{
  "slug": "aalen-johansen-estimator",
  "term": "Aalen-Johansen estimator",
  "aliases": [
    "AJ estimator"
  ],
  "definition": "The Aalen-Johansen estimator combines observed transitions and risk sets to estimate state probabilities, including competing-risk cumulative incidence.",
  "note": "Method: An ordered product of event-time transition matrices updates a stated starting distribution across states. With a single terminal event the estimate reduces to Kaplan-Meier; with competing first-event types it gives cumulative incidence for each type. [survival-aj]\n\nFirst-event calculation: At an event time t, the incidence increment for type k is S(t-) * d_k(t) / Y(t). Here S(t-) is the preceding event-free probability, d_k counts type-k events and Y is the observed event-free risk set immediately before t. All event types reduce S. Right-censoring changes later risk sets without adding event probability. Tied event types share the pre-event risk set. [survival-aj] [etm-aj]\n\nObservation: Entry and exit times can represent left-truncation. Population interpretation requires appropriate observation assumptions; informative follow-up is not automatically repaired. State occupation and general conditional transition probabilities are different targets with different model requirements. [etm-aj] Exact-event-time calculations do not turn an interval-censored inspection record into an observed onset time. [survival-aj]\n\nModel scope: Consistent estimation of marginal state-occupation probabilities need not require a Markov process when censoring and left-truncation are unrelated to that process and the stated regularity conditions hold. General conditional transition probabilities are a different target; landmarking conditions on membership in a specified state at a specified time. Neither result reconstructs unknown retirement reasons. [niessl-aj]\n\nStarting probabilities: A consistent initial distribution is also required. With left-truncation, justify a common initial state or external evidence; enrollment composition need not match installation. [niessl-aj]",
  "related": [
    "cumulative-incidence-function",
    "competing-risks",
    "kaplan-meier-estimator",
    "left-truncation",
    "interval-censoring",
    "right-censoring"
  ],
  "citations": [
    {
      "id": "survival-aj",
      "title": "R survival 3.8-6: survfit.formula, Details and multi-state construction",
      "url": "https://stat.ethz.ch/R-manual/R-patched/library/survival/html/survfit.formula.html",
      "accessed": "2026-11-05"
    },
    {
      "id": "etm-aj",
      "title": "R etm 1.1.1: empirical transition matrix, entry/exit data and model assumptions",
      "url": "https://search.r-project.org/CRAN/refmans/etm/html/etm.html",
      "accessed": "2026-11-05"
    },
    {
      "id": "niessl-aj",
      "title": "Niessl et al. (2020): State occupation and transition probabilities in non-Markov models, sections 1-2.1, arXiv:2004.06514v1",
      "url": "https://arxiv.org/html/2004.06514v1",
      "accessed": "2026-11-05"
    }
  ],
  "status": "shipping",
  "advantages": [],
  "limitations": [],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "updated": "2026-11-05",
  "provenance": [
    "Primary implementation documentation supports the estimator and observation scope. Failure before permanent retirement is a storage application, not a fitted fleet result.",
    "The first-event recurrence specializes the documented matrix construction. It is not a general recurrent-event fitting procedure or a claim that competing latent times are independent."
  ]
}