Signature
S_star = w_f * S_f + (1 - w_f) * S_m; h_star = (w_f * S_f * h_f + (1 - w_f) * S_m * h_m) / S_star
| Inputs | Definition | Unit |
|---|---|---|
w_f | Proportion of the cohort that is female at model start | proportion |
S_f | Survival of the female group to time t from the female life table at attained ages | probability |
S_m | Survival of the male group to time t from the male life table at attained ages | probability |
h_f | Background hazard for women at their attained age at time t | deaths per person-year |
h_m | Background hazard for men at their attained age at time t | deaths per person-year |
S_star | Expected survival of the whole cohort to time t when background mortality alone acts | probability |
|---|---|---|
h_star | Background hazard of the surviving cohort at time t | deaths per person-year |
Function
Background mortality function for the cycle death probability of a modelled cohort
Maps the attained age of a cohort, the cycle length and a disease-specific death probability to the probability of dying of other causes in each model cycle and to the all-cause probability of death, and relates all-cause survival to the survival expected in the general population. The split of the all-cause hazard into a background and an excess hazard is live as HE-FM-BTH-005 and is not restated; the formulas here give its survival-scale form (relative survival), its cycle-probability form for two independent risks, and the expected hazard of a cohort that mixes sexes or ages. A background probability for one cycle comes from a life-table annual probability q_x through the rescaling formula HE-FM-TP-003 with an observed interval of one year (HE-CF-TP-002 performs the same step), and HE-FM-TP-004 splits a cycle's deaths between background and disease death states. A multiplier on background mortality, such as a standardised mortality ratio, enters a cycle probability in the same way as the hazard ratio of HE-FM-HR-004. Related live records cover cure models in the relative survival form (HE-FN-CRM-001 and HE-FM-CRM-001 to HE-FM-CRM-004), cycle probabilities from a cumulative hazard (HE-FN-CUMH-001, HE-FM-CUMH-001 to HE-FM-CUMH-003), general-population QALYs from a life table (HE-FM-ASF-002), the cohort trace that carries the probabilities (HE-FN-CSIM-001) and tunnel states, whose clock restarts on entry while attained age keeps running (HE-FN-TUN-001). Years of life lost (HE-FN-YLL-001) use a normative standard life table, a different purpose from the national table used here. Notation follows the Background Mortality article.
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Implementations
Excel
Marginal expected survival and background hazard from named group cells
With the female share in FemaleShare, the group survivals in FemaleExpSurv and MaleExpSurv and the group hazards in FemaleBgHazard and MaleBgHazard, the first formula returns the value held in MarginalExpSurv and the second the value held in MarginalBgHazard. Filled down a column of model times with lookups on attained age, they give the cohort's background curve.
=FemaleShare*FemaleExpSurv+(1-FemaleShare)*MaleExpSurv; =(FemaleShare*FemaleExpSurv*FemaleBgHazard+(1-FemaleShare)*MaleExpSurv*MaleBgHazard)/MarginalExpSurv
Assumptions
Group shares fixed at model start
w_f is the share at time zero, as each subject carries the weight 1/N in TSD 21 equation 18. The share of women among survivors at time t, w_f S_f divided by S_star, follows from the formula and is not entered.
Group survival and hazard from each group's own life table
S_f, S_m, h_f and h_m come from life tables for each sex, which ONS constructs separately for males and females, at each group's attained age at time t, with S_f and S_m accumulated from model start. The rates are fixed and known, as for HE-FM-BGM-001.
Worked examples
Illustrative mixed-sex cohort in which men die sooner
Illustrative values at a point some years into the model, with half the cohort female at the start, expected survival of 0.80 for women and 0.70 for men, and background hazards of 0.020 and 0.030 a year, give a marginal expected survival of 0.75 and a marginal hazard of about 0.024667, below the fixed blend of 0.025 because women now make up about 53% of survivors.
w_f = 0.5; S_f = 0.80; S_m = 0.70; h_f = 0.020; h_m = 0.030; S_star = 0.75; h_star = 0.024667
Marginal background hazard at model start
At time zero both survivals are 1, so the marginal hazard is the simple weighted blend, 0.025 for the same hazards, a limiting case.
w_f = 0.5; S_f = 1; S_m = 1; h_f = 0.020; h_m = 0.030; S_star = 1; h_star = 0.025
Mixed-sex cohort that is 60% female at the start
With 60% of the cohort female at the start and the same illustrative survivals and hazards, the marginal expected survival is 0.76 and the marginal hazard about 0.023684.
w_f = 0.6; S_f = 0.80; S_m = 0.70; h_f = 0.020; h_m = 0.030; S_star = 0.76; h_star = 0.023684
Common errors
Fixed equal-weight blend of female and male background hazards
Averaging the two hazards with the starting shares at every time ignores that men die sooner. In the first worked example the blend gives 0.025 against 0.024667, and the gap widens as the cohort ages; TSD 21 section 6.2 weights each expected hazard by expected survival instead.
Background hazard read at the mean starting age
Mortality rises steeply with age, so the hazard at the mean age is below the average hazard of a cohort with a spread of ages. With illustrative hazards that double every 10 years, 0.010 at 60, 0.020 at 70 and 0.040 at 80, a cohort split equally between ages 60 and 80 has an average entry hazard of 0.025, while the hazard at the mean age of 70 is 0.020 (computed here for illustration).
Sources
Marginal expected survival and hazard in DSU TSD 21
Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield, Decision Support Unit, ScHARR, University of Sheffield; 2020 (updated March 2022). Section 6.2, equations 18 and 19 (page 75), which give marginal expected survival as the average of the expected survival of the N subjects and the marginal expected hazard as a weighted average of the subjects' expected hazard rates, with expected survival as the weights, as a reference because the mortality rate in patients with the disease of interest is unlikely to be lower than in the general population.
Separate life tables for males and females in the ONS methodology
Office for National Statistics. National life tables QMI. Last revised 10 December 2025. Section 6, methods used to produce the national life tables data, which states that life tables are usually constructed separately for males and females because of their different mortality patterns, and defines the mortality rate between age x and x + 1.
Canonical Identity
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