Relative Survival

Relative Survival in Cancer Registries and Survival Models

Relative survival measures how much a disease, usually a cancer, shortens survival by setting the survival seen in patients against the survival expected for a similar group of people in the general population. Any shortfall is credited to the disease, so no death has to be classified by cause, which is why population-based cancer registries use it as their standard measure. For health economists it matters twice over. Registry relative survival can supply baseline and long-term survival for models, and its split of the death hazard into a background part and an excess part underlies relative survival models used for extrapolation. The result depends on the matched population and on the estimator chosen.

Observed, expected and relative survival

Observed survival is all-cause survival in the patient group, with every death counted as an event. Expected survival is the survival the same patients would have had at general population death rates, taken from life tables matched on age, sex, calendar year and race, and on area where available, as a National Cancer Institute (NCI) SEER*Stat presentation describes. NICE Decision Support Unit Technical Support Document (TSD) 21 writes the link on the hazard and survival scales.

$$h_i(t) = h_i^{*}(t) + \lambda_i(t), \qquad R_i(t) = \frac{S_i(t)}{S_i^{*}(t)}$$

where $h_i(t)$ is the all-cause mortality rate for patient $i$ at time $t$ since diagnosis, $h_i^{*}(t)$ is the background mortality rate from population life tables, $\lambda_i(t)$ is the excess mortality rate due to the disease, $S_i(t)$ is all-cause survival, $S_i^{*}(t)$ is expected survival in the background population and $R_i(t)$ is relative survival.

Mariotto and colleagues at the NCI describe relative survival as a measure of excess mortality in cancer patients that needs no cause of death information, first motivated by the idea of cure, the point at which patients' survival returns to that of the general population and the excess hazard is zero. The NCI sets out two working assumptions. Competing causes of death are independent, and because a cancer-free comparison group is hard to obtain, general life tables are used on the assumption that cancer deaths are a negligible share of all deaths.

Net survival and the question relative survival answers

Relative survival estimates net survival, which Mariotto and colleagues define as the chance of surviving if the cancer were the only possible cause of death. The NCI calls this the policy-based statistic and notes that, as it is not influenced by changes in mortality from other causes, it suits tracking survival over time and comparing registries.

A patient's prognosis is a different question. The NCI's crude probability of death is the probability of dying of cancer while other causes are also acting, and Mariotto and colleagues judge such crude measures the most valuable for predictive tools, clinical decisions and cost-effectiveness analyses. A decision model needs all-cause survival, as in overall survival, so relative survival enters a model only once expected survival has been put back. Cause-specific survival is the other route to net survival, censoring deaths from other causes and relying on recorded cause of death, and the wider framework is that of competing risks.

Ederer, Hakulinen and Pohar Perme estimators

Registries usually estimate relative survival with a life table in follow-up intervals of a month or a year, as in the life table method. Interval observed survival is divided by interval expected survival and the ratios are multiplied to give cumulative relative survival. Mariotto and colleagues explain that the estimators differ in how long the matched people from the general population are treated as at risk.

The appendix to Dickman, Coviello and Hills gives the formulae. The original Ederer method of 1961 averages expected survival over every patient alive at diagnosis, so matched people stay in the expected group whatever happens to the patient. Ederer II averages over the patients still alive and under follow-up at the start of each interval.

$$\bar{p}_j^{*} = \frac{1}{l_j} \sum_{h=1}^{l_j} p_j(h), \qquad {}_1\bar{p}_i^{*} = \prod_{j=1}^{i} \bar{p}_j^{*}$$

where $p_j(h)$ is the expected probability that a general population person matched to patient $h$ on age, sex and calendar time survives interval $j$, $l_j$ is the number of patients alive at the start of interval $j$, $\bar{p}_j^{*}$ is the Ederer II expected interval survival and ${}_1\bar{p}_i^{*}$ is cumulative expected survival between diagnosis and the end of interval $i$. The Hakulinen method instead uses each patient's potential follow-up time, so matched people leave the expected group when the patient could no longer have been observed.

The SEERStat presentation states that Ederer II, under which matched people are at risk until the patient dies or is censored, is the default, and that the original Ederer and Hakulinen methods are no longer used in SEERStat. Mariotto and colleagues report that most methods give very similar five-year estimates.

Pohar Perme, Stare and Estève noted that the standard estimators do not give cancer mortality independent of national general population mortality, so they are not suited to comparisons between countries, and proposed a net survival estimator that needs no modelling. The SEER presentation explains that it weights each death by the inverse of the expected probability of surviving, so older people carry more weight. Mariotto and colleagues call it the only unbiased estimate of net survival, with a larger variance than Ederer II that can make long-term estimates in small data unstable. In a tutorial by Dickman and Coviello on localised colon cancer, five-year estimates were 0.7755 by Ederer II and 0.7797 by Pohar Perme, and the authors note that Ederer II bias grows with the spread of expected survival among patients, so estimating within age groups brings the two closer.

Cohort, period and age-standardised estimates

The SEER presentation describes a cohort approach, which includes only years of diagnosis with potential follow-up for the full duration, and a period approach, which uses only the most recent interval survival of patients diagnosed in different years. The period estimate is cross-sectional, so it reflects recent survival rather than the experience of a single diagnosis cohort.

Survival depends on age at diagnosis, and age mixes differ between areas and periods, so registries often age-standardise. SEER*Stat uses direct standardisation.

$$R_{\text{std}}(t) = \sum_{a} w_a \, R_a(t)$$

where $R_a(t)$ is relative survival in age group $a$, $w_a$ is the share of age group $a$ in the standard population, and $R_{\text{std}}(t)$ is age-standardised relative survival. The SEER presentation notes that standardised figures help comparison but may not represent survival for a specific population. A model of a particular patient group needs age-specific estimates instead.

Worked example with an illustrative registry cohort

All numbers here are illustrative. A registry cohort with one cancer is followed for three years in annual intervals. At the start of year 1 half the patients are aged about 60, with expected one-year survival of 0.99, and half about 85, with 0.91, so the Ederer II expected interval survival is their average.

$$\bar{p}_1^{*} = 0.5 \times 0.99 + 0.5 \times 0.91 = 0.95$$

where 0.5 is each age group's share of patients alive at the start of year 1 and 0.99 and 0.91 are their expected one-year survival probabilities.

1. Interval ratios. Observed interval survival is 0.80, 0.85 and 0.90 in years 1 to 3 and expected interval survival is 0.95, 0.94 and 0.93, so interval relative survival is 0.80 / 0.95 = 0.8421, 0.85 / 0.94 = 0.9043 and 0.90 / 0.93 = 0.9677.

2. Cumulative curves. Cumulative observed survival is 0.80, then 0.80 × 0.85 = 0.68, then 0.68 × 0.90 = 0.612. Cumulative expected survival is 0.95, then 0.95 × 0.94 = 0.893, then 0.893 × 0.93 = 0.8305.

3. Cumulative relative survival. The product of the interval ratios gives the three-year figure.

$$R(3) = 0.8421 \times 0.9043 \times 0.9677 = 0.7369 = \frac{0.612}{0.8305}$$

where $R(3)$ is relative survival at three years, equal to cumulative observed divided by cumulative expected survival. About 74% of patients would survive three years if the cancer were the only cause of death, while 61.2% actually survive.

4. Excess hazard. Treating each ratio as $\exp(-\lambda_j)$ over one year, the average excess hazard is $-\ln 0.8421 = 0.1719$ in year 1, $-\ln 0.9043 = 0.1006$ in year 2 and $-\ln 0.9677 = 0.0328$ in year 3. A falling excess hazard is the pattern that motivates a cure model.

5. Use in a model. A decision model for a younger cohort has expected interval survival of 0.97, 0.965 and 0.96, so cumulative expected survival is 0.97, 0.9361 and 0.8986. If the excess hazard carries over unchanged, modelled all-cause survival is $R(t) \times S^{*}(t)$.

$$S(3) = 0.7369 \times 0.8986 = 0.6622$$

where $S(3)$ is modelled all-cause survival at three years. The one-year and two-year values are 0.8421 × 0.97 = 0.8168 and 0.7615 × 0.9361 = 0.7128. The model cohort does better than the registry cohort (0.6622 against 0.612) only because its background mortality is lower, and a common excess hazard at younger ages is an assumption to test.

Relative survival in health economic models

TSD 21 explains that excess mortality models split all-cause mortality into other-cause and disease parts, fit a parametric model to the excess part and add population mortality back for long-term extrapolation, so that the external rates capture the rise in hazard that comes with population ageing. This is the approach behind the relative survival model and the net survival model. TSD 21 reports that such models are typically applied to population-based cancer registry data where cause of death is missing or unreliable, and in 2020 its authors knew of no use of the approach in health technology assessment.

Because registries do not include patients on new treatments, TSD 21 judges registry data likely to be of most use for extrapolation of the control group, after comparing the trial, disease and registry populations.

A registry estimate is net of background mortality, so it becomes an all-cause input only after multiplying by expected survival for the modelled cohort, as in step 5. Adding life-table deaths on top of an all-cause curve counts the same deaths twice, an error the background mortality page covers.

When relative survival fails and cause-specific survival is preferred

Relative survival is only as good as its life table. Mariotto and colleagues call it the preferred method for comparing registries and countries, because cause of death may be missing or recorded unevenly, but warn that it can be biased when patients differ from the general population on factors that affect deaths from other causes, such as socio-economic status, health status and smoking.

The bias runs both ways. If expected survival is too low, relative survival is overestimated and can exceed 100%, and Mariotto and colleagues give cancers found largely by screening, such as localised prostate and breast cancers, as the clearest case, citing PLCO screening trial participants with 30% to 50% lower mortality than expected from heart disease, injury and kidney disease. If expected survival is too high, as for smoking-related cancers such as lung cancer, relative survival is underestimated. Mariotto and colleagues favour cause-specific survival where suitable life tables do not exist and for factors such as comorbidity that life tables do not reflect.

Common misreadings

Most errors come from reading a net, ratio-based measure as an ordinary survival probability. Each one can change a model input or a comparison between registries.

1. Reading it as a patient's chance of survival. Mariotto and colleagues note that relative survival is often used this way, although it represents the net effect of the cancer alone. A patient's actual prognosis needs crude probabilities that include other causes.
2. Comparing Ederer II estimates across countries or age mixes. The standard estimators carry the influence of national population mortality, and Ederer II bias grows with the spread of expected survival, so a gap between registries may reflect the life table and estimator more than the cancer.

Sources

• Dickman PW, Coviello E. Comparison of life table estimators of relative/net survival. Tutorial, 2 March 2019. https://pauldickman.com/software/stnet/comparestrs/
• Dickman PW, Coviello E, Hills M. Estimating and modelling relative survival, appendix: formulae for estimation of expected survival. Undated, accessed 28 October 2026. https://www.pauldickman.com/software/sas/expected.pdf
• Mariotto A, Howlader N, Scoppa S, Green D. An overview of survival statistics in SEER*Stat. NCI Analytic Tools SEERies presentation, 28 October 2021. https://seer.cancer.gov/news/SEER-Relative-Survival-Overview.pdf
• Mariotto AB, Noone AM, Howlader N, et al. Cancer survival: an overview of measures, uses, and interpretation. Journal of the National Cancer Institute Monographs. 2014;2014(49):145-186. https://doi.org/10.1093/jncimonographs/lgu024
• National Cancer Institute, Surveillance Research Program. Measures of cancer survival. Accessed 28 October 2026. https://surveillance.cancer.gov/survival/measures.html
• Pohar Perme M, Stare J, Estève J. On estimation in relative survival. Biometrics. 2012;68(1):113-120. https://doi.org/10.1111/j.1541-0420.2011.01640.x
• 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. Decision Support Unit, University of Sheffield; 2020. Sections 3.7 and 4.3.6. https://doi.org/10.15131/shef.data.33849118