  Here, we explain and illustrate a geometric perspective on causal inference
in cohort studies that can help epidemiologists understand the role of
standardization in causal inference as well as the distinctions between
confounding, effect modification, and noncollapsibility. 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 risk in the unexposed
on the x-axis and risk in the exposed on the y-axis. The crude risks define one
point in the unit square, and the stratum-specific risks define two other
points in the unit square. These three points can be used to identify
confounding and effect modification, and we show briefly how these concepts
generalize to confounders with more than two levels. We propose a simplified
but equivalent definition of collapsibility in terms of standardization, and we
show that a measure of association is collapsible if and only if all of its
contour lines are straight. We illustrate these ideas using data from a study
conducted in Newcastle upon Tyne, United Kingdom, where the causal effect of
smoking on 20-year mortality was confounded by age. We conclude that causal
inference should be taught using geometry before using regression models.
