hesela.devNelson-Aalen estimator
The Nelson-Aalen estimator estimates cumulative hazard by summing observed event counts divided by their corresponding risk-set sizes.
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. [1] [2]
This 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. [2]
Two 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. [2]
Tied 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. [1]
If 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. [2]
Population interpretation requires appropriate censoring assumptions. Changing the estimator alone does not correct selective withdrawal of drives following warning telemetry. [3]
Relations
- related
- Cumulative hazard, Kaplan-Meier estimator, Right-censoring, Informative censoring, Interval-censoring, Hazard rate
Citations
- R survival: survfit.formula, stype/ctype and separate survival/cumulative-hazard estimates
- R survival: survfitkm.c at 0605821347ef, Nelson-Aalen addition and Kaplan-Meier multiplication
- Overgaard and Hansen (2019): On the assumption of independent right censoring, sections 1-2
Sources & caveats
- 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.