Bayesian index of superiority and the p-value of the conditional test for Poisson parameters
We consider the problem of comparing two Poisson parameters from the Bayesian perspective. Kawasaki and Miyaoka (2012b) proposed the Bayesian index $P(λ_1 < λ_2 | X_1,X_2)$ 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 $H_0: λ_1 \geq λ_2 $ versus an alternative hypothesis $H_1: λ_1<λ_2 $. Additionally, we investigate the generalized relationship between $P(λ_1/λ_2 <c | X_1, X_2)$ and the $p$-value of the conditional test with the null hypothesis $H_0: λ_1/λ_2 \geq c$ versus the alternative $H_1: λ_1/λ_2 < c$. We illustrate the utility of the Bayesian index using analyses of real data. Our finding suggests that the Bayesian index can potentially be useful in an epidemiology and in a clinical trial.