BNT162b2 mRNA Covid-19 Vaccine efficacy after 1st dose - explain the statistics

Question

Pfizer's paper (Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine) published recently states the following in the Efficacy subsection of the Results section:

Between the first dose and the second dose, 39 cases in the BNT162b2 group and 82 cases in the placebo group were observed, resulting in a vaccine efficacy of 52% (95% CI, 29.5 to 68.4) during this interval and indicating early protection by the vaccine, starting as soon as 12 days after the first dose.

Can anyone explain in simple terms the calculation of 52%? I am guessing that this number was obtained from:

But why is that the calculation? Does it depend on the fact that the vaccine and placebo groups were in equal numbers, such that by observing 82 COVID-19 cases in the placebo group, we match 82 subjects in the vaccine group, and since only 39 confirmed cases were observed in the vaccine group, the authors conclude that the "complementary (to 82)" 43 subjects did not become sick, and thus the vaccine is efficient in 52% of the subjects?

Answer 1

The paper itself describes how this is calculated (see the Methods section), but also see this Q&A at Biology.SE talking more broadly about how efficacy has been defined in these vaccine trials:

https://biology.stackexchange.com/q/96941/27148

They define efficacy as the fraction of infected in the vaccine compared to placebo categories, normalized for observation time (the integral of time by number observed). Quoting from the paper you linked:

Vaccine efficacy was estimated by 100×(1−IRR), where IRR is the calculated ratio of confirmed cases of Covid-19 illness per 1000 person-years of follow-up in the active vaccine group to the corresponding illness rate in the placebo group

giving the equations

IRR = (VaccineInfections/VaccinePersonYears/1000)/(PlaceboInfections/PlaceboPersonYears/1000)

VE = 100 * (1 - IRR)

If VaccinePersonYears and PlaceboPersonYears are equal, then IRR reduces to:

IRR = VaccineInfections/PlaceboInfections= 39/82 = 0.48

100 * (1-0.48) = 52% effective

So yes, this is algebraically equivalent to the equation you give, but it's originally calculated based on the actual data, not the simplified formula. It's just likely that VaccinePersonYears and PlaceboPersonYears are sufficiently similar (remember, they're giving the vaccine and placebo 50:50 to people in the same time frame), and it's a large study with thousands of participants so drop outs and such are likely to even out statistically and reduce to the simplified form you give.

Answer 2

The formula used for vaccine efficacy is as follows:

VE = (ARU - ARV) / ARU

(VE: vaccine efficacy, ARU: attack rate in unvaccinated participants, ARV: attack rate in vaccinated participants)

This is equivalent to:

VE = 1 - RR

(RR: Relative risk)

The attack rate is simply the number of new cases divided by the total group size.

ARV = 39 / 21,314 = 0.00182978

ARU = 82 / 21,258 = 0.00385737

VE = 0.5256 which is approximately 52%

These numbers are from figure three in the linked paper, showing the number of cases in each group between the first dose and the second dose (i.e. ignoring the effect of a second dose). The vaccine efficacy rose to approximately 95% by seven days after the second dose.

Your calculation produced a similar result due to the very similar group sizes (21,314 and 21,258).

Importantly, the authors note:

The study was not designed to assess the efficacy of a single-dose regimen.

This places caveats on the conclusions that can be drawn from this study with regard to a single-dose regimen of this vaccine.

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The Wikipedia page on vaccine efficacy provides a good summary of the measure and the associated caveats.