How do I calculate Holm–Bonferroni Correction by hand?
Holm-adjusted p for rank k = max over j≤k of p(j) × (m − j + 1), floored at the previous value is the entire method. Work the inputs in that order — Sort p-values ascending; smallest gets multiplier m = 12 — and you will reach 0.072 on the default figures, the same number this tool prints.
What happens to Holm–Bonferroni Correction if smallest raw p-value changes by 10 %?
The engine was re-run with that field moved by ten percent while everything else stayed fixed: the holm-adjusted p (smallest) goes across 0.065 to 0.079, a swing of roughly 10.0 % around the 0.072 baseline. That ratio is the practical definition of how much this input matters.
Is the Holm–Bonferroni Correction result of 0.072 good or bad?
Neither, on its own — the label “Even the smallest p fails; the whole family stops here” is the band this figure falls in for the example inputs, and your own inputs will place you elsewhere on the same ladder. Judge the number against your history and your alternatives, and remember that a ten-percent error in Smallest raw p-value alone moves it by ≈10.0 %.
How many significant figures are reliable in Holm–Bonferroni Correction?
About as many as your least precise input. The tool computes at full precision and displays rounded output; report 0.072 to a decimal place your inputs can actually support.
Is Holm–Bonferroni Correction accurate?
The arithmetic is exact: Holm-adjusted p for rank k = max over j≤k of p(j) × (m − j + 1), floored at the previous value is evaluated with full floating-point precision and every substitution is printed in the worked example. What limits accuracy is input quality and the model’s assumptions, so treat the 0.072 result as the correct answer to the question you typed — and check the assumptions section before acting on it.
When do Holm and Bonferroni agree?
Exactly at the first (smallest) p — the multipliers are both m there. Divergence grows deeper in the list.
Why is Holm valid?
It is a closed testing shortcut; the intersection-closure principle proves its step-down procedure controls FWER.
Does this Holm–Bonferroni Correction tool store my numbers?
No. The entire calculation runs in your browser session; nothing entered above is transmitted or saved, which is the correct behaviour for sensitive Science & Research figures and the reason you can re-run this page as often as you like.
What the labels on this page mean
- Comparison tests left after this rank
- 11 for the Holm–Bonferroni Correction worked case; it sits next to the main result because it carries context the headline leaves out.
- Alpha for the second-smallest p
- 0.005 for the Holm–Bonferroni Correction worked case; it sits next to the main result because it carries context the headline leaves out.
- Same p under plain Bonferroni
- 0.072 in the default Holm–Bonferroni Correction calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Holm-adjusted p (smallest)
- The primary output of this tool: 0.072 for the default inputs, computed exactly from Holm-adjusted p for rank k = max over j≤k of p(j) × (m − j + 1), floored at the previous value.
- Worked example
- Every substitution in order, from the example inputs to 0.072 — the audit trail for the headline figure.
- Verdict band
- The plain-language label — here “Even the smallest p fails; the whole family stops here” — mapping the computed figure onto experience-based ranges for Science & Research.
- Sensitivity (±10 %)
- How far the Holm–Bonferroni Correction moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Method, sources and what this tool cannot tell you
The model is published arithmetic implemented exactly as written: Holm-adjusted p for rank k = max over j≤k of p(j) × (m − j + 1), floored at the previous value. Source of record: Holm (1979), A simple sequentially rejective multiple test procedure. Where a professional version of this calculation differs, it differs in inputs and jurisdiction-specific coefficients, not in the algebra — which is why re-checking your inputs beats hunting for a second calculator.
Nothing here replaces a qualified professional, and nothing here is approximate either — the arithmetic is exact, and the honesty is in the inputs you supply. What that buys you is reproducibility — two people, two browsers, the same arithmetic, and a printed trail from inputs to result that either of them can audit.
After the number: what to do with it
Save the number together with its inputs and today’s date; a Holm–Bonferroni Correction figure without context is unusable the moment an input changes.
Whoever you are (people who need one trustworthy number before their next decision), the next step is the same: replace the example inputs with your own, note the date, and re-run whenever a figure moves. In Science & Research, a dated figure is a trend, and an undated one is a rumour.