Connecting and contrasting the Bayes factor and a modified ROPE procedure for testing interval null hypotheses

There has been strong recent interest in testing interval null hypotheses for improved scientific inference. For example, Lakens et al (2018) and Lakens and Harms (2017) use this approach to study if there is a pre-specified meaningful treatment effect in gerontology and clinical trials, instead of a point null hypothesis of any effect. Two popular Bayesian approaches are available for interval null hypothesis testing. One is the standard Bayes factor and the other is the Region of Practical Equivalence (ROPE) procedure championed by Kruschke and others over many years. This paper connects key quantities in the two approaches, which in turn allow us to contrast two major differences between the approaches with substantial practical implications. The first is that the Bayes factor depends heavily on the prior specification while a modified ROPE procedure is very robust. The second difference is concerned with the statistical property when data is generated under a neutral parameter value on the common boundary of competing hypotheses. In this case the Bayes factors can be severely biased whereas the modified ROPE approach gives a reasonable result. Finally, the connection leads to a simple and effective algorithm for computing Bayes factors using draws from posterior distributions generated by standard Bayesian programs such as BUGS, JAGS and Stan.

近年来，学界对基于区间零假设检验以优化科学推断的研究兴趣日益浓厚。例如，Lakens等人（2018）以及Lakens与Harms（2017）便采用该方法，在老年学与临床试验中检验是否存在预先设定的具有实际意义的治疗效应，而非仅检验存在效应的点零假设。针对区间零假设检验，目前已有两种主流的贝叶斯方法：一是标准贝叶斯因子法，二是由Kruschke等人多年来倡导的实用等价区域（Region of Practical Equivalence, ROPE）检验流程。本文梳理了这两种方法中的核心量化指标间的关联，借此对比二者间两项具有重要实际影响的核心差异。第一，贝叶斯因子高度依赖先验设定，而经改进的ROPE检验流程则具备极强的稳健性。第二，当在竞争假设的公共边界上取中性参数值生成数据时，两种方法的统计性质存在差异。在此情形下，贝叶斯因子可能会出现严重偏倚，而经改进的ROPE方法则可得到合理的检验结果。最后，基于二者的关联，本文提出了一种简便高效的贝叶斯因子计算算法，可借助标准贝叶斯工具（如BUGS、JAGS与Stan）生成的后验分布抽样完成计算。