Hybrid-based confidence intervals for the ratio of two treatment means in the over-dispersed Poisson data

In many clinical trials and epidemiological studies, comparing the mean count response of an exposed group to a control group is often of interest. This type of data is often over-dispersed with respect to Poisson variation, and previous studies usually compared groups using confidence intervals (CIs) of the difference between the two means. However, in some situations, especially when the means are small, interval estimation of the mean ratio (MR) is preferable. Moreover, Cox and Lewis [4 D.R. Cox and P.A.W. Lewis, The Statistical Analysis of Series of Events, Methuen, London, 1966.[Crossref] , [Google Scholar]] pointed out many other situations where the MR is more relevant than the difference of means. In this paper, we consider CI construction for the ratio of means between two treatments for over-dispersed Poisson data. We develop several CIs for the situation by hybridizing two separate CIs for two individual means. Extensive simulations show that all hybrid-based CIs perform reasonably well in terms of coverage. However, the CIs based on the delta method using the logarithmic transformation perform better than other intervals in the sense that they have slightly shorter interval lengths and show better balance of tail errors. These proposed CIs are illustrated with three real data examples.