Mid-p confidence intervals for the poisson expectation

Conventionally a confidence interval (CI) for the standardized mortality ratio is set using the conservative CI for a Poisson expectation, μ. Employing the mid-P argument we present alternative CIs that are shorter than the conventional ones. The mid-P intervals do not guarantee the nominal confidence level, but the true coverage probability is only lower than the nominal level for a few short ranges of μ. The implications for mid-P confidence intervals of various proposed definitions of two-sided tests for discrete data are discussed.