Hoffman: How You Can Tell Vaccines Are Working

Los Alamos

Maybe you heard the news recently that 19 percent of New Mexico’s new COVID-19 cases are happening to vaccinated people. And maybe, like me, your first impulse was to say, “What the heck? That sounds bad! I thought the vaccine was supposed to protect us!”

If that’s the case, let me ask you to think through the numbers a bit further, and you’ll realize that the evidence shows that vaccines really are protecting us. But you need a bit more information to see that.

As is so often the case with statistics, you need more than one number to understand what it all means. In this case, the other number you need to know is the fraction of all New Mexicans who have been vaccinated. That number – the fraction of all New Mexican adults who have had two shots – is 66 percent, almost two-thirds.

So stop and think what would be happening if the vaccines were doing no good at all. In that case, your odds of getting COVID-19 would be the same whether you were vaccinated or not. So, since 2/3 of everybody is vaccinated and 1/3 are unvaccinated, you would expect 2/3 of all the new cases to be happening in vaccinated people, and 1/3 to be happening in unvaccinated people.

But that is NOT what is going on. The headlines don’t say that 66 percent of all new cases are happening to vaccinated people. They say that only 19 percent of new cases are in vaccinated people. So it’s obvious from those two numbers that the vaccines are definitely reducing your odds of getting COVID-19.

How much are the odds reduced? Here’s a little math. If the vaccine reduces your chances of getting sick by a factor of f, then the numbers I just mentioned tell us that (0.66/f)/(0.66/f + 0.34) = 0.19. (Sorry to just pop this equation on you without explanation, but I’ll be glad to give you more details if you’re curious.)  Solving that equation for f gives a value a little larger than 8. So the vaccines are reducing our chances of getting COVID-19 by more than 8 times – about what we heard when the vaccines first came out. 

We also heard the news that 13 percent of hospitalizations were for fully vaccinated individuals – again, much less than 66 percent. Doing the same analysis again for this number tells us that the vaccines reduce your chances of going into the hospital with COVID-19 by more than 12 times. I.e., your chances of being hospitalized drop down to less than 8 percent of what they would be without the vaccine. That’s pretty good medicine!

Now, to be honest, the picture is a little more complicated than I’ve described it here, because many new cases are in young people 12-17 years old, and only 43 percent of them are vaccinated. If you re-do the analysis assuming the whole state was only 43 percent vaccinated, then you would conclude that the vaccines are cutting your odds of being infected or hospitalized by only about 3 times and 5 times, respectively – not as good as when we assumed a 66 percent vaccination rate. The truth must be somewhere in between. You would have to know the fraction of new cases and hospitalizations among young people, both vaccinated and unvaccinated, to work it out correctly. 

But either way you do it, it’s clear that the vaccines are having a good effect. I hope it’s also clear that statistics always need to be thought about carefully before we jump to any conclusions.


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