= Solution
Use the three-state <competing risks model> shown below. State 0 is alive after surgery, state 1 is dead from lung cancer, and state 2 is dead from other causes. Both death states are absorbing. Study censoring is an observation mechanism, not a biological transition to a fourth state.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-37-competing-risks.png]
{title=Competing deaths with treatment-specific transition intensities}
For treatment $z=0$ (radiotherapy) or $z=1$ (chemotherapy), define the <transition intensity>
$$
q_{0k}(t\mid z)=\lim_{h\downarrow0}\frac{\mathbb P(X(t+h)=k\mid X(t)=0,z)}{h},\qquad k=1,2.
$$
The fitted <Cox proportional-hazards models> are
$$
\boxed{\widehat q_{01}(t\mid z)=\widehat\lambda_{10}(t)e^{1.3113z},\qquad
\widehat q_{02}(t\mid z)=\widehat\lambda_{20}(t)e^{0.5903z}.}
$$
Here $\widehat\lambda_{10}$ and $\widehat\lambda_{20}$ represent the estimated baseline <cause-specific hazards> for radiotherapy. More formally, the baseline estimates can be expressed as <cumulative hazard> measures with increments $d_k(t)/\sum_{i\in R(t)}e^{\widehat\gamma_k z_i}$, rather than ordinary smooth functions. All outgoing intensities from states 1 and 2 vanish, and $q_{00}=-(q_{01}+q_{02})$. The treatment <hazard ratios> are 3.7111 and 1.8045 respectively. This model describes the two observed causes of death, not identifiable latent failure times after death has already occurred.
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