@article{c7f65d0dbc31437d949434c457e1b9e5,

title = "Borrowing strength and borrowing index for Bayesian hierarchical models",

abstract = "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.",

keywords = "Bayesian hierarchical model, Borrowing index, Borrowing strength, Clinical trials, Mallow's distance",

author = "Ganggang Xu and Huirong Zhu and Lee, {J. Jack}",

note = "Funding Information: Ganggang Xu{\textquoteright}s research was supported by National Science Foundation Award SES-1902195 . JJL{\textquoteright}s work was supported in part by grant CA016672 , 1P50CA221703 from the National Cancer Institute, USA and RP160668 from the Cancer Prevention and Research Institute of Texas (CPRIT), USA . We appreciate the editorial assistance from Jessica Swann. Appendix Proof of Theorem 1 For any permutation ( l 1 , … , l B ) of ( 1 , … , B ) , the definitions of d 2{\^ } F μ j | Y M 1 , F μ j | Y M 2 in (3.8) ensure that d 2{\^ } 2 F μ j | Y M 1 , F μ j | Y M 2 ≤ 1 B ∑ i = 1 B | Θ i , j M 1 − Θ l i , j M 2 | 2 , j = 1 , … , J , with Θ 1 M k , … , Θ B M k being posterior samples of Θ | Y under model M k , whose j th elements are μ j , i M k {\textquoteright}s for k = 1 , 2 and j = 1 , … , J . Therefore, we have ∑ j = 1 J d 2{\^ } 2 F μ j | Y M 1 , F μ j | Y M 2 ≤ 1 B ∑ j = 1 J ∑ i = 1 B | Θ i , j M 1 − Θ l i , j M 2 | 2 = 1 B ∑ i = 1 B ‖ Θ i M 1 − Θ l i M 2 ‖ 2 for any permutation l 1 , … , l B , which further implies that ∑ j = 1 J d 2{\^ } 2 F μ j | Y M 1 , F μ j | Y M 2 ≤ d 2 F{\^ } Θ | Y M 1 , F{\^ } Θ | Y M 2 , where the inequality follows from the definition of d 2 F{\^ } Θ | Y M 1 , F{\^ } Θ | Y M 2 in (3.7) . Next, by the inequality (6) in Dowson and Landau (1982) , we have that d 2 N ( μ{\^ } M 1 , Σ{\^ } M 1 ) , N ( μ{\^ } M 2 , Σ{\^ } M 2 ) ≤ E ‖ X M 1 − X M 2 ‖ 2 , where X M k {\textquoteright}s are random variables with mean μ{\^ } M k and covariance matrix Σ{\^ } M k , k = 1 , 2 , and the expectation is taken over all joint distributions of X M 1 and X M 2 . By Dowson and Landau (1982) , the above equality holds when X M k {\textquoteright}s are normally distributed. Since by definition, the empirical distributions F{\^ } Θ | Y M k {\textquoteright}s have means μ{\^ } M k {\textquoteright}s and covariance matrices Σ{\^ } M k {\textquoteright}s, we have that d 2 N ( μ{\^ } M 1 , Σ{\^ } M 1 ) , N ( μ{\^ } M 2 , Σ{\^ } M 2 ) ≤ d 2 F{\^ } Θ | Y M 1 , F{\^ } Θ | Y M 2 , which completes the proof of inequality d{\^ } 2 ∗ F Θ | Y M 1 , F Θ | Y M 2 ≤ d 2 F{\^ } Θ | Y M 1 , F{\^ } Θ | Y M 2 . □ Publisher Copyright: {\textcopyright} 2019 Elsevier B.V.",

year = "2020",

month = apr,

doi = "10.1016/j.csda.2019.106901",

language = "English (US)",

volume = "144",

journal = "Computational Statistics and Data Analysis",

issn = "0167-9473",

publisher = "Elsevier",

}