Sample Size for Survival Studies

Overview

Survival or time-to-event studies analyse both whether an event occurs and when it occurs.

Examples include:

• time to death;
• time to disease recurrence;
• time to hospital readmission;
• time to device failure; and
• time to treatment discontinuation.

Participants may be censored because the event has not occurred by the end of observation or follow-up ends for another reason.

For a two-group log-rank or Cox proportional-hazards comparison, statistical information depends strongly on the number of observed primary events. Total recruitment matters because it determines how many events are expected during accrual and follow-up.

Why this matters

A trial may enrol 1,000 participants but observe few events if prognosis is good or follow-up is short. Its treatment comparison may then be imprecise.

Another study with fewer participants but longer follow-up may observe more events and have greater power.

Therefore, survival-study planning requires:

• effect measure, often a hazard ratio;
• significance level and power;
• allocation ratio;
• control-group event pattern;
• accrual duration;
• additional follow-up;
• dropout and other censoring;
• competing events;
• timing of treatment effect; and
• planned analysis.

A conventional proportional-hazards formula may be inappropriate when the treatment effect is delayed, changes direction or is better represented by restricted mean survival time.

Events drive information

Under proportional hazards, Schoenfeld derived an approximate event requirement for a Cox-model treatment coefficient [1].

For allocation proportion $\pi$ to one group:

With equal allocation, $\pi=0.50$, so:

$$D = \frac{ 4\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2 }{ \left[\ln(HR)\right]^2 }$$

This event formula is an approximation based on proportional hazards and regular large-sample conditions.

Worked example: required events

Translating events into participants

A simple approximation is:

$$N = \frac{D} {\Pr(\text{primary event observed})}$$

If 60% are expected to experience an observed primary event:

$$N=247/0.60=411.67$$

Round upward to 412 participants.

If 10% are additionally expected to be lost before contributing the needed outcome information, a simplified inflation gives:

$$N_{\text{recruit}} = \frac{412}{0.90} = 457.78$$

Recruit 458 participants.

This two-step approximation is educational. A rigorous calculation should integrate the anticipated accrual, survival and censoring distributions rather than multiply broad percentages mechanically. Standard sample-size references provide time-aware methods for this conversion [2], [3].

Accrual and follow-up

Suppose recruitment lasts 24 months with 12 additional months of follow-up.

Participants enrolled first can contribute up to 36 months. Those enrolled last contribute only 12 months before administrative censoring.

Expected event counts require integration over entry times.

Planning should state:

• accrual start and end;
• accrual distribution;
• additional follow-up;
• event distribution in each group;
• dropout distribution;
• administrative censoring; and
• expected calendar time to reach the event target.

If accrual is slower than planned, the event target may be reached later even if final enrolment is unchanged.

Unequal allocation

The information term $\pi(1-\pi)$ is largest at equal allocation.

At 2:1 allocation, $\pi=2/3$:

$$\pi(1-\pi)=\frac{2}{9}=0.222$$

compared with 0.25 for equal allocation.

More events and participants are therefore generally required for the same power.

Hazard ratio assumptions

A hazard ratio compares instantaneous event rates under a model. It is not a risk ratio.

The planning hazard ratio should be supported by:

• clinically meaningful survival differences;
• credible historical data;
• expected median survival under a suitable distribution;
• survival probabilities at clinically relevant times; and
• consideration of non-proportional hazards.

The same hazard ratio can correspond to different absolute survival benefits depending on baseline hazard.

A sensitivity table should vary both the hazard ratio and control event pattern.

Non-proportional hazards

Proportional hazards may fail when:

• treatment benefit is delayed;
• benefit diminishes;
• hazards cross;
• treatment has early toxicity and later benefit; or
• cure fractions differ.

A log-rank design based on one constant hazard ratio may then have poor power or an unclear estimand.

Alternatives include planning for:

• weighted log-rank tests;
• milestone survival;
• restricted mean survival time;
• average hazard ratios;
• piecewise hazards; or
• simulation under clinically plausible curves.

The primary estimand, test and sample size must remain aligned. Restricted mean survival time is one interpretable alternative when proportional hazards are doubtful [4].

Competing risks

A competing event prevents the event of interest from occurring.

For disease recurrence, death without recurrence may be a competing event if recurrence is the primary event rather than a composite outcome.

Planning differs depending on whether the primary estimand concerns:

• cause-specific hazard;
• subdistribution hazard;
• cumulative incidence;
• composite event; or
• restricted mean time lost.

Treating competing deaths as ordinary independent censoring may be inappropriate.

Loss to follow-up versus event-free administrative censoring

Administrative censoring at study end is expected and may be non-informative under the design.

Loss to follow-up occurs when outcome status becomes unavailable before the planned end. It may reduce events and introduce bias if related to prognosis.

The adjustment should model timing where possible. Dividing by a dropout proportion is only an approximation.

Event-driven planning summary

simulated data

Table 1. Planning quantities

Quantity	Assumption
Hazard ratio	0.70
Two-sided significance level	0.05
Power	80%
Allocation	1:1
Required primary events	247
Expected observed event proportion	60%
Approximate participants before loss adjustment	412
Recruitment after 10% loss adjustment	458

Note: The participant conversion is a simplified teaching approximation.

Prognostic Cox models

Planning a study to estimate one Cox coefficient differs from developing a multivariable prognostic model.

Prediction-model development requires adequate:

• total participants;
• events;
• candidate predictor parameters;
• follow-up;
• model fit;
• calibration precision; and
• internal validation.

Riley and colleagues provide criteria for binary and time-to-event prediction-model development rather than a fixed events-per-variable rule [5].

Assumptions and sensitivity analysis

Event-driven monitoring MMed extension

An event-driven trial may continue until the prespecified number of primary events is observed, subject to calendar, safety and operational limits.

Blinded event-count monitoring can assess whether the target is likely to be reached without comparing treatment effects.

Unblinded sample-size re-estimation or adaptive changes require prespecified procedures to protect type I error and trial integrity.

Suggested reporting

Common mistakes

Knowledge check

Why do survival studies often plan around the number of events?

Answer and reasoning

Participants who remain event-free contribute follow-up information, but the precision of many treatment-effect estimates depends strongly on the number of observed events. Total enrolment alone does not determine this information.

What does a hazard ratio of 0.70 mean under proportional hazards?

Answer and reasoning

The treatment-group instantaneous event rate is 30% lower than the control-group rate at each time under the proportional-hazards model. It is not a statement that cumulative risk is 30% lower at every time.

Why must accrual duration be included?

Answer and reasoning

Participants entering at different calendar times have different maximum follow-up before study end, which changes the expected number of observed events.

Why may a delayed treatment effect invalidate a simple Schoenfeld calculation?

Answer and reasoning

The calculation assumes a constant log hazard ratio. With delayed effects, the chosen test and information accumulation differ, so simulation or another estimand-specific method may be needed.

What is the difference between a competing event and loss to follow-up?

Answer and reasoning

A competing event is observed and prevents the event of interest. Loss to follow-up means later event status is unavailable. They require different interpretation and methods.

Key take-home messages

References

1. Schoenfeld DA. Sample-size formula for the proportional-hazards regression model. Biometrics. 1983;39(2):499–503. https://doi.org/10.2307/2531021
2. Chow SC, Shao J, Wang H, Lokhnygina Y. Sample size calculations in clinical research. 3rd ed. Boca Raton (FL): CRC Press; 2018.
3. Machin D, Campbell MJ, Tan SB, Tan SH. Sample sizes for clinical, laboratory and epidemiology studies. 4th ed. Hoboken (NJ): Wiley; 2018. https://doi.org/10.1002/9781118874905
4. Royston P, Parmar MKB. The use of restricted mean survival time to estimate the treatment effect in randomized clinical trials when the proportional hazards assumption is in doubt. Stat Med. 2011;30(19):2409–2421. https://doi.org/10.1002/sim.4274
5. Riley RD, Snell KIE, Ensor J, Burke DL, Harrell FE Jr, Moons KGM, et al. Minimum sample size for developing a multivariable prediction model: Part II—binary and time-to-event outcomes. Stat Med. 2019;38(7):1276–1296. https://doi.org/10.1002/sim.7992
6. Hopewell S, Chan AW, Collins GS, Hróbjartsson A, Moher D, Schulz KF, et al. CONSORT 2025 statement: updated guideline for reporting randomised trials. BMJ. 2025;389:e081123. https://doi.org/10.1136/bmj-2024-081123