Comment by tzs

The Israel data shows excellent effectiveness of vaccines. For every age group it shows that the chances of severe COVID are greatly lower in the vaccinated than they are in the unvaccinated.

If you just look at the totals for their whole population it may not look that way because you run into a Simpson's paradox situation. That's because older people are both more likely in general to have their COVID cases be severe, and older people in Israel or more likely to be vaccinated.

To illustrate, imagine a population that has 2000 young people and 2000 old people, and both groups are 50% vaccinated with a vaccine that reduces your chances of severe disease by 90%, and image we are dealing with a disease that is 20x as likely to be severe in old people than it is in young people. Let's say that 1% of young, unvaccinated people get severe disease.

Here are the number of severe cases we'd expect in that population.

    1 in the 1000 young vaccinated people
   10 in the 1000 young unvaccinated people
   20 in the 1000 old vaccinated people
  200 in the 1000 old unvaccinated people

If we look at the overall numbers we get 21 severe cases in vaccinated people and 210 severe cases in unvaccinated people.

9.1% of severe cases would be in vaccinated people.

Now consider a population with 2000 young people and 2000 old people, with the same vaccines that reduces your chances of severe disease by 90% and where the diseases is 20x as likely to be severe in old people, but in this population 90% of the old people are vaccinated but the young are still only 50% vaccinated.

Here are the number of severe cases we'd expect in that population.

    1 in the 1000 young vaccinated people
   10 in the 1000 young unvaccinated people
   36 in the 1800 old vaccinated people
   40 in the  200 old unvaccinated people

Now for the overall population we get 37 severe cases in vaccinated people and 50 severe cases in unvaccinated people.

42.6% of cases would be in vaccinated people.

If the young people also get vaccinated at the same 90% rate as the old people, here are the numbers for severe cases.

   1.8 in the 1800 young vaccinated people
   2   in the  200 young unvaccinated people
  36   in the 1800 old vaccinated people
  40   in the  200 old vaccinated people

That gives for the overall population 37.8 severe cases in vaccinated people and 42 in unvaccinated people.

47.4% of cases would be in vaccinated people.

If no one was vaccinated, the numbers would be

   20 in the 2000 unvaccinated young people
  400 in the 2000 unvaccinated old people

for a total of 420 severe cases. 0% would be in vaccinated people.

If everyone was vaccinated, the numbers would be

    2 in the 2000 vaccinated young people
   40 in the 2000 vaccinated old people

for a total of 42 severe cases. 100% of cases would be in vaccinated people.

It is useful to look at the severe case totals in our population of 2000 young people and 2000 old people.

  420   severe cases when no one is vaccinated
  231   severe cases when 50% of both young and old are vaccinated
   87   severe cases when 50% of young and 90% of old are vaccinated
   79.8 severe cases when 90% of both young and old are vaccinated
   42   severe cases when everyone is vaccinated