Wald-Type Testing and Estimation Methods for Asymmetric Comparisons of Poisson Rates

This article considers the problem of asymmetric comparisons, that is, instances where one treatment serves as the compelling standard treatment for a certain disease or health condition, where the asymmetric comparisons are performed in terms of the Poisson rates. We propose asymptotic Wald-type tests and confidence intervals suitable for demonstrating superiority and noninferiority for a given margin. The testing methods for the difference and ratio are based on the unconstrained (MLE) and constrained (CMLE) maximum likelihood estimator. The resulting CMLE-based tests for asymmetric comparisons are equivalent to the standard (i.e., those for symmetric comparisons) CMLE-based tests. The asymmetric tests as well as the standard MLE-based tests are evaluated via simulations. The CMLE-based asymmetric tests are shown to adequately control the Type I error in most settings, while the MLE-based asymmetric tests are shown to have this property only in settings with relatively large sample sizes and means. In all settings, the MLE-based asymmetric tests are more powerful than both the asymmetric CMLE-based and the standard MLE-based tests. We also propose asymptotic confidence intervals that can be used to estimate the difference or ratio of the two rates (the presentation of this content includes supplementary material that is available online), and discuss how power and sample size estimation can be done to aid in study planning. Supplementary materials for this article are available online.