Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 7

Choose an individual uniformly from the population. The law of total probability gives the population density and survivor function as mixtures:A hazard function is a ratio of density to survival, so averaging the densities does not average those ratios. Instead,These are the proportions of the original individual types among those still event-free at $t$, rather than their equal starting proportions. For example, a mixture of different exponential distributions progressively favours the lower-rate individuals. This population hazard of a survival mixture generally differs from $n^{-1}{\sum }_i{h}_i(t)$.

In the counting-process intensity in survival analysis, the counting process $N_i(t)$ jumps once if an observed event occurs; the at-risk process $Y_i(t)$ is one immediately before $t$ only while individual $i$ is still observed and event-free. Conditional on that information, the chance of an event in a short interval is $h_i(t)dt+o(dt)$ for an at-risk individual and zero otherwise. This is the meaning ofThe history ${\mathcal{H}}_{t-}$ is the filtration generated by the observed event and censoring histories, entry information, and available covariates strictly before $t$. In particular, it determines the current risk set but does not reveal future event times. The right-hand side is the infinitesimal compensator of a counting process; the equality is interpreted as an intensity statement to first order in $dt$.

Without censoring, $Y_i(t)=1_{T_i\ge t}$, so its unconditional expectation is survival:The distinction between $>$ and $\ge$ is immaterial for a continuous event time.

With independent censoring, the at-risk process is $Y_i(t)=1_{T_i\ge t,C_i\ge t}$. Independence and the common censoring survivor function $G$ give

Apply the tower property of conditional expectation to the intensity equation. Since $h_i(t)$ is a specified individual hazard function,As in the conditional equation, this describes the first-order expected increment of the observed counting process.

Let $N(t)={\sum }_i{N}_i(t)$ and $Y(t)={\sum }_i{Y}_i(t)$ be the total observed counting process and size of the risk set. When the individual cumulative hazard functions are all $H_0$, summing the conditional intensity equations givesReplace the expected increment by its observed value and divide by the predictable risk set size. Thus the Nelson–Aalen estimator in integral form iswith the integrand defined as zero when $Y(u)=0$. This is an integral against the step counting process, so each event time contributes its event count divided by the risk set just before that time. The Nelson–Aalen estimator estimates integrated hazard over times with observable risk; it does not supply information after observation has ended.

Use the allowed ratio-of-expectations approximation where $G(u)>0$. The unconditional numerator and denominator from part (b) giveThe common censoring survivor function cancels in this approximation. By differentiating the logarithm, its value is the population integrated hazard:This is not generally the arithmetic mean of the $H_i(t)$; it is the cumulative hazard function of the population mixture from part (a). The cancellation requires a common independent censoring law and adequate observation of the time range.

The approximation establishes approximate unbiasedness, not exact finite-sample unbiasedness. For a direct counterexample, take one uncensored individual with exponential distribution of rate $\lambda$. The Nelson–Aalen estimator is then $1_{T\le t}$, whose expectation is $1-e^{-\lambda t}$, whereas $\overline{H}(t)=\lambda t$. Thus it is not an unbiased estimator in general.

More explicitly, for independent individuals with a common event hazard $h$, the exact expected estimator isThis exhibits finite-sample bias of Nelson–Aalen estimation and dependence on $G$ when empty risk sets are possible. In a heterogeneous population, replacing the random ratios by ratios of their means introduces an additional approximation. On adequately observed intervals with large risk sets, the population-mixture expression is the appropriate large-sample target.