Nelson-Aalen Cumulative Hazard Estimator

Use this estimator when you want to understand how risk accumulates over time rather than how survival probability decreases. It is also used as a foundation for smoothed hazard rate estimation and as an input to other survival models.

• Censoring is non-informative.
• Events are independent across subjects.

• The cumulative hazard H(t) represents the total accumulated risk up to time t. It starts at 0 and increases monotonically.
• A steep segment in the cumulative hazard curve indicates a period of high event rate. A flat segment indicates a period of low risk.
• The Nelson-Aalen estimator and the Kaplan-Meier estimator are closely related: S(t) is approximately exp(-H(t)). For small hazards, they give nearly identical results.
• The smoothed hazard estimate reveals how the instantaneous risk changes over time, which the cumulative hazard and survival curves obscure.

• The cumulative hazard can exceed 1.0 (unlike survival probability), which sometimes confuses researchers unfamiliar with hazard functions.
• Smoothed hazard estimates depend on the choice of bandwidth. Too narrow a bandwidth produces noisy estimates; too wide smooths over real features.
• With few events, the cumulative hazard estimate has wide confidence intervals, especially at later time points when few subjects remain at risk.

1. At each event time, estimate the instantaneous hazard as the number of events divided by the number at risk: d_i / n_i.
2. Cumulate these hazard increments: H(t) = sum of d_i / n_i for all event times up to t.
3. Compute variance using Greenwood-type formulas and construct confidence intervals.
4. Optionally smooth the hazard increments using a kernel smoother to estimate the instantaneous hazard rate function.