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

Empirical likelihood assigns unknown probability masses $p_j\ge 0$ to data-supported event times or intervals, imposes ${\sum }_j{p}_j=1$, and maximizes the product of each observation's probability. For event-time data, write $F(t)=\mathbb{P}(T\le t)$ and $S(t)=1-F(t)$ for the survivor function. Exact, right-censored, left-censored, interval-censored, and truncated observations contribute the probability of their respective compatible sets.

The maximization uses that $S$ is nonincreasing and right-continuous, $S(0)=1$, and $S(t)\to 0$ as $t\to \infty$. Probability mass need only be placed at endpoints that change an observation's compatible set; moving mass within any observationally indistinguishable interval leaves the likelihood unchanged. Maximizing over those masses gives the nonparametric maximum-likelihood estimator of the survivor function.

The death time is right-censored at 36 months, so the contribution is

An exact death at 42 months contributes the probability mass at 42,For a continuous model this is represented by the event density $f(42)$ rather than a point mass, but empirical likelihood permits discrete masses.

The awakening time is left-censored at 15 minutes, so its contribution is

The secondary cancer appears in the interval $(6,9]$, producing the interval-censored contribution

The bus may have arrived before observation began or after observation ended. Its compatible event set is $(-\infty ,2)\cup (20,\infty )$, so, with endpoint conventions chosen to match whether arrivals exactly at 2 or 20 would be seen, the contribution isThis is doubly censored data, because neither the side nor the event time is known.

Measure time in years after age 70. Residence in the care home begins at time 3, so inclusion is conditional on $T>3$: this is left truncation, also called delayed entry. Exact death at time 13 contributes

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 $\log Y_k$.