Frailty makes a population hazard ratio mix the conditional treatment effect with changing survivor composition. Suppose
$$
h(t\mid U=u,Z=z)=u\theta e^{\beta z}
$$
and $U\sim\operatorname{Exponential}(1)$. Part (a)(ii), with $\theta$ replaced by $\theta e^{\beta z}$, gives
$$
\bar h_z(t)=\frac{\theta e^{\beta z}}
{1+\theta e^{\beta z}t}.
$$
Although conditional hazards are proportional with ratio $e^\beta$, the marginal ratio is
$$
\frac{\bar h_1(t)}{\bar h_0(t)}
=e^\beta\frac{1+\theta t}{1+\theta e^\beta t},
$$
which varies with time and tends to one. Thus an ordinary marginal <proportional hazards> interpretation can be misleading in the presence of unobserved heterogeneity.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-207/6/c/solution