Data in public health often arrive as messy 2×2 tables of real life: smokers versus non-smokers, cases versus controls, treated versus untreated. Clinicians eye odds ratios, statisticians chase regressions, and everyone hopes the numbers tell a clean story about cause and effect. But reality rarely behaves like a neat equation. Subgroups diverge. A covariate such as age or sex can bend the measured association in surprising directions. A new way of thinking is to rotate the problem into a geometric portrait that makes the tensions visible instead of buried in a regression summary.
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 magic is what the points do there. This approach, championed by Eben Kenah of the University of Florida, gives a vivid map of three ideas that usually get tangled: confounding, association measure modification, and collapsibility. The paper expands this geometric lens to untangle how these concepts relate and, crucially, to show that two of them can drift apart even when the third is quiet. It is a reminder that data can be more like a landscape than a single line on a chart, and that reading that landscape carefully can change what we think we know about public health risk.
Why does this matter? Because misreading these relationships can lead to wrong conclusions about whether an exposure truly causes outcomes, or whether an apparent effect is simply a byproduct of who you studied. When data speak through subgroups, the story they tell can change shape. The Newcastle study cited in the work is a perfect illustration: researchers found that age was a powerful confounder when looking at smoking and 20-year mortality, and the Rothman diagram helps you see exactly how the stratum by age interacts with different measures of association. The result is not a single line of best fit but a geometric fabric that reveals where averages are trustworthy and where they are not.
Kenah is not merely reprinting old diagrams; he is sharpening a framework. He reframes association measure modification as a normal feature of data rather than a curious edge case. He also reframes collapsibility not as a regression quirk but as a property that emerges when you standardize away the distribution of the stratifying variable. In that light, the geometry becomes a compass for causal inference, guiding us toward conclusions that hold across subgroups and those that do not. The paper invites a shift in how we teach and practice causal analysis, from a fixation on coefficients to a discipline of reading landscapes.
A map of risk and the geometry of association
On a Rothman diagram, every point in the unit square represents two risks: the risk in the unexposed on the x axis and the risk in the exposed on the y axis. If you picture a cohort study, a stratum is a slice of the population defined by a covariate such as age group. The four possibilities of a binary covariate create clusters of points that reveal what is happening in each subgroup. The unit square is not just a picture; it is a playground where common measures of association can be drawn as contour lines.
From there, you can trace how a chosen measure behaves across the square. The risk difference is a simple line of slope one crossing the square; its contours are straight lines. The risk ratio also carves straight lines in the plane, so equal changes in exposure yield predictable proportional changes. In contrast the odds ratio and the cumulative hazard ratio bend and curve. Their contours curve away from the straight line, and that curvature reveals a subtle truth: even when two subgroups share the same point near the middle, the journey to get there on the curved contours can diverge, hinting at noncollapsibility or modification that depends on scale.
The Newcastle example used in the paper involves stratifying by age groups. Younger participants had a different pattern of the association between smoking and mortality than older participants. When you plot the stratum-specific points on the Rothman diagram, they do not all sit on the same contour line for a given measure. That is what association measure modification looks like in this geometric language. It is not a sign of a mistake; it is a natural feature of how real-world risk interacts with demographic structure. The key move is to compare the stratum-specific points with the standard or marginal point your model would produce if you averaged over age. The diagram makes it clear where the average diverges from the subgroups and why.
One more geometric gem: the null line where x equals y. Along this line, all measures of association reach their baseline of no effect. It is the diagonal of the unit square, the line that marks the moment when the exposed and unexposed face identical risks. It is the anchor point that calibrates our intuition and reminds us that different scales can tell slightly different stories about the same data.
Modification and collapsibility on different scales
When a covariate C modifies the association between exposure and outcome, the stratum-specific values of the chosen measure M do not line up on a single contour. In the Rothman diagram, you see stratum-specific points resting on at least two different contour lines. The intuition is simple: if the effect depends on the level of C, then the effect is not uniform across strata, and any single overall estimate is an approximation at best.
The Newcastle data illustrate that association measure modification depends on the scale you choose. On the risk difference and the risk ratio scales, the evidence for modification can look strong or weak depending on sample size and the exact groups you compare. On the odds ratio scale, the same data can appear to show modification, but the location of the stratum-specific points on curved contours makes the common estimate less representative of the subgroups. The visual language helps us understand why p values for interaction terms can flip depending on the measure used and how large each stratum is.
Another thread is the claim that confounding and association measure modification are logically independent. You can have a crude estimate that looks wrong because of confounding yet have no genuine modification when you examine the scales that matter for the question at hand. Conversely, you can observe clear modification while there is no confounding. The geometry makes that independence visible where regression coefficients often blur the distinction. The Newcastle example helps to illustrate this: the crude average of smoking on 20-year mortality may mislead you if you assume no modification, but once you plot the stratum-specific points the pattern becomes explicit.
Kenah emphasizes that a model with no interaction is not a statement about the world but a useful approximation. This is not a claim that interaction terms are optional; instead it is a reminder that the validity of a single estimate depends on the scale and the population you care about. A contour that runs through multiple strata is a sign you should consider interaction terms if your goal is to predict well or to understand causal mechanisms. The geometric language invites researchers to test for modification in the way that best serves their question, not simply to chase a fixed coefficient in a single regression model.
Standardization and what collapsibility means in practice
Collapsibility is a tricky term in epidemiology, but Kenah offers a crisp reframing that makes it easier to grasp. A measure is collapsible if, after standardizing to a common distribution of the stratifying covariate C, the marginal measure equals the stratum-specific measure. In plain terms: if you could equalize age, sex, or any other dividing line across the whole dataset, would the overall effect you measure be the same as the effect in each subgroup?
On the Rothman diagram, this criterion becomes geometric. A thread runs along the standardized hull the line or polygon that connects the stratum-specific points after standardization. If all the contour lines of the measure M are straight, this standardized hull sits along a single contour, and you can slide it and it stays on that contour. That is collapsibility in action. If you have curved contours, the hull tugs away from the single contour and you edge toward noncollapsibility. In plain language, noncollapsibility means that the act of averaging across subgroups changes the measure even when there is no confounding. It is not a bug in your model; it is a property of the measure itself.
The paper maps which common measures are collapsible in practice. The risk difference and the risk ratio are collapsible. The odds ratio and the cumulative hazard ratio are not, at least in general. This is why standardized odds ratios can drift toward the null when mixing populations with different age structures. The same drift shows up for cumulative hazard ratios, a reminder that noncollapsibility is baked into how some measures behave under standardization.
What does this imply for data analysis? It is a strong case for standardization as a first-class tool in causal inference, not an afterthought to be glossed over by a single regression coefficient. The geometry makes clear that the achievements and limits of a given measure depend on the scale and the population in view. It nudges researchers to report stratum-specific effects and to show what those effects look like after standardizing to a common covariate distribution. In short, it pushes for more robust, transparent reporting that better reflects the world outside the data desk.
Looking ahead, the Rothman diagram approach is not a rejection of regression but a companions light: a different lens that makes the logic of causal claims legible. Think of it as switching from a narrow path through a forest to a panoramic overlook. The math stays; the view broadens. If geometry becomes a standard part of how we teach causal thinking, the next generation of scientists may navigate questions about public health with more grace, and a lot more clarity.
The core idea is both elegant and practical. Geometry can illuminate the messy logic of causal inference, showing that association measure modification and collapsibility are not the same thing as confounding. The framework comes from Eben Kenah at the University of Florida, and it asks a broader audience to see the world of epidemiology with a painterly eye—where two risks and the lines that connect them reveal far more than a single coefficient ever could.
