Kaplan–Meier and Nelson–Aalen Estimation › Advantages

Kaplan-Meier and Nelson-Aalen make no parametric assumption about the shape of the survival distribution. We do not need to choose an exponential, Weibull, or log-normal model before looking at the data. The estimated curves follow the observed event times, so they are excellent first descriptions of a time-to-event sample. This makes them useful for exploratory analysis, for checking whether a parametric model is plausible, and for communicating results to audiences who want an empirical survival curve rather than a regression coefficient.

Both estimators handle right censoring in a principled way. A censored subject is not treated as if they survived forever, and they are not dropped from the analysis at time zero. Instead, they remain in the risk set until their censoring time and then leave. Kaplan-Meier curves are a standard way to summarize right-censored follow-up in clinical trials. The same risk-set construction can describe equipment lifetimes or time to customer churn under the stated censoring and sampling assumptions.

The estimators are also simple and fast. A production implementation sorts the observed times and computes risk-set counts and event counts, which is $O(n\log n)$ in the sample size. The output is easy to interpret: a step function for survival, a step function for cumulative hazard, pointwise confidence intervals, and medians or quantiles when the curve reaches them. The Breslow baseline cumulative-hazard estimator in a Cox model generalizes the Nelson-Aalen construction by weighting each risk-set member by its fitted relative risk; it reduces to Nelson-Aalen when all relative risks are one. Cumulative-hazard estimates also enter counting-process diagnostics, such as Cox martingale residuals, through the fitted cumulative hazard.