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The survival function of a left-censored sample: lifelines' fit_left_censoring.

public static KaplanMeierCurve EstimateLeftCensored(ReadOnlySpan<double> durations, ReadOnlySpan<bool> eventObserved, double confidenceLevel = 0.95)

Parameters — durations holds one non-negative time per subject. eventObserved is true where that time is the event's own, and false where the event happened at some unknown time before it; the two spans must be the same length. confidenceLevel lies strictly inside (0, 1).

Returns — a KaplanMeierCurve on the steps Estimate would build: one minus the reverse Kaplan-Meier estimate of the cumulative density, with its bounds.

Exceptions — ArgumentException when the spans differ in length, the sample is empty, or a duration is negative or NaN. ArgumentOutOfRangeException when confidenceLevel is not strictly inside (0, 1).

Example — ten readings, three of them only known to lie below a detection limit.

using Lodestar.Survival;

double[] readings = [3, 5, 2, 8, 4, 6, 1, 7, 5, 9];
bool[] exact = [true, false, true, true, false, true, false, true, true, true];

KaplanMeierCurve curve = KaplanMeier.EstimateLeftCensored(readings, exact);

double atFive = Math.Round(curve.Survival[5], 6);   // => 0.4
double lower = Math.Round(curve.Lower[5], 6);       // => 0.172779
double upper = Math.Round(curve.Upper[5], 6);       // => 0.747331

Remarks — the time axis is read backwards. At each step the cumulative density is the product of 1 − d/n over the later times, n counting the subjects whose duration is at most that time, and its log-log interval takes Greenwood's sum over the same times. The survival is one minus that density. A subject enters n at its own time and stays for every later one, the mirror of a right-censored subject leaving the risk set.

The curve need not start at one. When the smallest duration is itself left-censored, some mass lies before every observed time and the density at zero is already positive: above, the reading censored at 1 puts the curve at 0.833 at time zero.

Lower is the smaller bound. The survival's bounds are one minus the density's, and lifelines names the larger of them its lower bound; the two columns hold the same numbers, swapped.

Applies to — net10.0, netstandard2.0.

See also — KaplanMeier, KaplanMeier.Estimate, ParametricSurvival.FitLeftCensored.