Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Background mortality function for the cycle death probability of a modelled cohort

b(x_t, Delta, p_dis) = (p_bg, p_all)

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.

  • Relative survival as all-cause survival over expected background survival

    R_t = S_t / S_star_t

    Divides the all-cause survival of the patient group at time t by the survival expected at the same time in a comparable group of the general population, matched on age, sex and calendar year. This is equation 14 of NICE DSU TSD 21, the survival-scale form of the additive hazard HE-FM-BTH-005. Because all-cause survival is the product of expected survival and survival from the excess hazard, relative survival equals exp of minus the cumulative excess hazard (HE-FN-CUMH-001), so it falls only while the excess hazard is above zero and levels off at the cure fraction in a mixture cure model (HE-FM-CRM-001). A value above 1 means that cumulative modelled mortality up to t is below that of the general population.

  • All-cause cycle death probability from independent background and disease-specific death probabilities

    p_all = 1 - (1 - p_bg) * (1 - p_dis)

    Combines the probability of dying of other causes in a cycle, read from the life table at the cohort's attained age, with the probability of dying of the disease in the same cycle. A person survives the cycle only by escaping both, so the all-cause probability is 1 minus the product of the two survival probabilities. When both come from constant rates over the same cycle with HE-FM-TP-001, the result equals the probability from the summed hazard of HE-FM-BTH-005, the total compound rate that Sonnenberg and Beck form by adding the excess rate to the baseline rate. HE-FM-TP-004 splits the result between background and disease death states where a model keeps them apart.

  • Marginal expected survival and background hazard of a cohort mixing two sexes

    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

    Averages expected survival over the members of a cohort and weights each group's background hazard by its expected survival, the grouped form of equations 18 and 19 of NICE DSU TSD 21 for two groups such as women and men. Because the group with the higher hazard dies sooner, its weight in the marginal hazard shrinks over time, and the marginal background hazard falls below a fixed blend of the two hazards. The same form applies to age groups in a cohort with a spread of entry ages, where it avoids reading one hazard at the mean age. S_star is the expected survival of HE-FM-BGM-001 and h_star the background hazard of HE-FM-BTH-005 for the whole cohort.

Background Mortality — Functions & Formulae | HealthEconomics.wiki