Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 6 / b

This is administrative censoring. Under the usual assumption that entry calendar time is independent of the subsequent event time, it is an example of independent censoring, not informative censoring: the planned study closure, rather than a patient's prognosis, determines the available follow-up. Uniform entry by itself does not prove that assumption; if entry date is associated with prognosis, independence must instead be justified conditionally on the relevant covariates. The calculations below use independence.

For administrative censoring with uniform entry, let entry time $A\sim \mathrm{Uniform}({\tau }_a,{\tau }_b)$. Then $C={\tau }_c-A$ has the uniform distribution on $[\ell ,r]$, where $\ell ={\tau }_c-{\tau }_b$, $r={\tau }_c-{\tau }_a$, and $d=r-\ell ={\tau }_b-{\tau }_a$. Its density and survivor function areDividing density by survival and integrating gives the censoring hazard and integrated hazard:The hazard function is undefined once no uncensored individuals remain. As $t\uparrow r$, it diverges like $1/(r-t)$ and the cumulative hazard function diverges logarithmically. This reflects the certainty of administrative censoring by the longest possible follow-up time, not dependence of censoring on the event process.

An individual has neither died nor been censored by time $t$ exactly when $T>t$ and $C>t$. Under independent censoring, this has probability $F_T(t)G(t)$. Thus the probability of leaving observation by either route isThe two pieces agree at $t={\tau }_c-{\tau }_b$. The expression counts the first of death and censoring, so it does not double-count individuals who would eventually experience both.