{
  "slug": "cumulative-hazard",
  "term": "Cumulative hazard",
  "aliases": [
    "Cumulative hazard function",
    "Integrated hazard"
  ],
  "definition": "Cumulative hazard is the time integral of a hazard rate for a specified event and clock.",
  "note": "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)). [nist-hazard]\n\nFor 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. [nist-hazard] [nist-survival]\n\nWith 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. [therneau-competing]\n\nFor 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. [survfit-estimators]",
  "status": "shipping",
  "advantages": [],
  "limitations": [],
  "related": [
    "hazard-rate",
    "survival-function",
    "cause-specific-hazard",
    "cumulative-incidence-function",
    "afr"
  ],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "citations": [
    {
      "id": "nist-hazard",
      "title": "NIST e-Handbook 8.1.2.3: Failure (or hazard) rate; integral and negative-log identities",
      "url": "https://www.itl.nist.gov/div898/handbook/apr/section1/apr123.htm",
      "accessed": "2026-11-05"
    },
    {
      "id": "nist-survival",
      "title": "NIST e-Handbook 8.1.2.2: Reliability or survival function",
      "url": "https://www.itl.nist.gov/div898/handbook/apr/section1/apr122.htm",
      "accessed": "2026-11-05"
    },
    {
      "id": "therneau-competing",
      "title": "Therneau, Crowson and Atkinson: Multi-state models and competing risks (12 January 2026), section 2.1, equation 3",
      "url": "https://stat.ethz.ch/R-manual/R-devel/library/survival/doc/compete.pdf",
      "accessed": "2026-11-05"
    },
    {
      "id": "survfit-estimators",
      "title": "R survival: survfit.formula documentation, stype/ctype and Details (Kaplan-Meier versus exponentiated cumulative hazard)",
      "url": "https://stat.ethz.ch/R-manual/R-devel/library/survival/html/survfit.formula.html",
      "accessed": "2026-11-05"
    }
  ],
  "updated": "2026-11-05"
}