Rothman diagrams: the geometry of confounding and standardization

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.