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

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.