Help me understand the math of "11x more likely to die if unvaccinated"

This is an example of a recent article that quoted that factor between unvaccinated folks dying from the virus vs vaccinated.

Unvaccinated People Are 11 Times More Likely To Die Of COVID-19 Than Vaccinated : NPR

Now the math “mental gymnastics” I’m seeing across Facebook is “vaccinated people have a 99.99% survivability, so .01% will die of Covid. Multiply that .0001 by 11 and you get .0011. Subtract that from 1 and you get 99.89% survivability for the UNvaccinated!”.

I know there’s a flaw in that bullshit math somewhere, but I can’t find a cite that explains it in a way I can understand. I seriously doubt that the “11x as likely” means a difference between 99.99% and 99.89%.

I’d like to see the raw numbers that article was based on.

716,000 Americans have died of Covid. If those figures are correct it would mean that around 60,000 vaccinated people have died. And that seems far too high a figure. The most recent figure I could find say that around 8000 vaccinated people have died of Covid19.

These figures suggest the actual ratio is somewhere around 88:1.

500,000+ died before there were any vaccinated people.

Statistically, it doesn’t matter if you were unvaccinated by choice or because there was no vaccine available. The ratio just compare the number of unvaccinated people who died of Covid to the number of vaccinated people who died of Covid.

The title of the study cited in the link is Monitoring Incidence of COVID-19 Cases, Hospitalizations, and Deaths, by Vaccination Status — 13 U.S. Jurisdictions, April 4–July 17, 2021 . The numbers you are using are from the beginning of the pandemic to the present, a time period that mostly encompasses a period when vaccinated vs unvaccinated was irrelevant since there was no vaccine.

It matters if you are doing what you did, which is to use total deaths over a time period that includes when no vaccine was available as the denominator for a number that can only include the time when the vaccine was available.

That math would check out, but that’s not what the studies being cited are claiming. Also, those survival rates are very low compared to what is actually observed.

From the study (covering April-June 2021)

Among the unvaccinated, case rates in these 13 locales were 112 per 100,000. Death rates were 1.6 per 100,000. That is a case fatality rate of about 1.4%

Among the vaccinated, case rates were 10.1 per 100,000 and deaths were 0.1 per 100,000. That is a CFR of about 1%.

So the liklihood of dying if you get symptomatic COVID (assuming you won’t get tested if you are vaccinated and have no symptoms) is maybe 50% higher than if you get symptomatic COVID and are unvaccinated. But the overall death incident rate was more like 16x higher. (ETA: This is very age-dependent, hence the use of age-standardized metrics below. This number is pretty meaningless without considering age.)

The 11x number specifically comes from a more complicated calculation called “age-standardized incident rate ratios” which take those cruder numbers I quoted and buckets them based on age groups. This makes sense because vaccination rates, hospitalization rates, and death rates are all strongly correlated with age.

From the paper:

My reading of that is that the vaccines started out reducing the IRR of death from 16.6 (meaning you were 16.6 times more likely to die if unvaccinated) to 11.3 as delta became more prevalent.

It also points out that the IRR of actually getting COVID dropped to only 4.6 once delta became dominant. This matches what I’m seeing locally where there are a good number of symptomatic (but not serious) breakthrough cases. But the ratios of hospitalizations are still running >80% unvaxxed.

Something is wrong there. Or maybe not wrong but very confusing

Hmm… maybe.

What I think I’m reading is that for every 100,000 unvaccinated people in the study, 112 of them had a positive test result (a “case”). 1.6 of them died. 1.6 / 112 = 1.4% case fatality rate.

For every 100,000 vaccinated people in the study, 10.1 of them had a “case” and 0.1 of them died. 0.1 / 10.1 = 1% case fatality rate.

Obviously the ratio of death rates would be 1.6/0.1 = 16x more likely to die of COVID if you are unvaccinated. But that hides a lot of confounding factors (age in particular, but also other mitigation steps and the fact that at-risk individuals are more likely to be vaccinated).

Really the overall numbers aren’t as relevant as the IRR calculated in the paper. That standardizes for age and compares the incident rates of cases/hospitalization/death across the vaccinated and unvaccinated cohorts. And that is where the 11x number from the threat title comes from. It has nothing to do with the (erroneous) “survival rates” that the OP’s Facebook posts cite.

But you also have to consider the base rate.

Suppose there are 100 people in the world: 99 are vaccinated, and 1 is unvaccinated.

Two of the vaccinated people and the one unvaccinated person all die of COVID.

Then twice as many vaccinated people died of COVID than unvaccinated people, but it would be wildly misleading to say that vaccinated people are twice as likely to die.


That is why the ratio that matters is not a ratio of the number who died, it is a ratio of the rate of death.

In your example, the death rate for vaccinated people is 2/99 = 2.02 per hundred, while the death rate for unvaccinated people is 1/1 = 100 per hundred. The IRR (incident rate ratio) would be 100 / 2.02 = 49.5. So you would say that the unvaccinated people are about 50 times more likely to die.

Now obviously the sample is really bad for that type of analysis in your example, but that’s how the math would work out using IRR.

Your point about base rates is well observed - many of the anti-vaccination stories rely on this fundamental misunderstanding by crowing about how, for example, over half of the deaths in a certain region are of vaccinated patients. They fail to mention that if a population is >95% vaccinated (the very elderly, for example) you may expect the majority of deaths to be vaccinated even if the vaccine is 80+% effective at preventing death. In fact, if the vaccine has no effect at all you would expect the percentage of vaccinated deaths to match the vaccination rate.

Okay, I see your point there. A small percentage could be deceptive if it’s the product of an even smaller sample group.

But I don’t feel this is the case here. Current reports are saying just over fifty percent of Americans (56.5% is the exact figure from the CDC as of October 12) are fully vaccinated so we have two sample groups of comparable sizes. cite

To counter the arguments that we shouldn’t consider the deaths that occurred before vaccination was possible, I’ll offer figures that focus on a single month after the vaccination program was up and running. The most recent figures I could find were from June, which were reporting the deaths from May 2021. Approximately 18,000 people died of Covid in that month (in the United States). Of this total approximately 150 had been fully vaccinated.

These figures suggest that the ratio of unvaccinated deaths to vaccinated deaths is over 100:1. cite

This may have been true in May, but it almost certainly wasn’t true in September. Per the paper linked above, Delta didn’t become the majority strain until June (the week of June 20, to be exact). In the “prior to June 20” period, the percentage of deaths that were unvaccinated was 91%. So more like 10-to-1, not 100-1. For the period June 21-July17 it was 16% of deaths were fully vaccinated.

Now it is certainly possible that the areas in the study are not representative of the nation as a whole, but it generally aligns with local data I have seen which showed that fully vaccinated hospitalizations and deaths are generally in the 15% range currently. Fortunately rates are dropping quickly, which will hopefully continue.