Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 4 / d

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.