Freedman method — Sample Size Calculator

Determines the required number of events and sample size for a log-rank comparison of two survival curves using the Freedman formulation, which expresses the requirement through the ratio (1 + HR)/(1 - HR).

Key takeaways

What is calculated: Determine the required number of events and the corresponding participants.
Quantities you must supply: Hazard ratio to detect; Survival probability — control at the end of study; 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.

Method used for Freedman method

Freedman log-rank formula Alternative event-based formula, expressed through the ratio (HR+1)/(HR−1). It is the form tabulated in several regulatory guidance documents, so reviewers may expect it.

When to use

Use when the primary objective is to determine whether survival differs between two groups and the design assumptions are available as survival probabilities rather than as median survival times.

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Sample size

Hazard ratio to detect (HR)

Survival probability — control at the end of study

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 — Freedman log-rank formula

Formula used Freedman log-rank formula

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

Required events

S_{2} = S_{1}^{H R}

Treatment-arm survival

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

where —

HR: Hazard ratio to detect. > 0; must differ from 1
S₁: Control-arm survival probability at the end of study. 0 – 1 exclusive
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

How this method works

Freedman's derivation of the required events for the log-rank test. It differs modestly from Schoenfeld and is slightly more conservative, particularly for hazard ratios far from one.

Freedman's alternative to Schoenfeld, with the event probability computed from control-arm survival and the proportional hazards relation S₂ = S₁^HR.

The two differ modestly; Freedman is slightly more conservative, particularly for hazard ratios far from 1, and is the form tabulated in several guidance documents. State which was used, since a reviewer recomputing with the other will get a different figure.

Null hypothesis. H0: HR = 1.

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

Calculation procedure

State the hazard ratio $H R$ worth detecting and the survival expected in the control group at the end of the study.
Compute the events required by the Freedman form: $d = {(z_{1 - α / 2} + z_{1 - β})}^{2} {(\frac{1 + H R}{1 - H R})}^{2}$ .
Work out the overall probability of an event from the two survival probabilities.
Convert to enrolment: $N = d / \Pr (event)$ .
Round each arm up, then inflate for dropout. Freedman gives slightly more events than Schoenfeld, so name the method in the protocol.

Exact or approximate

Approximate — an asymptotic result for the log-rank statistic.

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

Slightly conservative relative to Schoenfeld.
Tabulated in several guidance documents.
Requires only the HR and control survival.

Limitations

Gives a different figure from Schoenfeld, so the method used must be stated.
Assumes proportional hazards.
Diverges from Schoenfeld most where the HR is far from one.

Assumptions

Hazards are proportional over follow-up.
The control-arm survival probability is correctly specified.
Censoring is non-informative.
Treatment-arm survival follows from proportional hazards as S2 = S1^HR.

Applicable adjustments

Supports allocation ratio and dropout inflation.

Conclusion

Report events, participants and the control-arm survival assumed. Always state which log-rank formula was used, since a reviewer recomputing with the other will get a different number.

References

Freedman LS. Tables of the number of patients required in clinical trials using the logrank test. Stat Med. 1982;1(2):121–129. Link
Machin D, Campbell MJ, Tan SB, Tan SH. Sample Size Tables for Clinical Studies. 3rd ed. Chichester: Wiley-Blackwell; 2009.
Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: Chapman & Hall/CRC; 2015.