A thread for fellow statistical analysis plan nerds (warning: math ahead).

From Pfizer's protocol, vaccine efficacy will be estimated by the incidence rate ratio. A tutorial on how this corresponds to their planned beta-binomial analysis. 1/6
https://pfe-pfizercom-d8-prod.s3.amazonaws.com/2020-09/C4591001_Clinical_Protocol.pdf

Even though vaccine efficacy is estimated by the incidence rate ratio VE=1-IRR=1-(m1/T1)/(m0/T0), the underlying test is an exact test for a single binomial proportion. The proportion in question is what fraction of total endpoints are in the vaccine arm p=(m1/(m1+m0)). 2/6

Under 1:1 randomization, we have roughly equal follow-up time across both arms (T1=T0). If our null hypothesis was VE=0, we would expect p=50%, with the same number of events across arms. This null can be expressed as a function of the follow-up time p=T1/(T1+T0). 3/6

In fact, we are interested in a different null hypothesis of VE<=0.30, to rule out a lower bound of 30% efficacy. So we can calculate, if VE=0.30, what proportion of events are expected in the vaccine arm? It is a bit lower - 41% - since the vaccine has some effect. 4/6

Under a frequentist paradigm, we can compare p_hat = m1/(m1+m0) to the null hypothesis value p using an exact test for a one-sample binomial proportion. This is equivalent to a test of IRR=1 because the test itself is built using the total follow-up T1, T0. 5/6

Pfizer is using a Bayesian analysis, hence the beta-binomial model with beta as the conjugate prior. They will assess the posterior probability P[VE>0.30|data]. Natural next question then is... what prior have they selected? Hope this thread was helpful! 6/END

Since folks are recirculating this thread in light of emerging data, I'll append that the prior was included in the protocol (I just hadn't read far enough at the time). https://twitter.com/biosbenk/status/1307099282063982592?s=20