Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 7 / 9 / Solution

Applying death-only rate correction under exponential censoring to the death fraction and combined-rate estimate, the corrected plug-in death rate isThe corresponding censoring-rate estimate is $0.1{\text{year}}^{-1}$. This correction is consistent under the independent exponential competition model: the death-only conditional rate estimates $s$ and the observed fraction estimates $\lambda /s$.

There is an important distinction from the full-data maximum-likelihood estimate. If $T_D$ and $T_C$ are total observed follow-up for the 100 deaths and 50 censored subjects, the full likelihood for both rates isThe reported death-only rate determines $T_D=100/0.3$, but supplies no $T_C$, so the exact full-data estimate cannot be reconstructed from the numerical summaries alone. The value $0.2$ is the requested corrected rate using those summaries; it need not equal the realised full-data maximum-likelihood estimate. The full analysis should use the actual censored follow-up whenever available.