Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 32 / 4

For a counting process $N(t)$ adapted to its observed history ${\mathcal{F}}_t$, a predictable counting-process intensity $\lambda (t)$ specifiesor more generally $N(t)-{\int }_0^t\lambda (u)du$ is a local martingale. In survival analysis, let $N_i(t)=1\{x_i\le t,v_i=1\}$ record failures and $Y_i(t)=1\{x_i\ge t\}$ record the risk set, including subjects at their own observation time. Under independent right censoring and a common hazard function $h$, the aggregate intensity is $Y(t)h(t)$, where $N={\sum }_i{N}_i$ and $Y={\sum }_i{Y}_i$.

Writing $H(t)={\int }_0^{t}h(u)du$, the conditional increment equation is $E(dN\mid {\mathcal{F}}_{t-})=YdH$. Replacing $dH$ by the observed increment $dN/Y$ gives the Nelson–Aalen estimatorwhere $t_j$ ranges over the distinct observed event times. Censored observations remove subjects from subsequent risk sets but do not produce hazard jumps.

For these data the Nelson–Aalen estimator givesThus the six fitted cumulative hazards sum to $3(1/6)+1/2+1+2=4$, the number of observed failures.

The event-count identity for Nelson–Aalen cumulative hazards follows by exchanging the finite sums. With everyone entering at time zero,This includes each failure subject in its own risk set. A delayed-entry dataset requires a different risk indicator and is not covered by this particular identity.

Under a constant hazard survival model, $\hat{H}(x_i)=\hat{\theta }{x}_i$. Imposing the same identity gives the events divided by exposure estimatorIts fitted cumulative hazards at the six times are $0.3,0.4,0.5,0.6,0.9,1.3$, again summing to four. It is also the maximum-likelihood estimate under independent right censoring, since the parameter-dependent likelihood is ${\theta }^d{e}^{-\theta {\sum }_i{x}_i}$. This is a sensible estimate if the exponential survival model is appropriate, but four failures give little precision. The event-count identity by itself does not validate constant hazard; the large final jump in the Nelson–Aalen estimator also reflects a risk set of one, rather than by itself proving an increasing hazard.