A hazard ratio of 0.7 at 80% power and a two-sided 5% level needs 247 events. With half the control arm and forty per cent of the treated arm expected to have the event, the average event probability is 0.45, so 549 subjects are needed — 275 per arm. The number to carry forward is 247, not 549: if recruitment is easy and follow-up short, the trial will fail for want of events however many people are enrolled.
247 events — which takes 549 subjects at these event rates
The event count is what the power calculation returns. The subject count is that divided by an average event probability of 0.4500, so it moves entirely with how common the event is: halve the event rate and the enrolment doubles while the 247-style requirement does not change at all. Per arm: 275 control and 275 treated.
Events required
247
what power depends on
Subjects
549
275 control, 275 treated
Event probability
0.4500
allocation-weighted average
Power achieved
0.8003
at 247 events
What the effect size costs
Events and subjects required at a range of hazard ratios
Hazard ratio
Events
Subjects
Against this design
0.5
66
146
0.27×
0.6
121
268
0.49×
0.67
196
436
0.79×
0.75
380
844
1.54×
0.8
631
1402
2.55×
0.85
1189
2642
4.81×
0.9
2829
6285
11.45×
Events scale with the inverse square of the log hazard ratio, which is why the column rises so steeply. Moving the target from 0.7 to 0.8 more than doubles the trial; moving it to 0.9 multiplies it by more than eleven. Choosing the hazard ratio to power for is the single most consequential number in a survival protocol.
Power against events
The power achieved at a range of event counts
Events
Subjects needed
Power
124
276
0.5103
186
414
0.6816
247
549
0.8003
309
687
0.8800
371
825
0.9299
494
1098
0.9775
This is the table a trial should be monitored against. A study that has enrolled its full target and accrued half its events has roughly 0.51 power, not the 0.80the protocol claimed — and extending follow-up is usually cheaper than enrolling more people.
This is Schoenfeld’s formula, which assumes proportional hazards throughout. If the effect is delayed or reverses — anything the proportional-hazards test would reject — the required event count is not what this returns, and a trial powered from it will be underpowered for the effect it actually has.
The event probabilities are assumptions about the future and are usually optimistic. Recruitment often selects healthier subjects than the registry data the estimate came from, so the real event rate comes in low and the enrolment target turns out to be the wrong number — while the event target stays right.
Events drive power Sensitivity to the effect Power at every event count Assumes proportional hazards
What this tool shows
A hazard ratio of 0.7 at 80% power needs 247 events. Whether that takes 549 subjects or 1411 depends on how common the event is — and the event requirement does not move. The second and third presets are the same comparison in two populations: identical 247 events, 2.6 times the enrolment, identical power. A survival trial that has recruited its full target and accrued half its events is at roughly 51% power, not the 80% the protocol claimed.
Schoenfeld’s required event count, and the enrolment that produces it at your event rates
The power achieved at a range of event counts, which is how a running trial should be monitored
Events required across seven hazard ratios, so the cost of the effect size is visible
Unequal allocation and what it costs in events
A preset where the event rate changes and the event requirement does not
Why the formula assumes proportional hazards, and what that costs when it fails
Events drive power Sensitivity to the effect Power at every event count Assumes proportional hazards
The event probabilities are assumptions about the future.
Updated 13 September 2026 · Works in any browser, no installation
A survival comparison’s power depends on the number of events observed, not the number of subjects enrolled. Schoenfeld’s formula returns an event count from the hazard ratio, the significance level, the power and the allocation ratio — nothing else enters it. The subject count is a second step, obtained by dividing by how often the event is expected to occur, and it moves while the event requirement stays where it is.
At a glance
Formula shown
d = (z_{α/2} + z_β)² / (p₁p₂·(ln HR)²), where p₁ and p₂ are the allocation proportions. Nothing about the sample size appears. The enrolment target is d divided by the allocation-weighted average event probability. Because the log hazard ratio is squared in the denominator, the requirement grows as the effect shrinks: 66 events at HR 0.5 and 2829 at HR 0.9.
Scenario support
Sizing a time-to-event trial, checking whether a protocol’s enrolment target matches its event assumptions, monitoring a running trial against events rather than recruitment, and judging whether a published survival study was powered for the effect it reports.
Educational estimate
Planning support from the values you enter — not professional advice.
The same trial, twice the enrolment, identical power
The second and third presets are the same comparison in two populations, and the contrast is the whole argument for thinking in events.
Both need exactly 247 events. Same hazard ratio, same power, same significance level — and Schoenfeld’s formula contains none of the event rates.
One needs 549 subjects and the other 1411. An average event probability of 0.45 against 0.175.
Power is identical in both. The extra 862 subjects buy nothing except the events they eventually supply.
Which reframes what to do when a trial is behind. Extending follow-up produces events from people already enrolled; recruiting more people produces events only after their own follow-up. The first is usually cheaper and always faster.
The effect size is the expensive decision
The requirement grows with the inverse square of the log hazard ratio, which makes the protocol’s target effect the most consequential number in it.
HR 0.5 needs 66 events. HR 0.7 needs 247. Nearly four times, for a target 0.2 closer to 1.
HR 0.9 needs 2829. Forty-three times the HR 0.5 trial, and 5955 subjects at the shipped event rates.
All three sound reasonable when written down. “A 10% reduction in the hazard” is a defensible clinical target and a very different study from “a halving”.
So the sensitivity table matters more than the point answer. A trial powered for 0.7 that turns out to have an effect of 0.85 was never going to detect it, and the table says so before the protocol is signed rather than afterwards.
Monitor against events, not recruitment
The power table is the one a running trial should be read against, and it usually is not.
At half the required events, power is about 51%. A coin flip, on a trial that has met its enrolment target in full.
At three-quarters it is about 68%. Still well short of the 80% the protocol promised.
Recruitment completion is the wrong milestone to celebrate, because it says nothing about whether the events will arrive.
And a trial that stops early at its enrolment target rather than its event target reports a non-significant result that was largely determined before the first patient was randomised.
Unequal allocation costs events
Allocating more subjects to one arm has reasons behind it and a price attached, and the price is computable in advance.
The formula carries a factor of 1/(p₁p₂). At equal allocation that is 4.0, which is its smallest possible value.
At 2:1 it is 4.5. On the fourth preset that is 182 events against 162 for a balanced design — 12.3% more.
At 3:1 it rises to 5.33, a third more events than balanced.
Sometimes the price is worth paying: more safety experience with a new treatment, or a control arm that is hard to recruit into. It should be a decision with a number attached rather than a default.
It assumes the hazard ratio is constant
Schoenfeld’s formula is derived under proportional hazards, and a trial powered from it inherits that assumption whether or not it holds.
A delayed effect breaks it. Immunotherapy trials where the curves overlap for months and then separate are the standard modern example.
Under a delayed effect the required events are higher than this returns, sometimes much higher, because the early events carry no signal.
If a delayed or reversing effect is plausible, size the trial by simulation rather than by this formula, and pre-specify an analysis that does not assume proportionality.
The event probabilities are guesses
The event count is arithmetic. The enrolment target rests on two numbers that are forecasts, and they are usually optimistic in a predictable direction.
Trial populations are healthier than registry populations. Eligibility criteria select for it, and consent selects for it again.
So the observed event rate comes in below the planned one, and the enrolment target that was derived from it turns out to be too small.
The event target stays correct throughout. It is the enrolment estimate that was wrong, which is another reason to treat the event count as the real design parameter.
The practical response is an event-driven design: pre-specify that the analysis occurs when the required events have accrued, and let the follow-up duration absorb the error in the rate assumption.
Reporting a survival sample size
Four items, and the first is the one whose omission makes the rest unverifiable.
Give the required events, not just the enrolment. The enrolment figure alone cannot be checked without the event-rate assumptions behind it.
Give both event probabilities and where they came from. They convert events into subjects and they are the part most likely to be wrong.
Give the hazard ratio you powered for, and say it is a target rather than an expectation. The sensitivity to it is steep enough that the distinction matters.
And say whether the analysis is event-driven. A fixed-duration trial and an event-driven one with the same enrolment are different designs with different power.
Method. The event requirement is Schoenfeld’s formula with the allocation factor written out, which means the answer depends on the hazard ratio, the two critical values and the allocation ratio and on nothing else — the event probabilities enter only when converting events into an enrolment target. The power achieved at a given event count is computed by inverting the same expression rather than by a separate approximation, so the two are guaranteed consistent. The verification suite checks the formula against its published behaviour: a hazard ratio of 0.7 at 80% power and a two-sided 5% level must give 247 events, the achieved power at that count must be 0.80, and halving the log hazard ratio must quadruple the requirement, which follows from the square in the denominator. It also asserts monotonicity on sixty generated designs — a hazard ratio closer to 1 always needs more events, and power always rises with the event count. That engine is verified on every change against 128 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Cox RegressionProportional hazards by partial likelihood with hazard ratios, concordance, and the Schoenfeld residual test computed on every fit rather than offered as an extra.
Kaplan-MeierSurvival with censoring handled, and the naive count printed beside it: five events in twenty subjects give 25.0000% by the plain count and a 30.1202% cumulative incidence by Kaplan-Meier.
Competing RisksAalen-Johansen cumulative incidence with the naive 1 − Kaplan-Meier beside it for every cause, and the probability identity printed as a check on both.
Log-Rank TestObserved minus expected accumulated at every event time, so the crossing-hazards blind spot is visible: a running total that peaks at +3.2652 and ends at -1.4717 gives p = 0.458065.
Cluster Sample SizeClusters needed for a target power, with the individual-randomisation comparison and the whole trade-off between more clusters and bigger ones.
Sample SizeResponses needed for a target margin of error, with the finite-population correction and a table of the whole cost curve — because n scales with 1/margin², so the last point of precision costs more than the first ten.
An educational tool. Schoenfeld’s formula assumes the hazard ratio is constant over follow-up; a delayed or reversing effect needs more events than it returns, and the analysis it sizes has the same problem. The enrolment figure rests on assumed event probabilities, which trial populations routinely undershoot because eligibility and consent select for healthier subjects.