Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 5

A proportional hazards family has hazard functions related bywhere the hazard ratio $c(z)$ is positive and independent of time. Equivalently, its cumulative hazard functions satisfy $H(t\mid z)=c(z)H_0(t)$ and its survivor functions satisfy $S(t\mid z)=S_0(t{)}^{c(z)}$.

Writing the two functions in the question as survivor functions, $S_2(t)=S_1(t{)}^{\lambda }$. Since $H_k(t)=-\log S_k(t)$,Differentiating at times where the hazard functions exist gives $h_2(t)=\lambda h_1(t)$. Their hazard ratio is therefore the constant $\lambda$, so they form a proportional hazards family.

The transformation is $T_k=a_k{U}_k^b$. Because $U_k$ has a unit-rate exponential distribution,Thus $T_k$ has a Weibull distribution, withConsequently $h_2(t)/h_1(t)=(a_1/a_2{)}^{1/b}$ is constant, proving proportional hazards.

Choose a parametric baseline hazard $h_0(t;\eta )$ and fitUnder independent right censoring, the full survival likelihood isEstimate $(\eta ,\beta )$ by maximum likelihood estimation and test $H_0:\beta =0$ with a likelihood-ratio test, Wald test, or score test. Equality of the two event-time distributions is exactly $\beta =0$ within this model.

A semiparametric proportional hazards model specifieswith finite-dimensional parameter $\beta$ but an unspecified baseline hazard $h_0$. A partial likelihood uses a component of the data likelihood that depends on $\beta$ while eliminating the nuisance function. In the Cox proportional-hazards model, conditioning on which member of each risk set experiences the event produces the Cox partial likelihood.

For each observed event $i$ with $v_i=1$, let $R_i=\{j:x_j\ge x_i\}$ be its risk set. The Cox partial likelihood isMaximize it to obtain $\hat{\beta }$, estimate its variance from the observed partial information, and test $H_0:\beta =0$ using a partial likelihood-ratio test, Wald test, or score test. A positive fitted coefficient means the group with $z=1$ has the larger hazard.

Letand let $C$ be the common partial likelihood contribution from the first $n-3$ observations. Since $t_{n-2}>x_{n-2}$, the three possible complete-data tail orderings and their partial likelihoods areTheir sum isWhen individual $n-2$ is right-censored at $x_{n-2}$, that individual leaves the risk set before the event at $x_{n-1}$, so the directly calculated Cox partial likelihood is also $C,a/(a+b)$. Summing over the unobserved compatible event orderings therefore reproduces the censored-data partial likelihood.