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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.

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.

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.

Define the individual background survivor function and common excess component byIntegrating the additive hazard function decomposition givesThe relative survivor function therefore compares actual survival with the survival expected from background mortality, and is common across individuals under this model. If $h_E\ge 0$, it is the survivor function of a hypothetical process having only the excess hazard. If excess mortality can be negative, the ratio may exceed one and lacks an ordinary survival-probability interpretation.

To estimate it, let $Y_i(u)$ indicate that individual $i$ is in the risk set, let $Y(u)={\sum }_i{Y}_i(u)$, and let $N(u)$ count all observed deaths. Assume appropriate independent censoring. The aggregate event intensity isConsequently the background correction must use the current risk-set averageonly on the observed range where $Y>0$. Dividing the event count by $Y$ estimates the total hazard there. Subtracting the known average background gives the offset-adjusted cumulative hazard estimatorHere $d_j$ is the event count at $t_j$ and $r_j=Y(t_j)$; under an untied scheme $d_j=1$. One possible estimate is ${\hat{F}}_{E,\exp }=\exp [-{\hat{H}}_E]$.

A product-limit version retains the full event fraction, which is especially relevant when the excess mortality is not small. Estimate the relative-survival differential equation $d{F}_E/F_E=-d{H}_E$ byAt an event time it multiplies the curve by $1-d_j/r_j$. Between events it solves ${\hat{F}}_E^{\prime }={\overline{h}}_B{\hat{F}}_E$. Starting at one therefore gives the risk-set-adjusted relative survivor estimatorThe Kaplan–Meier estimator factor accounts for observed all-cause mortality; the continuous multiplier corrects for known background mortality. The exponential cumulative-hazard version instead has event jumps $\exp (-d_j/r_j)$; it is close when event fractions are small but is not identical to this product-limit version.

Even if each background hazard is small, its accumulated contribution over long follow-up cannot simply be discarded. Nor should the excess contribution be linearized merely because the background is small. Using the known background integral inside the exponential avoids both errors. The average is over the evolving risk set: replacing it by an initial-cohort mean, or dividing by an arbitrary average of individual background survivor functions, is generally wrong when the background hazards differ. Censoring and deaths change which individuals contribute to the background correction.

Use the product-limit estimate in (b). Between consecutive event times there are no mortality jumps, soThus the estimated relative survivor curve rises between events when background mortality is positive, and is constant if it is zero. Censoring changes the risk set and can change the slope, but supplies no downward event jump. At the next event time the curve is multiplied by $1-d_k/r_k$.

Provided the curve is still positive, the ratio of the post-event estimates at the two consecutive event times isFor $d_k<r_k$, the later value is larger precisely whenFor example, suppose there are 100 individuals in the risk set after the earlier event, no intervening censoring, a 20-day gap, and one death at its end. If every background hazard is $0.001$ per day, then the integrated background contribution is $0.02$ andThe individual background hazards are small, but the interval contains less observed mortality than expected from that background. This can occur by chance in a small event sample or when the cohort is healthier than the life-table reference on other dimensions.

A nonnegative true excess hazard makes $F_E$ nonincreasing, but this unconstrained estimator need not inherit that restriction. An increase is therefore not proof of a clinical survival benefit. If the exponential cumulative-hazard version is chosen instead, its corresponding condition is ${\int }_{t_j}^{t_k}{\overline{h}}_B>d_k/r_k$; the qualitative explanation remains the same. If the risk set is exhausted by deaths, the product-limit factor is zero and no later increase is possible.

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.