I keep seeing references to the fraction of people with COVID related conditions that have been vaccinated. A recent story implied that the fact that about half of the infections in Israel were in vaccinated people was evidence that the vaccines aren't working. There is a nice post that looks at the stories about Israel. Half sounds like a lot only is you don't take into account the base rate of vaccination. Since 85% of the people have been vaccinated the fact that a far smaller fraction of the infected were vaccinated shows that the vaccines do work. Not taking the base rate into account, or deceptively not including it in the information provided, is the Base Rate Fallacy.
What is the relationship between vaccine efficacy, vaccination rate, and the fraction of those effected that are vaccinated? It turns out that it isn't as simple as you might expect.
Say the fraction of people that are vaccinated is v, and the fraction of illnesses the vaccine prevent is e. Then fraction of infected people that are vaccinated is:
Let's look at some examples. If we set both e and v at 0.85, meaning that 85% of the population has been vaccinated and that the vaccine is 85% effective in preventing disease. In that case the fraction of the cases that are vaccinated is expected to be just under half. For the current situation in the US let's take e=98% and v=30%. This gives us 0.008, which is a pretty good match for the oft repeated "99% of those hospitalized for COVID in the US have not been vaccinated".
I see a slight danger here. If we consider e=0.98 and v=0.5 we get 0.019. The vaccinated fraction more than doubles. The reason it goes up as the vaccination rate goes up is obvious, once it is pointed out. Take the extreme case, where everyone is vaccinated, v=1. Then the fraction of effected people that were vaccinated is also 100% since that's the only kind of people there are. This is the extreme case of the Base Rate Fallacy. I both hope and fear that when the US gets to v=0.5 we hear people using this doubling in the vaccinated fraction as evidence that vaccines aren't working. I hope for this because it will mean that v has gotten to 0.5 and I fear it will reduce future vaccinations. If we ever get to v=0.95 the fraction will be greater than 1/4, which will doubtless be trumpeted as a failure.
How do we get this equation?
Let's define some factors. Let's say that if G (for group) unvaccinated people are exposed we get Gc infections. That defines the degree of contagion c. If G vaccinated people are exposed there will be Gc(1-e) infections where e is the efficacy of the vaccine so we have a modified contagion factor of c(1-e).
If we have a group of N people then the number of cases expected in the vaccinated group, Cv, is Nv (the number of vaccinated people) times the modified contagion factor c(1-e).
The number cases in the unvaccinated group, Cu is N(1-v) (the number of unvaccinated people) times the contagion factor c.
So the fraction of cases that are vaccinated is the number of cases in vaccinated people divided by the total number of cases Cv/(Cu+Cv) or
There is a common factor of Nc so that cancels. So the number of people and the contagion factor don't enter into the final result. So we get:
We could also divide through by (1-e) and get the expression at the top.
