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

Risk set — where it appears

The units still being observed and still at risk at a given moment, which is the denominator every survival estimate is built from. Letting censored units contribute for as long as they were observed is what keeps that denominator honest.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The same study read three ways. At time 2 the truth is 0.497. Kaplan–Meier gives 0.532; dropping the censored subjects gives 0.180; treating the censoring time as the event time gives 0.392. Both naive readings understate survival, because the subjects they mishandle are the ones doing well.

The data that stops early

A subject still event-free when a study ends is not missing and not observed. It is known to exceed something, which is a third state most tools have no slot for — and the two obvious ways of forcing it into one are wrong by 31 and 13 percentage points.

survival · Censoring Kaplan–Meier from 120 subjects, 59 of them censored. The step curve is the estimate, the smooth curve is the truth it is trying to recover. 59 of 120 subjects were still event-free when observation stopped; they are not dropped, and they are not counted as events — they leave the risk set at the time they were last seen.

The curve that survives censoring

Kaplan–Meier recovers the true survival curve to within a fraction of a point at every censoring level from 37% to 71%, where dropping the censored subjects is off by 28 and then by 43. The estimator is a running product and the reason it works is in its denominator.

survival · Censoring How often the split is taken, and by which rule. Over 400 draws on each of five laws. The first two rows have no break in them at all, the last two have one at row 60, and the middle one is a moving average. A criterion that counts a fitted two-regime model's parameters and nothing else takes the split on 99% of draws where there is no break. Counting the break point as one more parameter brings that to 67%. Charging what the search actually manufactures — 5.16 units, measured on a law with no break — brings it to 16%, and still takes the split on 61% of draws where there is one.

Choosing whether to break

Charging what the search manufactures takes a rule from splitting a stationary sample on 99% of draws to 16%. It also costs regret, because the two mistakes a rule can make are not the same size.

charged · Break point One cohort of 40, and two intervals around the end of its curve. A single simulated study of 40 subjects with exponential survival at rate 0.35, dropout at rate 0.15 and follow-up to 6 — the first seed from 8811 upward whose plain band reaches below −0.05, chosen to show the failure rather than its frequency. The step curve is Kaplan–Meier and the smooth curve the truth. The plain band, the estimate plus and minus 1.96 Greenwood standard errors, first dips below zero at t = 3.78 and reaches −0.052; early on it also rises to 1.023, above one. At t = 5 the estimate is 0.069 with 1 subject still under observation, the plain interval runs from −0.052 to 0.191 and the log-log interval from 0.006 to 0.251, against a truth of 0.174. The log-log band is built on a scale that cannot leave [0, 1], and it bends away from the edge rather than through it.

The interval at the end of the curve

The interval most software prints around a survival curve covers 89.7% at five years, where 3.3 of forty subjects are still being watched and where the curve is actually read. The same variance carried on a log–log scale covers 94.8% there — and the failure was never the width.

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 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.

CensoringKaplan–MeierSurvival curveEstimandGreenwood's formulaHazardSurvival analysisAalen–JohansenBinomial proportionCause specific hazardChange pointClosed form

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