{
  "slug": "nelson-aalen-estimator",
  "term": "Nelson-Aalen estimator",
  "aliases": [
    "Nelson-Aalen cumulative hazard estimator",
    "NA estimator"
  ],
  "definition": "The Nelson-Aalen estimator estimates cumulative hazard by summing observed event counts divided by their corresponding risk-set sizes.",
  "note": "For an unweighted right-censored sample, H_NA(t) sums d_i/n_i over event times up to t, where d_i counts events and n_i is the risk set immediately before that time. Censoring removes units from later risk sets without adding an event increment. Specify the event, clock and same-time convention. [survfit-na] [survfit-code]\n\nThis dimensionless estimate can exceed one. It is neither a failure probability nor an annualized event rate. Kaplan-Meier multiplies 1 - d_i/n_i instead; exp(-H_NA) is a different finite-sample survival estimate. [survfit-code]\n\nTwo hypothetical drives are both observed before the first failure and fail at distinct times, with no censoring: H_NA = 1/2 + 1 = 1.5 and S_KM = 0, yet exp(-1.5) is about 0.22313. The positive transformed estimate does not mean any observed drive survived. This is arithmetic, not fleet evidence. [survfit-code]\n\nTied events: R survival distinguishes ordinary Nelson-Aalen (ctype=1) from the Fleming-Harrington tie correction (ctype=2), separately from its survival-estimator choice stype. Record both settings. These options do not convert interval-censored onsets into exact event times. [survfit-na]\n\nIf the same two hypothetical failures share one recorded event time, ordinary Nelson-Aalen adds d/n = 2/2 = 1, while the Fleming-Harrington tie correction adds 1/2 + 1 = 1.5. Kaplan-Meier still reaches zero; the two exponentiated estimates (stype=2) are about 0.36788 and 0.22313. These are estimator conventions for the recorded tie, not evidence about the order of the actual failures. [survfit-code]\n\nPopulation interpretation requires appropriate censoring assumptions. Changing the estimator alone does not correct selective withdrawal of drives following warning telemetry. [censoring-assumptions]",
  "related": [
    "cumulative-hazard",
    "kaplan-meier-estimator",
    "right-censoring",
    "informative-censoring",
    "interval-censoring",
    "hazard-rate"
  ],
  "citations": [
    {
      "id": "survfit-na",
      "title": "R survival: survfit.formula, stype/ctype and separate survival/cumulative-hazard estimates",
      "url": "https://stat.ethz.ch/R-manual/R-devel/library/survival/html/survfit.formula.html",
      "accessed": "2026-11-08"
    },
    {
      "id": "survfit-code",
      "title": "R survival: survfitkm.c at 0605821347ef, Nelson-Aalen addition and Kaplan-Meier multiplication",
      "url": "https://github.com/therneau/survival/blob/0605821347efb8630d15710a6bf24ff64396c4d3/src/survfitkm.c",
      "accessed": "2026-11-08"
    },
    {
      "id": "censoring-assumptions",
      "title": "Overgaard and Hansen (2019): On the assumption of independent right censoring, sections 1-2",
      "url": "https://arxiv.org/html/1905.02508v1",
      "accessed": "2026-11-08"
    }
  ],
  "status": "shipping",
  "advantages": [],
  "limitations": [],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "updated": "2026-11-08",
  "provenance": [
    "The two-device example applies the documented formulas to invented observations. It is not a measured fleet result, a confidence bound or a native R execution.",
    "The implementation citation is pinned to a source commit; the R-devel documentation URL may change. The note describes the ordinary unweighted right-censored case."
  ]
}