Testing the equality of two Poisson means using the rate ratio

In this article, we investigate procedures for comparing two independent Poisson variates that are observed over unequal sampling frames (i.e. time intervals, populations, areas or any combination thereof). We consider two statistics (with and without the logarithmic transformation) for testing the equality of two Poisson rates. Two methods for implementing these statistics are reviewed. They are (1) the sample-based method, and (2) the constrained maximum likelihood estimation (CMLE) method. We conduct an empirical study to evaluate the performance of different statistics and methods. Generally, we find that the CMLE method works satisfactorily only for the statistic without the logarithmic transformation (denoted as W(2)) while sample-based method performs better for the statistic using the logarithmic transformation (denoted as W(3)). It is noteworthy that both statistics perform well for moderate to large Poisson rates (e.g. > or =10). For small Poisson rates (e.g. <10), W(2) can be liberal (e.g. actual type I error rate/nominal level > or =1.2) while W(3) can be conservative (e.g. actual type I error rate/nominal level < or =0.8). The corresponding sample size formulae are provided and valid in the sense that the simulated powers associated with the approximate sample size formulae are generally close to the pre-chosen power level. We illustrate our methodologies with a real example from a breast cancer study.