Bayes factor functions for reporting outcomes of hypothesis tests

Abstract

Bayes factors represent a useful alternative to P -values for reporting outcomes of hypothesis tests by providing direct measures of the relative support that data provide to competing hypotheses. Unfortunately, the competing hypotheses have to be specified, and the calculation of Bayes factors in high-dimensional settings can be difficult. To address these problems, we define Bayes factor functions (BFFs) directly from common test statistics. BFFs depend on a single noncentrality parameter that can be expressed as a function of standardized effects, and plots of BFFs versus effect size provide informative summaries of hypothesis tests that can be easily aggregated across studies. Such summaries eliminate the need for arbitrary P -value thresholds to define “statistical significance.” Because BFFs are defined using nonlocal alternative prior densities, they provide more rapid accumulation of evidence in favor of true null hypotheses without sacrificing efficiency in supporting true alternative hypotheses. BFFs can be expressed in closed form and can be computed easily from z , t , χ 2 , and F statistics.