In a first step, we investigate the effect of demographics on the CFR estimates. To do so, we gathered weekly data on cumulative age-specific case and death numbers from various sources for the study countries between early March and the end of 2020. The data for Germany have been downloaded from the Robert Koch Institute (RKI)'s Github database (Robert Koch Institut, 2021). For Italy, the data have been collected from early press releases and then regular reports published by the Istituto Superiore di Sanità (ISS) (2020a, 2020b, 2020c, 2020d). For the United Kingdom (UK), we downloaded data as provided from the website of the UK's national health authority (Public Health England, 2021a). For France, we found, after comparing different sources, that a combination of national reports from Santé publique France (2020a, 2020b) until calendar week 32 and data from the new COVerAGE database (Riffe et al., 2021a, 2021b) for the following period provided the best coverage of the age-specific case and death data, as we have observed significant irregularities and missing data for France. All of the data for the remaining seven countries came from the latter database. In our comparison of crude and age-adjusted CFRs, we use the European Standard Population (European Union, 2013) for standardisation. We use the following notation: for each country j, d.jk is the number of deaths over all age groups observed during week k. Similarly, n^.jk is the number of observed new cases over all age groups during week k. Summing up the deaths up to week w, we obtain the total cumulative number of deaths in country j over all ages D.jw=∑k=1wd.jk. Similarly, summing up the cumulative number of cases up to week w, we obtain N^.jk=∑k=1wn^.jk, the total cumulative number of cases for country j and up to week w.

While the numbers of cases and deaths in week k, for each country j and age group i, are observed (respectively, N^ijk and Dijk), the number of new infections in age group i, country j and week k (denoted by nijk) is latent (unknown). The number of cumulative infections in this group up to week w is Nijk=∑k=1wnijk. Our first aim is to identify the role of the age structure of the cases in the overall CFRs, and to derive age-specific and age-standardised CFR estimates for all study countries. Based on the cumulative age-specific case numbers N^ijk of the 11 European countries, we calculate age-specific CFR estimates (3) CFRijk=DijkN^ijk. Multiplying these estimates with population weights derived from the European Standard Population, we obtain age-adjusted CFR estimates (4) CFRjkage=∑iwi×CFRijk.

In the second step, we investigate the impact of the surveillance of cases, as represented by time series data of weekly national tests provided by the European Centre for Disease Prevention and Control (ECDC) (2021b). As the ECDC does not provide corresponding test data for Switzerland and the UK, we obtained these data for these two countries from their respective national health services (Federal Office of Public Health, 2021; Public Health England, 2021b). However, age-specific data on testing for the first year of the pandemic were not available in time series form. Since symptoms vary by age (Davies et al., 2020), the detection rates of cases are age-specific. We deal with this data restriction indirectly by conducting principal component analysis (PCA). PCA is a data reduction technique that transforms the original, correlated variables into linear combinations that are uncorrelated, and are referred to as principal components (PCs) (Vanella, 2018). Working with PCs allows us to cover indirectly simultaneous trends and sensitivities of the case numbers to the testing strategy. As we lack age-specific data on testing for the first year of the pandemic, we cannot directly estimate the sensitivities of the age-specific case numbers to the testing strategy on a population level. Therefore, our approach is to approximate the different sensitivities of the age-specific case numbers to the testing strategy indirectly by estimating the sensitivity of the country-specific PCs to adjustments in the test numbers (irrespective of the age groups). We assume that the case numbers are influenced by the test numbers, but to differing degrees depending on age, as both the symptoms and the detection rates of infections vary by age (Gornyk et al., 2021). According to that hypothesis, n^ijk are functions of the weekly test numbers of the corresponding age group and week in the same country tijk, say n^ijk=f(tijk). This could be quantified by fitting a generalised linear model. However, since we do not know tijk, but only the overall test numbers tjk, we cannot derive age-specific models. To approximate the connection between random testing (irrespective of age) and age-specific case numbers, PCA is used. We perform PCA for each country separately on all square root transformed age-specific case time series² as follows: (5) Pzjk=∑iλzijn^ijk, with

Pzjk: zth PC of country j in week k

PCA allows us to cover the common trends in the case numbers across all age groups with a small number of indices that are linear combinations of the original variables.

Figure 2 illustrates the loadings of the first PC for each country. Except in the Netherlands, the loadings have a similar bathtub shape, with smaller loadings, in terms of absolute values, for the children and the elderly age groups and larger loadings for the working-age population. For the Netherlands, we observe a monotonously decreasing trend by age. Large absolute loadings represent a high correlation between the PCs and the corresponding age groups, and vice versa. There is an inverse relationship between the PCs and age-specific case numbers, which is represented by the negative signs of the loadings. Hence, increases in these PCs are associated with decreasing weekly case numbers of the corresponding age groups, and vice versa.³ For all countries, the respective first PCs⁴ explain 93–99% of the variance in the square roots of age-specific weekly new case numbers over the year 2020, and are therefore sufficient to explain the major age-specific trends in case numbers.

We see that decreases in the PCs⁵ are associated with increases in case numbers to varying degrees for different age groups. Looking at the example of Greece, we see (absolute) smaller loadings in the 0–17 and 64 and older age groups, and larger loadings for the 18–64 age group. Therefore, a decrease in that PC is associated, ceteris paribus, with more distinct increases in the case numbers among individuals aged 18–64 than among the younger and the elderly populations. As P1jk is a linear combination of the age-specific case numbers according to (5), we can quantify some connection P1jk=g(tjk), which is used to investigate the statistical association of P1jk to the overall weekly test numbers in country j. PCs account for the differences in detection rates indirectly, as we first quantify the test number coefficient in a regression of P1jk on the tests. After deriving the coefficient of tjk in P1jk=g(tjk) by the maximum likelihood, we can then predict the expected value of P1jk based on a given number of tests. By plugging in the predicted values of all Pzjk in (5), while holding all Pzjk with z≠1 fixed, we have a system of linear equations, which, given that we predict all Pzjk after test adjustment and know all λzij from singular value decomposition, we can derive predicted age-specific case numbers n^ijk from (5) after test adjustment without having access to age-specific test numbers.

The impact of testing on the case numbers and the detection rates of infections is investigated in a causal regression, as described above. However, it is important to separate increases in case numbers due to increases in infection numbers from those caused by more testing, and, hence, higher detection rates. Increases in test rates might be caused by a shift in the political agenda, such as a move to increase the number of random tests in order to detect more asymptomatic infections, or a response to higher numbers of infections that includes more testing of suspected cases (e.g., of individuals who have come into contact with confirmed cases). We investigate this connection through regression analysis as follows: for each country, the first PC is regressed on the first lag of official COVID-19 cases together with the weekly tests: (6) P1jk=αj+βjn^.jk−1+γjtjk, with

P1jk: value of PC1 for country j in week k

The kth residual from (6) shall be named εjk.

The model assumes that the observed cases are affected by the number of tests performed in the current week, which is, however, affected by both politically driven decisions and concrete increases in infection activity in the previous week. Therefore, n^.jk−1 serves as a control variable in the analysis that includes increases in both the numbers of infections and the share of positive tests. By using this approach, we mitigate potential bias in our interpretation of the connection between contemporaneous increases in both test and case numbers. After fitting country-specific models following (6), we adjust the observed case numbers for underestimation, and, thus, underdetection, by holding the control variable n^.jk−1 as observed and adjusting the test variable tjk to a specific value. This enables us to predict the number of cases we would, ceteris paribus, have expected to find given a fixed number of random tests each week that are not connected to observed positive cases. That value is in essence arbitrary. However, it appears plausible to set country-specific constants to account for the population size of each country. According to the ECDC (European Centre for Disease Prevention and Control, 2021b), among the study countries, the maximum outcome of the tests performed was around 5.7% of the estimated population, found for Portugal in calendar week 47. Therefore, we set our maximum of hypothetical weekly test numbers to t~j=max(tPT)⋅BjBPT, with max(tPT) being the maximum outcome of the weekly tests performed in Portugal, BPT being the population estimate for Portugal and Bj being the population estimate for country j. Let us define P~1jk as the hypothetical value we would have observed for P1jk for a test number fixed at t~j. The prediction of P~1jk is then: (7) E[P~1jk]:=E[P1jk|tjk=t~j]=α^j+β^jn^.jk−1+γ^jt~jk, where α^j, β^j and γ^j are country-specific parameter estimates derived by OLS regression according to (6). As infection time series are not stationary, but instead move in waves of peaks and troughs, a simple adjustment according to (7) would not include these seasonal patterns in the development of infections. To incorporate this seasonality into our adjustment, we add the residuals extracted from (6), and adjust our prediction from (7) to infection trends above or below expectations caused by the wave-like development of infections. We assume that the observed derivations from the case numbers expected from our model under the observed test and the previous case numbers would carry over, even under a specified number of tests. The adjustment of the PCs is therefore (8) P~1jk=α^j+β^jn^.jk−1+γ^jt~j+εjk, For each country, the remaining PCs are unchanged. Let P~j be the matrix of test-adjusted PCs for country j. We then derive the square roots of the test-adjusted weekly case numbers for country j by (9) M~ij=P~j×Λj−1, with

M~ij: the matrix of the square roots⁶ of the test-adjusted age-specific weekly case numbers of country j,

Finally, we compute the squares for all elements of (9), which are then test-adjusted case numbers, say n~ijk=(mijk)2, with mijk being the element in the ith row and the kth column of the matrix M~ij.

We will then compute the test-adjusted age-specific CFRs by (10) CFR~ijk=DijkN~ijk, with N~ijk being the sum of the weekly test-adjusted age-specific case numbers up to week k in country j. Using this, we derive the age- and test-adjusted CFR estimates over time, similar to (3): (11) CFRjkage,test=∑iwi×CFR~ijk.

In the third step, we investigate the effects of different lags between case reports and deaths on the CFR estimates. The unknown distribution of the time lag of Δ weeks between the reporting of a case and death is considered. Verity et al. (2020) estimated the average time from infection to death to be about 14 days. For instance, the age-specific and lag-adjusted CFR of age group i, in country j, in week k, based on a lag of the cases of Δ weeks is (12) CFR⌣ijk−Δ=DijkN^ijk−Δ. The age- and lag-adjusted CFR⁷ is then similar to (4): (13) CFRjk,Δage,lag=∑iwi×CFR⌣ijk−Δ. We compare this measure for the pandemic over the study period and for all countries with Δ=0,1,2,3; with Δ=0 being the case of the age-adjusted CFR, as in (4). We will provide the results in Section 3.3.

For our investigation of the impact of healthcare system capacity on CFRs, we use estimates of the available intensive care unit (ICU) beds per 1,000 inhabitants provided by the OECD (2021a), and the weekly means of daily ICU bed occupancies I.jk provided by the ECDC (European Centre for Disease Prevention and Control, 2021a) for seven of our study countries. For the countries for which these data are not available, we instead use hospital beds per 1,000 inhabitants together with weekly counts of new cases. Hospital admissions due to COVID-19 for the previous γ weeks are addressed as h.jk,γ. First, we give a qualitative assessment using graphical analysis of the connection between CFRs and healthcare capacity and occupancy. To compare the hospital bed capacities in the study countries, we adjust the weekly numbers of new hospitalisations, as provided by the ECDC (European Centre for Disease Prevention and Control, 2021a), by the bed capacities per 1,000 inhabitants. These figures are provided as static estimates by the OECD (2021b). As both variables are measured relative to each country's population, dividing them leads to an admission-per-hospital bed measure in percent, defined as (14) h.jk,γadj=h.jk,γb¯j, with b¯j denoting the per 100,000 inhabitant number of available hospital beds in country j. While this measure has its merits, as it is static and thus does not change over time, it should be seen as a rough adjustment parameter that accounts for the differences between national healthcare system capacities. Similarly, the connection between age-adjusted CFRs and ICU occupancies is investigated as (15) I.jkadj=I.jkc¯j, with c¯j being the estimates of the national ICU beds available. As the latter estimates are not available from the OECD, we use the latest available estimates provided by Our World in Data (2021).

Age-standardised CFRs for the study countries are illustrated in Figure 3, with Figure 3(a) again displaying estimates for the study countries in central and northern Europe, and Figure 3(b) providing estimates for the study countries in the Mediterranean region.

The courses of the age-standardised curves are more stable than those of the crude CFR curves. A large share of the decreases in the crude CFRs observed since spring 2020 vanishes when accounting for the age structure of the cases. In general, the weekly investigation smooths out some of the variations that appear in daily monitoring. In particular, some of the peaks shown in Figure 1 are smoothed out to a large extent. While international differences are still observable, the curves converge to a greater degree than is the case for the crude CFRs. The strong peak for France is a statistical artifact caused by the change in the input data in calendar week 33. The horizontal line again represents the mean of all observations of the CFR with an age structure according to the European Standard Population. The variance between this line and the age-standardised CFR curves has, compared to that of the crude CFR, decreased substantially. We understand that the crude CFR curves between countries are skewed due to the age structure of the cases, especially in the early stages of the pandemic. While cv(M1)⁸ of the initial crude CFRs is around 72%, the age standardisation decreases this value to cv(M2)⁹ ≈ 51%. Hence, a large share of the international and intertemporal variance in the CFR is explained by the demographics of the cases.

Regarding test adjustment, our regression models estimated following (6) for all countries shows a highly statistically significant effect of testing on the PCs, and thus on the weekly numbers of new cases, even when controlling for the first lag of the new case numbers. Figure 4 illustrates this finding via quantile-quantile plots of the model fits for each country.¹⁰

However, the adjustment of cases to testing leads to a considerable worsening of the CFR model, with cv(M3)¹¹ ≈ 124%. We conclude that the testing strategy has an effect on the case numbers. However, our approach of including this finding in CFR estimation does not lead to improvements. Therefore, in our further analysis, we proceed without a test adjustment. However, our results imply that the differences found in case fatality estimates stem at least in part from differences between countries in levels of underdetection.

The lag adjustments according to (12) and (13) do not provide improvements in the CFR estimates, but instead worsen them, with, e.g., a cv(M4)¹² ≈ 107%. employing the first weekly lag of cases. This pattern is especially apparent for the early stages of the pandemic, for which the CFRs are highly overestimated. Therefore, pure age standardisation, as done in M2, gives the most stable CFR estimates.

Figures 5 and 6 display scatterplots of the age-standardised CFRs, regardless of the country¹³ against I.jkadj and h.jk,γadj, respectively, over the study period without any lags, with lags of no and one week, with lags of the previous two weeks and with lags of the previous three weeks, respectively. Regardless of which lags are chosen for the COVID-19-related hospitalisations, or of whether we use the daily ICU occupancy, the age-standardised CFRs do not show any statistical correlation between the chosen healthcare hospital burden variable and the age-adjusted COVID-19-specific fatality rates.