The estimation of causal effects is a fundamental goal in the field of causal inference. However, it is challenging for various reasons. One reason is that the exposure (or treatment) is naturally continuous in many real-world scenarios. When dealing with continuous exposure, dichotomizing the exposure variable based on a pre-defined threshold may result in a biased understanding of causal relationships. In this paper, we propose a novel causal inference framework that can measure the causal effect of continuous exposure. We define the expectation of a derivative of potential outcomes at a specific exposure level as the average causal derivative effect. Additionally, we propose a matching method for this estimator and propose a permutation approach to test the hypothesis of no local causal effect. We also investigate the asymptotic properties of the proposed estimator and examine its performance through simulation studies. Finally, we apply this causal framework in a real data example of Chronic Obstructive Pulmonary Disease (COPD) patients.

翻译：因果效应的估计是因果推断领域的一个基本目标，然而，由于多种原因，这一工作颇具挑战。其中一个原因是，在许多真实场景中，暴露（或处理）是自然连续的。在处理连续暴露时，基于预定义阈值将暴露变量二分化可能导致对因果关系的理解产生偏倚。本文提出了一种新颖的因果推断框架，能够衡量连续暴露的因果效应。我们将特定暴露水平下潜在结果导数的期望定义为平均因果导数效应。此外，我们提出了一种适用于该估计量的匹配方法，并引入了一种排列检验方法来检验无局部因果效应的假设。我们还研究了所提估计量的渐近性质，并通过模拟研究评估其性能。最后，我们将这一因果框架应用于慢性阻塞性肺疾病患者的真实数据示例中。