Kaplan–Meier and Nelson–Aalen Estimation › Summary

Kaplan-Meier and Nelson-Aalen are the two foundational nonparametric estimators in survival analysis. Both are built from risk sets. At each event time $t_j$, $n_j$ subjects are at risk and $d_j$ events occur. Kaplan-Meier multiplies conditional survival factors $1-d_j/n_j$ to estimate $S(t)$. Nelson-Aalen adds hazard increments $d_j/n_j$ to estimate $H(t)$.

The estimators are closely related but not identical. Kaplan-Meier is the product-limit estimator, and Nelson-Aalen is the additive cumulative-hazard estimator. The exponential transform $\exp(-\hat H_{\mathrm{NA}})$ approximates Kaplan–Meier over a time range where event fractions and their accumulated second-order error are small, but it is not the same estimator. With ties or small risk sets, the difference can be material.

We used Greenwood’s formula for Kaplan-Meier and a Greenwood-type binomial plug-in variance for Nelson-Aalen, alongside the alternative Aalen variance. Intervals based on them are approximate and pointwise; transformed limits are undefined at the boundary values discussed above. The median is the first time the survival curve reaches or falls below $0.5$; if the curve stays above $0.5$ throughout follow-up, the median is not estimable. Tail estimates are limited by the number at risk, the amount of censoring, and the last follow-up time. Independent censoring and adequate follow-up support are key conditions for consistent estimation under the sampling assumptions; finite-sample bias can remain. Estimating cumulative incidence in the presence of competing events requires an appropriate competing-risk method, and delayed entry requires risk sets that account for entry times.