# On assessing excess mortality in Germany during the COVID-19 pandemic

Giacomo De Nicola 111Corresponding author<EMAIL_ADDRESS>muenchen.de Department of Statistics, LMU Munich, Germany  Göran Kauermann
Department of Statistics, LMU Munich, Germany  Michael Höhle  Department of
Mathematics, University of Stockholm, Sweden

###### Abstract

Coronavirus disease 2019 (COVID-19) is associated with a very high number of
casualties in the general population. Assessing the exact magnitude of this
number is a non-trivial problem, as relying only on officially reported
COVID-19 associated fatalities runs the risk of incurring in several kinds of
biases. One of the ways to approach the issue is to compare overall mortality
during the pandemic with expected mortality computed using the observed
mortality figures of previous years. In this paper, we build on existing
methodology and propose two ways to compute expected as well as excess
mortality, namely at the weekly and at the yearly level. Particular focus is
put on the role of age, which plays a central part in both COVID-19-associated
and overall mortality. We illustrate our methods by making use of age-
stratified mortality data from the years 2016 to 2020 in Germany to compute
age group-specific excess mortality during the COVID-19 pandemic in 2020.

##  1 Introduction

First identified in Wuhan, China, in December 2019, the Coronavirus disease
2019 (COVID-19) caused by the SARS-CoV-2 virus developed into a worldwide
pandemic during the spring of 2020 (Velavan and Meyer, 2020). One of the
challenges for scientists has been to evaluate its impact in terms of life
loss across different countries and regions of the world. A possible way to do
this is through directly looking at the number of people who died while they
were confirmed to be infected. This measure, often defined as
COVID-19-associated mortality, is certainly more robust than other pandemic-
related quantities such as e.g. the number of reported COVID-19 cases, for
which it has become clear that there is a non-negligible discrepancy between
cases detected through tests and the number of individuals who were infected
(Lau et al., 2021; Schneble et al., 2021). Nonetheless, the raw number of
COVID-related fatalities can be subject to biases and interpretative issues as
well. In particular, this number might also be biased downwards, as COVID-19
cases can still remain unreported until and after the point of death.
Moreover, it is not always straightforward to identify if COVID-19 was the
primary cause of death: Some patients might have a SARS-CoV-2 infection, but
the actual contribution of the virus to the death might be minimal (Vincent
and Taccone, 2020). To deal with these issues, comparing all-cause mortality
is generally considered a more robust alternative for assessing the damage
done by the pandemic, and to compare its impact between regions or countries.
A first look at this matter for Germany was provided by Stang et al. (2020),
who looked at data from the first wave ranging from calendar weeks 10 to 26 in
2020. The authors came to the conclusion that a moderate excess mortality was
observable for this period in Germany, in particular for the elderly. Morfeld
et al. (2021) consider regional variation in mortality in Germany during the
first wave (see also Morfeld et al., 2020). A calculation of the years of life
lost over the course of the pandemic in Germany in 2020 was pursued by Rommel
et al. (2021). International analyses on excess mortality due to COVID-19
include e.g. Krieger et al. (2020), looking at data from Massachusetts,
Vandoros (2020) who focuses on England and Wales, and Michelozzi et al. (2020)
investigating mortality in Italian cities. Global analyses in this direction
were pursued by Karlinsky and Kobak (2021) and Aburto et al. (2021).

Monitoring excess mortality has a long tradition as part of analysing the
impact of pandemics (Johnson and Mueller, 2002; Simonsen et al., 2013). With
the EuroMOMO project, Europe also runs an early-warning system specifically
dedicated to mortality monitoring (Mazick et al., 2007). However, no unified
methodological definition exists for deciding if the currently observed death
counts are higher than what would be expected. A very simple approach is to
compare the currently observed deaths for a selected time-period with the
average of death counts for a similar period in previous
years222https://www.destatis.de/DE/Themen/Gesellschaft-
Umwelt/Bevoelkerung/Sterbefaelle-Lebenserwartung/sterbefallzahlen.html.
Alternatively, the expected value can be computed by an underlying time-series
model based on past values, e.g. including seasonality and excluding past
phases of excess, as done in the EuroMOMO project (see e.g. Vestergaard et
al., 2020; Nørgaard et al., 2021). These approaches, however, do not come
without problems, as the age structure within a population can change
significantly over time. Given that both general and COVID-related mortality
are heavily dependent on age (Dowd et al., 2020; Levin et al., 2020), raw
comparisons not taking age into account will often lead to biased estimates.
More sophisticated approaches thus need to adjust for different or changing
age structures in the population. The latter point is of particular relevance
when looking at aging populations (Kanasi et al., 2016) and the infectious
risks for the elderly (Kline and Bowdish, 2016). Such age-adjustments have a
long tradition in demography when comparing mortality across different regions
with different age-structure (Keiding and Clayton, 2014; Kitagawa, 1964). A
general discussion on aging populations and mortality can be found in Crimmins
and Zhang (2019).

In this paper, we build on existing methodology to propose two ways of
calculating expected mortality taking age into account, respectively at the
weekly and at the yearly level. These methods are compared to the existing
benchmarks on data from Germany over the years 2016-2019, for which age-
stratified information is available. We furthermore apply those methods to
assess age group-specific excess mortality in Germany during the COVID-19
pandemic in 2020. The remainder of the manuscript is structured as follows. In
Section 2 we look at yearly expected mortality, while the weekly view is
pursued in Section 3. Section 4 ends the paper with some interpretative
caveats and concluding remarks.

##  2 Yearly Excess Mortality

We first look at yearly data and tackle the question of whether there was
excess mortality in Germany in 2020. In order to obtain an age adjustment for
mortality data we calculate expected deaths based on official life tables.
Life tables give the probability $q_{x}$ of a person who has completed $x$
years of age to die before completing their next life-year, i.e. before their
$x+1^{th}$ birthday. In our analysis we consider the death table provided for
the year 2017/2019 from the Federal Statistical Office of Germany (Destatis,
2020). The calculation of a life table, as simple as it sounds, is not
straightforward and is an age-old actuarial problem. First references date far
back, to Price (1771) and Dale (1772). A historical digest of the topic is
provided by Keiding (1987). Over the last decades, the calculation of the
German life-tables made use of different methods proposed in Becker (1874),
Raths (1909) and Farr (1859). We will come back to this point and demonstrate
that further adjustments are recommendable to relate the expected number of
deaths to recently observed ones. In particular, with increasing life
expectancy, the average age of the German population has been steadily
increasing (see e.g. Buttler, 2003), and this has some effect on the validity
of life tables, as discussed in Dinkel (2002). Generally, an aging population
leads to increasingly high yearly death tolls (see e.g. Klenk et al., 2007).
To quantify excess mortality one therefore needs to account for age effects,
e.g., leading to the standardized mortality ratio (SMR, see e.g. Rothman et
al., 2008). The SMR is defined as the ratio of observed death counts over
expected deaths and thus allows for an age adjusted view, meaning that instead
of pure death counts one takes the (dynamic) age structure into account.

Figure 1: Lexis Diagram

Calculating excess mortality on a yearly basis requires to calculate expected
fatalities using life tables provided by the relevant statistical bureau. We
make use of data provided by the Federal Statistical Office of Germany
(Destatis, 2020). A straightforward way of obtaining the expected number of
deaths for age group $A$ in year $y$ is to calculate

$\displaystyle e_{A,y}=\sum_{x\in A}q_{x}P_{x,y}$ (1)

where $P_{x,y}$ is the population size of individuals aged $x$ years at the
beginning of year $y$, and $q_{x}$ are, e.g., the age-specific death
probabilities in the most recent German life table from the years 2017/19,
calculated following Raths (1909). More specifically, let $D_{x}$ be the
cumulated number of individuals that died aged $x$ year old, i.e. before their
$x+1$-th birthday in the considered years 2017 to 2019. Let $P_{x,y}$ denote
the population size of $x$ year old individuals on December 31st in year
$y\in\\{2016,2017,2018,2019\\}$. Then $q_{x}$ provided in the German life-
tables is defined as

$\displaystyle
q_{x}=\frac{D_{x}}{\displaystyle\sum_{y=2016}^{2018}\frac{P_{x,y}+P_{x,y+1}}{2}+\frac{D_{x}}{2}}$
(2)

We label (1) in combination with (2) as Method 1 below. We will see that this
quantity is biased for estimating the expected number of deaths of $x$ year
old people in year $y$. To motivate this we look at the Lexis diagram in
Figure 1, and for simplicity we replace the calculation in (2) by looking at a
single year only, i.e from $y=t$ to $y=t+1$. This leads to $D_{x}=I+II$, where
$I$ and $II$ refer to the observed deaths in the two triangles in Figure 1.
Note that following the calculation principle (2) of the Statistisches
Bundesamt we would obtain $q_{x}$ as

$\displaystyle
q_{x}=\frac{D_{x}}{\displaystyle\frac{P_{x,t}+P_{x,t+1}}{2}+\frac{D_{x}}{2}}$
(3)

where $P_{x,t}$ and $P_{x,t+1}$ are the population sizes of $x$ year old
indicated in Figure 1. That is, $q_{x}$ is the probability of dying in
triangles $I$ and $II$. Let us define with $\tilde{q}_{x}$ the probability of
an individual aged $x$ years old at the beginning of year $t$ (i.e. on
December 31st in year $t-1$) to die before year $t+1$ starts. In other words
$\tilde{q}_{x}$ is the probability of dying in triangles $II$ and $III$. In
fact, this is the probability we are interested in. It is easy to see that
$\tilde{q}_{x}\neq q_{x}$. Assuming that the probability of dying in triangle
$I$ is roughly equal to the probability of dying in triangle $II$, and
assuming the same relationship for triangles $III$ and $IV$ holds, we can
conclude the approximate equivalence

$\displaystyle\tilde{q}_{x}=\frac{1}{2}q_{x}+\frac{1}{2}q_{x+1}$ (4)

which leads to the expected number of deaths

$\displaystyle\tilde{e}_{A,y}=\sum_{x\in A}\tilde{q}_{x}P_{x,y}.$ (5)

We label (5) as Methods 2 below. The adjustment is still not complete, and in
fact it can be shown that (5) is biased (see Hartz et al., 1983). Note that
individuals dying in triangle $III$ count as $x+1$ years old, so that part of
the deaths contributes to a different age group. We may now assume for
simplicity that the probability of dying in triangles $II$ and $III$ is
roughly the same, which leads to the following calculation. Let
$A=[a_{l},a_{r}]$

$\displaystyle\hat{e}_{A,y}=0.5\cdot\tilde{q}_{a_{l-1}}P_{a_{l-1},y}+\sum_{x=a_{l}}^{a_{r-1}}\tilde{q}_{x}p_{x,y}+0.5\cdot\tilde{q}_{a_{r}}P_{a_{r},y}$
(6)

where $\tilde{q}_{-1}=\tilde{q}_{0}$ and $P_{-1,y}=P_{0,y}$ gives the
approximation for the youngest age group. Accordingly, for $a_{r}=\max(x)$ we
take the full fraction of the last year, that is we add an additional
$0.5\cdot\tilde{q}_{a_{r}}p_{a_{r},y}$ to the formula above. We label (6) as
Method 3 below.

Based on these three methods we can now compare expected and observed
fatalities over the last years using the same 2017/2019 life-table as basis.
Note that, when looking at different years, one may more accurately also take
different life tables to account for changing life expectancy. We omit this
point for simplicity since we only look at five years, and changes in life
expectancy over this short period were moderate (Wenau et al., 2019). Figure 2
gives a first overview of the results for all age groups combined. We plot the
observed death counts (black dots), and the expected counts based on the
different methods are visualised as dashed lines in different colours. We can
see that Method 1, which uses (1), clearly underestimates the expected death
counts. Method 2 and Methods 3 perform equally well, which is not surprising,
since we do not take an age-specific view. The latter is carried out in Figure
3 for all different age groups available from the data. This age-specific view
shows how Methods 2 and 3 differ, and that overall Method 3 shows the better
fit. We can quantify this discrepancy by calculating the root mean squared
error for the different age groups, where we explicitly exclude year 2020 due
to the COVID-19 pandemic. The results of this can be found in Table 1.

Having established that Method 3 performs better than the other two over past
years, we can use the expected number of fatalities computed with this method
for 2020 to quantify the excess mortality during the first calendar year of
the corona pandemic in Germany. Table 2 contains expected and observed
mortality for all age groups in 2020, as well as the absolute and percentage
variations between the two. We see from the table that, for the entire
population, the age-adjusted excess mortality was in the order of 1% in 2020.
We stress that these results in terms of COVID-19 impact need to be
interpreted with utmost care: We here focus on the methodological aspects, and
defer the subject-matter discussion of the results to Section 4.

Table 1: Age-specific root mean squared error for expected yearly fatalities calculated with different methods over the years 2016 to 2019. Year 2020 is excluded due to the COVID-19 pandemic. The smallest value for each age group is highlighted in bold. | 0-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90+ | Overall
---|---|---|---|---|---|---|---|---|---
Method 1 | 302.4 | 121.9 | 413.8 | 2221.8 | 2801.3 | 2112.7 | 24244.9 | 18374.0 | 47942.7
Method 2 | 273.0 | 189.2 | 1052.8 | 2648.0 | 2969.6 | 10362.0 | 7038.7 | 18374.0 | 13676.7
Method 3 | 358.2 | 97.7 | 471.6 | 1775.9 | 1207.1 | 1760.1 | 7570.3 | 3413.2 | 13670.8
Age group | Expected 2020 | Observed 2020 | Absolute diff. | Relative diff.
---|---|---|---|---
$[00,30)$ | 7471 | 7298 | -173 | -2%
$[30,40)$ | 6663 | 6832 | 169 | +3%
$[40,50)$ | 15420 | 15704 | 284 | +2%
$[50,60)$ | 58929 | 57606 | -1323 | -2%
$[60,70)$ | 118047 | 118547 | 500 | +0%
$[70,80)$ | 199569 | 201844 | 2275 | +1%
$[80,90)$ | 379917 | 378404 | -1513 | +0%
$[90,\infty)$ | 193238 | 199761 | 6523 | +3%
Total | 979255 | 985996 | 6771 | +1%
Table 2: Expected and observed yearly mortality in 2020 for each of the six
age-groups, computed with Method 3. Figure 2: Expected deaths computed by year
with the three different methods described, for all age groups combined.
Realized fatalities are shown as black dots. Methods 2 and 3 are visually
indistinguishable, as age groups are pooled together.

Figure 3: Expected deaths by calendar year and age group computed with the
three different methods described. Realized fatalities are shown as black
dots.

##  3 Weekly Excess Mortality

The yearly view presented in the previous section does not allow to take
within-year seasonality into account for the expected deaths. We therefore now
look at weekly excess mortality statements. Classical standardization
approaches such as direct and indirect standardization can be used to adjust
the observed values for age effects, see e.g. Kitagawa (1964). We will focus
on indirect standardization, but given an appropriate choice of reference
population, direct standardization approaches are straightforward adaptations.

Figure 4: Weekly mortality probability estimates $\hat{q}_{t,a}$ as well as
the range (min-max) of the corresponding mortality probablities of the past
four years and their mean $\overline{q}_{t,x}$.

Let $q_{t,x}$ be the mortality probability specific to age $x$ and time period
$t$. In what follows, the considered time period will be one International
Organization for Standardization (ISO) week, but other intervals (e.g. months)
are also imaginable. We estimate $q_{t,x}$ by dividing the number of observed
deaths at age $x$ during time period $t$, defined as $D_{t,x}$, by the
corresponding population at the beginning of the time period, i.e. $P_{t,x}$.
To be specific, we define

$\displaystyle\hat{q}_{t,x}=\frac{D_{t,x}}{P_{t,x}}.$ (7)

Since the age-stratified population is only available as a point estimate for
the 31st of December of each year, we use linear interpolation to estimate
$P_{t,x}$. Furthermore, the exact population of the current year, i.e. on
December 31st, 2020, is not known at the time of analysis. We thus use a
corresponding population projection: Similarly to Ragnitz (2021), we use the
Destatis variant G2-L2-W2333https://www.destatis.de/DE/Themen/Gesellschaft-
Umwelt/Bevoelkerung/Bevoelkerungsvorausberechnung/Publikationen/Downloads-
Vorausberechnung/bevoelkerung-bundeslaender-2060-5124205199024.html. The
corresponding estimates of weekly mortality probabilities (7) are shown in
Figure 4. We see that in age groups $\geq 50$ years a substantial weekly
excess mortality is observable from week 45 on, with more pronounced excess
mortality for the elderly.

A weekly SMR-based excess mortality measure for the entire year 2020 can now
be computed as follows. Let $t$ denote a specific ISO week in 2020, i.e. this
will serve as notational shorthand for ISO week 2020-W$t$, where
$t=1,\ldots,53$. We form the expected age-time mortality probability for this
week by computing the average of the mortality of the same week over the last
4 years, i.e.

$\overline{q}_{t,x}=\frac{1}{4}\sum_{y=2016}^{2019}\hat{q}_{y\text{-W}t,x},\quad
t=1,\ldots,53.$

Because the years 2016-2019 do not have an ISO week 53, we define
$y\text{-W}53$ for $y=2016,\ldots,2019$ as
$\frac{1}{2}(q_{y\text{-W}52}+q_{(y+1)\text{-W}01})$. The indirect
standardization now computes the expected number of deaths for week $t$ as

$\overline{e}_{t,x}=\overline{q}_{t,x}\cdot P_{t,x}$

This corresponds to the expected number of deaths in week $t$ at age $x$, if
the current population would have been subject to the average death
probability over the past 4 years. Since fatalities are not given with exact
ages but rather by age group, we indicate this by using $q_{t,A}$, $P_{t,A}$
and $e_{t,A}$, where $A$ denotes the age classes. For the available Destatis
mortality data the six groups are
$[00-30),[30-40),[40-50),[50-60),[60-70),[70-80),[80,\infty)$. Fig. 4 shows
$\hat{q}_{t,A}$ as well as $\overline{q}_{t,x}$ for Germany. Note that the
comparison for week 53 with the past year is done using the imputation scheme
described above. Also note that this computation is equivalent to computing,
for each reference year $y$, the expected number of deaths for the relevant
week in 2020, and then taking the average of the expected deaths. In other
words: by applying the mortality probabilities for the same week of the
reference year $y$ to our study population (i.e. 2020-W$t$) and then averaging
the four expected fatalities, we get:

$\overline{e}_{t,x}=\frac{1}{4}\sum_{y=2016}^{2019}q_{y\text{-W}t,x}\cdot
P_{t,x}.$

One can now define the absolute excess mortality in week $t$ and age-group $A$
as $D_{t,A}-e_{t,A}$. Instead of focusing on absolute differences, it is
better in terms of interpretation to look at relative estimates of excess
mortality given by the standardized mortality ratio (SMR)

$\displaystyle SMR_{t,A}=\frac{D_{t,A}}{\overline{e}_{t,A}}.$ (8)

Figure 5: Weekly SMR estimates for the eight different age groups.

We plot the corresponding weekly estimate resulting from (8) for all age
groups in Figure 5. As already seen in the incidence plots, we note that in
the older age groups the first approx. 10 weeks of the year had a rather low
SMR, followed by a small increase consistent with the first COVID-19 wave.
Furthermore, substantial increases are then seen in in the $\geq 50$ year old
age groups starting from week 45, coinciding with the 2nd wave, and reaching
up to 40% more deaths than expected in certain weeks.

If we instead aggregate observed and expected counts per year, we could also
generate yearly excess-mortality statements similar to Tab. 2 (see e.g. Höhle,
2021 for comparison).

### Direct standardization

Whereas the indirect standardization strategy, pursued above, extrapolates the
average death probability from the past to the current population, an
alternative is to apply the mortality probabilities from each reference year
to a common standard population and then compare these numbers. This approach
is, e.g., used by Statistics
Austria444https://www.statistik.at/web_de/presse/125475.html and uses the
Eurostat 2013 population as common
reference555https://ec.europa.eu/eurostat/documents/3859598/5926869/KS-
RA-13-028-EN.PDF/e713fa79-1add-44e8-b23d-5e8fa09b3f8f:

$\overline{e}_{y\text{-W}t,A}^{\text{s}}=q_{y\text{-W}t,A}\cdot
P_{a}^{\text{s}},$

where $P_{A}^{\text{s}}$ denotes the size of the standard population in age-
group $A$ and the expected number of deaths for the week $t$ in 2020 is given
by
$e_{t,A}^{s}=\sum_{y=2016}^{2019}\overline{e}_{y\text{-W}t,A}^{\text{s}}/4$.

##  4 Discussion

The COVID-19 pandemic posed numerous challenges to scientists. One of those
challenges lies in estimating the number of casualties brought upon by the
pandemic. To tackle this issue, we pursued an approach based on comparing
observed all-cause mortality in 2020 with the number of fatalities that would
have been expected in the same year without the advent of COVID-19. Building
on existing methodology, we proposed two simple ways of computing expected
mortality at the yearly and at the weekly level. We then put those method to
work to obtain estimates for excess mortality in 2020 in Germany. The two
approaches yield similar results at the aggregate level, and highlight how
2020 was characterized by an overall excess mortality of approximately 1%. The
light excess mortality was apparently driven by a spike in fatalities related
to COVID-19 at the end of the year in older age groups.

Interpreting COVID-19 mortality has become a politically sensitive issue,
where the same underlying data are used to either enhance or downplay the
consequences of COVID-19 infections. We therefore stress that our interests
are methodological, and that the presented results are restricted to the
calendar year 2020 for Germany as a whole. Altogether, the mild mortality in
the older age groups during the first weeks (e.g. due to a mild influenza
season) balanced the excess in the higher age groups which came later in the
year. Clearly noticeable is the second wave during Nov-Dec 2020, which also
continued in the early months of 2021. To better account for such seasonality,
excess mortality computations for influenza are often pursued by season
instead of calendar year, i.e. in the northern hemisphere for the period from
July in Year $X$ to June in Year $X+1$ (Nielsen et al., 2011). Similarly, the
impact of COVID-19 cases and fatalities was not only temporally, but also
spatially heterogeneous, with strong peaks in Dec 2020 in the federal states
of Saxony, Brandenburg and Thuringia (Höhle, 2021). Hence, using mortality
aggregates over periods and regions only provides a partial picture of the
impact of COVID-19. Furthermore, the mortality figures observed in 2020
naturally incorporate the effects of all types of pandemic management
consequences, which include changes in the behavior of the population
(voluntary or due to interventions). Disentangling the complex effects of all-
cause mortality and the COVID-19 pandemic is a delicate matter, which takes
experts in several disciplines (demographers, statisticians, epidemiologists)
to solve. Timely analysis of all-cause mortality data is just one building
block of this process; Nevertheless, the pandemic has shown the need to do
this in near real-time based on sound data while adjusting for age structure.

Our analysis was motivated by the fact that many of the methods that have been
applied to tackle this issue so far fail to take the changing age structure of
the population into account. This can lead to biased results, and especially
so for the rapidly aging developed countries. In the case of Germany, for
example, the absolute number of people aged 80 or more increased by
approximately 20% from 2016 to 2020. Such a remarkable increase will naturally
have an effect on overall mortality, and as such direct comparisons in the
number of casualties across different years will lead to significant
overestimation of the excess mortality. Our approaches are instead robust to
such changes in population structure, and can be used regardless of the
demographic context. Note that, for both of our approaches, it would also be
possible to obtain confidence intervals through imposing simple distributional
assumptions. The same methodologies could be used to pursue a similar analysis
for any country in which mortality data and a mortality table are available,
for any given year. A natural use for the proposed methodology would also be
to assess the overall damages caused by the pandemic when it will be finally
considered a thing of the past. All in all, we hope the proposed methods will
help shedding light on the issue of computing the expected number of
fatalities, and in the assessment of potential excess mortality.

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