An analysis of eight 95 per cent confidence intervals for a ratio of Poisson parameters when events are rare

We compared eight nominal 95 per cent confidence intervals for the ratio of two Poisson parameters, both assumed small, on their true coverage (the probability that the interval includes the ratio of Poisson parameters) and median width. The commonly used log-linear interval, justified by asymptotic considerations, provided coverage and relatively narrow intervals, despite small numbers of arrivals. However, the uniform and scores intervals, defined in the text, come very close to providing coverage while providing substantially narrower intervals. These intervals might have practical applications. In a sensitivity analysis, none of the intervals maintained coverage for negative binomial data, indicating that distributional assumptions should be checked before taking our recommendations.