The Pearson Bayes factor: An analytic formula for computing evidential value from minimal summary statistics

TL;DR

This paper introduces the Pearson Bayes factor, a simple analytic formula that allows researchers to compute exact Bayesian evidence using minimal summary statistics like t or F scores, facilitating easier hypothesis testing.

Contribution

The paper presents a novel, closed-form Bayes factor formula based on a specific prior choice, enabling computation from minimal summary data without complex integrations.

Findings

The Pearson Bayes factor accurately estimates evidential value in simulations.

It allows exact Bayes factor computation from common summary statistics.

The method simplifies Bayesian hypothesis testing for published research.

Abstract

In Bayesian hypothesis testing, evidence for a statistical model is quantified by the Bayes factor, which represents the relative likelihood of observed data under that model compared to another competing model. In general, computing Bayes factors is difficult, as computing the marginal likelihood of data under a given model requires integrating over a prior distribution of model parameters. In this paper, I capitalize on a particular choice of prior distribution that allows the Bayes factor to be expressed without integral representation and I develop a simple formula -- the Pearson Bayes factor -- that requires only minimal summary statistics commonly reported in scientific papers, such as the $t$ or $F$ score and the degrees of freedom. In addition to presenting this new result, I provide several examples of its use and report a simulation study validating its performance. Importantly, the Pearson Bayes factor gives applied researchers the ability to compute exact Bayes factors from minimal summary data, and thus easily assess the evidential value of any data for which these summary statistics are provided, even when the original data is not available.