Estimating excess deaths in the UK, methodology changes: February 2024 – Office for National Statistics

Calculation of excess deaths

The number of excess deaths in each period and age-sex-geography stratum is calculated as the difference between the observed and expected number of deaths:

Where E-hat[i,t] is the estimated number of excess deaths, d[i,t] is the observed number of deaths and d-hat[i,t] is the expected number of deaths in age-sex-geography stratum i in period t.

The estimated total number of excess deaths in each period is obtained by summing estimated excess deaths across age groups, sexes, and geographies:

This "bottom-up" approach ensures additivity throughout the aggregation structure. For example:

estimated excess deaths by age group for males in a particular UK country will sum to the total estimated excess deaths across all age groups for males in that UK country

estimated excess deaths for males and females in a particular UK country will sum to the total estimated excess deaths for both sexes combined in that UK country

estimated excess deaths in individual UK countries (England and Wales combined, Scotland, and Northern Ireland), including deaths among non-residents, will sum to the total estimated excess deaths for the UK

Temporal additivity between monthly and annual estimates of excess deaths is also achieved by summing the estimated excess deaths obtained from the monthly model to derive annual totals. However, weekly estimates will not necessarily sum to annual estimates, as weeks may straddle calendar years at the beginning and end of each year.

It will always be the case that the number of excess deaths in a period is an estimate rather than a known value, because the number of expected deaths is a counterfactual quantity that must be estimated from observed data using statistical techniques. To reflect this uncertainty inherent in expected and excess deaths estimates, 95% confidence intervals are constructed around the excess deaths estimates using the following formula:

Where E-hat[i,t] is the estimated expected deaths in age-sex-geography stratum i in period t, and SE(E-hat[i,t]) is the standard error of the estimate, which is the square root of the variance of the estimate, V(E-hat[i,t]). See our Uncertainty and how we measure it methodology for more information on confidence intervals and standard errors.

The number of excess deaths is estimated as the difference between the observed and expected number of deaths, so the variance of the estimated excess deaths is a combination of the variances of both these quantities. However, the observed number of deaths is a known quantity rather than an estimate, so it has no variance. Therefore:

Where d-hat[i,t] is the expected number of deaths in age-sex-geography stratum i in period t and V(d-hat[i,t]) is its variance, approximated through the Delta method.

The overall variance of the expected total number of deaths across age groups, sexes and geographies in each period can be found by summing the stratum-specific variances within periods:

The population denominators used to calculate mortality rates in each period and age-sex-geography stratum are derived from mid-year population estimates. These population estimates are not timely enough to feed into contemporary estimates of excess deaths. For example, estimates relating to mid-2022 were not published until August 2023 for Northern Ireland, on the Northern Ireland Statistics and Research Agency (NISRA) website, and November 2023 for England and Wales in our 2021-based National population projections bulletin. They have not yet been published for Scotland.

In the future, the Dynamic Population Model and resulting admin-based population estimates may provide more timely estimates (see our Admin-based population estimates: local authorities in England and Wales article). For the time being, the mid-year population estimates are extrapolated with population projections in each age-sex-geography stratum. Historical estimates of excess deaths will be revised whenever population projections for a given year are replaced by the mid-year population estimate.

National population projections are typically updated once every two years but subnational projections (needed for population denominators in the English regions) are only updated once every four years. These are published several months after the corresponding national update. For example, our 2021-based National population projections bulletin was published in January 2024, and before this our 2020-based National population projections bulletin was published in January 2022.

However, our latest available Subnational population projections bulletin (2018-based) was published in March 2020. Therefore, contemporary population sizes for the English regions are obtained by applying the regional proportions from the latest mid-year population estimates to the latest available national population projections for England. This ensures that the population denominators used for calculating mortality rates across the English regions sum to the national population denominator for England.

Population estimates and projections relate to the estimated population size at the mid-point of each year, but population denominators are needed on weekly and monthly bases for excess deaths calculations. Therefore, weekly and monthly population estimates are linearly interpolated between the mid-year estimates.

The pandemic saw a large increase in death registrations, particularly in certain weeks and months that coincided with "waves" of infection (for example, when new COVID-19 variants became widespread in the population). To avoid these periods affecting estimates of expected deaths in subsequent periods, they are removed from the dataset when the model is fitted so that they do not contribute to the mortality baseline. This means that estimates of excess deaths in subsequent periods relate to the additional deaths registered in the period, over and above what would be expected from previous periods had they not been extraordinarily affected by the pandemic.

We define periods extraordinarily affected by the direct mortality impacts of the pandemic as being those where COVID-19 was given as the underlying cause of death for at least 15% of all deaths registered in the period across the UK. This threshold gives the greatest coherence between the weekly and monthly data in terms of periods excluded from the model fitting. These periods are April and May 2020, and November 2020 to February 2021 for monthly data; they are Weeks 14 to 22 of 2020, and Week 45 of 2020 to Week 8 of 2021 for weekly data.

The annual calendar on which we report our weekly mortality statistics usually comprises 52 seven-day weeks and is 364 days in length. By contrast, the Gregorian calendar year (used by most countries across the world) is 365 days long for non-leap years and 366 days long for leap years. This means that the reporting calendar slips out of alignment with the Gregorian calendar by one or two days each year. To avoid this misalignment becoming too severe, there is international agreement that a "Week 53" should periodically be added to the reporting calendar.

Week 53 occurs infrequently (it was last added to the mortality calendar in 2020, and before that in 2015), so it is not practical to estimate a separate seasonal term for it when fitting models to five years of data. Instead, any instances of Week 53 are re-labelled as Week 52 when fitting models and obtaining expected numbers of deaths. This assumes that the mortality rate in a typical Week 53 is similar to a typical Week 52.

In the future, it is anticipated that we will publish estimates of excess deaths in each of the four UK countries as well as the total excess deaths in the UK as a whole. National Records of Scotland (NRS) and NISRA will also separately publish estimates of excess deaths for Scotland and Northern Ireland, respectively, using the same methodology as the Office for National Statistics (ONS). This will ensure consistent and comparable estimates across all parts of the UK.

For consistency with the death registrations data we publish and the devolved administrations, the following models are fitted to estimate excess deaths:

deaths registered in England or Wales, including those for non-residents

deaths registered in Scotland, including those for non-residents

deaths registered in Northern Ireland, including those for non-residents

deaths registered in England or Wales among residents of England

deaths registered in England or Wales among residents of Wales

The total number of estimated excess deaths across the UK is then derived by summing the outputs from the first three models listed. The fourth model listed includes English region of residence as an explanatory variable.

In practice, 10 models are fitted to obtain estimates of excess deaths: five for weekly data and five for monthly data. In addition, five models are fitted to the annual data to obtain standard errors and confidence intervals around the annualised estimates (monthly excess deaths estimates can be summed within years to obtain annual estimates, but this is not possible for the standard errors because of the existence of correlation between successive monthly estimates, which is generally the case with any time series data). To obtain the variance of the annualised estimate, we assume that its coefficient of variation is the same as that of the estimate from the model fitted to annual data.

The models fitted to annual data include age group, sex, English region (only in the model for deaths registered in England or Wales among residents of England), a trend component and the number of weekdays in the year.

In our current approach to estimating excess deaths in England and Wales, and that of the devolved administrations of Scotland and Northern Ireland, the expected (baseline) number of deaths is estimated as the average number of deaths registered in a recent five-year period. In contrast, our new methodology is based on age-specific mortality rates rather than death counts, so trends in population size and age structure are accounted for. Furthermore, the five-year average mortality rate is adjusted for a trend, so historical changes in population mortality rates are also accounted for.

Before the pandemic, the five-year period used in the current methodology was the five years preceding the current year. For example, the expected number of deaths in 2019 was estimated as the average number of deaths registered from 2014 to 2018 (inclusive). Weekly and monthly expected deaths were estimated as the average number of deaths registered in the same week or month over the past five years. For example, the expected number of deaths in Week 1 of 2019 was estimated as the average number of deaths registered in Week 1 from 2014 to 2018 (inclusive).

The expected number of deaths in 2021 was estimated as the average of deaths registered from 2015 to 2019 rather than 2016 to 2020, to avoid the pandemic distorting the excess deaths calculation. The expected number of deaths in 2022 was estimated as the average of deaths registered in 2016, 2017, 2018, 2019 and 2021.

In contrast, individual weeks and months, rather than whole years, that were substantially affected by the immediate mortality impact of the pandemic are removed from the expected deaths calculation under the new methodology.

Other improvements brought about by the change in methodology include:

use of a statistical model means that multiple demographic, trend, seasonal and calendar effects can be included simultaneously in the estimation of expected deaths, and confidence intervals can readily be obtained

a "bottom-up" approach to aggregation means that estimates of excess deaths are additive across age groups, sexes, and high-level geographies, and between months and years

having a common methodology for all four UK countries means that estimates of excess deaths are consistent and comparable across all parts of the UK, and the new methodology is largely coherent (though not identical) to that used by the Office for Health Improvement and Disparities (OHID) to estimate excess deaths in English local authorities.

Excerpt from:

Estimating excess deaths in the UK, methodology changes: February 2024 - Office for National Statistics