Prior effective sample size in conditionally independent hierarchical models

Prior effective sample size (ESS) of a Bayesian parametric model was defined by Morita, et al. (2008, Biometrics, 64, 595-602). Starting with an ε-information prior defined to have the same means and correlations as the prior but to be vague in a suitable sense, the ESS is the required sample size to obtain a hypothetical posterior very close to the prior. In this paper, we present two alternative definitions for the prior ESS that are suitable for a conditionally inde-pendent hierarchical model. The two definitions focus on either the first level prior or second level prior. The proposed methods are applied to important examples to verify that each of the two types of prior ESS matches the intuitively obvi-ous answer where it exists. We illustrate the method with applications to several motivating examples, including a single-arm clinical trial to evaluate treatment response probabilities across di®erent disease subtypes, a dose-finding trial based on toxicity in this setting, and a multicenter randomized trial of treatments for affective disorders.