Let
$$
m=\pi\gamma+(1-\pi)\beta,
$$
and use constant baseline hazard $h_0=m$. Define the two-point frailty
$$
U=
\begin{cases}
\gamma/m,&\text{with probability }\pi,\\
\beta/m,&\text{with probability }1-\pi.
\end{cases}
$$
Then $\mathbb EU=1$, and the conditional rates are exactly $\gamma$ and $\beta$. The experimental-treatment population has
$$
S_E(t)=\pi e^{-\gamma t}+(1-\pi)e^{-\beta t}
$$
and
$$
h_E(t)=
\frac{\pi\gamma e^{-\gamma t}+(1-\pi)\beta e^{-\beta t}}
{\pi e^{-\gamma t}+(1-\pi)e^{-\beta t}}.
$$
Since standard treatment has hazard $\beta$, the population hazard ratio is
$$
\boxed{
R(t)=\frac{h_E(t)}\beta
=\frac{\pi\gamma e^{-\gamma t}+(1-\pi)\beta e^{-\beta t}}
{\beta\{\pi e^{-\gamma t}+(1-\pi)e^{-\beta t}\}}.}
$$
At zero,
$$
R(0)=1-\pi\left(1-\frac\gamma\beta\right),
$$
whereas for $\pi>0$,
$$
R(t)\longrightarrow\frac\gamma\beta
\qquad(t\to\infty).
$$
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.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-207/6/b/solution