Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 37 / 5 / a

Right censoring occurs when a subject's exact failure time $T$ is unobserved but it is known that $T>C$, where $C$ is the censoring time. The observed pair is $U=\min (T,C)$ and $\Delta =1_{\{T\le C\}}$. For example, a living participant at study closure contributes a survival lower bound rather than a recorded death at closure. Right censoring differs from simply omitting the subject: time spent in the risk set remains informative.

A standard sufficient non-informative censoring assumption is independent censoring: $T$ and $C$ are independent, or conditionally independent given the covariates used in the analysis. Their distributions need not be equal, have the same shape, or have proportional hazards. In a likelihood analysis the censoring distribution should also carry no information about the survival parameters being estimated, allowing its factor to be omitted. In the more general risk-set formulation, the censoring history must not alter the subsequent failure hazard conditional on the observed history used by the model. Informative censoring can arise if worsening unrecorded health causes withdrawal.

Among subjects alive at any given time and with the same relevant covariates or observed history, those censored at that time have the same subsequent failure distribution as those who remain under observation. Equivalently, the observed risk set represents the failure hazard of comparable subjects still alive, rather than selectively retaining lower- or higher-risk subjects. This is a statement about their future conditional risk, not that the observed failure-time distribution among censored people equals the distribution among people who eventually fail: their different observation patterns make that comparison inappropriate.