What’s the difference between vaccine efficacy and effectiveness, and what do vaccine makers mean when they say their therapies are “95% effective”?

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Vaccine efficacy refers to the percentage change in having a disease in people who received a vaccine in a clinical trial, compared to people who did not receive the vaccine, i.e by what percentage does the vaccine lower the chance of getting a disease
On the other hand, vaccine effectiveness is a measure of how well a vaccine works in the community outside of clinical trials, i.e. in real life
The difference between the two parameters should be immediately obvious. Vaccine efficacy is calculated under optimal conditions, these may include the exclusion of the elderly and the immunocompromised, for example. The testing is also done under very strict supervision...
The general health status of clinical trial participants in efficacy measurements is known and the close monitoring means that generally only "suitable" candidates are taken on as trialists
Vaccine effectiveness on the other hand measures how the same vaccine performs on everyone, under imperfect conditions. Meaning the vaccine has to deal with people with underlying conditions that were not allowed in the clinical trials stage. This then affects how well it works
Onto how the vaccine developers measure "efficacy", i.e how is "vaccine X has 97% efficacy" measured, and what it means...
For this let's look at how Pfizer-BioNTech concluded on "95% efficacy" after their phase 3 clinical trials ...
First, how Pfizer ran their tests for their COVID-19 gene therapy, which is not actually a vaccine, but let's call it that for simplicity's sake.

They had 43661 participants from the U.S, Germany, Turkey, South Africa, Brazil and Argentina who were divided into 2 groups...
Half of the participants (21830) were given the vaccine in 2 doses, and the other half were given a placebo. The results and goals, what they call "endpoints" were evaluated 7 and 14 days after the second vaccine dose.
14 days after the second dose, 170 people from both groups tested positive for COVID-19. They call this "the attack rate"; 8 were from the vaccine group, and 162 from the placebo group
Now they took the 8 who tested positive from the vaccinated group and divided that number by the total of the vaccinated group. 8/21830*100 = 0.04%. So, Pfizer says from this, 0.037% of vaccinated people were infected.

Next is the unvaccinated group...
The "attack rate" here is calculated similarly: 162/21830*100 = 0.74%. Again, Pfizer concludes that 0.74% of unvaccinated people were infected with COVID-19
Next, they take the two numbers to determine the "reduction" in infection, i.e "reduction in attack rate" between the two groups.

Simple arithmetic again: 0.036/0.74*100 = 5%. So, from 0.74% to 0.036% is a 95% reduction! Voilà!
In broad terms, this is how the Pfizer & BioNTech phase 3 study concluded that their vaccine has 95% efficacy. This is basically how they all do it, with small variations here and there, including exceedingly complex mathematics which seems solely aimed at inducing confusion.
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