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

Frailty — where it appears

An unobserved factor multiplying each person's hazard, so that some are more prone to the event than others with nothing recorded to say so. Survival then selects the less frail, and a population's hazard changes over time even when no individual's does.

Named by 2 essays across 2 fields — 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 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

Named alongside it

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

Cause specific hazardCensoringCompeting risksCox modelDependent censoringEstimandHazard ratioKaplan–MeierLog-rank testNelson–AalenNon-collapsibilityNon-identifiability

All concepts