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In the evaluation of treatment effects, it is of major policy interest toknow if the treatment is beneficial for some and harmful for others, aphenomenon known as qualitative interaction. We formulate this question as amultiple testing problem with many conservative null $p$-values, in which theclassical multiple testing methods may lose power substantially. We propose asimple technique---conditioning---to improve the power. A crucial assumption weneed is uniform conservativeness, meaning for any conservative $p$-value $p$,the conditional distribution $(p/\tau)\,|\,p \le \tau$ is stochastically largerthan the uniform distribution on $(0,1)$ for any $\tau$. We show this propertyholds for one-sided tests in a one-dimensional exponential family (e.g.\testing for qualitative interaction) as well as testing $|\mu|\le\eta$ using astatistic $X \sim \mathrm{N}(\mu,1)$ (e.g.\ testing for practical importancewith threshold $\eta$). We propose an adaptive method to select the threshold$\tau$. Our theoretical and simulation results suggest the proposed tests gainsignificant power when many $p$-values are uniformly conservative and loselittle power when no $p$-value is uniformly conservative. We apply our methodto two educational intervention datasets.

Multiple testing when many $p$-values are uniformly conservative, with application to testing qualitative interaction in educational interventions