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 $P(\lambda_1 < \lambda_2 | X_1,X_2)$ 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 $H_0: \lambda_1 \geq \lambda_2$ versus an alternative hypothesis $H_1: \lambda_1<\lambda_2$. Additionally, we investigate the generalized relationship between $P(\lambda_1/\lambda_2 <c | X_1, X_2)$ and the $p$-value of the conditional test with the null hypothesis $H_0: \lambda_1/\lambda_2 \geq c$ versus the alternative $H_1: \lambda_1/\lambda_2 < c$. 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.