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KaplanMeier.RestrictedMean

The restricted mean survival time of a curve, with its variance: lifelines' restricted_mean_survival_time.

public static RestrictedMeanResult RestrictedMean(KaplanMeierCurve curve)

public static RestrictedMeanResult RestrictedMean(KaplanMeierCurve curve, double horizon)

The first overload integrates without a limit, lifelines' default t=inf.

Parameters — curve is a curve from KaplanMeier.Estimate. horizon is the upper limit of the integral, lifelines' t: non-negative, and infinity allowed.

Returns — a RestrictedMeanResult: the area under the curve up to horizon, and its variance.

Exceptions — ArgumentNullException when curve is null, and ArgumentException when it holds no step or not one estimate per step. ArgumentOutOfRangeException when horizon is negative or NaN.

Example — Freireich's control arm, up to week 10 and without a limit.

using Lodestar.Survival;

double[] control = [1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23];
bool[] controlObserved = [true, true, true, true, true, true, true, true, true, true, true,
                          true, true, true, true, true, true, true, true, true, true];
KaplanMeierCurve curve = KaplanMeier.Estimate(control, controlObserved);

RestrictedMeanResult tenWeeks = KaplanMeier.RestrictedMean(curve, 10);
RestrictedMeanResult whole = KaplanMeier.RestrictedMean(curve);

double mean = Math.Round(tenWeeks.Mean, 6);           // => 6.619048
double variance = Math.Round(tenWeeks.Variance, 6);   // => 11.283447
double unrestricted = Math.Round(whole.Mean, 6);      // => 8.666667
double spread = Math.Round(whole.Variance, 6);        // => 39.84127

Remarks — the mean is ∫ S up to the horizon and the variance 2 ∫ τ S(τ) dτ less the squared mean, as lifelines defines them, both summed exactly over the steps.

A measured divergence in lifelines' variance. lifelines sums the mean the same way, but integrates the second moment by scipy.integrate.quad over the step function, which lands up to 1.5 % relative from the exact integral over 300 random curves: on the whole control arm above it reports 39.786 where the area is 39.841. This method returns the exact integral, the corpus freezes both, and the mean matches lifelines to the last digit.

A curve still above zero at its last step never closes its area, so an infinite horizon gives an infinite mean and variance unless the curve reaches zero, as this one does.

Applies to — net10.0, netstandard2.0.

See also — RestrictedMeanResult, KaplanMeier.CompareAt.