The problem occurs when the sample odds ratio is Inf, such as in the
following example. Given the fact that both upper bounds of the two 95%
confidence intervals are Inf, I would have expected that the two lower
bounds be equal, but they aren't.
x <- matrix(c(9,4,0,2),2,2)
x
# [,1] [,2]
#[1,] 9 0
#[2,] 4 2
rbind("two.sided.95CI"=fisher.test(x)$conf.int,
"greater.95CI"=fisher.test(x,alt="greater")$conf.int)
# [,1] [,2]
#two.sided.95CI 0.2985103 Inf
#greater.95CI 0.4625314 Inf
Using the noncentral hypergeometric distribution, we can calculate the
probability mass of each possible table with same marginals as x.
Ref.: Alan Agresti (1990). Categorical data analysis. New York: Wiley.
Page 67.
Hence, the result below suggests that the two-sided confidence interval
has a confidence level of 97.5% as opposed to 95%.
n11 <- 7:9
theta <- 0.2985103
choose(9,n11)*choose(15-9,13-n11)*theta^n11/
sum(choose(9,n11)*choose(15-9,13-n11)*theta^n11)
#[1] 0.67344877 0.30154709 0.02500414
The 95% confidence interval with one-sided (greater) alternative appears
to be correct.
theta <- 0.4625314
choose(9,n11)*choose(15-9,13-n11)*theta^n11/
sum(choose(9,n11)*choose(15-9,13-n11)*theta^n11)
#[1] 0.5608724 0.3891316 0.0499960