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

Cause specific hazard — where it appears

The rate at which subjects still at risk of every ending event have one particular ending, with the other endings treated as censoring. It is estimated correctly by censoring the competing events, and turning it into a risk by one minus its exponential overstates the chance of having that event by a stated time.

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

Also named here as competing risks, dependent censoring, nelson–aalen — the same set of essays touches all of them, so they are one junction rather than several.

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 risk of one cause, estimated two ways. Two causes of an ending event with constant hazards 0.2 (the one of interest) and 0.3 (the competitor), random dropout at 0.1 and follow-up to 6. The lower line is the cumulative incidence, (0.2/0.5)(1 − e^(−0.5t)), the chance of actually having had this event by t; the dots on it are the Aalen–Johansen estimate over 2000 studies of 300, 0.3670 at t = 5 against 0.3672. The upper line is 1 − e^(−0.2t), and the dots on it are one minus Kaplan–Meier with the competing event treated as censoring: 0.6318 at t = 5 against 0.6321. The second is larger by a factor of 1.722 at t = 5, and it is not an error of estimation. It estimates, correctly, the risk in a population where the competing cause does not exist.

One minus Kaplan–Meier is not a risk

With two ways for observation to end, one minus Kaplan–Meier for one cause reads 0.6318 at t = 5 where the chance of actually having had that event is 0.3670. Added across the two causes, the complements reach 1.4088 — more than the whole cohort. Nothing is estimated badly: the complement estimates, correctly, the risk in a world where the other cause does not exist.

survival · Censoring

Named alongside it

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

CensoringCompeting risksDependent censoringEstimandKaplan–MeierNelson–AalenNon-identifiabilityAalen–JohansenClosed formCumulative incidenceFrailtyHazard

All concepts