An improved definition of official excess winter mortality statistics as the basis for detailed analysis and monitoring

2.1Time series decomposition

At face value the expression ‘excess winter mortality’ suggests that we would like to measure the number of deaths which occur additionally during winter compared with some ‘normal’ number of deaths for other periods. The question is therefore how best to construct a reference ‘normal’ number against which the current season’s deaths can be compared. This is not as simple a process as seasonally adjusting the time series of recorded deaths, since in this case the normal winter peak would be detected by the seasonal adjustment algorithm and removed. We can however use the calculated adjustment as a part of our estimate.

Figure 2.

Monthly deaths series for England and Wales (January 1970–July 2019), with seasonally adjusted and trend-cycle estimates.

Take as an example the monthly time series of registered deaths in England and Wales (Fig. 2, black/top line). The monthly mortality data are compiled from national daily death occurrences derived from death registrations, available from the latest Excess Winter Mortality Bulletin [21]. This series is clearly seasonal, with peaks in mortality in the winter months. A trend is therefore needed as an estimate of the “normal” number of deaths, and in standard time series decompositions this is called the trend-cycle. If we use 1/2(da+ds) as an estimator, it will be lower than the long-term average level of the series as depicted by the trend-cycle component in Fig. 2, because it does not include the winter period. Therefore, the current “normal” number of deaths used as a reference level in the definition of EWM will cause it to overstate excess deaths slightly. This is illustrated in Fig. 3, which depicts the deaths occurring during the defined winter period (December to March) against the reference level (average of non-winter deaths).

Figure 3.

Number of deaths as a function of winter period (black/top line) and the average of non-winter months (red/bottom line). The dotted line represents the mean number of deaths for the whole span.

A traditional time series decomposition [28] renders an observed time series into three components, the trend T, seasonal S and irregular I. These components can either sum

or multiply

to form the original series Y.

The England and Wales mortality series is best fitted by a multiplicative model, so we use this in our description below, although it can straightforwardly be translated to an additive model instead. Decomposition is achieved either by the iterative application of various types of moving averages (a non-parametric approach, typically using the X-11 filter [29]) or using a model-based approach [30].11

National statistical institutes (NSIs) around the world present seasonally adjusted series which have the estimated seasonal component removed, leaving

The monthly seasonally adjusted time series of deaths, shown with a red (medium saturation) line in Fig. 2, removes some of the volatility in the original series, but is still rather variable. To measure excess winter mortality, however, we do not want this traditional seasonally adjusted estimate. The concept most nearly corresponding to the current EWM definition would be to take the seasonal and the irregular, that is a trend adjusted series

This now incorporates the seasonal fluctuation above the long-term average level of the series represented by the trend – in other words the amount that mortality is regularly higher in the winter periods, and the irregular – which contains an estimate of the part of the original series which is not the regular pattern, and which is therefore of primary interest in detecting unusual mortality events. These elements of the decomposed time series are shown diagrammatically in Fig. 4. Note that E⁢(I)= 1 for a multiplicative decomposition, so unusual mortality, represented by the I component, could be < 1 so that T×S×I may be smaller than T×S.

Figure 4.

Diagrammatic representation of decomposition of the mortality time series into trend T, seasonal, S and irregular I components. (W) and (S&A) respectively refer to winter and combined spring and autumn components respectively. The descriptions show how the components relate to measures of winter mortality – see text for more details. There is an irregular term for the non-winter period too, but as the primary interest is in winter mortality, it is not included in the diagram.

YTA captures excess deaths in the winter period, which is composed of a regular seasonal component and an irregular component that captures unexpected mortality. Seasonality can be stable, or can evolve, but when it evolves, this typically happens rather slowly. Therefore, the S element in a series of YTA estimates will be quite similar from year to year. Its effect will be largely to change the level of the YTA series, where the interesting shorter term changes will be in the irregular. While it would be possible to gather almost all the same information from just the irregular, including the seasonal improves the picture of deaths relative to the trend, and is necessary for comparisons over long periods.

We will define YTA=S×I as excess winter mortality and submit that this is closer to the intended meaning than the existing measure. To distinguish it, however, we label it TEWM (for trend-based excess winter mortality). It measures the extra mortality in winter relative to a long-run average (measured here by the trend). The winter period irregular I(W) is then defined as unusual winter mortality, that is the extra (proportional, in the multiplicative case) mortality compared to a typical winter (which may be less than 1 in a winter with low mortality).

2.2Properties of TEWM

TEWM has several advantages. It is minimally affected by unusual numbers of deaths in non-winter periods, as we illustrate more fully in Section 3, and therefore gives a more objective measure of excess winter deaths. It is based on a time series decomposition, which means that it is not necessary to wait for the comparator periods to become available as in the EWM measure, which cannot be calculated until the spring (Apr–July) period following the winter of interest has elapsed and its data have been processed. TEWM can be calculated as soon as March data have been collected and processed, and would therefore be available around four months earlier than EWM.

TEWM also has some disadvantages. Time series decomposition is achieved with the iterative application of complex moving averages, or using a model-based approach. Therefore, the values for the irregular, seasonal and trend elements are directly dependent on the filters or fitted model parameters used for the decomposition procedure. Importantly, this will lead to revisions in TEWM as new data points are added. Revisions in the existing EWM measure are minimal and occur only when provisional numbers are used for the most recent estimate.

Another issue that affects TEWM is whether any data points are identified as outliers; the standard approach to outliers in time series decomposition is to make a temporary prior-adjustment, where the outlier is replaced (wholly or partially) by an imputed value for the purposes of analysing the series, and then the prior adjustment is reversed in the output [29]. The identification of outliers can be subjective (based on prior knowledge about the series), or may be automated using a suitable algorithm. There are various methods for automatic identification of outliers, which may lead to different outcomes [31]. In addition, unusual data points at the beginning and end of a series cannot always be clearly classified as either an outlier or a level shift. This classification can have a big impact on the decomposition, because outliers are assigned to the irregular element, while level shifts are assigned to the trend-cycle [32].

3.1Method

Monthly time series of deaths in England and Wales spanning August 1970 to July 2019 were aggregated into four-month sections (quadrimesters) reflecting the dw, da and ds season periods described above (148 observations). The data were decomposed into T, S and I elements using TramoSeats [30] and automatic ARIMA model selection, which resulted in a (011)(011) model. The model order refers in each bracket to an autoregressive (AR) component, the differencing and a moving average (MA) component respectively, and the first bracket shows the quadrimestral information and the second bracket the annual information. The 2017/18 winter period was identified as an outlier, which is in line with the comment from the Excess Winter Mortality publication identifying it as the highest excess since the 1970s. Thus in this case the model is

where y^t is the estimate of log(deaths), yt are the observed log(deaths) values, εt are the residuals (on the log scale), all for quadrimester t, θ1 and Θ1 are the non-seasonal and seasonal MA parameters, and t-3 is one year earlier than t because we deal with quadrimesters. The final variable, xW⁢2017⁢β, represents the outlier for the winter 2017 value, with its estimated coefficient β= 0.13, and the zt error term is white noise.

Figure 6.

Comparison of standardised values for EWM (black/darkest line), EWM as measured by YTA (green/lightest line), and the Irregular factor only (red/medium saturation line) for four-month winter periods.

3.2Results

The current ONS method of calculating EWM is compared to the I and I×S (the latter is TEWM) components (as the decomposition was multiplicative) for the winter periods in Fig. 5. It can be seen that the patterns of the estimates are similar, but the level has shifted in line with the new definition which is now relative to the mean of the whole series (as in Fig. 4). Figure 6 shows standardised versions of the series from Fig. 5 (by subtracting the mean and dividing by the standard deviation of the series for the period considered) to better demonstrate the differences. There are a few obvious points of divergence (e.g. winters 1993/94, 2002/03, 2003/04, indicated by dotted vertical lines in Figs 5 and 6).

Figure 7.

Top panel: Irregular terms from the time series decomposition, shown separately as series for winter, spring and autumn quadrimesters. Bottom panel: The irregular in the winter quadrimester and the mean irregular for the spring and autumn quadrimesters.

Figure 7 provides a comparison of the irregular elements for the winter period relative to spring and autumn (top panel), as well as relative to the average of the non-winter periods (i.e. the baseline used for estimating EWM, bottom panel).

The variance of unusual deaths is larger for winter periods. The top panel of Fig. 7 reveals instances of peaks of unusual deaths in the autumn period, sometimes higher than the corresponding winter period peaks (for example in 1993/94). It is these periods which cause the EWM measure to underestimate excess winter deaths and make it less suitable as a monitoring and analysis measure.

Figure 8.

Comparison of seasonal factors for winter months, spring (green/lightest line) and autumn (red/ darkest line) months, and the average of non-winter months. Dotted lines represent the mean value for the respective seasonal components.

Although the general pattern of EWM and TEWM generally agree, as well as the differences caused by peaks in autumn deaths, there are differences in the evolution of the series. Specifically (see Fig. 6), EWM grows more slowly than TEWM. As expected, the estimated seasonal components for non-winter periods are notably lower than the winter period (Fig. 8, in line with the diagrammatic representation in Fig. 4). There is also a gradual reduction in the winter seasonal, reflecting a general decrease in the numbers of deaths occurring in winter, and an offsetting upward trend for the autumn seasonal factors. This shows a change in the balance of deaths between autumn and winter, while spring is largely unaffected. This induces the observed differences in the levels of EWM and TEWM.

3.3Revisions analysis

As described in Section 2.1, a potential limitation of TEWM is its susceptibility to revisions as new data points are added. A revisions history analysis (“revisions triangle”) of the last four years of data was undertaken (Table 1, see also supplementary material), examining the revision as a percentage of the previous estimate for that reference period as more data become available. The ARIMA model was held constant, and the analysis was performed with each successive March being the last available datapoint, providing a complete winter period. The automatic detection for outliers (one-off unusual values) implemented in TRAMO-SEATS was used as an illustration. One of the limitations of using automatic outlier detection for the regression part of the ARIMA model is that what is estimated to be an outlier may change from one year to the next. In a more formal production setting, outliers should be judged with expert knowledge, not entirely depending on automatic detection procedures. Also, it is likely that when an outlier is modelled based on prior knowledge, it would be kept in the model as new data is added. This will reduce the size of future revisions. For the current purposes, the assumption of no prior knowledge and use only of the automatic outlier detection was compared to revisions when an outlier is identified sooner and kept in the model.

Figure 9.

Estimated TEWM series as it evolves from first publication in 2002-3, using automatic outlier detection. Each shade represents a different vintage of the series, and where they overlie each other revisions are minimal.

To illustrate the impact of “unstable” outliers, in Table 1(a), for publication year 2015–16, winter 2014–15 was estimated to be an outlier value as it was unusually high for the local values at the time. However, as more data became available, it was no longer considered an outlier. As a result, the analysis suggests that when 2016–17 data became available, TEWM for period 2015–16 was revised down by 2.73% and the TEWM for the previously outliered value of 2014–15 dropped by 2.14%. Furthermore, in publication year 2018–19 (the latest used in this analysis), the winter period 2017–18 was estimated to be an outlier value, but this was not the case when winter 2017–18 was the latest period. Modelling winter 2017–18 as an outlier led to an overall 5.96% upward revision of TEWM with the latest data, and also revised previous periods’ estimates upwards. It is generally expected that revisions would be greatest for the latest data points, as the table illustrates (see also the full revisions history in the supplementary material: Revisions_tables_autoAO.xlsx). Identifying an outlier adjusts the decomposition so that the seasonal (what is considered as typical for the series) is decreased in favour of the irregular component. In this case the irregular goes up substantially, as the 2017–18 winter was higher than usual. In comparison, the revisions for winter 2017–18 are lower if it is modelled as an outlier as soon as that data point becomes available (rather than a year later), and it is kept as an outlier in the model in the next publication year (denoted by 2017–18ao14ao17 and 2018–19ao14ao17 in panel (b) of Table 1, see also supplementary material: Revisions_tables_fixedAO.xlsx). As expected, the revisions for the 2014–15 estimate are also reduced when the outlier for that period is kept in the model. It is worth noting that the revision for 2017–18 is still relatively high (3.3% upwards), mainly because the number of deaths in winter 2018–19 was quite low, so the seasonal component would be evolving downwards as well.

Figure 9 shows a “porcupine plot” of the successive estimates of TEWM as time progresses from 2002–3. It is clear that the model is stable in the earlier years of this plot, but that there is more volatility in the most recent years. Nevertheless, the revisions are small. The average revision one year after first publication (which is expected to be the largest because the model can be affected more by the addition of new data points) is 0.97% for the 89–90 to 17–18 estimates of TEWM, and 2.60% in the last four years (14–15 to 17–18).

In comparison, Fig. 10 illustrates the same data but with stable outliers in the model (corresponding to Table 1(b)). The differences between the vintages are visibly smaller, and the revision one year after first publication for the period 89–90 to 17–18 is 0.81% and for the last 4 years it is 1.48%.

Figure 10.

Estimated TEWM series as it evolves from first publication in 2002–3, with fixed outliers for winter 2014 and 2017. Each shade represents a different vintage of the series, and where they overlie each other revisions are minimal.