Sample size to estimate a median survival time

When to use it

A single-arm study or cohort reporting median survival or median time to relapse.

Not for

Survival that is clearly not exponential, such as a high early hazard that then levels off; plan survival at a fixed time instead.

Notation

$n$: Number of participants
$m$: Expected median survival time
$\lambda$: Event hazard, ln 2 / m
$t$: Follow-up per participant
$\eta$: Loss-to-follow-up hazard
$P$: Share with an event during follow-up
$f$: Precision factor: the interval runs from m / f to m × f
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α (1.960 for 95%)
$q$: Non-response rate

Formulas

Events set the precision

n = \left\lceil \frac{z_{1-\alpha/2}^{2}}{P\, (\ln f)^{2}} \right\rceil, \qquad P = \frac{\lambda}{\lambda + \eta}\left(1 - e^{-(\lambda + \eta) t}\right)

Under exponential survival the median is ln 2 / λ, and the log of the estimated hazard, events over total time at risk, has variance 1/D for D events [1]. A median beyond the follow-up is extrapolated from the model.

Non-response

n_{\text{approach}} = \left\lceil \frac{n}{1 - q} \right\rceil

The number to approach so that n respond. Applied last, after any clustering.

Parameters

Confidence level $1-\alpha$

Values: 90%, 95% or 99%

95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

Precision factor $f$

Values: Above 1

1.25 means the interval runs from m / 1.25 to m × 1.25.

Expected median survival time $m$

Values: Above 0, in the follow-up’s time unit

From earlier studies of a similar population.

Follow-up per participant $t$ Optional

Values: Above 0; 1 by default

Use one time unit for every field, such as years.

Loss-to-follow-up rate $\eta$ Optional

Values: 0 or more, per unit of time

Participants lost before the time are censored, which leaves fewer events and widens the interval.

Expected non-response rate $q$ Optional

Values: 0% up to, but not including, 100%

The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

Assumptions

Survival is exponential, with a constant hazard, and losses are unrelated to the outcome.
Non-response is unrelated to the outcome.

Worked examples

A two-year median, three years of follow-up

A median of 2 years followed for 3: 64.6% have an event, so n = 1.96² / (0.646 × (ln 1.25)²) = 119.4, rounded up to 120.

Sample size: 120

Open this example in the calculator

A longer median and losses

A median of 5 years followed for 2 with losses at 0.1 a year: only 22.0% have an event, so n = 350.

Sample size: 350

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
Exponential fits to simulated samples at the planned sizes gave median interval factors of 1.2485 to 1.2503 against 1.25 and coverage of 95.2% to 95.3%. An independent implementation in R recomputes these sizes and is compared with the engine’s.

How to cite

StatWisely. Sample size to estimate a median survival time. Method document, engine version 1.0.0, reviewed 2026-10-10. https://statwisely.com/methods/precision-median-survival/

Cite the original references for the formulas your design uses as well.

References

1.Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: CRC Press; 2015.