当许多$p$ -值一致保守时的多重检验，并应用于检验教育干预中的定性交互作用
Multiple testing when many $p$-values are uniformly conservative, with application to testing qualitative interaction in educational interventions
摘要：
在评估治疗效果时，了解治疗是否对某些人有益而对另一些人有害，这是一个重大的政策问题，这种现象被称为质量互动。我们将这个问题表述为一个具有许多保守的空$p$ -值的多重检验问题，其中经典的多重检验方法可能会大大失去功效。我们提出了一个简单的技术-空调--以提高功率。我们需要的一个关键假设是一致保守性，这意味着对于任何保守的$p$ -值$p$，条件分布$(p/τ)\,|\,p \le τ$随机大于任何$τ$的$(0,1)$上的均匀分布。我们证明了这个性质适用于一维指数族中的单侧检验（例如\定性交互测试）以及使用统计量$|μ|\leη$（例如，用阈值$η$测试实际重要性）。我们提出了一种自适应的方法来选择阈值$τ$。我们的理论和仿真结果表明，提出的测试获得显着的权力时，许多$p$ -值是一致保守的，失去很少的权力时，没有$p$ -值是一致保守的。我们将我们的方法应用于两个教育干预数据集。
英文摘要：
In the evaluation of treatment effects, it is of major policy interest to know if the treatment is beneficial for some and harmful for others, a phenomenon known as qualitative interaction. We formulate this question as a multiple testing problem with many conservative null $p$-values, in which the classical multiple testing methods may lose power substantially. We propose a simple technique---conditioning---to improve the power. A crucial assumption we need is uniform conservativeness, meaning for any conservative $p$-value $p$, the conditional distribution $(p/τ)\,|\,p \le τ$ is stochastically larger than the uniform distribution on $(0,1)$ for any $τ$. We show this property holds for one-sided tests in a one-dimensional exponential family (e.g.\ testing for qualitative interaction) as well as testing $|μ|\leη$ using a statistic $X \sim \mathrm{N}(μ,1)$ (e.g.\ testing for practical importance with threshold $η$). We propose an adaptive method to select the threshold $τ$. Our theoretical and simulation results suggest the proposed tests gain significant power when many $p$-values are uniformly conservative and lose little power when no $p$-value is uniformly conservative. We apply our method to two educational intervention datasets.
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