Survival Sample Size with Loss to Follow-Up

Study Design

Specify the expected survival, accrual, follow-up, and loss-to-follow-up assumptions.

α — significance level

Power

Test

Exponential survival assumptions

Reference time

Control survival

Treatment survival

Enter the expected proportion surviving at the reference time. Under the exponential model, λ = −ln(S)/t.

Accrual and follow-up

Accrual period

Maximum follow-up

The maximum follow-up is the total calendar length of the study. Patients entering at the end of accrual receive shorter follow-up.

Loss to follow-up

Loss by horizon (%)

Loss horizon

A 5% loss by 6 months corresponds to an exponential dropout hazard of −ln(0.95)/6.

Required Sample Size

STT2-style exponential survival calculation with accrual and exponential loss to follow-up.

Enter the study assumptions and click Calculate Sample Size.

Methodology

This calculator implements the two-group exponential-survival sample-size method corresponding to STT2 procedure: Test based on exponential survival, accrual period, dropouts. The method assumes exponential event times, uniform accrual over the specified accrual period, and exponential loss to follow-up with a common dropout hazard.

Convert survival proportions to hazard rates

For an exponential survival distribution, an expected survival probability S(t) at time t corresponds to the constant hazard rate

λ = −ln[S(t)] / t

The treatment-to-control hazard ratio is then HR = λ_E / λ_C.

Exponential loss to follow-up

If a proportion W is expected to be lost by a specified loss horizon L, the common exponential dropout hazard is

d = −ln(1 − W) / L

Expected event probability during the study

For group i, with event hazard λ_i, dropout hazard d, accrual period T₀, and total study follow-up T, the expected proportion experiencing the event is

E(P_i) = [λ_i / (λ_i + d)] × [1 − {e^{−(λ_i+d)(T−T₀)} − e^−(λ_i+d)T \over (λ_i+d)T₀}]

This averages the event probability over uniformly distributed entry times while accounting for the competing possibility of loss to follow-up.

Sample-size formula

For equal allocation, the required number in each group is calculated as

n = [ (z_1−α/s + z_1−&beta) / {ln(λ_E) − ln(λ_C)} ]² × [1/E(P_E) + 1/E(P_C)]

Here s = 2 for a two-sided test and s = 1 for a one-sided test. The reported enrollment target is rounded up to the next whole patient per group.

Approximate number of required events

The corresponding approximate total number of events is

E = 4(z_1−α/s + z_1−&beta)^{2>
/
[ln(HR)]²}

The event requirement depends on the significance level, power, and hazard ratio; accrual and loss to follow-up determine how many patients are needed to generate that event information.

Interpretation

Longer accrual generally increases the required enrollment because later-enrolled participants have less observation time.
Loss to follow-up reduces the expected event contribution from each enrolled participant and therefore increases the required enrollment.
The method assumes exponential survival and exponential, independent loss to follow-up. More complex survival, accrual, or dropout patterns require simulation or a more general survival-design method.

References

Lakatos, E. & Lan, K.K.G. (1992). A comparison of sample size methods for the logrank statistic. Statistics in Medicine, 11, 179–191.

Elashoff, J.D. (2007). Advisor Version 7.0 User's Guide. Statistical Solutions Ltd. The STT2 procedure is documented as the exponential-survival method allowing an accrual period and exponential dropouts.

Rubinstein, L.V., Gail, M.H., & Santner, T.J. (1981). Planning the duration of a comparative clinical trial with loss to follow-up and a period of continued observation. Journal of Chronic Diseases, 34(9–10), 469–479. doi:10.1016/0021-9681(81)90007-2.

Schoenfeld, D.A. (1983). Sample-size formula for the proportional-hazards regression model. Biometrics, 39(2), 499–503. doi:10.2307/2531021.