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Assume I have a high school statistics class under my belt:

One common misunderstanding is that 95% efficacy means that in the Pfizer clinical trial, 5% of vaccinated people got COVID. But that's not true; the actual percentage of vaccinated people in the Pfizer (and Moderna) trials who got COVID-19 was about a hundred times less than that: 0.04%.

What the 95% actually means is that vaccinated people had a 95% lower risk of getting COVID-19 compared with the control group participants, who weren't vaccinated. In other words, vaccinated people in the Pfizer clinical trial were 20 times less likely than the control group to get COVID-19.

-Reference: https://www.livescience.com/covid-19-vaccine-efficacy-explained.html

I would like to understand with a "concrete" example how the these numbers are calculated, so I can understand the contours, limitations and assumptions of what it means to 95% (mRNA C19 vaccines) vs ~70% (JNJ).

A concrete example or a pointer to such an example written for at a college freshman level is appreciated.

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2 Answers 2

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If you move to a hermit hut in a middle of Alaskan forest to grow and hunt your own food, and wear an N99 mask 24/7, you have 0% chance of getting COVID-19 no matter if vaccinated or not. The referred risk is a risk for some exposed to COVID-19, so one that corrects for such factors using a statistical model. It’s an estimate of efficacy of the vaccine that is independent of other factors. As you can see, the actual risk may differ because other factors (e.g. wearing a mask, working at a hospital) would impact it.

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Imagine we observe $n$ people who got vaccinated and $n$ people who received a placebo vaccination. Because everything is double blinded and the world ist beautiful, both groups are exposed to corona infection particles to the exact same amount. Depending on that amount, the immune system, wheather, infectious properties of the virus etc only a fraction $d$ of the unvaccinated get the disease. The number of non-vaccinated probands who get the disease is thus

$ill_{placebo}= n * d$

we assume, that the vaccinated have some protection. Let 's call that protection $p$ and

$ill_{vaccinated}=n * d* p$

Does that sound convincing on the high school statistics class level?

So your question was, how to compute $p$ and the simple answer is: We observe how many out of $n$ placebo recipients and how many out of $n$ vaccinated get the disease and can then compute

$p = \frac{ill_{vaccinated}}{ill_{placebo}} = \frac{n*d*p}{n*d}$

As p is a value between $0$ and $1$ it is suitable to give in in percent, such as 95% for Biontech/Pfizer and 70% on AstraZeneca.

However you noticed how I assumed that exposition and immune system an wheather and whatnot are the same between the vaccination group and the placebo group. They will almost certainly not have been the same between two studies performed by differing companies in different countries. So the comparison of two such fractions is a difficult matter.

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