Confidence intervals for directly standardized rates using mid-p gamma intervals

Directly standardized rates continue to be an integral tool for presenting rates for diseases that are highly dependent on age, such as cancer. Statistically, these rates are modeled as a weighted sum of Poisson random variables. This is a difficult statistical problem, because there are k observed Poisson variables and k unknown means. The gamma confidence interval has been shown through simulations to have at least nominal coverage in all simulated scenarios, but it can be overly conservative. Previous modifications to that method have closer to nominal coverage on average, but they do not achieve the nominal coverage bound in all situations. Further, those modifications are not central intervals, and the upper coverage error rate can be substantially more than half the nominal error. Here we apply a mid-p modification to the gamma confidence interval. Typical mid-p methods forsake guaranteed coverage to get coverage that is sometimes higher and sometimes lower than the nominal coverage rate, depending on the values of the parameters. The mid-p gamma interval does not have guaranteed coverage in all situations; however, in the (not rare) situations where the gamma method is overly conservative, the mid-p gamma interval often has at least nominal coverage. The mid-p gamma interval is especially appropriate when one wants a central interval, since simulations show that in many situations both the upper and lower coverage error rates are on average less than or equal to half the nominal error rate.