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

Log-rank test — where it appears

The standard test that two survival curves are the same, comparing at each event time the events observed in one group with the number a common hazard would predict. It is most powerful against a constant hazard ratio, and it can see nothing when an early benefit and a late harm cancel.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Two worlds, one Kaplan–Meier curve, two truths. World A gives each subject a frailty with mean one and variance 1, and multiplies both its event hazard (0.35) and its dropout hazard (0.5) by it, so the subjects likeliest to leave are the ones likeliest to fail. World B has independent event and dropout times whose hazards are world A's crude hazards. Kaplan–Meier over 1000 studies of 400 gives the same curve from both — 0.7750 and 0.7763 at t = 1; 0.6627 and 0.6641 at t = 2; 0.5924 and 0.5932 at t = 3; 0.5047 and 0.5042 at t = 5 — and that curve is world B's truth, 0.5052 at t = 5. World A's truth is 0.3636 there. The dashed lines are the two bounds that assume nothing, from every dropout failing on leaving (0.1905 at t = 5) to none ever failing (0.6667).

A dropout the data cannot see

Two worlds leave the same record to the last detail a study can write down — the same times, the same share ending in the event, the same share leaving first — and a log-rank test between them rejects at its own 5% level at every sample size from a hundred to sixteen hundred. Kaplan–Meier converges on 0.5052 at t = 5 from both. The truth is 0.5052 in one and 0.3636 in the other, and what is left to argue about is where between two bounds to stand.

survival · Censoring One treatment, a different hazard ratio at every follow-up. The hazard ratio a Cox model converges to, found as the root of its expected score by numerical integration, as the trial runs longer; dropout at 0.1 throughout. The proportional treatment reads 0.5 at every τ. The waning treatment reads 0.5000 at τ = 1, 0.6362 at τ = 3 and 0.7890 at τ = 8 — the same two arms, the same effect in the same first year, and a number that drifts towards one as later, effect-free events are added to the average. The dots are the mean of 400 Cox fits with 400 subjects an arm: 0.5014 at τ = 1, 0.7020 at τ = 2, 0.7635 at τ = 3, 0.8034 at τ = 5, 0.8208 at τ = 8. The crossing treatment reads 0.3429 at τ = 1, exactly 1 at τ = 3 by construction, and 1.0611 at τ = 8: beneficial, null or harmful according to when the trial stopped.

The hazard ratio the follow-up chose

A treatment that halves the hazard for one year and then does nothing has a Cox hazard ratio of 0.5000 if the trial stops at one year, 0.7617 at three and 0.8194 at eight. Nothing about the treatment differs between those numbers. When hazards are not proportional the hazard ratio is an average, and the length of follow-up and the dropout rate choose its weights.

survival · Censoring The difference in restricted mean survival at every horizon, in three worlds. Treatment minus control, in closed form, with dropout irrelevant to the truth. The proportional treatment's difference grows to 0.4766 at τ = 3 and the waning treatment's to 0.2675. The crossing treatment's rises to 0.1776 at τ = 2, near where the two survival curves cross, and falls back to 0.1366 at τ = 3. The ticks along the bottom are the eleven horizons, from 0.5 to 3 in quarters, at which a trial below reads its differences.

A horizon chosen after looking

A difference in restricted mean survival read at whichever of eleven horizons looks most convincing rejects 11.24% of trials in which the treatment does nothing, against 4.70% at a horizon fixed in advance. The correlation of the differences across horizons is closed, and the Gaussian process it defines prices the choice at a critical value of 2.317 — which brings the counted size back to 4.99% and keeps 96.92% of the power that a horizon nobody could have known to fix would have had.

survival · Censoring 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.

Kaplan–MeierCensoringEstimandStatistical powerSurvival curveCritical valueHazard ratioMonte CarloProportional hazardsRestricted mean survivalCause specific hazardClosed form

All concepts