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

Cox model — where it appears

A regression for survival data in which each covariate multiplies the hazard by a constant factor, leaving the baseline hazard unspecified. When the factors are not constant, or patients differ in unmeasured risk, it reports a weighted average ratio.

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 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

Named alongside it

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

Hazard ratioCensoringEstimandFrailtyHazardKaplan–MeierLog-rank testModel misspecificationNon-collapsibilityProportional hazardsRandomisationRestricted mean survival

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