0
$\begingroup$

Summary: the placebo group had 1129 subjects and 18 infections. The vaccine group had 1131 subjects and 0 infections. More info here: https://www.pfizer.com/news/press-release/press-release-detail/pfizer-biontech-announce-positive-topline-results-pivotal

I'm interested in examining Pfizer's claim (same link) that this is "100% efficacy". Of course, one interpretation of that phrase is "nobody in the vaccinated group got COVID," which is trivially true, because that was directly observed. But I think people would tend to interpret it as something more like, "based on the study results, if I get the vaccine, my probability of getting COVID will be reduced by 100% (i.e. to 0%)." No finite sequence of vaccinated people not getting COVID can prove that vaccinated people get COVID exactly 0% of the time, so that doesn't sound right to me.

What is the posterior distribution for efficacy, given Pfizer's data, and given the definition "expectation of COVID probability reduction due to the vaccine" for "efficacy"? I'm taking the prior distribution for efficacy as the uniform distribution on [0,1]. Is that a reasonable prior distribution to use, in this context?

In case it's helpful in calibrating answers: I have a BS in math that covered more calc and algebra, and not so much statistics, and I'm currently working through Probability Theory by Jaynes in my free time. My intuition here is that the posterior distribution is the beta distribution with shape parameters 1 and 19.032, but I'm not confident in it.

$\endgroup$

1 Answer 1

0
$\begingroup$

There isn’t going to be a “clean” answer to this question because a posterior depends on the prior. In addition, there is the open question as to whether or not it is better to model the two groups as having different parameters or one parameter alone. Likewise, if you want a point estimate, then you need to state your utility function. If you change the loss you experience by being wrong, then you change your point estimate.

For example, imagine that all vaccines killed any participant if it is bad. Well, that would imply an all-or-nothing loss function. It may also be an asymmetric loss function if the injury from taking it has a different slope from not taking it.

If the Haldane prior is used, then the MAP estimator is that it is 100% effective. However, let us use the efficacy estimates of the Pfizer and Moderna vaccines in the wild among American adults as our prior, which is roughly 90%. So we will start with a beta(1,9). We could choose a stronger prior such as beta(10,90) or actual research data. However, I will use beta(1,9) because it is still pretty agnostic while assuming that a vaccine that will be competitive.

That leaves a posterior of $1140\theta^{1139}$. All values with any density are quite close to 100%. The 99.99% interval is roughly [.99596,1].

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.