Suppose we have two independent Poisson-distributed variables $X_1$ and $X_2$. We want to test whether the Poisson parameters are equal, i.e. whether $\lambda_1=\lambda_2$.
Now we have 4 distinct statistical exact tests to choose:
- E-test (see Krishnamoorthy and Thomson, A more powerful test for comparing two Poisson means, Journal of Statistical Planning and Inference 119 (2004) 23–35; see also Checking if two Poisson samples have the same mean on Cross Validated)
poisson.exact(tsmethod="balker")(from exactci R package)
Now, given all those tests are labeled as "exact", one would expect all yield the same p-values. Contrary to this, the quoted paper clearly illustrates that methods 2-4 give different significance. Furthermore, I personally implemented the E-test and found that this test gives yet another, distinct result. Why is that?