Bayesian index of superiority and the p-value of the conditional test for Poisson parameters

TL;DR

This paper explores Bayesian methods for comparing Poisson parameters, providing new expressions for the Bayesian index and examining its relationship with p-values in conditional tests, with applications in epidemiology and clinical trials.

Contribution

The paper introduces four new expressions for the Bayesian index of Poisson parameters and investigates its connection to p-values in conditional hypothesis testing.

Findings

Bayesian index expressed via beta, F, binomial, and negative binomial distributions.

Relationship established between Bayesian index and p-value of conditional tests.

Demonstrated utility of Bayesian index with real data analyses.

Abstract

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.