In survival analysis, which statement describes the Nelson-Aalen cumulative hazard function?

• A. The cumulative hazard function (i.e., the accumulated hazard over time).
• B. The hazard at a specific time t.
• C. The survival probability at time t.
• D. The probability density of event times.

Answer: (A)

The Nelson-Aalen cumulative hazard function is a nonparametric estimate of the cumulative hazard H(t), which represents the accumulated risk of the event occurring up to time t. It sums the observed contributions to hazard at each event time, capturing how much hazard has accumulated as time progresses.

Specifically, the estimator adds, for each distinct event time t_i, the amount d_i divided by the number at risk Y_i just before t_i. That is, Ĥ(t) = Σ_{t_i ≤ t} (d_i / Y_i). This yields a nondecreasing function that starts at zero and grows as time advances.

This is different from the hazard at a single time point h(t), which is the instantaneous risk, or from the survival probability S(t), which relates to the hazard through S(t) = exp(-∫_0^t h(u) du) in continuous time. In practice, the Nelson-Aalen estimator provides Ĥ(t), and survival can be approximated by exp(-Ĥ(t)) or estimated directly with methods like Kaplan-Meier.