Achieved power for the log-rank test (Freedman)

When to use it

Assess a time-to-event trial when Freedman’s method is the one the protocol or a regulator expects.

Not for

For the same events, Freedman gives slightly different power from Schoenfeld, more so the further the hazard ratio is from 1.

Notation

$D$: Events across both arms.
$N$: Participants across both arms, when given instead of events.
$S$: Control survival at the end of follow-up.
$\mathrm{HR}$: Hazard ratio, treatment to control.
$k$: Allocation ratio, treatment to control.

Formulas

Events from participants

D = N\left[\frac{1-S}{1+k}+\frac{k\,(1-S^{\mathrm{HR}})}{1+k}\right]

When participants and control survival are given instead of events, the expected events follow as in the survival sample-size calculator: control survival S at the end of follow-up and S^HR under proportional hazards, weighted by the arms’ shares.

Events with recruitment and follow-up periods

P(\lambda) = \frac{\lambda}{\lambda+\eta}\left[1 - \frac{e^{-(\lambda+\eta)F}\left(1-e^{-(\lambda+\eta)A}\right)}{(\lambda+\eta)A}\right], \qquad \lambda_0 = \frac{-\ln S}{A+F}, \quad D = N\left[\frac{P(\lambda_0)}{1+k}+\frac{k\,P(\mathrm{HR}\,\lambda_0)}{1+k}\right]

When the accrual period A and follow-up period F are given, participants enter uniformly over A and are followed for at least F, with exponential survival and losses at rate η. Control survival S is then read at A + F, and each arm’s event share is averaged over entry [2]. This is how the survival sample-size calculator converts events to participants, so the power at its answer reaches its target.

Power, inverting the sample-size formula

\text{Power}=\Phi\!\left(\frac{\frac{\sqrt{Dk}\,|1-\mathrm{HR}|}{1+k\,\mathrm{HR}}-z_{1-\alpha/\text{tails}}}{1}\right)+I_{\text{tails}=2}\,\Phi\!\left(\frac{-\frac{\sqrt{Dk}\,|1-\mathrm{HR}|}{1+k\,\mathrm{HR}}-z_{1-\alpha/\text{tails}}}{1}\right)

Freedman’s events formula, D = (z_α + z_β)²(1 + kHR)²/[k(1 − HR)²], solved for z_β [1].

Parameters

Number of events, or of participants $D$

Values: Integer, at least 2

Events expected across both arms; or, with control survival given, the participants, from which the events follow.

Control survival (optional) $S$ Optional

Values: Between 0 and 1

Give it to enter participants instead of events: the share of controls still event-free at the end of follow-up.

Accrual period (optional) $A$ Optional

Values: 0 or more, with the follow-up period

How long recruitment lasts; participants are assumed to enter evenly over it.

Follow-up period (optional) $F$ Optional

Values: 0 or more, with the accrual period

Follow-up after recruitment closes, so the last participant is followed for F and the first for A + F.

Loss to follow-up rate (optional) $\eta$ Optional

Values: 0 or more per unit of time

On the same time scale as the periods; it needs both of them.

Allocation ratio $r$ Optional

Values: Positive; 1 for equal groups

Treatment participants per control participant.

Hazard ratio $\mathrm{HR}$

Values: Positive, not 1

Treatment hazard divided by control hazard.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

Proportional hazards: the hazard ratio is constant over follow-up. Power depends on events, so too few events, not too few participants, is what makes a trial underpowered.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 250 events, 2:1 allocation, and HR = 0.70 at α = 0.05: √(250 × 2) × 0.3/(1 + 1.4) = 2.80.

Needed for analysis: 250 · To approach: 750

Open this example in the calculator

Validation

Power is the log-rank sample-size formula solved for z_β, with the far tail added for a two-sided test. At the size that calculator returns it reaches the target, one fewer falls short, and it agrees with an independent calculation in R.

How to cite

StatWisely. Achieved power for the log-rank test (Freedman). Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-logrank-freedman/

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

References

1.Freedman LS. Tables of the number of patients required in clinical trials using the logrank test. Statistics in Medicine. 1982;1(2):121–129.
2.Lachin JM, Foulkes MA. Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics. 1986;42(3):507–519.