It is often argued that because the number of infections is greater in the vaccinated fraction of a population, compared to the unvaccinated fraction, the corresponding vaccine is not effective.
But this is a simple logical error.
The point being that while the number of infections in the vaccinated can be greater, because of a high vaccinate rate, the percentage will be lower unless the vaccine really is ineffective.
Another way to think about it, is to consider that a small percentage of a large fraction of a population can easily be greater than a large percentage of a small fraction of a population.
Imagine a school with a 1000 pupils and a 95% vaccination rate, so 950 children are vaccinated and 50 are not vaccinated. A highly infectious disease is brought to the school by a visitor that infects say 40 of the 50 unvaccinated children. No vaccine has a 100% seroconversion rate and for the purposes of illustration let’s say that 80 of the 950 vaccinated children become infected.
There are twice as many infected vaccinated children as infected unvaccinated children. Does this mean that vaccination doesn’t work? No, of course not, if vaccination didn’t work, there would be around 760 infected vaccinated children.
A much better and intellectually honest way to compare the groups is to use the fraction or percentage that become infected. 80% of the unvaccinated children are infected, while only 8.4% of the vaccinated children are.
Thus in this context vaccination reduces infection by (1 – 8.4/80), i.e. by almost 90%. This could be considered to be the contextual or point vaccine effectiveness. A confidence interval for this value could be calculated if required to make sure the scenario is statistically sound.