In an observational study, the treatment received and the outcome exhibited may be associated in the absence of an effect caused by the treatment, even after controlling for observed covariates. Two tactics are common: (i) a test for unmeasured bias may be obtained using a secondary outcome for which the effect is known and (ii) a sensitivity analysis may explore the magnitude of unmeasured bias that would need to be present to explain the observed association as something other than an effect caused by the treatment. Can such a test for unmeasured bias inform the sensitivity analysis? If the test for bias does not discover evidence of unmeasured bias, then ask: Are conclusions therefore insensitive to larger unmeasured biases? Conversely, if the test for bias does find evidence of bias, then ask: What does that imply about sensitivity to biases? This problem is formulated in a new way as a convex quadratically constrained quadratic program and solved on a large scale using interior point methods by a modern solver. That is, a convex quadratic function of N variables is minimized subject to constraints on linear and convex quadratic functions of these variables. The quadratic function that is minimized is a statistic for the primary outcome that is a function of the unknown treatment assignment probabilities. The quadratic function that constrains this minimization is a statistic for subsidiary outcome that is also a function of these same unknown treatment assignment probabilities. In effect, the first statistic is minimized over a confidence set for the unknown treatment assignment probabilities supplied by the unaffected outcome. This process avoids the mistake of interpreting the failure to reject a hypothesis as support for the truth of that hypothesis. The method is illustrated by a study of the effects of light daily alcohol consumption on high-density lipoprotein (HDL) cholesterol levels. In this study, the method quickly optimizes a nonlinear function of $N=800$ variables subject to linear and quadratic constraints. In the example, strong evidence of unmeasured bias is found using the subsidiary outcome, but, perhaps surprisingly, this finding makes the primary comparison insensitive to larger biases.

中文翻译：

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观察性研究中偏倚检验的敏感性分析

在一项观察性研究中，即使在控制观察到的协变量后，接受的治疗和显示的结果也可能在没有治疗引起的影响的情况下相关联。常见的策略有两种：(i) 可以使用效果已知的次要结果来测试未测量的偏差，以及 (ii) 敏感性分析可以探索需要存在的未测量偏差的大小，以解释观察到的结果协会作为治疗引起的效果以外的东西。这样的未测量偏差测试可以为敏感性分析提供信息吗？如果偏差测试没有发现未测量偏差的证据，那么问：结论是否因此对更大的未测量偏差不敏感？相反，如果偏见测试确实找到了偏见的证据，那么问：这对偏见的敏感性意味着什么？这个问题以一种新的方式表述为凸二次约束二次规划，并通过现代求解器使用内点法大规模求解。也就是说，一个凸二次函数否变量被最小化受限于这些变量的线性和凸二次函数。最小化的二次函数是主要结果的统计量，它是未知治疗分配概率的函数。限制此最小化的二次函数是辅助结果的统计量，它也是这些相同的未知治疗分配概率的函数。实际上，对于由未受影响的结果提供的未知治疗分配概率的置信集，第一个统计量被最小化。这个过程避免了将未能拒绝假设解释为支持该假设真实性的错误。该方法通过一项关于每日少量饮酒对高密度脂蛋白 (HDL) 胆固醇水平影响的研究加以说明。$否=800$受线性和二次约束的变量。在该示例中，使用辅助结果发现了未测量偏差的有力证据，但也许令人惊讶的是，这一发现使主要比较对更大的偏差不敏感。