Relative survival

Relative survival compares observed survival in a diseased cohort with expected survival derived from a suitable reference population. A useful individual model adds a known background mortality hazard to a common excess hazard. Reference mortality can depend on age, sex and calendar time. The comparison avoids classifying each death by cause, but its interpretation depends on the reference population being appropriate. An excess-survival ratio is not automatically the effect of a causal intervention removing one cause of death.

Under the additive hazard function model $h^{(i)}=h_B^{(i)}+h_E$, the survivor function factorization gives $F_E=F^{(i)}/F_B^{(i)}$, independently of $i$. If the excess hazard is nonnegative, this has a survival-probability interpretation for a hypothetical process with just that hazard. If excess hazards are negative, relative survival can exceed one and is a comparison ratio rather than an ordinary survivor function.

With a common excess hazard, let ${\overline{h}}_B={\sum }_i{Y}_i{h}_B^{(i)}/{\sum }_i{Y}_i$ be the known background hazard averaged over the current risk set. Estimate the excess cumulative hazard by subtracting $\int {\overline{h}}_B$ from the Nelson–Aalen estimator. Applying the product-limit construction gives the displayed estimator: the continuous background correction multiplies the ordinary Kaplan–Meier estimator. Its event jump is $1-d_j/r_j$, while between events it grows at rate ${\overline{h}}_B{\hat{F}}_E$. Hence the unconstrained curve can increase, even if the true excess hazard is nonnegative. Replacing its jump factor by $\exp (-d_j/r_j)$ gives the alternative exponential cumulative-hazard estimator; these versions are close for small event fractions but are not identical.

An excess hazard measures mortality above the background hazard. When it is common across individuals, each observed survivor function factors into its individual background survivor function and a common relative survivor function. A nonnegative excess hazard gives a nonincreasing relative survivor function; a negative excess hazard can instead reflect better survival than expected in the reference population.

In relative survival, the background hazard is an individual's expected mortality hazard in a suitable reference population. Its dependence on age, sex and calendar time can be obtained from population life tables. In a heterogeneous cohort, estimating a common excess hazard requires averaging these known background hazards over the current risk set, rather than over the original cohort.