A novel borrowing strength measure and an overall borrowing index to characterize the strength of borrowing behaviors among subgroups are proposed for a given Bayesian hierarchical model. The constructions of the proposed indexes are based on the Mallow's distance and can be easily computed using MCMC samples for univariate or multivariate posterior distributions. Consequently, the proposed indexes can serve as meaningful and useful exploratory tools to better understand the roles played by the priors in a hierarchical model, including their influences on the posteriors that are used to make statistical inferences. These relationships are otherwise ambiguous. The proposed methods can be applied to both the continuous and binary outcome variables. Furthermore, the proposed approach can be easily adapted to various settings of clinical trials, where Bayesian hierarchical models are deem appropriate. The effectiveness of the proposed method is illustrated using extensive simulation studies and a real data example.

中文翻译：

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贝叶斯分层模型的借用强度和借用指数

对于给定的贝叶斯分层模型，提出了一种新颖的借贷强度度量和整体借贷指数，以表征子组之间借贷行为的强度。建议指标的构造基于 Mallow 距离，并且可以使用单变量或多变量后验分布的 MCMC 样本轻松计算。因此，所提出的指标可以作为有意义和有用的探索工具，以更好地理解先验在分层模型中所扮演的角色，包括它们对用于进行统计推断的后验的影响。否则，这些关系是模棱两可的。所提出的方法可以应用于连续和二元结果变量。此外，所提出的方法可以很容易地适应各种临床试验环境，其中贝叶斯分层模型被认为是合适的。所提出的方法的有效性通过广泛的模拟研究和一个真实的数据例子来说明。