Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 41 / 4

Let $N(t)$ count observed events and $Y(t)$ count individuals still in the risk set just before $t$. Under a common hazard function $h(t)$ and independent censoring, the expected event count over a short interval, conditional on the observed past, isHere $H(t)={\int }_0^{t}h(u)du$ is the integrated hazard. Thus an estimating increment for $dH$ is $dN/Y$, on the range where the risk set is nonempty. Accumulating these increments gives the Nelson–Aalen estimatorThe observation times $t_j$ include both events and censorings, $r_j=Y(t_j)$, and ${\delta }_j$ is one for an event and zero for censoring. Without ties each event adds $1/r_j$; censoring changes later risk sets but adds nothing immediately. Equivalently, the local event fraction estimates the local hazard increment, and summing those estimates produces the integrated hazard. The counting-process intensity in survival analysis justifies this estimating equation; it does not imply exact finite-sample unbiasedness after the risk set can become empty.

Assume the usual cohort setup: all individuals enter at time zero, each contributes one event or censoring time, and every such observation is included. Order these distinct observation times as $t_1<\cdots <t_m$. The risk set size just before $t_k$ is then $r_k=m-k+1$. Write ${\delta }_k=1$ for an event and zero for censoring. Including the jump at an event time, the Nelson–Aalen estimator isExchange the two finite sums:Thus the sum equals the number of observed events. Each event increment appears once for each individual still at risk at that event, exactly cancelling its denominator. This is the event-count identity for Nelson–Aalen cumulative hazards. The cohort and entry convention matter: delayed entry would invalidate the simple equation $r_k=m-k+1$ and require a modified identity.

One method breaks a tied group into a sequence of distinct artificial times, by a specified or random ordering. Process tied events before tied censorings if their common recorded time means the censored individuals were still under observation when the events occurred. At each artificial event, add the reciprocal of the current risk set size and then remove that individual. This applies the untied Nelson–Aalen estimator to the resulting grid. For example, two tied failures among two individuals give successive increments $1/2$ and $1$.

On that expanded grid there is one observation time per individual. The cancellation argument in (b) is unchanged, so breaking ties preserves the identity when the sum includes every artificially separated observation. The chosen ordering can affect the estimated curve within a tied group and must not be presented as an observed event order.

A second method retains the recorded tied times. For a group with $d_j$ events and pre-time risk set size $r_j$, the grouped Nelson–Aalen estimator adds $d_j/r_j$ once. All events in the group use the same denominator; censorings in the group are removed afterwards. If two individuals fail at the same time and there are no other observations, the curve has one jump of $2/2=1$. Its sum over the one distinct time is $1$, whereas the event count is $2$. Hence the unweighted distinct-time identity does not generally survive grouping ties. On the artificially separated grid in the first method, the two post-observation values would instead be $1/2$ and $3/2$, whose sum is $2$.

There is nevertheless an exact weighted version for the grouped method. Let $n_j$ count all observations, event or censoring, at distinct time $t_j$. With everyone entering at zero, $r_k={\sum }_{j\ge k}{n}_j$. ThereforeThis grouped Nelson–Aalen event-count identity sums over individuals, with repeated values for tied times, rather than once per distinct time. Keeping that distinction resolves the apparent conflict between a valid grouped estimator and failure of the literal unweighted formula.