Normal Distribution
nothing is called 95% until it has been counted
Concept

Restricted mean survival time — where it appears

The average time a subject survives up to a stated horizon, equal to the area under the survival curve from zero to that horizon. It needs no proportional-hazards assumption, and a difference between two arms reads as time gained or lost before the horizon.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The ratio of the arms' hazards over time, when treatment halves every patient's hazard. Every patient's hazard ratio is 0.5. With two risk groups — half at 0.1 a year, half at 1 — the ratio of the arms' hazards rises from 0.5 to 0.929 at 3.5 years and falls back towards 0.5. With a gamma frailty of variance 1 it rises steadily, to 0.857 at ten years.

Every patient at one half

A randomised trial in which the treatment halves every patient's hazard, and patients differ only in their baseline risk. The unadjusted Cox hazard ratio is 0.524 after six months of follow-up, 0.635 after three years and 0.679 after seven, and the ratio of the arms' hazards at a single moment climbs from 0.50 to 0.929 and falls back. Nothing is confounded and nothing is wrong. Survival removes high-risk patients from the control arm faster, and a hazard ratio compares whoever is left.

reversal · Simpson Power of four tests of two survival curves, in three shapes of treatment effect, at 50, 100 and 200 a side. 6,000 trials at each setting, dropout 0.1 a year, follow-up to three years. hazard halved throughout, 50 a side: log-rank 64.9%, best of eleven horizons 51.7%, restricted mean at 3 61.1%, one test for all twelve 56.6%; hazard halved throughout, 100 a side: log-rank 91.8%, best of eleven horizons 81.0%, restricted mean at 3 88.4%, one test for all twelve 86.0%; hazard halved throughout, 200 a side: log-rank 99.7%, best of eleven horizons 98.4%, restricted mean at 3 99.4%, one test for all twelve 99.3%; halved for a year, then nothing, 50 a side: log-rank 18.5%, best of eleven horizons 32.1%, restricted mean at 3 26.6%, one test for all twelve 30.8%; halved for a year, then nothing, 100 a side: log-rank 29.3%, best of eleven horizons 51.7%, restricted mean at 3 42.5%, one test for all twelve 50.2%; halved for a year, then nothing, 200 a side: log-rank 51.4%, best of eleven horizons 81.4%, restricted mean at 3 70.3%, one test for all twelve 80.2%; curves that cross, 50 a side: log-rank 5.1%, best of eleven horizons 45.7%, restricted mean at 3 10.4%, one test for all twelve 44.0%; curves that cross, 100 a side: log-rank 5.2%, best of eleven horizons 76.7%, restricted mean at 3 16.4%, one test for all twelve 75.5%; curves that cross, 200 a side: log-rank 5.2%, best of eleven horizons 96.7%, restricted mean at 3 26.5%, one test for all twelve 96.6%.

One test for a curve that might cross

The log-rank statistic and the differences in restricted mean survival at eleven horizons are one Gaussian vector under no effect, and its correlations are closed — from −0.461 at half a year to −0.949 at three, where the log-rank statistic is nearly the restricted mean itself. So one critical value, 2.374, prices the largest of all twelve. At 100 a side it keeps 93.6% of the log-rank test's power when the hazard is halved throughout and 98.5% of the best horizon's when the curves cross, where the log-rank test has none. Running both tests and keeping either rejection rejects 8.67% of trials with no effect.

survival · Censoring

Named alongside it

The objects these essays reach for when they reach for this one.

Closed formCox modelCritical valueFrailtyGaussian processHazard ratioKaplan–MeierLog-rank testMonte CarloMultiple testingNon-collapsibilityProportional hazards

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