Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 6

A frailty random variable is an unobserved positive multiplicative risk factor. A proportional frailty model hasThe scale of $U$ is not separately identifiable from $h_0$: multiplying $U$ by a constant and dividing $h_0$ by it leaves the model unchanged. We may therefore normalize $\mathbb{E}U=1$, which lets $h_0$ represent the mean initial hazard multiplier and makes relative frailty interpretable.

If $U\sim \mathrm{Exponential}(1)$ and $h_0(t)=\lambda$, then the Laplace transform of $U$ givesThus $h(0)=\lambda$, while $h(t)\to 0$ as $t\to \infty$. Survivors become increasingly enriched for low-frailty individuals.

Letand use constant baseline hazard $h_0=m$. Define the two-point frailtyThen $\mathbb{E}U=1$, and the conditional rates are exactly $\gamma$ and $\beta$. The experimental-treatment population hasandSince standard treatment has hazard $\beta$, the population hazard ratio isAt zero,whereas for $\pi >0$,The treatment effect is therefore non-proportional and strengthens among later survivors as the high-rate subgroup is depleted. A trial should allow adequate follow-up, avoid relying only on a constant-hazard-ratio Cox model, and prespecify survival-curve, milestone-risk, restricted-mean-survival, or time-varying-effect analyses. Its power and interpretation will depend materially on follow-up duration.