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Simpson's paradox strikes again: Refuting reports vaxxed have 5x case fatality rate than unvaxxed

Updated: Sep 15, 2021

I have seen a lot of people obsessing with Table 5 of the August report put out the UK Public Health England focusing on the Delta variant

In Table 5 on page 17 of this report, you can see that:

  • There were 402 Covid deaths among 47,008 fully vaccinated for case fatality rate (CFR) of 0.86%, meaning 0.86% of confirmed Delta COVID-19 cases resulted in death.

  • There were 253 Covid deaths among 151,054 unvaccinated for CFR=0.17%

In these data, we see that the CFR for vaccinated was >5-fold higher than for unvaccinated (0.86/0.17=5.02). Is the case fatality rate really 5x higher for vaccinated than unvaccinated? I've heard some people say this, claiming it is evidence of Antibody Dependent Enhancement (ADE), in which an immune response in fact leads to worse outcomes if the person gets infected.

Fortunately, in this report they also split data out by age (<50/>50).

For the older group, the CFR for vaccinated (1.81%) is 3.3-fold LOWER than CFR for unvaccinated (5.96%). This suggests, for the group at real risk of death from COVID-19, with Delta the risk of death was more than 3-fold lower in vaccinated breakthrough infections than unvaccinated infections. For younger group, CFR for vaccinated (0.05%) is 1.57-fold higher than CFR for unvaccinated (0.03%). This may raise some concerns and is worth looking into. However, note how low the death numbers are -- only 13 in the entire vaccinated cohort of >25,000 infections, and there are no details on the characteristics of these individuals in the report. This is an extra 5 deaths over what we would expect if the CFR was no higher than in unvaccinated. Here we see another clear illustration of Simpson's paradox, given we have:

  • A confounding factor (age) that is strongly associated with both:

  • the outcome (death), since older people are MUCH higher risk to die of COVID-19, and

  • the exposure (vaccination), since older people had MUCH higher vaccination rates.

This phenomenon can produce a wildly distorted view if analysis is done without stratifying or otherwise adjusting for the confounding factor (here, age).

This phenomenon pops up ALL THE TIME right now, given how age is strongly associated with both vaccination status and severe disease outcomes in most places.

Now, to illustrate how strong this Simpson's effect is here, suppose instead of 13 deaths in the <50yr vaccinated group, there were 2 deaths. See the following table.

In this case, we would have that for both age groups, the CFR was much lower for vaccinated than unvaccinated by a factor of 4.2x for <50yr and 3.3x for >50yr, but if you compute it without stratification, you find the CFR for vaccinated is 5x HIGHER in vaccinated than unvaccinated. This shows how extreme this effect can be when you have a confounder (age) so strongly associated with both the outcome (death) and exposure (vaccination). Thus, I hope people realize soon that any estimate of vaccine effectiveness must adjust for age by stratification or some other approach or it will give very confusing and misleading results.

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