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NelsonAalen

The Nelson-Aalen estimator of a cumulative hazard function.

public static class NelsonAalen

Example — the same trial arm, read as accumulated risk rather than as survival.

using Lodestar.Survival;

double[] durations = [6, 6, 6, 7, 10, 13, 16, 22, 23, 6, 9, 10, 11, 17, 19, 20, 25, 32, 32, 34, 35];
bool[] observed = [true, true, true, true, true, true, true, true, true,
                   false, false, false, false, false, false, false, false, false, false, false, false];

NelsonAalenCurve curve = NelsonAalen.Estimate(durations, observed);

double bySix = curve.CumulativeHazard[1];  // => 0.150250…
double bySeven = curve.CumulativeHazard[2];  // => 0.209074…

Remarks — a sum of hazard increments rather than a product of survival fractions. The two estimators share a timeline because they share a risk table, and they are two readings of it.

The difference shows at the end. In a sample whose last duration is observed, Kaplan-Meier reaches zero and can fall no further, while the hazard keeps the size of that last step. A curve that has hit zero has lost the ability to distinguish "the study ended" from "everyone died"; the hazard has not.

Reference behaviour is lifelines.NelsonAalenFitter 0.30.3 at its default nelson_aalen_smoothing=True, whose tie increment is Σ 1/(n − i) rather than d/n. Matched over 9 samples.

Applies to — net10.0, netstandard2.0.

See also — the estimators index, KaplanMeier.

Members

Member	What it does
NelsonAalen.Estimate	The cumulative hazard of a right-censored sample.