Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 1

Write $S(t)=\mathbb{P}(T>t)$ for the survival function; this is the function denoted $F_T$ in the survival notation. For a proper continuous event-time distribution with hazard function $h$, its cumulative hazard function is $H(t)={\int }_0^{t}h(s)ds=-\log S(t)$. The probability integral transform makes $S(T)$ uniform on $(0,1)$, including when $S$ has flat intervals: those intervals have zero event probability. ConsequentlyThus the cumulative hazard probability transformation gives $\boxed{H(T)\sim \mathrm{Exponential}(1)}$. No strict monotonicity of $H$ is needed on intervals that carry no event mass. The usual proper continuous survival law is essential; an atom of subjects who never fail would instead require separate treatment of the mass at infinity.

Under independent censoring, order the distinct observed failure times as $t_1<\cdots <t_m$. Let $r_j=\#\{i:x_i\ge t_j\}$ be the size of the risk set just before $t_j$, and let $d_j=\#\{i:x_i=t_j,v_i=1\}$ be the number of failures there. The estimated conditional probability of surviving that event time is $1-d_j/r_j$. Multiplying these conditional survival probabilities gives the Kaplan–Meier estimatorA censored observation leaves the risk set after its censoring time but does not create a downward survival jump. With tied failure and censoring times, the displayed risk-set convention includes those censored at the time while processing failures, then removes them.

Since $H=-\log S$, the Kaplan–Meier estimator of the cumulative hazard function isThis is not exactly the Nelson–Aalen estimator ${\sum }_{t_j\le t}{d}_j/r_j$, although the two are close when the individual fractions are small. If a jump exhausts the risk set, $\hat{S}$ becomes zero and ${\hat{H}}_{\mathrm{KM}}$ becomes infinite; transformed residuals beyond that point require a finite-tail modelling convention or restriction of follow-up.

The transformed values are Cox–Snell residuals. Under a correct common survival function and a good fitted cumulative hazard function, the complete transformed failure times should have unit exponential distribution. Transform the observed times and retain their original censoring indicators, then estimate residual survival with a Kaplan–Meier estimator. Its target isso a plot of its logarithm against $u$ should lie approximately on a line of slope $-1$. Censored residuals must not be treated as fully observed failure times.

There is a useful caution for this particular common nonparametric fit. The shared-fit Cox–Snell residuals reproduce the fitted survival curve property means that the pooled residual plot is largely built into the transformation. With event ordering and compatible tie processing preserved, at a transformed failure time $u_j=\hat{H}(t_j)$ its residual Kaplan–Meier estimator equals $\hat{S}(t_j)=e^{-u_j}$. Finite steps, coalesced censoring times and estimation uncertainty still matter, and an infinite final residual arises if the fitted survivor becomes zero. Separate group plots are more informative about a common-distribution assumption than this pooled fit alone.

If both groups share the same survival function and the common fit is adequate, their complete Cox–Snell residuals have the same unit exponential distribution. Under independent censoring in each group, both groupwise Kaplan–Meier estimators should therefore be close to $e^{-u}$ and to each other, subject to ordinary sampling variation and the number of subjects still at risk. Different censoring patterns can change precision without changing the residual survival target.

Let $S_A,S_B$ be the distinct group survival functions, and let $H_{\ast }$ denote the common pooled cumulative hazard function used for the residuals. At a point where its inverse is defined, residual survival in group $g$ isThus the group residual curves generally differ. On intervals where $S_A(t)>S_B(t)$, the same common increasing transformation gives $S_{u,A}(u)>S_{u,B}(u)$: the better-surviving group has more large residual failure times. If the original survival curves cross, the residual curves can also cross, so no fixed ordering or proportional hazards assumption is implied.

Distinct group distributions generally produce distinct residual survival curves, even when the pooled residual curve looks exponential. Separate correctly fitted group hazards would restore the unit-exponential target within each group, but that is a different transformation from the common fit under discussion.

For the population identity, write $X=\min (T,C)$, $V=1_{\{T\le C\}}$, and $G(t)=\mathbb{P}(C\ge t)$. Assume independent censoring, with $f_T(t)=h(t)S(t)$. Thenwhile the Tonelli theorem and the cumulative hazard function giveThis is mean accumulated hazard under independent censoring, proving $\mathbb{E}[V-H(X)]=0$. For the fitted martingale residual $Y=V-\hat{H}(X)$,Hence its expectation is approximately zero when the fitted hazard error at the observed time is small in mean. This spells out the required sense of a good estimate; pointwise consistency alone does not control a divergent tail. In particular, a Kaplan–Meier estimator with zero final survival gives infinite transformed times there and cannot supply finite residuals without a tail restriction. Neither exact finite-sample zero mean nor validity under informative censoring is being assumed.

To check an omitted explanatory variable $Z$, plot the martingale residuals against $Z$ or compare groupwise mean residuals, using a smooth trend or appropriate uncertainty intervals. Under a correct conditional survival model and conditionally independent censoring, $\mathbb{E}[Y\mid Z]$ should be approximately zero. A systematic positive trend indicates more observed failures than the fitted accumulated hazard predicts; a negative trend indicates fewer. Such patterns suggest adding $Z$, a nonlinear term or an interaction and reassessing the fit. The bounded-above, often long negative tail of martingale residuals means that the diagnostic is a mean-pattern check, not a requirement for symmetric Gaussian residuals.