Sample Size for Survival Analysis: Why Events, Not People, Drive Power

The single most common mistake in planning a time-to-event study is treating it like any other two-group comparison and running a standard binary- or continuous-outcome power calculation. Survival-analysis power does not scale with total enrollment the way a t-test or a chi-square test does — it scales with the number of events observed. A trial that enrolls 2,000 patients but sees only 60 deaths is powered like a 60-event study, not a 2,000-person one. Get this distinction wrong early and the rest of the sample-size justification, however carefully the arithmetic is done, is built on the wrong quantity.

This matters most at the exact moment a researcher has to defend a number to someone else — a grant reviewer asking why the proposed cohort is large enough, or an IRB/ethics committee checking that the study is neither underpowered (wasting participants’ time on an unanswerable question) nor needlessly overpowered (exposing more people than necessary to study burden). The formula that answers both concerns correctly is the one David Schoenfeld derived for the Cox proportional-hazards model, and it starts from events, not people.

Why Events, Not Enrollment, Drive Power

The log-rank test and the Cox proportional-hazards model share the same underlying statistic — when there is a single binary covariate and no tied event times, the Cox score test, the Mantel-Haenszel test, and the log-rank test are mathematically the same test (see the Kaplan-Meier and Cox proportional hazards guides for how those pieces fit together). The precision of that test — how tightly it can distinguish two hazard functions — depends on how many event times were actually observed, because each event is what contributes information to the partial-likelihood comparison. Two participants who are censored (lost to follow-up, or reach study end event-free) contribute almost nothing to that comparison, regardless of how long they were followed.

That is why a survival-analysis sample-size justification has to answer two separate questions in order: first, how many events are needed to detect the hazard ratio of interest at the desired power; second, how many people need to be enrolled, followed for how long, to actually accumulate that many events given realistic event and dropout rates. Skipping straight to the second question — picking a round enrollment number and hoping — is the single most common defect reviewers flag in a study-section or IRB sample-size paragraph.

The Schoenfeld Formula: How Many Events You Need

David Schoenfeld’s 1983 Biometrics paper, “Sample-Size Formula for the Proportional-Hazards Regression Model,” derived the number of events required to detect a target hazard ratio at a given significance level and power, for a Cox model with a single covariate (equivalently, a two-arm log-rank comparison):

d = (z_α/2 + z_β)² / [p(1−p)(ln HR)²]

where d is the required number of events, z_α/2 and z_β are the standard normal quantiles for the chosen two-sided significance level and power (1.96 and 0.84 for the conventional α=0.05, 80% power case), p is the proportion of the sample allocated to one arm (0.5 for equal allocation), and HR is the hazard ratio you want to be able to detect. Note what does not appear in the formula: total sample size, accrual duration, or follow-up length. Those all matter for turning the event count into an enrollment plan (next section) — they play no role in how many events the test itself needs.

Worked example: to detect a hazard ratio of 0.70 (a 30% relative reduction) with two-sided α=0.05 and 80% power, equal allocation (p=0.5): d = (1.96+0.84)² / [0.25 × ln(0.70)²] = 7.849 / (0.25 × 0.1272) ≈ 247 events. That is the number that goes into a grant’s power-analysis paragraph — not a total-N figure by itself.

From Events to Enrollment: Accrual, Follow-Up, and Dropout

Converting a target event count into an enrollment plan requires three additional, study-specific inputs, each of which needs its own justification (typically from pilot data, a comparable published cohort, or registry data — never an unsupported guess):

• Baseline event rate. The proportion of the control/reference group expected to experience the event within the planned follow-up window. This is the input reviewers scrutinize hardest, because the whole calculation is only as credible as this number’s source.
• Accrual period. How long recruitment runs. Participants enrolled early are followed longer (and so are more likely to have the event observed, censoring aside) than participants enrolled near the end of accrual.
• Dropout / loss-to-follow-up rate. Every participant lost before an event or before study end reduces the effective event yield without contributing information — a higher dropout rate means more people need to be enrolled to net the same number of events.

L.S. Freedman’s 1982 formula (an alternative derivation assuming exponential survival and uniform accrual) produces comparable enrollment estimates to Schoenfeld’s approach when group sizes are equal, and is the other formula most sample-size software implements alongside it. For designs where accrual and dropout are not well approximated as uniform or exponential — a staggered-entry trial, or a study with a sharply rising dropout curve — the Lakatos (1988) method extends the calculation via a Markov/quadrature approximation that can absorb an arbitrary accrual and dropout pattern; it is the more defensible choice when the simplifying assumptions behind Schoenfeld/Freedman clearly do not hold.

Worked Example: From Target Hazard Ratio to Enrollment Target

Continuing the 247-event target above: suppose the reference arm’s 3-year event rate (from a comparable published cohort) is 25%, accrual runs for 18 months, and total follow-up (accrual plus a further observation window) runs 36 months from first enrollment, with an assumed 10% cumulative dropout. Under these assumptions, roughly 55-60% of enrolled participants are expected to either have the event or be followed long enough for it to be observed with reasonable probability by the analysis date — the exact figure depends on the accrual/follow-up split and is what dedicated software (next section) computes rather than a hand formula, because it involves integrating the event-time distribution over a staggered-entry accrual window. Working backward from 247 required events at that effective yield gives an enrollment target in the 420-450 range before applying the dropout inflation — the kind of number a study-section reviewer expects to see derived step by step, with each input traceable to a source, rather than presented as a bare total.

Freedman, Lakatos, and the Tools Researchers Actually Use

Hand-computing the accrual/follow-up integration behind the worked example above is impractical past the simplest cases, which is why sample-size planning for survival studies is almost always done in dedicated software rather than a spreadsheet formula:

• R: powerSurvEpi — implements Schoenfeld’s and related formulas specifically for epidemiological and clinical time-to-event designs, including adjustment for a covariate-outcome correlation.
• R: gsDesign and rpact — built for group-sequential and adaptive survival trial designs, handling interim-analysis event targets as well as the final sample size.
• Stata: stpower — a built-in command family (stpower cox, stpower logrank, stpower exponential) covering the Schoenfeld, Freedman, and exponential-hazards approaches natively.
• PASS / NCSS — commercial software with a dedicated “Logrank Tests with Proportional Hazards (Schoenfeld and Wu)” procedure that implements the Lakatos method for non-uniform accrual and dropout, which the free R/Stata options handle less directly.

Whichever tool is used, the assumptions that go into it — baseline event rate, accrual length, follow-up length, dropout rate, and the target hazard ratio, along with the source each was drawn from — belong explicitly in the write-up. A software-generated number with no visible assumptions is not more defensible than a hand calculation; it is just harder to check.

Common Mistakes When Justifying Sample Size for a Grant or Protocol

• Running a two-sample-means or two-proportions power calculation for a time-to-event outcome. These formulas answer a different question (detecting a difference in a mean or a proportion measured at a single time point) and do not account for censoring or the accrual/follow-up structure of survival data — the resulting N is not meaningful for a log-rank or Cox analysis.
• Quoting total enrollment without the underlying event count. A reviewer who cannot see the number of events implied by an enrollment figure cannot check whether the study is actually powered for its stated hazard ratio.
• Picking a “typical” hazard ratio without a cited source. The target HR should come from a pilot study, a comparable published trial, or a minimum clinically important difference — not from working backward from a sample size that was fixed by budget or feasibility and calling the resulting HR “detectable.”
• Ignoring competing risks. When participants can leave the risk set for reasons other than the event of interest (e.g., death from an unrelated cause in a disease-specific-survival endpoint), a standard Schoenfeld/Freedman calculation that treats those exits as ordinary censoring can understate the required sample size; a competing-risks-aware design calculation is the more defensible choice in that setting.
• Assuming a flat, unrealistic accrual pattern. Real accrual is rarely uniform — it typically ramps up as sites open and activate. Overstating early accrual overstates how much follow-up time later-enrolled participants will accumulate, and inflates the projected event count.

Frequently Asked Questions

Is total sample size or number of events the “real” answer to give a reviewer?

Both, but in order: state the required number of events first (the quantity the statistical test is actually powered by), then show the enrollment number that is expected to produce that many events under the stated accrual, follow-up, and dropout assumptions. Giving only the enrollment total without the event count underneath it is the more common gap reviewers flag.

What if there is no pilot data to estimate the baseline event rate?

Use the most comparable published cohort or registry rate available, cite it explicitly, and consider presenting the sample-size table across a plausible range of event rates (a sensitivity table) rather than a single point estimate — this is generally viewed more favorably than an unsupported single assumption.

Does a larger hazard ratio always mean a smaller required sample?

Yes, all else equal — the further HR is from 1.0, the fewer events are needed to detect it, because (ln HR)² in the denominator of the Schoenfeld formula grows as HR moves away from 1. This is also why an unrealistically large assumed effect size is a common way underpowered studies get proposed: the smaller “required” N looks feasible on paper because the target effect was never realistic.

How does allocation ratio affect the calculation?

Unequal allocation (for example 2:1 favoring the experimental arm) always requires more total events than 1:1 allocation to achieve the same power, because the p(1−p) term in the denominator is maximized at p=0.5. Unequal allocation is sometimes chosen anyway for participant-recruitment or ethical reasons, but the resulting event-count penalty should be shown explicitly, not absorbed silently into a larger total-N figure.

For adjacent methodology, see how to interpret a hazard ratio, the general power analysis and sample size calculation guide for non-survival designs, and the research methods hub for the full study-design and sampling-and-power collection.