[ithkuil]

> so I might be missing something

You're surely calculating a useful number, but it's not the "vaccine efficacy". Definition from the CDC website:

    Vaccine efficacy/effectiveness is interpreted as the proportionate reduction in disease among the vaccinated group. So a VE of 90% indicates a 90% reduction in disease occurrence among the vaccinated group, or a 90% reduction from the number of cases you would expect if they have not been vaccinated.

Your calculation looks at the number of hospitalised people not in the people who got the disease.

So, on a first glance it seems you're computing a useful number: how effective is the vaccine from preventing your to get hospitalised assuming that you would have been hospitalised if you caught covid. Surely that's a useful number, it just shouldn't be confused with the >90% vaccine efficacy number, which measures a different thing.

Now, I do also have some qualms with your calculation of the "hospitalisation prevention efficacy" rate.

Let me rewrite your calculation symbolically:

    "Israelis >12 years are v/(100-v) vaccinated/not vaccinated. Among the hospitalised h/(100-h) are vaccinated/not vaccinated. So you have h/v = x and (100-h)/(100-v) = y Therefore the ratio of risks is x/y = z. Which gives you (1-z) as I mentioned."

Let's make a little thought experiment. Let's imagine that 98% of people were vaccinated with a vaccine with 90% efficacy. Since the efficacy is not 100%, some people will still get the disease and some will still get hospitalised. Since very few people in this scenario are unvaccinated, most of the people who end up in hospitals will be vaccinated. Let's imagine that 50% of those who get the disease get hospitalised. Since only 10% of those vaccinated that get infected contract the disease, only 5% of the vaccinated population will get hospitalized. OTOH (in this scenario) 50% of the infected unvaccinated people get hospitalized, but since only 2% of the people are unvaccinated, this means only 1% of the infected unvaccinated population get hospitalized; 99% of the hospitalized people are thus vaccinated.

    v = 98
    h = 99

    "People from our scenario are 98/2 vaccinated/not vaccinated. Among the hospitalised 99/1 are vaccinated/not vaccinated. So you have 99/98 = 1.01 and 1/2 = 0.5 Therefore the ratio of risks is 1.01/0.5 = 2.02. Which gives you -1.02 as I mentioned."

That formula doesn't seem to make sense.

[orbifold]

It does make sense from the numbers 99% of hospitalised vaccinated, 98% of the population vaccinated you should conclude that the vaccine increases the risk of hospitalisation. That is why the number comes out negative. It would be the same in https://en.wikipedia.org/wiki/Vaccine_efficacy if you swapped the numbers for vaccine / placebo. The number would come out negative if in a trial the vaccine worked worse than the placebo. In your scenario you confuse yourself by calculating with percentages, I think.

[ithkuil]

> you should conclude that the vaccine increases the risk of hospitalisation.

Not really. It just means that since in that scenario the vast majority of people are vaccinated the break-through cases (people who despite being vaccinated end up hospitalised) outnumbers the small percentage of unvaccinated people who end up hospitalised *despite* the fact that the vaccine efficacy rate remains constant.

I'm not sure if we're talking past each other, so let me take one step back and make another attempt:

I made an extreme example to make it easy to see. Let's make an even more extreme example without any numbers nor percentages:

Let's imagine that everyone, literally every single person in a population gets vaccinated. How many hospitalised people do you expect to see?

Since the vaccine is not perfect and some people get severely sick despite the vaccine, you'd expect to see some number of hospitalised people.

How many of those hospitalised people would be vaccinated?

Well, we just said that in this scenario everybody in the population has been vaccinated, thus everybody who got hospitalised is vaccinated!

Can we conclude that vaccines cause hospitalization? We clearly cannot conclude that from these numbers alone.

Can we agree on this before continuing to talk about which crucial metric we're ignoring in this discussion?

[orbifold]

I'm honestly not sure, maybe we are really just talking past each other. If there are no unvaccinated people left the relative risk reduction would be zero, that just follows from the definition. The absolute number of hospitalisations would still be meaningful. In the perfect world that everyone was vaccinated you would still see an age dependent decrease of effectiveness over time and relative to alpha. The real data from Israel seems to show that two vaccine doses in a very compliant population are not enough to effectively prevent the number of cases, hospitalisations and deaths to rise without additional other measures (boosters, masks, etc.).

[ithkuil]

I'm not arguing against what data from Israel may show in general, I didn't go into that rabbit hole.

What I'm arguing is that the argument as presented in this thread cannot support this conclusion since they only focus on the vaccinated/non vaccinated ratio of hospitalised cases, which by definition goes up as more people get vaccinated. The effect is also compounded by a skewed distribution in favour of older people "getting severely sick" (and thus hospitalized) versus the general lower "getting sick" bar set to measure vaccine effectiveness.

This is a bit counterintuitive and makes for an easy topic for journalists to create a sensationalistic piece.

Unfortunately that's how most articles of the subject that I see cited look like. Perhaps there are better articles that make a stronger case, with all the relevant number (such as hospitalizations / infected people in general population). Can you share one if you know about it?