Inferring causal relationships between variable pairs in the observational study is crucial but challenging, due to the presence of unmeasured confounding. While previous methods employed the negative controls to adjust for the confounding bias, they were either restricted to the discrete setting (i.e., all variables are discrete) or relied on strong assumptions for identification. To address these problems, we develop a general nonparametric approach that accommodates both discrete and continuous settings for testing causal hypothesis under unmeasured confounders. By using only a single negative control outcome (NCO), we establish a new identification result based on a newly proposed integral equation that links the outcome and NCO, requiring only the completeness and mild regularity conditions. We then propose a kernel-based testing procedure that is more efficient than existing moment-restriction methods. We derive the asymptotic level and power properties for our tests. Furthermore, we examine cases where our procedure using only NCO fails to achieve identification, and introduce a new procedure that incorporates a negative control exposure (NCE) to restore identifiability. We demonstrate the effectiveness of our approach through extensive simulations and real-world data from the Intensive Care Data and World Values Survey.

翻译：在观测研究中推断变量对之间的因果关系至关重要，但由于未测量混杂的存在而极具挑战性。虽然先前的方法采用阴性对照来调整混杂偏倚，但它们要么局限于离散设定（即所有变量均为离散），要么依赖于较强的识别假设。为解决这些问题，我们开发了一种通用的非参数方法，可同时适用于离散和连续设定，用于在未测量混杂下检验因果假设。通过仅使用单个阴性对照结果（NCO），我们基于新提出的连接结果变量与NCO的积分方程，建立了新的识别结果，仅需完备性及温和的正则性条件。随后，我们提出了一种基于核的检验方法，其效率优于现有的矩限制方法。我们推导了检验的渐近水平和功效性质。此外，我们考察了仅使用NCO时我们的方法无法实现识别的情况，并引入了一种结合阴性对照暴露（NCE）的新方法以恢复可识别性。我们通过大量模拟实验及来自重症监护数据和世界价值观调查的真实数据，验证了所提方法的有效性。