Location: Children's Nutrition Research CenterTitle: On the null distribution of Bayes factors in linear regression Author
|Zhou, Quan - Children'S Nutrition Research Center (CNRC)|
|Guan, Yongtao - Children'S Nutrition Research Center (CNRC)|
Submitted to: Journal of the American Statistical Association
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 5/19/2017
Publication Date: 8/1/2017
Citation: Zhou, Q., Guan, Y. 2017. On the null distribution of Bayes factors in linear regression. Journal of the American Statistical Association. doi:10.1080/01621459.2017.1328361.
Interpretive Summary: There are two schools of thought in the statistical community to infer association between a factor and an outcome (such as nutrition intake and obesity). One approach relies exclusively on data; the other borrows the past knowledge as prior to reweighs the evidence from data. The first approach measures the strength of association using the p-values; the second approach Bayes factors. To compromise between the two approaches we sought to compute p-values from Bayes factors. We demonstrated a surprisingly simple connection between Bayes factors and their corresponding p-values, and we provided a software to efficiently compute p-values of Bayes factors. Our findings unifies the two seemingly contradictory approaches to the same statistical problem, and eases the acceptance of Bayes factor by the scientific community whose members are predominantly trained to use p-values.
Technical Abstract: We show that under the null, the 2 log (Bayes factor) 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.