Health authorities want to eat their cake and have it, too.

The severe illness paradox.

The vaccine is x% effective against preventing cases, but y% effective against hospitalization or death. I am sure, by now, we have all heard this trope. In case you haven’t, consider this example quotes from a recent news article

“Protection from the two-dose vaccine against infection dropped to 33% compared with 80% before the emergence of omicron. However, two doses are still 70% effective at preventing hospitalization in omicron patients in South Africa, Fauci said.”

That is an interesting statement that, perhaps snobbishly, bothers me to no end. The phrase is purposefully ambiguous, and I suspect that many vaccine enthusiasts have been unwittingly tricked into reading it in its broadest possible meaning.

Confused? Let me explain.

We often hear the stories of people getting vaccinated then having a, increasingly common, rare breakthrough infection. These individuals wind up in the hospital, then parade themselves in the media as a vaccine success story. These stories are part of the “it would have been worse…” series of media articles that have, deservedly, become a right-wing meme.

Anyone that has ventured into an advanced statistics class (the kind that teaches quasi-experimental methods) should be able to spot the error in logic these individuals are making, not that you need a statistics background to see through their statements. The only reason I reference statistics is because a single, perfectly fitting phrase will, time and time again, be uttered by any professor worth their salt: “We do not observe the counterfactual”.

There is no way of knowing, especially in an individual case, what the health outcome that did not occur would have been. We. do. not. observe. the. counterfactual. In large experiments, not observing the counterfactual causes us to make assumptions. Often, we can feel confident about these assumptions, but often is not always. There are excellent experiments where I would feel confident betting on an assumption. But it would be a bet, nothing more. If I was playing heads-up poker, and had aces in the hole, and my opponent went all-in pre-flop, I would call with the understanding that I will not always win.

The thing about bets is you need to know how to calculate the odds; otherwise, you are just playing for fun. When it comes to these vaccinations, I, like yourself, am not playing for fun. I care about the odds. For that reason, the trope, 33% effective at preventing cases but 70% effective at preventing hospitalizations, is deceptive.

Let us consider the two meanings, and what the health authorities have told us.

Meaning #1:

A random vaccinated person in the population is x% less likely to be infected with the virus and a random vaccinated person in the population is y% less likely to end up in the hospital for the virus.

Meaning #2:

A random vaccinated person in the population is x% less likely to be infected with the virus, and having been infected, they are y% less likely to end up in the hospital for the virus.

These are, as you can tell, two completely different meanings, but both can be ascribed from Fauci’s statement above. Depending on how a person’s mind works, they may immediately assume one meaning or the other. At first glance, I would assume that the health authorities are trying to convey meaning #1, except they continually make a certain statement that implies they are actually trying to convey meaning #2.

Take the Center for Disease Control, say, “Fully vaccinated people with a vaccine breakthrough infection are less likely to develop serious illness than those who are unvaccinated and get COVID-19”. This statement, in combination with the statement that the vaccine is y% effective against hospitalization, implies meaning #2. If meaning #2 is true, then the odds of having a severe illness are far different than meaning #1.

Let’s us use a simply example to put these numbers in context.

rV = -VE*rU + rU

Where, 

rV = risk for the vaccinated population; and,
rU = risk for the unvaccinated population; and,
VE = Vaccine effectiveness

Assume a population of 1000 in each group, and assume 100 unvaccinated people were infected, ie., rU = 0.1. Also assume VE = 33%. 

rV = -0.1*33%+0.1 = 0.067

So 67 vaccinated individuals are infected. 

We can do the same for hospitalizations. rV = -0.1*70%+0.1 = 0.03, so 30 individuals end up in the hospital. This is using method #1.

But, using method #2, let's assume that 15 unvaccinated individuals end up in the hospital. The risk of ending up in the hospital for the unvaccinated then becomes 0.015. If we assume 70% VE at this point, rV = -0.015*70%+0.015 = 0.0045... or 4.5 people in a population of 1000.

Which is true?

Well, regular readers will remember I referenced a Canadian document with the (biased) statistics since the vaccination campaign began. Here is a screenshot from the document.

Do you see what I see? Let me adjust:

These are the same numbers with the odds of being hospitalized or dying per case. Apparently, people that are vaccinated have a higher chance of dying per case.

Option 1: Method #1 is being used to calculate VE and the Fauci’s of the world are lying about the fact that those who get the virus are at less likely to develop serious illness, or the

Option 2: Method #2 is being used and cases in the vaccinated are being underestimated by a lot¹.

If option 1 is true, then our omniscient friends that claim to know the counterfactual have been terribly misled. If option 2 is true, then VE is much lower than what is being reported, and the vaccinated are spreading the virus at a high rate, but surviving at an equally impressive rate. Which is it?

1. The Canadian document did not explicitly reference VE, but usually Canadian numbers have hovered around 90% VE for cases, death, and hospitalization, often even claiming higher VE for death and hospitalization.