Key Points:
I downloaded and analyzed data containing daily COVID-19 deaths and critical infection numbers from the Israeli ministry of health dashboard from August 10 and September 8, overall and split out by <60yr and >60yr, to assess effectiveness of boosters and vaccination without boosters relative to unvaccinated.
Because of age confounding, it is crucial to stratify by age groups, and further to remove children from the <60yr group; otherwise Simpson's paradox makes vaccines look like they are not effective in protecting against COVID-19 deaths at all.
After doing this adjustment, we see the vaccines and boosters are strongly protecting against both COVID-19 deaths and critical infections.
For the >60 population, we see ~70% effectiveness for vaccinated, which increases to >93% for boosted, agreeing with other public reports showing boosters reducing serious cases in the older population.
For the 12yr-60yr population, we see ~90% effectiveness for vaccinated, whether boosted or not.
However, there may still be a remaining "Simpson's effect" attenuating the estimated effectiveness of boosters in the 12yr-60yr group, since most boosted are in the 40-60yr group that has higher risk of death and critical disease, so the effectiveness of boosters might in fact be >>90%.
One limitation of these data are that the Israeli MoH did not separate out previously infected. These comprise a substantial proportion of unvaccinated (31.5% of 12-60 group and 22.2% of >60 group), which leads to substantial additional attenuation of effectiveness estimates.
Overall, these data agree with what we have seen in all careful analyses of vaccine data: that even with Delta and with some waning immune protection after some time, the vaccines are still strongly protecting against severe, critical, and fatal disease.
Boosters provide further protection but it is not yet clear whether they are necessary for the younger age groups.
In a previous blog post, I highlighted two key quantitative points that often confuse people when looking at COVID-19 data for vaccinated and unvaccinated populations:
It is meaningless to consider "percent of cases/hospitalizations/deaths" that are vaccinated without accounting for what percentage of the population is vaccinated. This can easily be done by normalizing the counts of cases/hospitalizations/deaths based on the overall number vaccinated or not in the population, but many fail to make this basic adjustment.
With observational data, the failure to account for confounding factors that are associated with both exposure (vaccination status) and outcome (cases/hospitalizations/deaths) can produce misleading results, sometimes dramatically misleading if the confounding is strong enough. In the previous post, "age" was the confounding factor. "full vaccination" the exposure, and "severe disease" the outcome, with older people much more likely both to be vaccinated as well as to have severe disease, causing the paradoxical result of vaccine effectiveness appearing much lower when computed overall (67.5%) than when computed separately by age groups (91.8% for <50yr, 85.2% for >50yr), illustrating a statistical artifact known as Simpson's Paradox.
The data analyzed in that post were downloaded from Israel's Ministry of Health (MoH) dashboard and comprised of all "currently active serious COVID-19 cases" as of August 15, 2021. These data provided a snapshot of active serious cases at that time, and the analysis involved an assessment of how the rates differed among vaccinated or unvaccinated. The dashboard did not contain information on COVID-19 deaths or critical cases, it did not contain data over a longer period of time, and it did not contain information on 3rd shot boosters that had just started being distributed on August 1.
This 3rd shot booster program, initially limited to immunocompromised and older residents, has now been made available to all adult Israeli residents, and in fact an Israeli resident whose second shot was >5 months ago is not considered "fully vaccinated" unless they have received the 3rd shot booster.
Recently, the MoH has updated the dashboard to include daily counts of deaths, new serious cases, and new critical cases, on September 9 sharing data containing daily counts from August 10, 2021 through September 8, 2021 split out by boosted, vaccinated but not boosted, and unvaccinated. These data contain both raw counts and normalized "counts per 100k" from which the denominators (total population boosted, vaccinated, or unvaccinated) can be inferred, plus they report both overall counts as well as split out by age group (<60, >60). Because they contain data over a fixed time period, these data enable estimation of effectiveness of vaccine and/or booster summarizing difference in rates for unvaccinated and vaccinated for that period of time, rather than for just a snapshot of currently active cases like the previous analysis, plus enables evaluation of difference in rates for those boosted vs. those fully vaccinated but not yet boosted. I have downloaded and collated these data together to assess what they can tell us about vaccine effectiveness (VE) vs. deaths, critical cases, or severe cases of COVID-19 for boosters, as well as for vaccination without boosters. The links to the data sets and transparent explanations for my calculations are provided at the end of this article.
As we will see, the same tricky quantitative nuances highlighted in the previous post are evident here for all three outcomes, so once again this post serves as a warning to be careful not to misinterpret simple summaries of the data, and to be sure to stratify by age to avoid the Simpson's paradox effect caused by strong age confounding that also is present for these outcomes. Warning: the initial tables make it appear that the vaccines and boosters are "not working" at preventing COVID-19 deaths, but please be sure to see the later tables that show once age is accounted for reasonably well the vaccines and boosters seem to be doing well in reducing risk of COVID-19 death.
Also, I must start with a disclaimer: While the following analysis is legitimate and the best I can do with the publicly available data on the MoH dashboard, I do not consider this to be a state-of-the-art analysis that is the end-all indicator of VE vs. death, critical disease, or severe disease in Israel, or effects of boosters. There are other important confounders other than age that should be accounted for but were not available in this data set, there are some questions about how previously infected or partially vaccinated were handled in this data set that may require additional adjustment, plus there is uncertainty about whether the recorded dates are date of event (death, hospitalization, ICU entry) or date of positive PCR test that shades precise interpretation. Someone with access to more complete data could do a much more complete and careful analysis with more advanced modeling, as has been done in excellent papers by various research groups inside of Israel. In spite of these limitations, I think this analysis has value not just for further illustrating how statistical nuances can distort proper interpretation of these types of data, but also in providing some information about how the vaccines and boosters appear to be impacting advanced outcomes like severe disease, critical disease, and deaths.
Effectiveness of Vaccination and Boosters against COVID-19 Deaths
I will start by focusing on the COVID-19 death data, to assess the estimated effectiveness of the booster and of vaccination without booster in preventing COVID-19 deaths relative to the unvaccinated group during this period of time.
Following are the total COVID-19 deaths for unvaccinated (Not Vax), Boosted (Boost, meaning they have received a 3rd shot booster), and vaccinated but not boosted (Vax (NB)) combining the counts from August 10 through September 8.
We see a total of 667 COVID-19 deaths during that time period, with only 41.5% (277) in the unvaccinated group, 12.5% in the boosted group, and 46.0% in the vaccinated but not boosted group. Thus, we see ~60% of COVID-19 deaths were at least fully vaccinated. These raw proportions do not paint a promising picture for the effectiveness of vaccines or boosters, but we know better than to interpret these raw numbers alone without accounting for proportion vaccinated and boosted, right, thus falling for the base rate fallacy? The MoH data set does not explicitly contain total number boosted, unvaccinated, or vaccinated but not boosted, but for various outcomes they include both raw numbers as well as normalized numbers per 100k population from which these raw numbers can be inferred as detailed at the end of this article. Since these numbers change every day as more people are vaccinated or boosted, for simplicity in this table we include the average numbers in each category over the period of August 10 through September 8:
We can see that looking across the entire Israeli population, during this time period we have an average of 62% vaccinated, with 13.2% already given 3rd shot booster and 48.8% being fully vaccinated but not yet receiving 3rd shot booster.
Given these total number boosted, unvaccinated, and vaccinated but not yet boosted, we can normalize the number of deaths in each group based on number per 100k population.
For example, we have 83 deaths in the boosted group from a population of 1,200,310 boosted, which means that in the time period of August 10 through September 8, the proportion of boosted who died of COVID-19 is 83/1,200,310 = 0.0000691.
As these small numbers are hard to interpret, we can multiply this rate times 100,000 by shifting the decimal place 5 places to get B = "rate of COVID-19 deaths per 100k among boosted." as 6.91, meaning that from August 10 through September 8, 6.91 out of every 100k boosted died of COVID-19.
We can do the same for N = "rate of COVID-19 deaths per 100k among unvaccinated" and
V = "rate of COVID=19 deaths per 100k among vaccinated but not boosted"
B = 83/1,200,310 x 100,000 = 6.91 per 100k
V = 307/4,446,814 x 100,000 = 6.90 per 100k
N = 277/3,456,060 x 100,000 = 8.01 per 100k.
Note that these numbers are proportion of the respective populations who die from a COVID-19 infection, not proportion of those with COVID-19 infection who die. The latter proportion is called the case fatality rate, which would be an interesting number to look at as well, but a number we can't estimate here without counts of confirmed COVID-19 cases in these groups and to be more precise, matching of cases and deaths.
Based on these normalized rates, we can compute vaccine effectiveness vs. COVID-19 death for boosted and for vaccinated (not boosted) relative to the unvaccinated for this time period:
VE_death for boosters: 1 - B/N = 1 - 6.91/8.01 = 13.7%
VE_death for Vax (NB): 1 - V/N = 1 - 6.90/8.01 = 13.7%
Ugh. This does not look good. A 13.7% effectiveness vs. COVID-19 deaths would suggest that the boosters (or vaccines without boosters) are only preventing 13.7% of the COVID-19 deaths that would have occurred without vaccination or boosters. If this reflected reality, this would be truly alarming as it would suggest that neither the vaccines nor even the boosters are substantially reducing the risk of death by COVID-19. However, please don't stop reading here and go away convinced the "vaccines don't work"
Just as it was for severe disease in our previous analysis, age is a major confounding factor here:
Older people are much more likely to be boosted (or vaccinated but not boosted) than younger people.
Older people are inherently much more likely to die if they get COVID-19 than younger people.
In fact, one would expect age to be an even stronger confounder here, since the disparate risk of death between young and old is even greater for deaths than severe disease, and the disparity in rates of boosters between young and old are even greater at this point in time than the rates of vaccination. Could it be that these overall effectiveness numbers are distorted by a Simpson's paradox effect involving age once again? Fortunately the MoH data is split out by age (they use 60 as cutpoint so we have the data split out by <60 and >60). Here are the counts and normalized counts for boosters and vaccination without boosters split out by the two age groups:
We can see the vast majority of deaths are from the >60 group, with 254/277=91.6% of unvaccinated deaths, 82/83=98.8% of boosted deaths, and 298/307=97.1% of vaccinated (no booster) deaths in those >60yr.
Here, the the normalized rates of COVID-19 deaths per 100k in the unvaccinated group is >230x higher in the >60 than <60 group, clearly showing the risk of death from COVID-19 is MUCH higher for older than that is for younger people (which we already knew, of course). From these numbers, we can compute age-specific effectiveness estimates for boosters and vaccination (no booster).
Note that the vaccine effectiveness numbers split out by age are much higher than the overall numbers without age stratification, suggesting once again a strong Simpson's effect.
These results suggest that for both the younger and older age groups, >2/3 of COVID-19 deaths that would have occurred sans vaccination were prevented by full vaccination even without boosters.
For the >60yr, this protection appears to be substantially stronger in the older group, with an estimated 5.2-fold reduction { (1-0.68)/(1-0.938) } = 5.2 } in risk of death with the boosted group relative to vaccinated but not boosted. This number agrees with some of the published papers (see here and here) in terms of effectiveness of boosters in reducing breakthrough infections and severe cases in the first few weeks after the booster program began, and provides some preliminary support that the booster program may help reduce risk of severe outcomes in older groups.
We do not see evidence of reduced risk of death from COVID-19 from boosters in the younger (<60yr) group, but note that only 5% of this age group has received boosters, they received the boosters more recently, and there is only a single death so far in this cohort, so there is really not enough data yet to make a strong determination of this effect.
While these age-stratified VE_death numbers are much higher than the unstratified numbers (13.7%), the numbers for the <60yr group both with boosters (62.7%) and without (66.6%) would comprise a substantial waning of protection vs. COVID-19 death relative to the previously reported >90% numbers.
Please don't stop reading here either -- this is also not the final story.
However, there are caveats and uncertainties in these data that we need to consider when trying to interpret these results. Note the total number unvaccinated, boosted, and vaccinated but not boosted computed from these data sum to 9,103,184, which is roughly the entire Israeli population. This raises a number of questions.
Unlike the data used in the previous blog post that only Israeli residents >12yr, these data clearly include all Israeli residents, including small children, and since vaccination is not yet available for any of those <12yr, these should be completely contained in the unvaccinated group. Since children have exceptionally low risks of deaths from COVID-19, failure to separate this group out would severely distort the effectiveness numbers for the <60yr cohort as illustrated below.
It is not clear how this dashboard counts partially vaccinated, which would include those who have either received only a single dose of vaccine or are within 7 days of their second dose. These are either included in the Vax (NB) or Not Vax counts, but I have not been able to find documentation of which one. If they are included in the vaccinated group, this would attenuate the vaccine effectiveness under the condition that a single dose provides less protection than a two-dose regimen, as is clear in other published data.
It is also not clear how the previously infected are counted in this table. Israel did not make vaccination available to those previously infected until March 2021, and then only offered a single dose, and this was optional, since previously infected were given "green pass" immune passports whether vaccinated or not. As a result, a substantial proportion of previously infected may remain unvaccinated, and these are likely included in the Not Vax group. Since we know that most previously infected retain robust immune protection, the failure to remove or stratify out the previously infected in these data has the potential to attenuate vaccine or booster effectiveness data. Note that most published papers on Israeli data that perform rigorous modeling remove these previously infected, but the dashboard does not provide this information so I was not able to remove them here.
These factors might have a strong impact on the effectiveness estimates. While I can't do anything about the partially vaccinated or previously infected with the information available to me here, we can try to separate out the <12yr old population from the <60yr numbers in the table.
There are other data sets shared on the dashboard that only include numbers for >12yr (including the one in the previous post) that suggest a total population >12yr of 6,876,056. Subtracting from the 9,105,184 total here, we get an estimated 2,229,128 Israeli residents <12yr in these data. Given children <12yr have not been eligible for vaccination yet, all of these are among the 3,300,794 unvaccinated in the <60yr group. We see 23 unvaccinated COVID-19 deaths in the <60 age group. While the MoH doesn't split out how many of these deaths are from individuals <12yr, it is clear from other data that the risk for death from COVID-19 is orders of magnitude less for children <12yr than adults 50-59, 40-49, or even 30-39. Based on this, we expect none of these deaths are in the <12yr group, or at most 1. So if we assume no COVID-19 deaths <12yr and pull out the children into their own age category, we see:
Note that >2/3 of the unvaccinated individuals (2,229,128/3,300,794) in the <60yr group and >64% of the total unvaccinated in the population (2,229,128/3,456,060) are children <12yr. Given the lack of deaths in this cohort, we see that the normalized rates per 100k in the unvaccinated group for adults 12-60yr increase substantially when the children are separated out.
While the rate of unvaccinated COVID-19 deaths in the full <60yr group was 0.70 per 100k, when the children <12yr are separated out, the rate of unvaccinated COVID-19 deaths in the 12-60yr group more than triples to 2.15 per 100k (even if we assume 1 of the COVID-19 deaths was in the <12yr group this number still increases to 2.05). Thus, if we recompute the effectiveness of vaccines/boosters in preventing COVID-19 deaths further sub-stratifying the children out of the <60yr group, we get the following numbers:
When we split the children out from the <60yr group, we see that the effectiveness of the booster in reducing the rate of COVID-19 deaths relative to unvaccinated is 87.9%, much closer to the 93.8% we saw in the >60yr group. Further, this 12-60yr fully vaccinated age group seems to have been strongly protected vs. COVID-19 death during this time period, even without boosting, with effectiveness of 89.3% suggesting that full vaccination prevents ~90% of COVID-19 deaths that would have occurred in this age group sans vaccination.
This high effectiveness was obscured in the combined <60yr analysis, since the systematic differences between the <12yr group and 12-60yr groups with respect to both the exposure (vaccination/booster) rates and outcome (risk of COVID-19 death) rates induce another Simpson Effect, and leads to greatly attenuated effectiveness estimates if we pool these groups together.
It appears that the failure to separate out the <12yr children contributed even more strongly to the low overall 13.7% effectiveness numbers than the failure to split into >60yr/<60yr, since the "overall" effectiveness vs. death for the >12yr combined group (69.4% for vaccinated, boosted or not) is much higher.
Thus, it is absolutely critical to separate children out as their own stratum when assessing vaccine effectiveness vs. COVID-19 related events.
Conclusions from COVID-19 Death Analysis:
These data affirm the points made in the previous blog post: that it is crucial to stratify by a strong confounding factor such as age when estimating vaccine (or booster) effectiveness vs. COVID-related events to avoid Simpson's paradox induced distortions, and these data further illustrate how crucial it is to further separate out children into their own stratum.
Thus, our final analysis of these data stratify into 3 age groups (<12yr, 12yr-60yr, >60yr). In the older >60yr group, we see higher VE_death for boosters than those fully vaccinated who have not received boosters yet. This agrees with a medRXiv paper from Israeli researchers presenting a rigorous analysis that shows that based on the first few weeks' data, the boosters greatly reduce risk of severe infections for older subjects. Apparently, in this older group, the observed waning immunity (also shown by this same group in a different medRXiv paper) extends to reduced protection vs. advanced outcomes like severe disease and death. This provides some support for the use of boosters in the older population, since it seems the extra flurry of circulating neutralizing antibodies induced by the booster may be effective in reducing risk of severe disease and death for this cohort.
The effectiveness numbers for >60yr vs serious illness, critical disease, and death (~70%) is considerably lower than what was seen by Israeli groups who removed previously infected from the unvaccinated and adjusted for other confounders, with 91% effectiveness vs serious disease for those vaccinated more recently and 86% for those vaccinated back in January or February. it is possible our numbers are attenuated by the inclusion of previously infected, especially in the unvaccinated group. Another blog post demonstrates that 22% of the unvaccinated in the >60yr group are previously infected, and their inclusion in the control group here would attenuate the effectiveness estimate.
However, for the younger 12yr-60yr group, we see high effectiveness in preventing COVID-19 death in the vaccinated group, whether boosted or not. With only 7.2% boosted in this age group and their booster shots being so recent, we do not yet have substantial evidence in these data to really evaluate the effect of boosters.
However, there is likely another latent Simpson's effect in the data for this age group. Given that most of those 7.2% boosted are in the older age groups, 40-59yr, and this age group has much higher risk for COVID-19 death than those 12-39yr, it is plausible that the effectiveness of boosters vs. COVID-19 deaths or critical disease is >>90%. Since the MoH dashboard does not split out in such fine age groups, we can't tell from these data. While it is possible that once we have more complete data, we will see a further reduction in risk of COVID-19 death in younger adults with booster, during this time period, it appears adults 12yr-60yr are strongly protected vs. COVID-19 death by vaccination (~90%), whether boosted or not.
Also, since this age group was vaccinated more recently the >60yr group, it is possible that the "waning" effect observed in the older Israeli residents has not kicked in yet but later will, further reducing the protection of 2-dose vaccination vs. COVID-19 deaths and possibly making boosters necessary to restore the same protection. However, since adults 40+ had access to vaccines starting January 19, and 16+ starting February 1, which is 6-7 months before this time period, it seems like many of the adults 16yr-60yr had been fully vaccinated >5m before this time period so should have begun to experience any potential waning effect.
While it is possible that we will see stronger booster effects even in the younger age groups once we have more complete data on boosters in this age group and more time for immune protection from the first two shots to potentially wane, it is still not clear to me whether a booster is needed to retain strong protection against COVID-19 death (or as we see later, against severe or critical disease) for the younger age groups -- data from the next couple months should tell us a lot as a higher proportion of this group receives 3rd shot boosters.
While these data are interesting to evaluate in light of decisions to roll out boosters or not, it is important not to draw firm conclusions given the short period of time since the booster program started, the small number of boosted in the younger adult age groups, and the time it takes for advanced outcomes like COVID-19 deaths to manifest. Nonetheless, these data show that vaccination, with or without boosting, still provides substantial protection vs. death from COVID-19, and any narrative suggesting the vaccines are not effectively preventing COVID-19 deaths is countered by these data.
Effectiveness of Vaccination and Boosters against Critical COVID-19 Infections
I performed these same analyses for the MoH data on what they call "critical COVID-19 infections." It is unclear to me how they define this, but it appears to be something very strict like ICU admissions given the relatively low numbers, comparable to the COVID-19 death numbers. Here are the numbers for critical COVID-19 disease:
We once again see a strong Simpson's effect with age, in that the estimated effectiveness vs. critical disease for both age groups are much higher than the overall number. And, one again, the inclusion of children <12yrs in the <60yr group can also distort those numbers.
Again it is not clear how many of the 195 unvaccinated critical cases in the <60yr group were <12yr, but given other data showing the risk of critical disease is so much less for <12 than others in the 30-60yr groups, it is likely safe to assume there are very few. Here we have assumed zero, but if you would like to see how the numbers would change if you assume a different number of critical cases in unvaccinated <12yr, it is straightforward to recompute the numbers given these changes.
Here are the numbers with children separated out, assuming for now no critical COVID-19 cases in children.
Similarly, we see the effectiveness estimates for the younger adult group greatly increase when the children are stratified out.
Overall, these data suggest very strong protection against critical COVID-19 disease in the 12yr-60yr group, with or without boosters, and for the >60yr group, the protection seems clearly stronger with boosters.
Effectiveness of Vaccination and Boosters against Serious COVID-19 Infections
Next we repeated these analyses for "severe cases." Again, it is not clear exactly how the MoH defines this, but it seems to include hospitalized cases as well as cases with certain advanced symptoms like very low pulseox. There are many more events in this group than the critical disease group. Here are the numbers for serious COVID-19 disease:
Once again, we see a strong Simpson's effect for age.
Here are the numbers with children separated out, assuming for now no serious COVID-19 cases in children (it is likely there are some serious cases in the <12yr age group, but again this should comprise a very small proportion of the total in the <60yr group -- and please feel free to recompute with different assumptions on number of serious cases in unvaccinated <12yr):
We also see very strong protection against serious COVID-19 disease in the 12-60yr group, with or without boosters, and for the >60yr group, the protection seems clearly stronger with boosters.
All of these data are consistent, suggesting the vaccines continue to provide strong protection vs. severe disease, critical disease or COVID-19 death.
For the older >60yr group, we see substantial protection vs. severe disease, critical disease or COVID-19 death from vaccination even with boosters (~70%), but this protection appears to have waned from the earlier reported numbers of >90%. However, a substantial proportion of this age group has already received boosters, and these have restored protection vs. severe disease, critical disease or death to very high levels (~93-96%) demonstrating a 4-fold to 6-fold reduction of risk of severe disease, critical disease, or COVID-19 deaths. Even though little time has passed since the booster program started, this provide support for their use in the older adult age group, especially at this time when the current Delta surge leads to high risk of exposure for this vulnerable population.
For the younger 12yr-60yr age group, we see vaccination provides a very high level of protection (~90%) vs. severe disease, critical disease, or COVID-19 death, whether boosted or not. These data do not provide any support for the benefit of boosters in these age groups, but given the low proportion boosted so far and the potential of a remaining Simpson's effect attenuating the estimated booster effectiveness numbers in this analysis , it is possible that as more time passes later data could reveal boosters provide a further reduction of risk.
Data and Details of Calculation:
I downloaded the data for this post from the Israeli MoH dashboard on September 9, 2021. The datasets I downloaded include:
Daily COVID-19 deaths from August 10 - September 8:
This spreadsheet was downloaded on the MoH dashboard from the section that is on the left side, second last section from the bottom.
Daily serious and critical cases from August 10 - September 8:
This data set was downloaded from the top middle section of the MoH dashboard.
Daily number vaccinated from August 10 - September 8:
This data set was downloaded from the top right section of the MoH dashboard.
All three of these data sets have the same format: date, age group (overall, <60, >60), and raw counts of the variable (death, vaccination, severe case, critical case) split by unvaccinated, boosted, and vaccinated (not boosted), as well as normalized counts per 100k. I could not find documentation for the MoH definition of "severe" and "critical" disease, but based on the numbers it seems "critical" is likely ICU stays and "severe" a much broader category that includes hospitalizations or perhaps also including other severe symptoms (e.g. very low blood oxygen levels).
For some strange reason the ordering of rows by age group is not consistent within or between spreadsheets, so it required care to collate these data sets together into a common spreadsheet.
I split the rows into 3 sections: overall, <60, and >60, and within each section each row was 1 day between Aug10 and Sept8. From these collated data, I did the following calculations:
1. Inferring of total numbers unvaccinated/boosted/vaccinated-not-boosted per day:
These numbers were not provided on the dashboard (as far as I could tell), but they could be inferred from the downloaded tables since each provided both raw counts and normalized counter per 100k for all outcomes, vaccination categories, age groups, and dates. For example, on September 6, the "vaccinations" spreadsheet reports a raw count of 412 in the "boosted" column with corresponding normalized rate of 19.1 in the "boosted per 100k" column. From this, based on some simple algebra based reversing the equation above for computing rates per 100k, we can infer the total boosted population on September 6:
Total boosted population on Sept6 = 412*100,000/19.1 = 2,157,068
To check the math, you can see 412 out of a population of 2,157,068 yields a proportion of 412/2,157,068 = 0.000191, which when shifting the decimal place 5 places to the right corresponds to a normalized rate of 19.1 per 100k.
This can be repeated for all days, age groups (all/<60/>60), and vaccination categories (boosted/unvaccinated/vaccinated but not yet boosted) to get corresponding daily counts in the population. We could compute this from any of the outcomes (deaths, severe cases, critical cases, vaccinations), and if you do this within the attached spreadsheets, you will see that they all yield similar population level results, subject to differential rounding errors since the rates per 100k are rounded to 1 decimal place. Thus, we used the data from the "daily vaccinations" spreadsheet to compute the population numbers, since these had the largest daily raw counts of all the outcomes and thus least affected by roundoff error and most accurate.
2. Estimating average population numbers over the August 10 - September 8 time period
Given that more people are vaccinated and boosted every day, the total number of boosted, unvaccinated, and vaccinated-not-boosted change from day to day. To compute effectiveness numbers, I averaged each of these numbers over all of the days in the time period, separately by age group, and used these numbers as "population numbers" reported in the tables.
3. Computing overall raw counts normalized rates/100k for each outcome/vaccination group
For each outcome (death, severe disease, critical disease) and vaccine group, I summed the counts over all days to get the raw numbers of each outcome to put into the table, and divided by the average population numbers in those categories to get the normalized rates per 100k for the table. These normalized rates were then used to compute the vaccine effectiveness numbers for boosters or vaccination-without-boosters as described above. This was repeated for each age group (overall, <60, >60).
Following is the analysis data set that includes the collated data and results after all of these calculations, and contains all of the numbers required to generate the tables presented above.