Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 6 / b

A period survival analysis estimates survival under the recent mortality conditions operating during a fixed calendar period. For calendar year 2022, each cancer patient's contribution begins at the later of diagnosis and 1 January 2022, which treats earlier diagnoses as left-truncated at the start of the year. Follow-up ends at death, loss to follow-up, or 31 December 2022, so surviving records are right-censored. Combining these risk-set contributions, commonly through a Kaplan–Meier estimator, constructs the 2022 period survivor function; records from several diagnosis cohorts may contribute different segments of follow-up.

Construct the two countries' period-specific risk sets with the same left truncation and right censoring rules, then compare their event counts by a Log-rank test. This is a nonparametric comparison because it does not specify the shape of either country's hazard function.

Under a country-specific constant hazard survival model, let $D_k$ be the number of deaths and $Y_k$ the total person-time at risk in country $k$. The likelihood is proportional toso the maximum-likelihood estimator is ${\hat{\lambda }}_k=D_k/Y_k$. Test $H_0:{\lambda }_0={\lambda }_1$ by a likelihood-ratio test, score test, or Wald test; equivalently, give $D_k$ a Poisson distribution with mean ${\lambda }_k{Y}_k$ and test the country coefficient in a Poisson regression with offset $\log Y_k$.