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.

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).