We outline a geometric perspective on causal inference in cohort studies that can help epidemiologists understand the role of standardization in controlling for confounding. For simplicity, we focus on a binary exposure X, a binary outcome D, and a binary confounder C that is not causally affected by X. Rothman diagrams plot the risk of disease in the unexposed on the x-axis and the risk in the exposed on the y-axis. The crude risks define a point in the unit square, and the stratum-specific risks at each level of C define two other points in the unit square. Standardization produces points along the line segment connecting the stratum-specific points. When there is confounding by C, the crude point is off this line segment. The set of all possible crude points is a rectangle with corners at the stratum-specific points and sides parallel to the axes. When there are more than two strata, standardization produces points in the convex hull of the stratum-specific points, and there is confounding if the crude point is outside this convex hull. We illustrate these ideas using data from a study in Newcastle, United Kingdom, in which the causal effect of smoking on 20-year mortality was confounded by age.

中文翻译：

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罗斯曼图：混杂和标准化的几何学。


我们概述了队列研究中因果推理的几何观点，可以帮助流行病学家了解标准化在控制混杂因素中的作用。为简单起见，我们关注一个二元暴露 X、一个二元结果 D 和一个不受 X 因果影响的二元混杂因素 C。Rothman 图在 x 轴上绘制了未暴露者的疾病风险，在 y 轴上绘制了暴露的风险。粗略风险定义单位平方中的一个点，C 每个级别的地层特定风险定义单位平方中的其他两个点。标准化沿连接特定于层的点的线段生成点。当 C 混杂时，粗点离开此线段。所有可能的粗略点的集合是一个矩形，在特定于层的点上具有拐角，并且边平行于轴。当有两个以上的地层时，标准化会在特定于地层的点的凸包中生成点，如果原始点位于此凸包之外，则会出现混淆。我们使用英国纽卡斯尔的一项研究的数据来说明这些想法，其中吸烟对 20 年死亡率的因果影响与年龄混淆。