Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 5

A proportional hazards model has $h(t\mid g)=h_0(t)r(g)$, with a time-independent hazard ratio. In the Cox proportional-hazards model, $r(g)=e^{\beta g}$ and the baseline hazard $h_0$ is unspecified. At each untied event time, conditioning on the identity of the person who fails given the current risk set produces the factorMultiplying these factors gives the Cox partial likelihood. It is called partial because it uses the conditional event ordering information that identifies $\beta$ while discarding the part of the full likelihood involving the unspecified baseline hazard.

Write $\theta =e^{\beta }$ for the group-1 to group-0 hazard ratio. At $t_1$, all four subjects are at risk, so subject 1 contributes $1/(2+2\theta )$. At $t_2$, subjects 2, 3, and 4 are at risk, so subject 2 contributes $\theta /(1+2\theta )$.

If $t_2<c<t_4$, subject 3 leaves before $t_4$ and subject 4, if it fails, is alone in its risk set. Henceindependently of $k$. Its log derivative vanishes when $1-2{\theta }^2=0$, giving

If $c=t_4$, subject 3 remains in the risk set at a failure of subject 4. ThereforeFor $k=0$, $\hat{\theta }=1/\sqrt{2}$. For $k=1$, the score equation is $1-\theta -4{\theta }^2=0$, soThus $\hat{\theta }$ is constant at $1/\sqrt{2}$ for $t_2\le c<t_4$. At $c=t_4$ it remains there when $k=0$ and drops to approximately $0.39$ when $k=1$, because only then does the censoring time change an event's risk set.

For independent exponential survival times with total observed person-time at risk $T_{\bullet }$ and $d$ events, the maximum-likelihood rate is $\hat{h}=d/T_{\bullet }$. A two-group exponential model is a proportional-hazards model because both hazards are constant, so their ratio is constant.

HereThus the group-1 to group-0 hazard-ratio estimate isIt decreases continuously as $c$ increases. The parametric exponential likelihood uses exact exposure times through each arm's person-time, whereas the Cox partial likelihood uses only which subjects belong to each event's risk set. For $t_2<c<t_4$, changing $t_4$ alters group 0 person-time and hence $\hat{\lambda }$, but subject 4 is alone if it fails at $t_4$, so that event contributes one to the partial likelihood and $\hat{\theta }$ is unchanged.