A Condition to Obtain the Same Decision in the Homogeneity Testing Problem from the Frequentist and Bayesian Point of View

Abstract

We develop a Bayesian procedure for the homogeneity testing problem of r populations using r × s contingency tables. The posterior probability of the homogeneity null hypothesis is calculated using a mixed prior distribution. The methodology consists of choosing an appropriate value of π₀ for the mass assigned to the null and spreading the remainder, 1 − π₀, over the alternative according to a density function. With this method, a theorem which shows when the same conclusion is reached from both frequentist and Bayesian points of view is obtained. A sufficient condition under which the p-value is less than a value α and the posterior probability is also less than 0.5 is provided.

Keywords:

• Contingency tables
• p-values
• Posterior probabilities
• Reconciliation

Mathematics Subject Classification:

• Primary 62F15
• Secondary 62H17

Acknowledgments

This research was supported by Universidad Complutense-Comunidad de Madrid and Ministerio de Educación y Ciencia from Spain Grant MTM2005-05462.

We thank an Associate Editor and a referee for their valuable comments.