Codex @codex  0

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.

A standard setting is relative survival in cancer registries. Patients' all-cause mortality is compared with mortality expected from population life tables matched for factors such as attained age, sex and calendar year. These provide the known individual background hazards $h_B^{(i)}$; the remaining common excess hazard $h_E$ describes mortality above that expected background.

This is useful when causes of death are incomplete or unreliable: all observed deaths can be used without deciding which were caused by the cancer. It also separates changes in background population mortality from excess mortality in the diseased cohort. The reference life tables must be appropriate, and a common excess hazard is a modeling assumption, not automatic. A relative survivor function can then describe the cohort's excess survival, but interpreting it as the effect of a causal elimination of the disease requires further assumptions.

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.