Population data for Spain stratified by age group and gender for 2019 are obtained from the National Institute of Statistics of Spain ⁴ ( Figure 1). We note the large percentage of older adults with over-60 males and females representing 11.41% and 14.16% of the whole population, respectively.

Data on the daily total confirmed cases and deaths, as well as daily confirmed cases and deaths by age group and gender from a subset of the population are reported by the Spanish Ministry of Health and provided by 5. Assuming this subset is representative of all cases, in terms of the relative distribution among age group and gender, we can estimate the daily number of confirmed cases in each group by multiplying the total number of cases by the proportion of cases in each group. Daily number of deaths in each age group and gender are calculated following a similar procedure. Figure 2 and Figure 3 show the proportion of cases and deaths, respectively, in each age group and gender. We observe a low proportion of confirmed cases in young people (under 20 years old) and a high proportion of deaths in older age groups and males. Figure 4 shows the total number of confirmed cases and deaths over time.

We can examine the relative risks in each age group and gender to compare the severity of the disease between population groups. The relative risk in each population group is obtained by dividing the number of deaths in a group by the total population in that group, and normalizing the values so the risk of males older than 80 is equal to 1. We observe a roughly tenfold increase in risk for every 20 year increase in age, consistent with an earlier smaller study of cases in China ⁶ .

At any point in time, the crude CFR is calculated by dividing the cumulative number of deaths by the cumulative number of reported cases. As noted, CFRs may be affected by preferential ascertainment of severe cases. This is likely to occur in COVID-19 where cases asymptomatic or with mild symptoms are less likely to seek medical care or be included in the surveillance data. This could result in an upward bias (or overestimate) of the crude CFRs by under-reporting of cases. We can partially correct this bias by calculating the adjusted daily number of confirmed cases following the procedure detailed in 6. Specifically, we calculate NC _a = pop _a /cases _a where pop _a and cases _a are the population and the number of cases, respectively, in group a, a ∈ { males 0–9, males 10–19, males 20–29, males 30–39, males 40–49, males 50–59, males 60–69, males 70–79, males 80+, females 0–9, females 10–19, females 20–29, females 30–39, females 40–49, females 50–59, females 60–69, females 70–79, females 80+ }. We assume perfect ascertainment in the group with maximum 1/NC _a value which is the group of males older than 80. Then, we assume the attack rate is the same in all groups and estimate the adjusted number of cases in each population group by multiplying the confirmed cases by NC _a /NC _{males 80+}. Figure 6 and Figure 7 show the cumulative confirmed cases and the cases adjusted for preferential ascertainment over time for each age group and gender. Finally, we calculate the CFRs adjusted for preferential ascertainment by dividing the cumulative number of deaths by the cumulative number of adjusted cases in each population group. We also calculate 95% confidence intervals using exact binomial tests ⁷ .

CFRs can also be biased due to the delay between disease onset and death. At any moment in time, the cumulative number of confirmed cases includes people who have not yet died but may do so in the future. Therefore, crude fatality rates may underestimate the true severity of the disease. We can correct this bias by replacing the denominator with an estimate of the cumulative number of cases with known outcomes by the time rates are calculated. Specifically, we adjust for this bias as follows. Let T be the point in time when the CFRs are calculated. The probability that a case confirmed at time t, t = 1, . . . , T, has a known outcome by time T is expressed as p t = ∫ m = 0 T − t g ( m ) d m , where g( m) denotes the probability density that a case has a known outcome m days after the disease onset. The cumulative number of cases in group a with known outcomes by time T can be calculated as case s a = ∑ t = 1 T ncase s a t × p t , where ncases _at denotes the new number of cases in population group a and date t. Here we calculate the number of adjusted cases assuming a log-normal distribution of the time from disease onset to death with mean equal to 13 days and a standard deviation equal to 12.7 ⁸ ( Figure 5). Figure 6 and Figure 7 show cumulative cases adjusted for preferential ascertainment of severe cases and time delay between confirmation and death for each population group. Then we calculate corrected CFRs using the adjusted cases as denominator and 95% confidence intervals using an exact binomial test.

The procedure we used to obtain adjusted CFRs requires strong assumptions that could greatly affect the results. First, we have adjusted crude CFRs by preferential ascertainment of severe cases by assuming complete ascertainment in the group with the highest attack rates (males older than 80). We then have assumed a uniform attack rate in all population groups, and used demography-adjusted under-ascertainment rates to obtain estimates of the number of infected individuals in each population group. However, there could also be under-ascertainment in the males older than 80 group due to extensive strain on the health system, and this fact could mean the CFR estimates are only an upper bound on the real values. We could correct this bias by further scaling the number of cases after the initial demographic adjustment. For example, we could multiply the adjusted cases by a value α > 1 to obtain a higher number of infected cases and lower CFRs. Moreover, the uniform attack rate assumption could be incorrect if certain population groups have more interactions with other people and are more exposed to the disease.

CFRs may also be biased due to the delay between disease onset and death. To correct this bias, we have considered a log-normal distribution with mean 13 days and standard deviation 12.7 days for the time from disease onset to death ⁸ , and estimated the CFRs using as denominator the cumulative number of cases that could have a clinical outcome by the time rates are calculated. However, other distributions may be considered that could change the results.

To illustrate these limitations, we conduct a sensitivity analysis where we calculate the CFRs using different levels of ascertainment and different distributions for the time from disease onset to death. Specifically, we estimate the adjusted number of cases in each population group by multiplying the confirmed cases by NC _a /NC _{males 80+}× α using α values equal to 1, 1.5 and 2. We also use delay distributions equal to a log-normal distribution with mean 13 days and standard deviation 12.7 days ⁸ ( Figure 5), and a Gamma with mean 18.8 days and coefficient of variation 0.45 days ⁹ ( Figure 9).

Analysis are performed with the statistical software R version 3.6.1 ¹⁰ . Plots are created with the R package ggplot2 version 3.3.0 ¹¹ .

Data on total confirmed cases and deaths, as well as confirmed cases and deaths by age group and gender from a subset of the population are reported by the Spanish Ministry of Health and provided by 5. Population data for Spain are obtained from the National Institute of Statistics of Spain ⁴ .

Data used is available from GitHub ( https://github.com/Paula-Moraga/coronavirus-cfr) and archived with Zenodo ¹² .

Zenodo: Paula-Moraga/coronavirus-cfr: First release. http://doi.org/10.5281/zenodo.3856444 ¹²

This project contains the following underlying data:

ccaa_covid19_casos2020-05-14.csv (number of confirmed cases in each of the 17 regions of Spain from 2020-02-21 to 2020-05-14)

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).