Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 4 (Analysis of Survival Data)

The risk set at $x_i$ contains every individual still under observation and event-free immediately before $x_i$. Since the observed times are strictly ordered increasingly, these are individuals $i,i+1,\dots ,n$, so its size is

At time $x_i$, the observed number of events is $v_i$ and the exposure to the common instantaneous hazard is the risk-set size $n-i+1$. The likelihood score for a hazard increment therefore equates observed and expected events:Thus the Nelson–Aalen estimator isIt estimates the cumulative hazard function by adding event count divided by current exposure at every observed event time.

A martingale residual is observed minus model-expected event count:where ${\hat{\Lambda }}_i$ is the fitted individual cumulative hazard. Under an adequate model it estimates the terminal value of a counting-process martingale.

Fit a model omitting the continuous explanatory variable $z$, plot ${\hat{M}}_i$ against $z_i$, and add a flexible smooth curve. A curve fluctuating around zero without structure supports omission. A monotone or curved trend indicates that event incidence still depends on $z$, suggesting inclusion of $z$ or a nonlinear transformation of it. The residuals are highly skewed, so the smoothed trend is more informative than an assumption of Gaussian scatter.

With a common hazard and the Nelson–Aalen estimator from part a,

Interchanging the order of the finite sums givesEach hazard increment is counted once for every individual exposed to it, exactly reproducing its event count.

For the fitted Cox proportional-hazards model,where ${\hat{H}}_0$ is the fitted baseline cumulative hazard, usually obtained with the Breslow estimator.

A residual $0.99$ must correspond to an event and fitted cumulative event count $0.01$. The event occurred much earlier than the model expected for that individual.

A residual $-4$ means that the fitted expected event count by the observed follow-up time exceeded the observed count by four. It may be a censored individual with fitted cumulative hazard four, or an individual whose event occurred only after fitted cumulative hazard five. In either case the individual remained event-free substantially longer than predicted.

For an observed event,Its supremum is one, approached when the fitted cumulative hazard is near zero, while there is no finite theoretical lower bound.

For a censored observation,Its supremum is zero and again there is no finite theoretical lower bound. In a finite fitted dataset the realized minima are of course finite.

The residual plot has two asymmetric clouds. Event residuals lie below the horizontal boundary $1$, while censored residuals lie at or below $0$. Because $\hat{\beta }>0$, fitted cumulative hazard contains the rapidly increasing multiplier $e^{\hat{\beta }z}$; individuals with large $z$ who remain event-free until censoring can therefore have very negative residuals. At small $z$, events tend to lie near $1$ and censored observations near $0$. The resulting scatter is wedge-shaped and increasingly spread toward negative values as $z$ grows, rather than homoscedastic or approximately normal.