= Solution
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.
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