hesela.devCumulative hazard
Cumulative hazard is the time integral of a hazard rate for a specified event and clock.
For an absolutely continuous event time with S(0) = 1 and S(t) > 0, H(t) = integral from 0 to t of h(v) dv = -ln S(t). It is dimensionless, nondecreasing and can exceed one; it is not a failure probability. F(t) = 1 - exp(-H(t)). [1]
For a surviving drive at age a, an additional u units of survival has probability exp(-(H(a + u) - H(a))) = S(a + u) / S(a), for u >= 0 and positive survival at both endpoints. Keep the population, event and time origin fixed. This identity does not require constant hazard. [1] [2]
With competing first events, H refers to the all-cause hazard when S means freedom from every event. A cause-specific cumulative hazard H_k(t) = integral from 0 to t of h_k(v) dv is not generally -ln(1 - CIF_k(t)): the cumulative incidence depends on all competing hazards through S. These are probability-model identities, not a fitted drive model or a rule equating finite-sample estimators. [3]
For ordinary right-censored samples, the Nelson-Aalen estimate sums event-count/risk-set fractions, while Kaplan-Meier multiplies their survival complements. Thus exponentiating minus the former need not reproduce the latter. Record which estimator and tie convention produced each reported curve; the continuous-model identity above does not require those estimates to coincide. [4]
Relations
- related
- Hazard rate, Survival function, Cause-specific hazard, Cumulative incidence function, Annualized failure rate (AFR)
Citations
- NIST e-Handbook 8.1.2.3: Failure (or hazard) rate; integral and negative-log identities
- NIST e-Handbook 8.1.2.2: Reliability or survival function
- Therneau, Crowson and Atkinson: Multi-state models and competing risks (12 January 2026), section 2.1, equation 3
- R survival: survfit.formula documentation, stype/ctype and Details (Kaplan-Meier versus exponentiated cumulative hazard)