I need help calculating the Mid-P of the confidence limits for the standardized mortality ratio, or any observed/expected ratio. I am familiar with the Mid-P for a binomial calculation but I am trying to understand how to calculate confidence intervals for a standardized mortality ratio under a Poisson distribution. I've found calculators with formulas such as OpenEpi but I haven't been able to replicate their results. Does anyone know of a source describing this calculation, or of a simple example that could help?
The exact p-value when the test statistic has a discrete distribution is calculated by summing all the probabilities of events equal to or more extreme than the value you are testing. For example, if your value is 8, the p-value is equal to: $1 - (P(Y = 7) + P(Y = 6) ...)$. To calculate the mid-p value, you also include half of the probability of the current value: $1 - (P(Y = 8)/2 + P(Y = 7) + P(Y = 6) ...)$.
There is an example of how to do these calculations in an "Introduction to Categorical Data Analysis" by Alan Agresti starting on page 13. It appears that this page is currently viewawble on Google Books. The example uses the binomial distribution which isn't exactly what you are looking for, but it should generalize to any discrete distribution (just change the function you use to calculate $P(Y)$).