Bayesian Index of Superiority and the p-Value of the Conditional Test for Poisson Parameters

Abstract

We consider the problem of comparing two Poisson parameters from the Bayesian perspective. Kawasaki and Miyaoka (2012b) proposed the Bayesian index θ = P(λ₁ < λ₂ | X₁, X₂) and expressed it using the hypergeometric series. In this paper, under some conditions, we give four other expressions of the Bayesian index in terms of the cumulative distribution functions of beta, F, binomial, and negative binomial distribution. Next, we investigate the relationship between the Bayesian index and the p-value of the conditional test with the null hypothesis H₀ : λ₁ ≥ λ₂ versus an alternative hypothesis H₁ : λ₁ < λ₂. Additionally, we investigate the generalized relationship between θ = P(λ₁/λ₂ < c | X₁, X₂) and the p-value of the conditional test with the null hypothesis H₀ : λ₁/λ₂ ≥ c versus the alternative H₁ : λ₁/λ₂ < c. We illustrate the utility of the Bayesian index using analyses of real data. Our finding suggests that θ can potentially be useful in an epidemiology and in a clinical trial.