Standardised Mortality Ratio

Standardised Mortality Ratio in Hospital Performance and Economic Models

A standardised mortality ratio (SMR) asks how many deaths a group would have had if its members, band by band, had died at the rates of a reference population, and then sets the deaths actually recorded against that figure. A ratio of 1, often written as 100, means mortality in line with the reference, while 1.25 or 125 means 25% more deaths than the group's own make-up predicts. In health systems the method sits behind English hospital mortality indicators such as the Summary Hospital-level Mortality Indicator (SHMI) and the Hospital Standardised Mortality Ratio (HSMR), which have been used to select NHS hospital trusts for investigation. In economic evaluation, an SMR estimated for people with a condition is a common way to scale general population death rates inside a decision model. Both uses rest on indirect standardisation, and the weighting it implies limits which comparisons between two SMRs are valid.

Indirect standardisation and the expected count

Crude death rates mostly reflect age and illness mix, so a comparison has to hold that mix constant. Indirect standardisation keeps the study group's own structure and borrows death rates from a standard population. Breslow and Day, in their International Agency for Research on Cancer (IARC) text on cohort studies, describe the SMR as comparing the observed number of deaths with an expected number obtained by applying the standard rates to the cohort's age structure.

$$\text{SMR} = \frac{D}{E^{*}} = \frac{\sum_{j} d_j}{\sum_{j} n_j \lambda^{*}_j}$$

where $D$ is the total observed number of deaths in the study group, $E^{*}$ is the expected number, $j$ indexes the strata (usually age bands, sometimes also sex, diagnosis or period), $d_j$ is the number of deaths observed in stratum $j$, $n_j$ is the person-years or admissions of the study group in stratum $j$, and $\lambda^{*}_j$ is the death rate of the standard population in stratum $j$.

Breslow and Day note that William Farr used the SMR in the 1855 annual report of the Registrar General to compare mortality in occupational groups. They show that the SMR is a weighted average of the stratum-specific ratios $d_j / (n_j \lambda^{*}_j)$, with weights equal to each stratum's expected deaths. When the true rate ratio is the same in every stratum, those weights make the SMR the minimum variance estimate of that common ratio.

Only total deaths are needed, so the SMR can be computed where deaths by age are not published, although without them the assumption of constant rate ratios cannot be tested.

Direct standardisation and the comparative mortality figure

The alternative is direct standardisation, which keeps the study group's own stratum rates and applies them to a fixed standard population. Dividing the resulting directly standardised rate by the standard population's rate gives what Breslow and Day call the comparative mortality figure (CMF).

$$\text{CMF} = \frac{\sum_{j} w_j \, d_j / n_j}{\sum_{j} w_j \lambda^{*}_j}$$

where $w_j$ is the standard population weight for stratum $j$ (for example, its person-years), $d_j / n_j$ is the study group's own death rate in stratum $j$, and $\lambda^{*}_j$ is the standard death rate in that stratum, as above.

The two measures trade bias against variance. The CMF needs reliable rates in every stratum and becomes unstable when some strata hold few deaths, while the SMR's standard error depends only on the total number of deaths. Breslow and Day give $\text{SE}(\log \text{SMR}) = 1/\sqrt{D}$, so an SMR built on 100 deaths has a standard error on the log scale of 0.1. They judge the SMR the more appropriate measure when the sample is small and statistical significance is at issue.

The price is comparability. Each SMR weights its stratum ratios by its own expected deaths, so two groups with different age or case mix are averaged with different weights. Breslow and Day show that the ratio of two SMRs may lie completely outside the range of the stratum-specific rate ratios when each group's own stratum SMRs vary across strata and the two groups have very different structures, even if the rate ratio between the groups is the same in every stratum. They also cite Kilpatrick's observation that the ratio of two CMFs is a CMF while the ratio of two SMRs is not an SMR. In practice the two measures usually give close results. The observed-to-expected ratio page sets out the same problem for risk-model-based indicators.

Hospital mortality ratios in the NHS

Hospital indicators apply indirect standardisation to admissions, adding diagnosis and admission type to age and sex, a form of case-mix adjustment. Jarman and colleagues, in their 1999 study of English hospital death rates, defined the hospital indirectly standardised mortality ratio as actual deaths divided by expected deaths, multiplied by 100. Expected deaths came from national death rates stratified by 10-year age group, sex and 85 primary diagnoses, which together accounted for 80% of in-hospital deaths.

The SHMI is the trust-level mortality indicator published by NHS England. NHS England's GOV.UK statistics page describes it as a monthly Accredited Official Statistic giving the ratio between the actual number of patients who die following hospitalisation at a trust and the number expected on the basis of average England figures, given the characteristics of the patients treated there. The 2015 indicator specification from the Health and Social Care Information Centre counts deaths in hospital or within 30 days of discharge and fits a separate logistic regression for each diagnosis group, with age, gender, admission method, year index and the Charlson Comorbidity Index as case-mix terms. That version banded trusts as higher than expected, as expected or lower than expected using 95% control limits from a random effects model that allows for over-dispersion between trusts.

These ratios have shaped regulatory action. In 2013 Sir Bruce Keogh reviewed 14 trusts selected because they had been outliers for two consecutive years on either the SHMI or the HSMR. The review was commissioned after high mortality at Mid Staffordshire NHS Foundation Trust had been linked with failures of care. His overview report noted that the two measures produced two completely different lists of outlier trusts, and called the use of such statistics to count avoidable deaths "clinically meaningless and academically reckless". It also observed that coding practice can move excess death rates up or down without any change in the number of lives saved.

The evidence that followed supported that caution. Hogan and colleagues reviewed 100 randomly selected deaths in each of 34 English acute trusts and judged 3.6% of deaths avoidable (95% confidence interval 3.0% to 4.3%). The association between the proportion of avoidable deaths and either the HSMR or the SHMI was small and not statistically significant. They concluded that any metric based on mortality is unlikely to reflect the quality of a hospital, and that SMRs are best restricted to conditions with high case fatality and good clinical data.

SMRs as inputs to decision models

Decision models need death rates for people who survive the period covered by trial data. A common approach takes age- and sex-specific rates from a national life table and multiplies them by an SMR for people with the condition. Carey and colleagues, evaluating tisagenlecleucel for relapsed or refractory diffuse large B-cell lymphoma in Ireland with a partitioned survival model, treated patients alive after 60 months as long-term survivors subject to age- and sex-matched general population mortality with an SMR of 1.36, taken from a Danish population-based study. The SMR entered their probabilistic sensitivity analysis with a log-normal distribution.

Three choices decide whether this is sound. The SMR must multiply rates or hazards and not annual probabilities, a point developed on the all-cause mortality page. The source cohort has to resemble the modelled patients in disease stage and time since diagnosis, because an SMR measured early after treatment usually differs from one measured among long-term survivors. A single SMR applied at every attained age also assumes a constant rate ratio across ages, the same assumption Breslow and Day say is needed to justify the summary. When the excess is large in young adults and small in old age, a summary SMR estimated in a younger cohort overstates mortality for an older modelled cohort, as the worked example shows.

Worked example with two regional cohorts and two age bands

All figures are illustrative. Two regional registers follow people with the same chronic condition for one year. The reference is national general population mortality of 0.5% per person-year under 65 and 4.0% per person-year at 65 and over.

Step 1, expected deaths. For Region A, 8,000 × 0.005 = 40 and 2,000 × 0.04 = 80, so $E^{*}$ = 120. For Region B, 2,000 × 0.005 = 10 and 8,000 × 0.04 = 320, so $E^{*}$ = 330.

Step 2, the SMRs. Region A records 80 + 96 = 176 deaths, an SMR of 176 / 120 = 1.47. Region B records 22 + 416 = 438 deaths, an SMR of 438 / 330 = 1.33. Read side by side, Region A appears to have the worse excess mortality.

Step 3, the band ratios. Region B has the higher death rate in both bands, 1.1% against 1.0% under 65 (a ratio of 1.10) and 5.2% against 4.8% at 65 and over (a ratio of 1.08). The reversal arises because the younger band, where the excess is greatest, carries one third of Region A's expected deaths (40 of 120) but only 3% of Region B's (10 of 330). The ratio of the two SMRs, 1.33 / 1.47 = 0.90, lies outside the range of the band ratios.

Step 4, the CMFs. With an illustrative standard of 5,000 person-years in each band, standard deaths are 5,000 × 0.005 = 25 plus 5,000 × 0.04 = 200, a total of 225. Region A's own rates give 5,000 × 0.010 = 50 plus 5,000 × 0.048 = 240, a total of 290 deaths and a CMF of 290 / 225 = 1.29. Region B's give 5,000 × 0.011 = 55 plus 5,000 × 0.052 = 260, a total of 315 and a CMF of 315 / 225 = 1.40. The CMF ratio, 315 / 290 = 1.09, lies between the band ratios of 1.08 and 1.10, and puts Region B correctly above Region A.

Step 5, precision. Observed deaths are treated as approximately Poisson counts, and Byar's approximation, set out on the observed-to-expected ratio page, gives limits for 176 deaths of 150.96 and 204.01, so Region A's 95% interval is 150.96 / 120 = 1.26 to 204.01 / 120 = 1.70. For 438 deaths the limits are 397.93 and 481.01, giving 397.93 / 330 = 1.21 to 481.01 / 330 = 1.46 for Region B. Both confidence intervals exclude 1, and they overlap. Step 3 has already shown that the two SMRs use different weights, so the SMRs give no basis for ranking the regions.

Step 6, transport into a model. A modeller simulating patients aged 65 and over in Region A who used the summary SMR would set the annual death rate at 1.467 × 0.04 = 0.0587, an annual probability of $p = 1 - e^{-0.0587} = 0.0570$. The band-specific ratio gives 1.2 × 0.04 = 0.048, so $p = 1 - e^{-0.048} = 0.0469$. The summary SMR overstates annual mortality for this group by about one fifth, which shortens modelled survival and tends to shrink the QALY gains of any treatment that extends life.

Interpreting SMRs and common errors

An SMR answers one narrow question, whether a group had more deaths than its own structure predicts under reference rates. Several readings go beyond that.

1. Ranking providers by SMR. Two SMRs use different weights, so a league table of SMRs is not a like-for-like comparison. Funnel limits from statistical process control handle chance better.
2. Reading excess deaths as avoidable deaths. Keogh's report and the Hogan study both show that excess deaths implied by a high ratio do not count deaths caused by poor care.
3. Forgetting what the expected count depends on. Clinical coding and the choice of reference population both move the expected count, so the ratio can change without any change in quality of care. Broader questions of model choice are covered under risk adjustment.
4. Treating a summary SMR as age-free. In a decision model, a single SMR assumes a constant rate ratio. Band-specific SMRs or a modelled excess hazard are safer when the ratio clearly varies with age.

Sources

• Breslow NE, Day NE. Statistical Methods in Cancer Research. Volume II, The Design and Analysis of Cohort Studies. IARC Scientific Publications No. 82. Lyon: International Agency for Research on Cancer; 1987. Chapter 2. https://publications.iarc.who.int/download/SP82_vol2-2.pdf
• Carey N, Leahy J, Trela-Larsen L, Mc Cullagh L, Barry M. Cost-utility and value of information analysis of tisagenlecleucel for relapsed/refractory diffuse large B-cell lymphoma in the Irish healthcare setting. Journal of Market Access and Health Policy. 2023;11(1):2166375. https://doi.org/10.1080/20016689.2023.2166375
• Health and Social Care Information Centre. Indicator Specification: Summary Hospital-level Mortality Indicator. Version 1.19, 10 June 2015. https://assets.publishing.service.gov.uk/media/5a7f5dd4ed915d74e622a06a/SHMI_specification.pdf
• Hogan H, Zipfel R, Neuburger J, Hutchings A, Darzi A, Black N. Avoidability of hospital deaths and association with hospital-wide mortality ratios: retrospective case record review and regression analysis. BMJ. 2015;351:h3239. https://doi.org/10.1136/bmj.h3239
• Jarman B, Gault S, Alves B, Hider A, Dolan S, Cook A, Hurwitz B, Iezzoni LI. Explaining differences in English hospital death rates using routinely collected data. BMJ. 1999;318(7197):1515-1520. https://doi.org/10.1136/bmj.318.7197.1515
• Keogh B. Review into the quality of care and treatment provided by 14 hospital trusts in England: overview report. NHS England; 16 July 2013. https://web.archive.org/web/20150417043907/http://www.nhs.uk/NHSEngland/bruce-keogh-review/Documents/outcomes/keogh-review-final-report.pdf
• NHS England. Summary Hospital-level Mortality Indicator (SHMI), Deaths associated with hospitalisation, England, June 2025 to May 2026. GOV.UK statistics page, published 8 October 2026. https://www.gov.uk/government/statistics/summary-hospital-level-mortality-indicator-shmi-deaths-associated-with-hospitalisation-england-june-2025-may-2026