Consider the Pfizer vaccine for COVID-19. The data from the phase 3 trials shows that they had 21720 people in the vaccinated group and 21728 people in the control group:
n1 = 21720 n2 = 21728
In the vaccinated group, 8 people were infected with the virus. In the control group, 162 people were infected with the virus.
x1 = 8 x2 = 162
The ratios of infection are:
r1 = x1 / n1 = 0.000368 r2 = x2 / n2 = 0.007456
So, if the control group gives us the infection rate in the absence of the vaccine, how many of those infections were eliminated by the vaccine in the other group? The difference (that was eliminated) is r2 — r1, and we have to compare that with the baseline r2 to obtain the efficiency of the vaccine:
E = (r2 — r1) / r2 = 95.06%
Okay. But what is the 95% confidence interval for E? If you apply the procedure from this question...
...then the CI95 you get is 89.96% ... 97.57%
But couldn't you use the Fisher exact test instead? (if no, why?). In R, this would be:
> vac = matrix(c(21720 - 8, 21728 - 162, 8, 162), nrow = 2) > vac [,1] [,2] [1,] 21712 8 [2,] 21566 162 > fisher.test(vac) Fisher's Exact Test for Count Data data: vac p-value < 2.2e-16 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 10.09795 48.05249 sample estimates: odds ratio 20.38808
But if you transform the CI for odds into percentages (
100 * odds / (1 + odds)), you get 90.989% ... 97.961%. The interval has shifted a little, compared to the previous technique.
What am I missing?