Independent censoring

A simple sufficient independent censoring assumption is that the potential observation-ending time $C$ and survival time $T$ are independent random variables. More generally, appropriate conditional independence given the covariates used in the analysis can suffice for analysis within those conditions. The requirement concerns the potential times, rather than independence of the observed duration $Y=\min (T,C)$ and event indicator $\delta$: those recorded quantities ordinarily depend on each other.

For an ordinary pooled Kaplan–Meier estimator, the observed risk set must represent the future event experience of everyone still event-free at each time. Conditional independence given a covariate does not alone justify ignoring that covariate in a pooled calculation. Also, there must be a positive probability of remaining observed through the horizons being estimated. If follow-up ends at a common closing time, it supplies no observations to identify the later survival function. Independent subjects with a common survival distribution and independent censoring form the basic model in Lu Tian's STAT331 notes, Unit 3, pages 1–4.

For example, subjects who stop attending because their condition is deteriorating can produce informative censoring. A predetermined common closing date can instead give administrative censoring, provided calendar entry and other relevant conditions do not select future survival times. The reason for stopping observation must be justified from study design and context; the recorded event indicators alone cannot verify independent censoring.

Suppose the recorded durations retain an exponential distribution of rate $\theta$, but an independent fraction $q$ are relabelled as failures and the rest as right-censored. Applying the ordinary survival likelihood gives $\hat{\theta }=d/{\sum }_i{y}_i\to q\theta$, rather than $\theta$. A censoring flag chosen independently of the recorded duration does not establish independent censoring of a latent failure time. Genuine simulation instead draws an event time and an independent censoring time and records their minimum.

For an event time with an exponential distribution of mean $\mu$ and an independent censoring time, let $X$ be their minimum. Memorylessness of the exponential distribution makes $X+\mu$ the conditional mean of the latent event time when it is censored. Keeping $X$ when the event is observed gives an unbiased imputed event time by the tower property of conditional expectation.

Administrative censoring ends follow-up at a fixed study or data-cutoff date. It is independently censoring when that cutoff and the resulting follow-up duration are unrelated to prognosis after conditioning on modeled variables.

When cohorts share one time-since-entry survival distribution, their observations can be pooled using the Kaplan–Meier estimator, even when later cohorts have shorter administrative follow-up. A short-follow-up cohort still improves estimation of early survival, which is a factor in survival at later times. Pooling requires comparable survival across cohorts and independent censoring; calendar-period or cohort effects can invalidate that assumption.

If entry is uniform on calendar interval $[{\tau }_a,{\tau }_b]$ and follow-up stops at ${\tau }_c$, the censoring duration is uniform on $[{\tau }_c-{\tau }_b,{\tau }_c-{\tau }_a]$. Its hazard function diverges at the longest possible follow-up. It is independent censoring when entry date is independent of prognosis after the required conditioning.