{"title":"The flaw of averages: Bayes factors as posterior means of the likelihood ratio.","authors":"Charles C Liu, Ron Xiaolong Yu, Murray Aitkin","doi":"10.1002/pst.2355","DOIUrl":null,"url":null,"abstract":"<p><p>As an alternative to the Frequentist p-value, the Bayes factor (or ratio of marginal likelihoods) has been regarded as one of the primary tools for Bayesian hypothesis testing. In recent years, several researchers have begun to re-analyze results from prominent medical journals, as well as from trials for FDA-approved drugs, to show that Bayes factors often give divergent conclusions from those of p-values. In this paper, we investigate the claim that Bayes factors are straightforward to interpret as directly quantifying the relative strength of evidence. In particular, we show that for nested hypotheses with consistent priors, the Bayes factor for the null over the alternative hypothesis is the posterior mean of the likelihood ratio. By re-analyzing 39 results previously published in the New England Journal of Medicine, we demonstrate how the posterior distribution of the likelihood ratio can be computed and visualized, providing useful information beyond the posterior mean alone.</p>","PeriodicalId":19934,"journal":{"name":"Pharmaceutical Statistics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pharmaceutical Statistics","FirstCategoryId":"3","ListUrlMain":"https://doi.org/10.1002/pst.2355","RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/1/28 0:00:00","PubModel":"Epub","JCR":"Q4","JCRName":"PHARMACOLOGY & PHARMACY","Score":null,"Total":0}
引用次数: 0
Abstract
As an alternative to the Frequentist p-value, the Bayes factor (or ratio of marginal likelihoods) has been regarded as one of the primary tools for Bayesian hypothesis testing. In recent years, several researchers have begun to re-analyze results from prominent medical journals, as well as from trials for FDA-approved drugs, to show that Bayes factors often give divergent conclusions from those of p-values. In this paper, we investigate the claim that Bayes factors are straightforward to interpret as directly quantifying the relative strength of evidence. In particular, we show that for nested hypotheses with consistent priors, the Bayes factor for the null over the alternative hypothesis is the posterior mean of the likelihood ratio. By re-analyzing 39 results previously published in the New England Journal of Medicine, we demonstrate how the posterior distribution of the likelihood ratio can be computed and visualized, providing useful information beyond the posterior mean alone.
贝叶斯因子(或边际似然比)作为频数法 p 值的替代方法,一直被视为贝叶斯假设检验的主要工具之一。近年来,一些研究人员开始重新分析著名医学期刊以及美国食品与药物管理局批准药物试验的结果,结果表明贝叶斯因子得出的结论往往与 p 值不同。在本文中,我们研究了贝叶斯系数可直接量化证据相对强度的说法。特别是,我们证明,对于具有一致先验的嵌套假设,零假设相对于备择假设的贝叶斯因子是似然比的后验平均值。通过重新分析之前发表在《新英格兰医学杂志》上的 39 项结果,我们展示了如何计算似然比的后验分布并将其可视化,从而提供了超越后验平均值的有用信息。
期刊介绍:
Pharmaceutical Statistics is an industry-led initiative, tackling real problems in statistical applications. The Journal publishes papers that share experiences in the practical application of statistics within the pharmaceutical industry. It covers all aspects of pharmaceutical statistical applications from discovery, through pre-clinical development, clinical development, post-marketing surveillance, consumer health, production, epidemiology, and health economics.
The Journal is both international and multidisciplinary. It includes high quality practical papers, case studies and review papers.