We show that the strength of an instrument is incompletely characterized by the proportion of compliers, and we propose and evaluate new methods that extract more information from certain settings with comparatively few compliers. Specifically, we demonstrate that, for a fixed small proportion of compliers, the presence of an equal number of always‐takers and never‐takers weakens an instrument, whereas the absence of always‐takers or, equivalently, the absence of never‐takers strengthens an instrument. In this statement, the strength of an instrument refers to its ability to recognize and reject a false hypothesis about a structural parameter. Equivalently, the strength of an instrument refers to its ability to exclude from a confidence interval a false value of a structural parameter. This ability is measured by the Bahadur efficiency of a test that assumes that the instrument is flawless, or the Bahadur efficiency of a sensitivity analysis that assumes that the instrument may be somewhat biased. When there are few compliers, the outcomes for most people are unaffected by fluctuations in the instrument, so most of the information about the treatment effect is contained in the tail of the distribution of the outcomes. Exploiting this fact, we propose new methods that emphasize the affected portion of the distribution of outcomes, thereby extracting more information from studies with few compliers. Studies of the effects of Catholic high schools on academic test performance have used ‘being Catholic’ as an instrument for ‘attending a Catholic high school’, and the application concerns such a comparison using the US National Educational Longitudinal Study. Most Catholics did not attend Catholic school, so there are few compliers, but it was rare for non‐Catholics to attend Catholic school, so there are very few always‐takers.

中文翻译：

---

在天主教高中产生的认知结果研究中以弱小的手段找到力量

我们表明，工具的强度不能完全由合规者的比例来表征，我们提出并评估了可以从某些设置中以较少的合规者提取更多信息的新方法。具体而言，我们证明，对于固定的一小部分合规者，始终存在和永不接受的人数相等会削弱一种工具，而始终存在的人的缺席或等效地，从未采取的人的存在会增强乐器。在此陈述中，工具的强度是指其识别和拒绝有关结构参数的错误假设的能力。等效地，工具的强度是指其从置信区间中排除结构参数的假值的能力。此能力通过假定仪器无缺陷的测试的Bahadur效率或假定仪器可能存在一定偏差的灵敏度分析的Bahadur效率来衡量。当合规者很少时，大多数人的结局不受仪器波动的影响，因此有关治疗效果的大多数信息都包含在结局分布的尾部。利用这一事实，我们提出了强调结果分布影响部分的新方法，从而从很少有合规性的研究中提取了更多信息。对天主教高中对学业成绩的影响的研究已使用“成为天主教徒”作为“参加天主教高中”的工具，并且该申请涉及使用美国国家教育纵向研究进行的比较。大多数天主教徒没有上天主教学校，因此很少有合规者，但非天主教徒很少上天主教学校，因此常去者很少。