The indirect effect of an exposure on an outcome through an intermediate variable can be identified by a product of regression coefficients under certain causal and regression modeling assumptions. In this context, the null hypothesis of no indirect effect is a composite null hypothesis, as the null holds if either regression coefficient is zero. A consequence is that traditional hypothesis tests are severely underpowered near the origin (i.e., when both coefficients are small with respect to standard errors). We propose hypothesis tests that (i) preserve level alpha type 1 error, (ii) meaningfully improve power when both true underlying effects are small relative to sample size, and (iii) preserve power when at least one is not. One approach gives a closed-form test that is minimax optimal with respect to local power over the alternative parameter space. Another uses sparse linear programming to produce an approximately optimal test for a Bayes risk criterion. We discuss adaptations for performing large-scale hypothesis testing as well as modifications that yield improved interpretability. We provide an R package that implements the minimax optimal test.

翻译：在特定因果与回归模型假设下，暴露变量通过中介变量对结果变量产生的间接效应可通过回归系数的乘积进行识别。在此背景下，无间接效应的零假设属于复合零假设，因为只要任一回归系数为零，该假设即成立。这导致传统假设检验在原点附近（即当两个系数相对于标准误差均较小时）检验效能严重不足。本文提出的假设检验方法具有以下特性：（i）保持α水平的第Ⅰ类错误；（ii）当两个真实效应相对于样本量均较小时，能显著提升检验效能；（iii）当至少一个效应不小时，仍保持检验效能。其中一种方法给出了闭式检验，其在备择参数空间上关于局部检验效能达到极小极大最优。另一种方法采用稀疏线性规划，针对贝叶斯风险准则生成近似最优检验。我们讨论了适用于大规模假设检验的改进方案，以及能提升结果可解释性的修正方法。本文提供了实现极小极大最优检验的R软件包。