Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 41 / 5 / a

Survival data describe the time $T$ from a specified origin to an event, together with covariates and information about how observation begins and ends. Some event times are observed exactly; others are subject to censoring or excluded by truncation. Here the time origin is the opening of a business and the event is its closure. An observed record contains an entry age, an exit age, an event indicator and the business characteristics. The time origin and observation mechanism are part of the data specification.

With right censoring, an event has not occurred by the last observation time $C$, so the data establish only $T>C$. One records $Y=\min (T,C)$ and $\delta =1_{\{T\le C\}}$, distinguishing an observed event from a censored exit. A business still operating at the follow-up deadline is administratively right-censored. Its observed age is a lower bound on its lifetime, not its eventual closure age. Standard survival analysis requires suitable independent censoring, possibly conditional on the fitted covariates.

With left truncation, a subject is observed only if it survives to its entry time $E$: subjects with $T\le E$ are absent from the sample. A business already operating when observation begins has this delayed-entry mechanism. It joins a risk set only after its recorded entry age. Left truncation excludes earlier failures entirely; left censoring instead includes subjects whose event is known to have happened before a specified time. These mechanisms must not be conflated.