Input variables

A few input variables in the sidebar allow to determine how the main figure displaying excess mortality in the main panel is built. As a first step, a ‘Country or region’ has to be selected out of the 38 countries and regions available in the STMF. The database provides death counts and death rates for all ages combined and by broad age groups, i.e., for 0-14, 15-64, 65-74, 75-84, 85+. One of these mortality measures has to be selected as the ‘Measure’ variable. The user has to define the ‘Sex’ variable as well, with its values being females, males and both sexes.

From the available years for the selected country or region a ‘Target year’ can be selected. The target year is the main focus of interest and it is compared to a reference level to quantify the magnitude of the excess mortality in the target year, i.e., the difference between the target year and the reference level. The reference level is calculated for a user-defined reference period, a subset of the available years in the dataset. The next section, ‘Estimation of excess mortality’ provides details on the calculation of the implemented ‘Reference level’ input variable.

It is possible to plot all available years for a country or region by clicking on the ‘Show other years’ checkbox. When checked, the available years appear in light gray color and information on all data points are accessed by the click point information (see details in the ‘Graphical and numerical presentation of excess mortality’ section).

Selecting the ‘Additional country or region’ checkbox reveals a new graphical panel below the main figure with the same features as the main figure and can be calibrated independently with input controls appearing additionally. This enables the user to compare two different countries and regions with each other or even different target years or reference levels, etc. of the same country or region.

With the help of the colored boxes next to the paintbrush icon the user can change the color scheme applied to the figures.

Estimation of excess mortality

Direct and indirect estimations of excess mortality during wars and epidemics have a long history. In order to estimate the excess flu or heatwave deaths, researchers usually rely on the difference between the observed and expected (according to the usual conditions) number of deaths. The crucial methodological challenge is to estimate the expected mortality pertaining to a given week.

In the literature, there are two major approaches to the estimation of the intra-annual excess mortality [9, 10]. The first one is focused on the variation of mortality across weeks within a year in question and expresses a notion of “seasonality” [11–14]. The second one (actively used for assessment of the COVID-19-related mortality) investigates the mortality deviations for certain weeks compared to the mortality experience of previous years [10, 15–17]. This estimation depends on two components: a general mortality trend during the last few years and seasonal fluctuations. Such an estimation can be done using regression analysis [18] or the method of time series analysis [19, 20]. In the web application, based on various assumptions we implemented six different methods of the reference level for excess mortality estimation: the week-specific averages, the week-specific trends, the week-specific lower quartiles, the yearly average-week, the summer average-week, and the yearly lower-quartile-week. The first three methods determine reference levels corresponding to the second estimation approach, while the latter three correspond to the first approach. Nevertheless, this set should not be considered as exhaustive and we implemented them to present the variability of estimates.

Let x_ij denote the value of the user-selected measure on week i in year j.

The Week-specific Averages for week i, equals the arithmetical mean over the years in the selected period and is given by (1) where the set P contains all years in the selected reference period and |P| denotes the number of elements in the set P. The reference period may or may not include the target year.

For the target year T and for week i, the reference level predicted by the Week-specific Trends, is the expected value of a linear model estimated by ordinary least squares (OLS) over the years in the selected reference period. If there are 22 weeks available in the target year then the reference level consists of the results of 22 independent linear models. The estimation is carried out via the lm function of the stats package in R [21]. The reference level is given by (2) where and are the estimated coefficients of the OLS regression over the reference period for week i.

The Week-specific Lower Quartiles for week i, is the average of the values of the selected measure below the lower quartile in the reference period for week i. We implemented the calculation of the lower quartile with the help of the quantile function from the stats package. If there are 34 weeks available in the target year then this reference level requires the calculation of 34 lower quartiles, one for each week separately. Therefore, this reference level consists of values defined by (3)

The Yearly Average-week, is the mean value of values defined by Eq (1) for the available weeks in the target year and is calculated by (4) where W is the set of available weeks in the target year. Please note, if not all weeks are available in the target year, then |W| could be lower than 52. As a consequence, denotes the expected level of mortality if every week had the same average level of mortality in the year. This reference level, in contrast to the week-specific averages, reflects the seasonal variation of mortality within a calendar year or a number of years.

The Summer Average-week, is similar to the above-mentioned yearly-average-week value, i.e. , but excludes from the calculation the winter weeks that tend to have higher mortality in general in comparison to summer weeks. For countries and regions in the Northern Hemisphere we define winter season as weeks from calendar week 1 to week 12 and from 48 to week 52 included, and weeks 22 and 38 for those situated in the Southern Hemisphere. Thus, generally, the value of is expected to be lower than . The formula for the Summer Average-week is given by (5) with W* denoting the set of non-winter weeks, i.e., available weeks between calendar weeks 13 and 47 for the Northern Hemisphere and weeks 1 to 21 and 39 to 52 in the Southern Hemisphere.

The Yearly Lower-quartile-week, is the average of the values not greater than the lower quartile based on all the values of the user-selected measure in the reference period. This reference level requires the calculation of only one lower quartile value over a single year, e.g., for the target year, or over multiple years (lower quartile of the reference period). The Yearly Lower-quartile-week is defined by (6)

Eqs 1, 2 and 3 correspond to the mainstream approach that compares the observed week-specific mortality with that during a certain reference period. Correspondingly, the reference level of mortality varies across weeks. Eq 1 determines the reference level as the average over the reference period (following the excess mortality monitoring procedures by the Office of National Statistics in England and Wales, the New York Times, the Financial Times, the Our World in Data, etc.). In some countries (especially countries of Eastern Europe) mortality was steeply decreasing over the last 15-18 years. To address these trends, Eq 2 determines the baseline as a continuation of the week-specific trends. To address the heat-wave outbreaks and other mortality elevations out of the winter season, Eq 3 defines the reference level of mortality as the average mortality for the lower quartile of the mortality distribution across weeks within the reference period.

According to the mainstream practice by the Office of National Statistics in England and Wales and others, the reference period includes several years (usually 4-7 years) preceding the target year. For example, the years 2015-19 with 2020 as a target year. However, depending on the purpose, the reference period can be longer or shorter and may also include the target year (e.g. interpolation instead of extrapolation).

Eqs 4, 5 and 6 correspond to the “seasonality” approach. The excess mortality expresses losses due to the mortality difference between different periods within a year (or a number of years). Here, the reference level of mortality is a week-independent constant. In Eq 4, the reference level is defined as the average across all available weeks. In Eq 5, this level is defined as an average of mortality over a lower mortality season (non-winter season). In Eq 6, the low mortality season is defined in a flexible way according to mortality across weeks based on the lower quartile.

Eqs 4 to 6 may be used with the reference period consisting of the target year only. In this case, the excess mortality expresses the mortality elevation compared to the average mortality for all 52 weeks or to the average mortality over the lower-mortality weeks of the target year. However, if weekly mortality experiences large random fluctuations (in countries with small populations), the reference period could include the target year and a few years before this year, e.g., the years 2014-18 with 2018 as a target year.

Graphical and numerical presentation of excess mortality

An important feature of the visualization tool is the possibility to estimate the intra-annual excess mortality. Upon opening the application, a country or region is presented with a focus on the excess mortality in 2020 compared to the reference level of week-specific averages over the whole available period of the previous years. There are three possible ways of interaction with the figure.

Firstly, hovering with the cursor over an interval that contains excess mortality (or deficit) polygon, information on this polygon is provided in a pop-up bubble next to the cursor, detailing the start and end weeks; and the length of this polygon. The example in Fig 2 gives information on a 25-week-long excess mortality polygon between weeks 1 and 25.

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Fig 2. Hover information of a polygon.

Information on the duration—the start, end and length—of an excess mortality polygon appears on hovering over a polygon.

https://doi.org/10.1371/journal.pone.0246663.g002

Second, selection in the figure by brushing with the cursor reveals summary information below the figure. The sum of death counts or the mean of the death rates over the selected weeks are shown by sex and the type of the difference between the target year and the reference level. The example in Fig 3 shows a selection between weeks 8 and 20 for the year 2020 compared to the reference period of 2010-2019 for England and Wales. During these weeks England and Wales faced a total of 61275 excess deaths. Please note that the selection is week-dependent and summary statistics might not correspond to the exact area of the selection on the other axis, e.g., in this figure the tip of the excess mortality polygon is not selected.

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Fig 3. Summary statistics information of a selection.

Information on death counts or death rates of the selected weeks by sex and type of difference are displayed below the figure.

https://doi.org/10.1371/journal.pone.0246663.g003

Third, when the checkbox ‘Show other years’ is selected clicking in the figure in the vicinity of a data point reveals information on the nearest data point. Fig 4 presents a click event for French females with target year 2020. The click point information identifies the data point from 2003 on week 33 that pertains to the unusually high amount of the total deaths due to the heatwave in that year. As a consequence, click point information can speed up outlier identification significantly.

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Fig 4. Click point information.

Clicking in the figure helps identifying the nearest data point by revealing which year and week it pertains to.

https://doi.org/10.1371/journal.pone.0246663.g004

All the above-mentioned interaction methods are available for the additional country or region panel, therefore, summary and click point information can be directly compared below the figures.