Assume the following experiment: $G$ people with certain disease are randomly assigned into 2 groups with equal probability. People in group $i$ will received medicine $m_i, i=1,2$. After a period of time, it is observed that in group $i$, $N_i$($i=1,2$) people have recovered. Note that we do not know how many people were assigned to each group.
The numbers can be summarized into the following 2x2 table
|group1||N1||G1 - N1||G1|
|group2||N2||G2 - N2||G2|
|total||N||G - N||G|
Here we assume that only the values N1, N2 and G are known. We don't have the full table. We are interested in which medicine has a higher recovery rate. How can we test it based on the information? I am aware of the exact test by Fisher and Barnard, but I am not sure if they could be directly applied in this simple case.
As an example, assume N1=40, N2=20, G=100. Since the patients are assigned into each group with equal probability, we are likely to see a distribution of ~50/50:
In such scenario it's intuitively clear(Barnard's test has p-value of 0.000039) that m1 is more effective that m2.
PS: If I simply assume that G1( and thus G2) are given, perform Barnard's test to get a p-value $p=p(G_1)$, and aggregate these p-values through $$ p = \sum p(G_1) P(G_1=g) $$ where the summation is over all possible value of $g$, and $G_1 \sim Bin(G,0.5)$, would I get something that makes sense?
Sorry if this is a noob question as I am not familiar in this area. Thanks in advance!