The question "Test of Equal Proportions with zero successes" showed me that using a test of equal proportions with very low or very high numbers of successes doesn't give an accurate p-value. Below is an example where I'm testing if three independent experiments have the same probability of success ($H_0: p_1 = p_2 = p_3$), for $n_1 = n_2 = n_3 = 10$ and $f_1 = f_2 = 0$ and $f_3 = 1$ (zero successes out of 10 trials in two of the three experiments, one success in the third experiment).
prop.test(c(0,0,1), c(10,10,10))$p.value
## [1] 0.3554102
## Warning message:
## In prop.test(c(0, 0, 1), c(10, 10, 10)) :
## Chi-squared approximation may be incorrect
The same with $f^\prime = n - f$:
prop.test(c(10,10,9), c(10,10,10))$p.value
## [1] 0.3554102
## Warning message:
## In prop.test(c(10, 10, 9), c(10, 10, 10)) :
## Chi-squared approximation may be incorrect
How to compute exact p-values in these cases?
