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Nelson - Aalen estimator
Definition
The Nelson - Aalen estimator estimates a cumulative hazard function from time-to-event data. At each observed event time, it adds the number of events divided by the number still at risk immediately before that time. It is nonparametric: the basic calculation does not impose a particular lifetime distribution such as an exponential or Weibull distribution.
What it means
Cumulative hazard accumulates event-rate contributions over time. It is not a probability and can exceed one. A right-censored observation contributes information up to its last observed event-free time; its later event time is unknown. Treating that observation as a failure, or deleting it entirely, changes the information used in the estimate.
How it works
For event times t-j, count events d-j and the corresponding risk-set sizes n-j. Add d-j/n-j over event times up to the requested time. Censoring reduces later risk sets without creating an event increment. The elementary interpretation assumes independent units and appropriate independent censoring. In the continuous-time survival framework, exponentiating the negative cumulative hazard gives a related survival estimate, distinct from Kaplan - Meier.
Three practical examples
- A reliability team studies time until an AI monitoring device first fails. Devices still working when observation ends are right-censored, so their known operating time remains useful.
- A service team examines time until a ticket is resolved. Unresolved tickets at the reporting date are censored. Resolution is the event, illustrating that a survival-analysis event need not be harmful.
- An evaluation team studies time until a deployed system first breaches a defined performance threshold. It specifies the threshold, monitoring schedule, and observation end date before interpreting the estimated curve.
Think of it this way
Imagine keeping a running total of failure contributions. One failure among many devices adds less than one failure among only a few remaining devices. A device leaving observation while still working changes who can contribute later, but it does not count as a failure at departure.
Worked example
Consider an original study of six independent devices, all entering observation at time zero. Device A fails at day 2. Device B is censored at day 3. Devices C and D both fail at day 4. Device E is censored at day 5. Device F fails at day 6. Assume the censoring mechanism is appropriate for the simple analysis. There are no event-censoring ties in this example.
Immediately before day 2, all six devices are at risk. One failure adds 1/6, so the cumulative hazard estimate becomes approximately 0.1667. Device A then leaves the risk set. At day 3, Device B leaves observation without a recorded failure. The estimate remains 1/6: censoring creates no upward jump.
Immediately before day 4, four devices remain at risk: C, D, E, and F. The two failures contribute 2/4 = 1/2. The cumulative estimate becomes 1/6 + 1/2 = 2/3, approximately 0.6667. Both failed devices leave the risk set. Device E is then censored at day 5, again without changing the cumulative estimate.
Immediately before day 6, only F remains at risk. Its failure contributes 1/1 = 1. The cumulative estimate becomes 2/3 + 1 = 5/3, approximately 1.6667. This is a valid cumulative-hazard value, not a claim that 166.7% of devices failed. The large final jump also makes the small remaining risk set conspicuous.
At day 4, exponentiating the negative estimate gives exp(-2/3), approximately 0.5134. Kaplan - Meier instead multiplies the retained fractions: (1-1/6)(1-2/4) = (5/6)(1/2) = 5/12, approximately 0.4167. These are distinct estimators. Their difference is visible here because the event fractions are large. For small event fractions, exp(-u) and 1-u are closer.
At day 6, Kaplan - Meier acquires a factor of zero because the final at-risk device fails. Its estimate becomes zero, whereas exp(-5/3) is approximately 0.1889. This endpoint contrast should not be hidden by calling the two methods identical or by interpreting the transformed estimate as evidence of observed devices surviving beyond day 6.
Now consider incorrectly deleting B and E because their lifetimes are incomplete. Only four records remain. The day-2 increment becomes 1/4, and the day-4 increment becomes 2/3, producing 11/12 by day 4 instead of 2/3. Deletion discards their genuine contributions to the original risk sets.
The two day-4 events also require a stated tie convention. The grouped increment used here is 2/4. A sequential tie correction would use 1/4 + 1/3 = 7/12 instead. Such corrections explain some software differences; they should not be silently mixed within a calculation.
Illustrative case study
This fictional case concerns Cedar Devices, an imaginary monitoring-equipment supplier. Its analyst initially imports only failed-device records. A reviewer compares the table with the deployment register and restores devices that were still working when their observation ended.
The revised analysis displays the at-risk counts beside the cumulative-hazard curve and labels the tie convention. The team also investigates whether devices removed from observation were unusually unreliable. If removal depends on developing failure symptoms, ordinary independent-censoring reasoning may be inappropriate. This case illustrates data preparation and interpretation; it reports no measured reliability improvement.
Common misconceptions and mistakes
A flat estimated segment means no recorded events occurred there, not that the true event rate must be zero. A cumulative curve is also different from a smoothed instantaneous hazard curve.
A device's observation ending is not automatically independent censoring. Reasons for ending follow-up matter. Competing events, delayed entry, and repeated events require explicitly appropriate risk-set definitions and methods; the elementary first-event example does not address them automatically.
Business use
Use the curve to summarize event-time evidence together with follow-up coverage. Operational reports should define the event, time origin, censoring reasons, and tie handling. When few units remain, a late jump can depend on very little information. Comparing two curves is not, by itself, a causal assessment of competing products or maintenance policies.
People also ask
Can the estimate decrease?
The basic estimator cannot: its event increments are nonnegative.
Is it a failure percentage?
No. The worked example reaches 5/3, while a probability must remain between zero and one.
Check your understanding
Why is the day-4 risk set four?
Answer: A failed earlier and B was censored earlier.
Why does censoring E create no jump?
Answer: No event was observed for E, although its departure changes the next denominator.
Related terms
- Risk set
- Survival function
- Cumulative hazard
- right censoring
- Kaplan–Meier estimator
- competing risks
- delayed entry
- event-time analysis
Learn next
Rebuild the six-device risk table, then compare the cumulative sum with the Kaplan - Meier product. Record exactly how censored observations and tied events enter each calculation.
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- Negatively skewed - A negatively skewed distribution is asymmetric with a tail extending farther toward lower values than toward higher values. It is also called left-skewed.
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Sources
Regression with R, Survival analysis, definitions 11.1–11.2 and censoring assumptions: https://rwr.nrhstat.org/11_survival.html . scikit-survival, official Nelson–Aalen estimator documentation: https://scikit-survival.readthedocs.io/en/v0.28.0/api/generated/sksurv.nonparametric.nelson_aalen_estimator.html . All device records and the Cedar Devices case are original illustrations.
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