Signature
R_t = S_t / S_star_t
| Inputs | Definition | Unit |
|---|---|---|
S_t | Probability that a member of the patient group is alive at time t, whatever the cause of death | probability |
S_star_t | Probability of surviving to time t in a comparable general population group of the same ages, sex and calendar years, from the life table | probability |
R_t | Relative survival at time t, the ratio of all-cause survival to expected survival | ratio, no unit |
|---|
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
Relative survival from named all-cause and expected survival cells
With all-cause survival in AllCauseSurvival and expected survival at the same time and attained age in ExpectedSurvival, the cell named RelativeSurvival returns the ratio. Filled down a column of model times it gives the relative survival curve.
=AllCauseSurvival/ExpectedSurvival
Assumptions
Expected survival based on fixed life-table rates matched to the patients
S_star_t comes from population mortality rates by age, sex and calendar year, which TSD 21 says are typically assumed to be fixed and known with no associated uncertainty. The general population must be exchangeable with the patients for other-cause mortality, which may need matching on smoking or other lifestyle characteristics, and for conditions with frequent comorbidity general population rates may be too low.
Common time origin and attained age for the two survival curves
Both survival probabilities start at the same time origin, and S_star_t accumulates the life-table rates at each patient's attained age, entry age plus elapsed time, up to t. For a cohort with a spread of ages or both sexes, S_star_t is the average of the individual expected survivals given by HE-FM-BGM-003.
Worked examples
Relative survival at 10 years in the article's illustrative cure model
In the article's illustrative cure model (cure fraction 0.40, uncured excess hazard 0.25 a year, constant background hazard 0.015 a year), all-cause survival of 0.386674 at 10 years over expected survival of 0.860708 gives relative survival of 0.449251, the article's R(10).
S_t = 0.386674; S_star_t = 0.860708; R_t = 0.449251
Relative survival at 30 years close to the cure fraction
At 30 years the same illustrative model gives 0.255263 over 0.637628, a relative survival of 0.400332, almost exactly the cure fraction of 0.40 because nearly all uncured patients have died, as in the article.
S_t = 0.255263; S_star_t = 0.637628; R_t = 0.400332
Relative survival equal to 1 with no excess mortality
When the patients die at the general-population rate the two curves are equal and relative survival is 1 at every time, a limiting case that checks an implementation.
S_t = 0.860708; S_star_t = 0.860708; R_t = 1
Common errors
Relative survival used in place of all-cause survival
Leaving out expected survival treats relative survival as the survival of the patients, so cured patients never die, which TSD 21 section 4.3.5 warns against. In the article's illustrative cure model life years to 30 years rise from 11.93 to 14.40, an overstatement of 2.47 years, or 21%.
Expected survival divided by all-cause survival
Inverting the ratio returns 2.2259 in place of 0.4493 at 10 years in the article's example (computed here for illustration). A relative survival above 1 for a group with excess mortality is the sign of the inversion.
Sources
Relative survival as the ratio of all-cause to expected survival 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 3.7.2, equation 14, which states that relative survival is the ratio of the all-cause survival and the expected survival in the background population, obtained by transferring equation 13, the additive split of the all-cause mortality rate, to the survival scale; the same section lists the assumption that life-table rates are fixed and known with no associated uncertainty, and section 3.7.3 asks for a population exchangeable in other-cause rates.
Plausibility of patient mortality against the general population 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 (pages 75 and 76), which states that it is unlikely that the mortality rate in patients with the disease of interest will be lower than that in the general population and that in nearly all cases it would not be expected to be lower; section 4.3.5, which states that other cause mortality must be incorporated, otherwise a proportion of patients will be predicted never to die; and section 4.2 on general population rates that may be too low when comorbidity is common.
Canonical Identity
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