Rothman diagrams: the geometry of association measure modification and collapsibility

Here, we outline how Rothman diagrams provide a geometric perspective that can help epidemiologists understand the relationships between effect measure modification (which we call association measure modification), collapsibility, and confounding. A Rothman diagram plots the risk of disease in the unexposed on the x-axis and the risk in the exposed on the y-axis. Crude and stratum-specific risks in the two exposure groups define points in the unit square. When there is modification of a measure of association $M$ by a covariate $C$, the stratum-specific values of $M$ differ across strata defined by $C$, so the stratum-specific points are on different contour lines of $M$. We show how collapsibility can be defined in terms of standardization instead of no confounding, and we show that a measure of association is collapsible if and only if all its contour lines are straight. We illustrate these ideas using data from a study in Newcastle, United Kingdom, where the causal effect of smoking on 20-year mortality was confounded by age. From this perspective, it is clear that association measure modification and collapsibility are logically independent of confounding. This distinction can be obscured when these concepts are taught using regression models.