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The survival function of a right-censored sample, whose subjects may enter late.

public static SurvivalCurve Estimate(ReadOnlySpan<double> durations, ReadOnlySpan<bool> eventObserved, double confidenceLevel = 0.95)

public static SurvivalCurve Estimate(ReadOnlySpan<double> durations, ReadOnlySpan<bool> eventObserved, ReadOnlySpan<double> entries, double confidenceLevel = 0.95)

The second overload takes lifelines' entry.

Parameters — durations holds one non-negative time per subject, and eventObserved is true where it ends in the event, false where the subject was censored at it. entries holds each subject's entry time, non-negative and at most its duration, or is empty for everyone entering at zero. confidenceLevel lies strictly inside (0, 1).

Returns — a SurvivalCurve at time zero, every duration and every entry time, with its bounds.

Exceptions — ArgumentException when the spans differ in length, the sample is empty, a duration is negative or NaN, or the entries are neither empty nor one valid time per subject. ArgumentOutOfRangeException when confidenceLevel is not strictly inside (0, 1).

Example — the first event, one in ten: exp(−1/10), and its interval.

using Lodestar.Survival;

double[] months = [5, 8, 12, 3, 15, 9, 20, 6, 11, 14];
bool[] died = [true, true, false, true, true, true, false, true, true, false];

SurvivalCurve curve = BreslowFlemingHarrington.Estimate(months, died);

double atThree = Math.Round(curve.Survival[1], 6);   // => 0.904837
double lower = Math.Round(curve.Lower[1], 6);        // => 0.49169
double upper = Math.Round(curve.Upper[1], 6);        // => 0.986012

Remarks — a step with d events among n adds 1/n + … + 1/(n − d + 1) to the hazard, and the squares of those to its variance, as lifelines' Nelson-Aalen fitter does with ties. The bounds are exp(−H e^{±z√V/H}), the log transform of the hazard's interval carried through exp(−·).

Lower is the smaller bound. lifelines labels the larger one its lower bound, being the image of the hazard's lower bound; the two columns hold the same numbers, swapped.

Late entrants join the risk set after the events at their entry time, as lifelines counts them, except at the first step; SurvivalStep.AtRisk reports them with the entrants, lifelines' at_risk. Each entry time is a step of its own even where nothing happens at it.

Applies to — net10.0, netstandard2.0.

See also — BreslowFlemingHarrington, NelsonAalen.Estimate, KaplanMeier.Estimate.