Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 6

A frailty model introduces an unobserved positive random effect $U$ that multiplies an individual's hazard. With $H_0(t)={\int }_0^t{h}_0(s)ds$, marginal survival isDifferentiation givesConsequently $\bar{h}(0)=h_0(0)$ exactly when $\mathbb{E}U=1$.

For equal point masses at $1/2$ and $3/2$ with $h_0=\theta$,Its initial value is $\theta$ because $\mathbb{E}U=1$, while $\bar{h}(t)\to \theta /2$ as the more frail survivors are progressively depleted.

For $U\sim \mathrm{Exponential}(1)$,Again $\bar{h}(0)=\theta$ because $\mathbb{E}U=1$, but now $\bar{h}(t)\to 0$ as $t\to \infty$.

Survival preferentially selects smaller frailties, so $\mathbb{E}[U\mid T\ge t]$ generally decreases with $t$. For the two-point distribution,At $t=0$ this equals $g(u)$, andIt decreases from $1$ to $1/2$, demonstrating survivor selection toward the less frail class.

Frailty makes a population hazard ratio mix the conditional treatment effect with changing survivor composition. Supposeand $U\sim \mathrm{Exponential}(1)$. Part (a)(ii), with $\theta$ replaced by $\theta e^{\beta z}$, givesAlthough conditional hazards are proportional with ratio $e^{\beta }$, the marginal ratio iswhich varies with time and tends to one. Thus an ordinary marginal proportional hazards interpretation can be misleading in the presence of unobserved heterogeneity.