Noninformative priors, credible sets and Bayesian hypothesis testing for the intraclass model

Abstract

The objective of this paper is twofold. The first is to find a Bayesian credible interval for the intraclass correlation coefficient in symmetric normal models based on some “default” or “noninformative” priors. Probability matching priors and reference priors along with Jeffreys’ prior are considered, and the one-at-a-time reference prior emerges as the “optimal” according to several criteria. The second objective is to compare two nested models such as the intraclass and independence models using the distance or divergence between the two as the basis of comparison. A suitable criterion for this is the “power divergence measure” as introduced by Cressie and Read (J. Roy. Statist. Soc. Ser. B 46 (1984) 440). Such a measure includes the two Kullback-Leibler divergence measures and the Hellinger divergence measure as special cases. For the specific problem, the power divergence measure turns out to be a function solely of ρ, the intraclass correlation coefficient. Also, this function is increasing for ρ>0, is decreasing for ρ<0, and the minimum is attained at ρ=0. Thus the model comparison problem in this case amounts to testing the hypothesis H 0 : ρ=0 . Due to the duality between hypothesis tests and set estimation, the hypothesis testing problem can also be solved by solving a corresponding set estimation problem. The present paper develops Bayesian methods based on the Kullback-Leibler and Hellinger divergence measures, rejecting H 0 : ρ=0 when the specified divergence measure exceeds some number d. This number d is so chosen that the resulting credible interval for the divergence measure has specified coverage probability 1− α. The length of such an interval is compared with the (i) equal two-tailed credible interval and (ii) the HPD credible interval for ρ with the same coverage probability which can also be inverted into acceptance regions of H 0 : ρ=0 . An example is considered where the HPD interval based on the one-at-a-time reference prior turns out to be the shortest credible interval having the same coverage probability.