Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 6 / b

Left truncation means that an individual enters observation only after a delayed entry time $L_i$ and is included only if the event time exceeds $L_i$. Earlier failures are absent from the dataset, so analysis must condition on survival to entry.

In the Kaplan–Meier estimator, replace the ordinary risk set at event time $t$ byThe product-limit factor remains $1-d_j/|R(t_j)|$, but only individuals who have entered and not yet exited contribute to its denominator.

Conditioning on survival to $L_i$ changes the constant-hazard likelihood contribution toThereforeusing only person-time accumulated after entry.

For period survival analysis over a calendar window, patients diagnosed earlier enter at the window's start only if still alive. Their survival times are left-truncated at the elapsed duration from diagnosis to that date, while follow-up is censored at the window's end. This estimates survival using mortality experience in the chosen recent period without pretending that earlier survivors were under observation from diagnosis.