Log-rank test — Sample Size Calculator

Determines the sample size required to detect a specified difference between the survival distributions of two groups using the log-rank test.

Key takeaways

What is calculated: Determine the required number of events and the corresponding number of participants.
Quantities you must supply: Median survival — control; Median survival — treatment; Accrual period; Additional follow-up after accrual; Significance level; Power; Allocation ratio.
Type of calculation: Power-based: it answers how much data is needed to detect a stated effect. Power 0.80 at α = 0.05 is the usual target; confirmatory studies use 0.90.
Calculation routes: 2 are available for this design; the one selected determines the formula, the quantities requested and the result.

Methods available for Log-rank test

Median survival times with accrual and follow-up (Schoenfeld) Two-arm survival comparison specified through median survival times, accrual and follow-up. Reducing a survival design directly to a participant count without accounting for event probability is the commonest error in trial planning.
Survival rates at the end of the study (Machin / Freedman) Sample size for a two-arm survival comparison stated through the survival rate expected in each group. Survival rates at a stated time are how clinical assumptions are usually available. Converting them by hand into a hazard ratio and an event probability is where planning errors creep in.

When to use

Use when the primary objective is to determine whether the time-to-event or survival experience differs between two groups.

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Sample size

Calculation method

Median survival — control

Median survival — treatment

Accrual period

Additional follow-up after accrual

Significance level (α)

Type I error rate.

Power (1 − β)

Probability of detecting the specified effect.

Alternative

A two-sided alternative splits α between both tails; a one-sided alternative places all of α in one tail, needs a smaller n, and must be pre-specified.

Allocation ratio (n₂ / n₁)

Set to 1 for equal group sizes.

Anticipated dropout / non-response (proportion)

Final n is inflated by 1/(1 − dropout).

Reset to defaults

Awaiting inputs

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Set the parameters and calculate.

Methodology — Log-rank test — Schoenfeld formula with exponential survival

Formula used Log-rank test — Schoenfeld formula with exponential survival

d = \frac{{(z_{1 - α / 2} + z_{1 - β})}^{2}}{q_{1} q_{2} {[\ln (H R)]}^{2}}

Required events

λ = \frac{\ln 2}{m e d i a n}

Exponential hazard rate

N = \frac{d}{\Pr (e v e n t)}

Total subjects

$z_{1 - α / 2} (two-sided) in place of z_{1 - α} (one-sided)$ Two-sided alternative selected: α is split between both tails, so the larger quantile is used and the requirement rises

where —

m₁: Median survival in the control arm. > 0
m₂: Median survival in the treatment arm; HR = m₁/m₂. Must differ from m₁
a: Recruitment period, with uniform entry assumed. ≥ 0
f: Additional follow-up after the last subject enters. ≥ 0
α: Probability of rejecting a true null hypothesis (Type I error). 0.0001 – 0.5; conventionally 0.05, or 0.025 one-sided for regulatory non-inferiority
1 − β: Probability of rejecting the null hypothesis when the specified alternative is true. 0.50 – 0.9999; conventionally 0.80 or 0.90
k: Allocation ratio n₂/n₁. 0.05 – 20; 1 gives equal groups
ℓ: Expected proportion of enrolled units lost before analysis. 0 – 0.95

Formulas for the 2 methods

Calculation method	Formula
Median survival times with accrual and follow-up (Schoenfeld)	$d = \frac{{(z_{1 - α / 2} + z_{1 - β})}^{2}}{q_{1} q_{2} {[\ln (H R)]}^{2}}$
Survival rates at the end of the study (Machin / Freedman)	$d = \frac{{(z_{1 - α / 2} + z_{1 - β})}^{2} {(1 + k φ)}^{2}}{k {(1 - φ)}^{2}}$

How this method works

Power for the log-rank test depends on the number of events, not the number of subjects. The required events follow from the hazard ratio; subjects follow by dividing by the probability of experiencing an event, computed here from exponential survival under uniform accrual.

Selected method - median survival with accrual and follow-up (Schoenfeld). Power depends on the number of events, not the number of subjects. Schoenfeld's result gives the events required; subjects follow by dividing by the probability of experiencing an event, computed from the median survival times with uniform accrual plus additional follow-up.

A trial short of events can be rescued by recruiting more subjects or by extending follow-up. The latter is usually cheaper, with diminishing returns once most subjects have had an event.

Null hypothesis. H0: HR = 1 — the two survival distributions are equal.

Alternative hypothesis. Two-sided H1: HR != 1. One-sided H1: HR < 1 or HR > 1.

Calculation procedure

State the median survival expected in each arm and convert to hazard rates: $λ = \frac{\ln 2}{median}$ .
The hazard ratio follows as $H R = λ_{2} / λ_{1}$ .
Compute the events required: $d = \frac{{(z_{1 - α / 2} + z_{1 - β})}^{2}}{q_{1} q_{2} {[\ln (H R)]}^{2}}$ .
Work out the probability an enrolled subject has an event, given the accrual period and the follow-up after it, and divide: $N = d / \Pr (event)$ .
Round each arm up, then inflate for dropout. Longer follow-up raises the event probability and lowers the enrolment needed.

Exact or approximate

Approximate — an asymptotic result for the log-rank statistic, standard in trial practice.

The formula shown always matches what was computed: when a one-sided test is selected the rendered quantile changes from z_1−α/2 to z_1−α, and where several methods exist, the formula follows the method selected in the calculator.

Advantages & limitations

Advantages

Reports events and participants separately.
Accounts for accrual and follow-up explicitly.
The standard method reviewers expect.

Limitations

Assumes proportional hazards and exponential survival.
Sensitive to the hazard ratio near the null: moving from 0.80 to 0.75 cuts required events by about a third.
Does not model loss to follow-up; use the Lakatos method for that.

Assumptions

Hazards are proportional over the whole follow-up period.
Survival is exponential, so a constant hazard applies within each arm.
Accrual is uniform over the recruitment period.
Censoring is non-informative.

Applicable adjustments

Supports allocation ratio and dropout inflation.

Conclusion

Report required events, participants per arm, and the assumed medians, accrual and follow-up. Extending follow-up is usually cheaper than recruiting, with diminishing returns once most subjects have had an event.

References

Schoenfeld DA. Sample-size formula for the proportional-hazards regression model. Biometrics. 1983;39(2):499–503. Link
Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: Chapman & Hall/CRC; 2015.
Machin D, Campbell MJ, Tan SB, Tan SH. Sample Size Tables for Clinical Studies. 3rd ed. Chichester: Wiley-Blackwell; 2009.
MedCalc Software Ltd. Sample size calculation: survival analysis (log-rank test). Link