Bayes factors are an increasingly popular tool for indexing evidence from experiments. For two competing population models, the Bayes factor reflects the relative likelihood of observing some data under one model compared to the other. In general, computing a Bayes factor is difficult, because computing the marginal likelihood of each model requires integrating the product of the likelihood and a prior distribution on the population parameter(s). In this paper, we develop a new analytic formula for computing Bayes factors directly from minimal summary statistics in repeated-measures designs. This work is an improvement on previous methods for computing Bayes factors from summary statistics (e.g., the BIC method), which produce Bayes factors that violate the Sellke upper bound of evidence for smaller sample sizes. The new approach taken in this paper extends requires knowing only the $F$-statistic and degrees of freedom, both of which are commonly reported in most empirical work. In addition to providing computational examples, we report a simulation study that benchmarks the new formula against other methods for computing Bayes factors in repeated-measures designs. Our new method provides an easy way for researchers to compute Bayes factors directly from a minimal set of summary statistics, allowing users to index the evidential value of their own data, as well as data reported in published studies.

翻译：贝叶斯因子作为一种日益流行的工具，用于量化实验中的证据强度。针对两个竞争性总体模型，贝叶斯因子反映了在某一模型下观测到特定数据的相对似然性。通常计算贝叶斯因子具有挑战性，因为计算每个模型的边缘似然需要积分似然函数与总体参数先验分布的乘积。本文提出了一种新的解析公式，可直接基于重复测量设计中的最小汇总统计量计算贝叶斯因子。本研究改进了基于汇总统计量计算贝叶斯因子的传统方法（如BIC方法），后者在小样本条件下会违反Sellke证据上限。本文的新方法仅需已知$F$统计量及其自由度——这两者均是大多数实证研究中常规报告的指标。除提供计算示例外，我们通过仿真研究将新公式与重复测量设计中其他贝叶斯因子计算方法进行了基准测试。该新方法为研究者提供了一种便捷途径，使其能够直接依据最小汇总统计量计算贝叶斯因子，从而量化自身数据及已发表研究的证据价值。