In instrumental variable (IV) settings, such as in imperfect randomized trials and observational studies with Mendelian randomization, one may encounter a continuous exposure, the causal effect of which is not of true interest. Instead, scientific interest may lie in a coarsened version of this exposure. Although there is a lengthy literature on the impact of coarsening of an exposure with several works focusing specifically on IV settings, all methods proposed in this literature require parametric assumptions. Instead, just as in the standard IV setting, one can consider partial identification via bounds making no parametric assumptions. This was first pointed out in Alexander Balke's PhD dissertation. We extend and clarify his work and derive novel bounds in several settings, including for a three-level IV, which will most likely be the case in Mendelian randomization. We demonstrate our findings in two real data examples, a randomized trial for peanut allergy in infants and a Mendelian randomization setting investigating the effect of homocysteine on cardiovascular disease.

翻译：在工具变量（IV）设置中，例如不完善的随机试验和采用孟德尔随机化的观察性研究，可能会遇到连续暴露变量，但其因果效应并非真正兴趣所在。相反，科学兴趣可能在于该暴露变量的粗化版本。尽管已有大量文献探讨暴露变量粗化的影响，且有多项研究专门针对IV设置，但该文献中提出的所有方法均需参数假设。然而，如同标准IV设置，可以通过无参数假设的边界进行部分识别。这一观点最初由Alexander Balke在其博士论文中指出。我们扩展并阐明其工作，在多种设置下推导出新边界，包括三级IV（这很可能是孟德尔随机化中的常见情况）。我们通过两个真实数据示例展示研究结果：一项关于婴儿花生过敏的随机试验，以及一项研究同型半胱氨酸对心血管疾病影响的孟德尔随机化设置。