On the Null Distribution of Bayes Factors in Linear Regression

• 成果类型: Article
• 署名作者: Zhou, Quan; Guan, Yongtao
• 署名单位: Baylor College of Medicine
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2017.1328361
• 发表日期: 2018
• 页码: 1362-1371
• 关键词: genome-wide association; open-angle glaucoma; intraocular-pressure; P-values; statistical-methods; variable selection; hypotheses; imputation; models; common

摘要

We show that under the null, the is asymptotically distributed as a weighted sum of chi-squared random variables with a shifted mean. This claim holds for Bayesian multi-linear regression with a family of conjugate priors, namely, the normal-inverse-gamma prior, the g-prior, and the normal prior. Our results have three immediate impacts. First, we can compute analytically a p-value associated with a Bayes factor without the need of permutation. We provide a software package that can evaluate the p-value associated with Bayes factor efficiently and accurately. Second, the null distribution is illuminating to some intrinsic properties of Bayes factor, namely, how Bayes factor quantitatively depends on prior and the genesis of Bartlett's paradox. Third, enlightened by the null distribution of Bayes factor, we formulate a novel scaled Bayes factor that depends less on the prior and is immune to Bartlett's paradox. When two tests have an identical p-value, the test with a larger power tends to have a larger scaled Bayes factor, a desirable property that is missing for the (unscaled) Bayes factor. Supplementary materials for this article are available online.