Censoring (statistics)

Censoring records that an observation lies in a known set without revealing its exact value. In survival analysis, it commonly bounds an unobserved event time.

Suppose conditionally independent latent values have normal distribution $N(\mu ,{\sigma }^2)$, with $\sigma >0$, but the observation only says which side of a known threshold $c_i$ contains each value. With $s_i=1$ for a value above the threshold and $s_i=-1$ for a value below, the likelihood is the displayed product, where $\Phi$ is the standard normal distribution function. Repeated thresholds give corresponding powers. These factors are tail probabilities, not probability densities evaluated at the thresholds. Multiplying normal densities at $c_i$ would describe a different observation process. The thresholds must be treated as fixed or otherwise modelled appropriately; informative selection of thresholds can change the likelihood.

Informative censoring occurs when censoring carries information about the future event time beyond the variables conditioned on in the analysis. It can invalidate ordinary Kaplan–Meier estimator and survival-model residual interpretations.

If a cohort is periodically confirmed event-free but exact event reports update only those who fail, censoring the remaining subjects at their last scheduled confirmation creates informative censoring. Between confirmations the nominal risk set retains future observed failures and removes nonfailures. An apparent final failure may therefore empty the recorded risk set even though other cohort members remain event-free. Use a common verified cutoff or a complete event register with a common administrative censoring date.

An unobserved survival time is right-censored when it is known to exceed the last event-free follow-up time. For a potential survival time $T$ and potential observation-ending time $C$, recordWhen $\delta =1$, the occurrence was observed at $Y$; when $\delta =0$, only $T>Y=C$ is known. An intact machine at the study's closing date and an event-free subject who stops attending follow-up can both yield right censoring, but the reasons for stopping matter for independent censoring. A subject right-censored at year two has contributed two years to person-time at risk and belongs to earlier risk sets. Neither recording a failure at year two nor discarding that subject preserves this information.

Under independent censoring and separate parameters for the observation mechanism, the part of the survival likelihood involving an absolutely continuous probability distribution for $T$ is $f(Y{)}^{\delta }S(Y{)}^{1-\delta }$. Thus a right-censored observation contributes the survival function $S(Y)$, rather than the event probability density function $f(Y)$. See Lu Tian's Stanford notes, Section 2.1, equations (2.1)–(2.2). The Kaplan–Meier six-subject cohort shows exactly how this distinction changes the calculation.

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.

An event time is left-censored when it is known only to be no greater than the first observation time.

An event is interval-censored when it is known only to have occurred between two observation times.

In the usual left-and-right-censoring sense, doubly censored data contain some left-censored and some right-censored observations, possibly alongside exact values. The record identifies which side is censored. This is the convention used for datasets mixing both sides of censoring.^[ref].

For known observation limits $L<R$, one can record $Y=\min \{\max (T,L),R\}$ with an indicator specifying $T\le L$, an exact interior event, or $T\ge R$. For a continuous event-time probability distribution with cumulative distribution function $F$, probability density function $f$ and survival function $S$, the respective likelihood contributions are $F(L)$, $f(T)$ and $S(R)$. A measurement instrument that reports “below its lower detection limit” or “above its upper detection limit” produces precisely this distinction.

If the record only says that $T$ lies outside $(L,R)$ without identifying the side, its contribution would be $F(L)+S(R)$. That is different recorded information and is not the defining convention above. Terminology also needs context when both the origin and endpoint of a duration are uncertain; an interval-censored origin together with an interval-censored endpoint describes a different observation scheme from simple left-and-right censoring of one event time.