Competing Risks Sample Size Calculator

Methodology

This calculator implements the competing-risks sample-size approach described by Melania Pintilie for testing a covariate effect on a cause-specific hazard when another event can pre-empt the event of interest. Pintilie's paper provides a sample-size method for competing-risk data; the same family of calculations is used by commercial sample-size procedures for competing risks. The implementation here follows the exponential, independent failure-time, uniform-accrual formulation documented in the statistical literature for its corresponding log-rank competing-risks procedure and is consistent with the STT4 competing-risks procedure in.

1. Convert cumulative incidences to cause-specific hazards

At the fixed time point T₀, let F_ev,1 be the control cumulative incidence for the event of interest and F_cr,1 the control cumulative incidence for the competing event. Under independent exponential event times, the total hazard is first obtained from the probability of experiencing either event:

The treatment event-of-interest hazard is then h_ev,2 = HR × h_ev,1. The competing-event hazard is held constant between groups, h_cr,2 = h_cr,1. Consequently, the treatment cumulative incidences are implied by the specified hazard ratio rather than entered independently.

2. Account for uniform accrual

Because patients enter throughout the accrual period, not everyone receives the full planned follow-up. For a subject whose total hazard is H = h_ev + h_cr, the event-of-interest probability averaged over uniform entry during accrual R is:

The overall event probability is the allocation-weighted average of the two group-specific probabilities. This is the quantity used to convert the required number of events into the number of participants.

3. Required number of events

For a two-sided log-rank test, the required number of events of interest is obtained from the Schoenfeld/Pintilie large-sample expression:

Here p is the proportion assigned to the control group and 1−β is the target power. The required total enrollment before loss to follow-up is E / Pr(ev).

4. Loss to follow-up and integer sample size

If W is the anticipated proportion lost to follow-up, the enrollment requirement is inflated by 1/(1−W). The calculator then searches upward from the continuous sample-size requirement and returns the smallest integer N whose achieved power is at least the requested target, while preserving the requested allocation as closely as possible.

Assumptions

• The event-of-interest and competing-event failure times are independent.
• Both failure-time distributions are exponential, implying constant cause-specific hazards.
• Subjects enter uniformly during the accrual period.
• The competing-event cause-specific hazard is assumed to be the same in both groups.
• The specified hazard ratio applies to the cause-specific hazard of the event of interest.
• Loss to follow-up is handled as a simple proportional inflation of enrollment.
• The calculation is a large-sample approximation and is intended for planning rather than replacing simulation when more complex accrual, censoring, non-proportional hazards, or treatment effects on the competing event are expected.

Worked Validation Example

The implementation was checked against Example 2 in the the software competing-risks documentation. Using 90% power, two-sided α = 0.05, equal allocation, 10% loss to follow-up, 4 years of accrual, 3 years of additional follow-up, T₀ = 3 years, control event CIF = 0.10, control competing-risk CIF = 0.65, and HR = 0.40, the published calculation gives a total sample size of 662 (331 per group). The corresponding required number of events is approximately 50.1. This calculator reproduces that result.

References

Pintilie, M. (2002). “Dealing with competing risks: testing covariates and calculating sample size.” Statistics in Medicine, 21(22), 3317–3324. DOI: 10.1002/sim.1271.

Pintilie, M. (2006). Competing Risks: A Practical Perspective. John Wiley & Sons, Chichester.

Machin, D., Campbell, M. J., Tan, S. B., & Tan, S. H. (2009). Sample Size Tables for Clinical Studies, 3rd ed. Wiley-Blackwell.

the software, LLC. this method Sample Size Software: Logrank Tests Accounting for Competing Risks, Chapter 716. The documented procedure states that its calculations are based on Pintilie (2006) and Machin et al. (2009), themselves based on Pintilie (2002), and assumes independent exponentially distributed event and competing-risk failure times with uniform accrual.

. Advanced User Manual, procedure STT4, “Two Sample Test of Survival Curves with Competing Risks.” The calculator is intended to reproduce the corresponding fixed-design competing-risks sample-size calculation in a native browser implementation.