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
pfe-pfizercom-d8-prod.s3.amazonaws.com/2020-09/C45910…
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
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