hesela.devAalen-Johansen estimator
The Aalen-Johansen estimator combines observed transitions and risk sets to estimate state probabilities, including competing-risk cumulative incidence.
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. [1]
First-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. [1] [2]
Observation: 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. [2] Exact-event-time calculations do not turn an interval-censored inspection record into an observed onset time. [1]
Model 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. [3]
Starting 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. [3]
Relations
- related
- Cumulative incidence function, Competing risks, Kaplan-Meier estimator, Left-truncation, Interval-censoring, Right-censoring
Citations
- R survival 3.8-6: survfit.formula, Details and multi-state construction
- R etm 1.1.1: empirical transition matrix, entry/exit data and model assumptions
- Niessl et al. (2020): State occupation and transition probabilities in non-Markov models, sections 1-2.1, arXiv:2004.06514v1
Sources & caveats
- 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.