Competing Risks and Cumulative Incidence

Theory: Competing Risks and Cumulative Incidence

Two constant cause-specific hazards lambda1,lambda2 define all-cause S(t)=exp(-(lambda1+lambda2)t). For positive total hazard compute CIF_j=lambda_j/(lambda1+lambda2)*(1-S), and compare CIF1 with naive1-exp(-lambda1*t). Show S+CIF1+CIF2=1 and the limiting cause1 incidence.

Worked Example

Consider lambda1 = 1, lambda2 = 1, and t = 1.

1. Calculate sum of hazards: sumLambda = 1 + 1 = 2
2. Calculate all-cause survival: S(1) = exp(-2 * 1) ≈ 0.135335
3. Calculate CIF1: CIF1 = (1 / 2) * (1 - exp(-2)) ≈ 0.432332
4. Calculate naive cause 1 failure: naiveFailure = 1 - exp(-1 * 1) ≈ 0.632121

Experiment

Observe how competing risks affect cumulative incidence in the application:

1. Set Cause 1 rate to 1 and Cause 2 rate to 0.
2. Observe that CIF1 and naive failure are identical because there is no competing risk.
3. Set Cause 2 rate to 1. Notice how CIF1 drops despite Cause 1 rate remaining unchanged.
4. Check the limiting incidence for Cause 1, showing it approaches lambda1 / (lambda1 + lambda2).

Limitations

This is a declared constant-hazard probability model, not fitting patient data, causal identifiability of latent failure times or a Fine–Gray regression.

References

The underlying theory is covered by the primary reference. Further details on how competing risks operate under constant hazards can be verified empirically in the experiment.

• Göran Broström, Event History Analysis with R2e, chapter 11