Concept

Proportional hazards — where it appears

The assumption that one group's hazard is a constant multiple of another's at every time, so a single hazard ratio sums up the difference. The log-rank test and Cox regression are built on it; when hazards cross or wane, that ratio becomes an average whose weights the follow-up chooses.

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

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

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–Meier Log-rank test Statistical power Censoring Closed form Cox model Critical value Estimand Gaussian process Hazard Hazard ratio Model misspecification

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