1. Thoughts on ratio variables and other algebraic manipulations of raw variables (Second draft) DAG-perspective Eyal Shahar April 2008

2. Consider ratios as effects of interest A B A/B Exposure (E) A/B is fully determined by A and B. If we estimate the effect of E on A and the effect of E on B, we have also estimated the effect of E on A/B. No new knowledge is gained. If E affects A alone (and not B), then its effect on A/B is the same as its effect on A times a constant. It is still legitimate to estimate the effect of E on A/B, but the reason for doing so must be the assumption that the variable A/B is “interesting”. Why would it be interesting, then? Maybe because it plays the role of a cause of other interesting effects.

3. The general case A B Any algebraic combinations of A and B A/B, A/B 2, A/B 3, A 2 /B, A/√B, w 1 A+w 2 B (index) (w 1 A+w 2 B)/(w 1 +w 2 ) (weighted average) (B-A)/Ax100 (percent change) A/B x 100 where A is part of B (A  B) Exposure Any of infinite algebraic combinations is fully determined by A and B. If we estimate the effect of E on A and the effect of E on B, we have also estimated the effect of E on any of those algebraic combinations. No new knowledge is gained. Why, then, is the ratio of particular interest? Why is it more interesting than all other expressions? If we can’t find arguments to single out one algebraic combination as “interesting”, maybe none is interesting? [Notice that BMI is not simply A/B; it is A/B 2. And no, height 2 is not the volume of the human body.]

4. Consider ratios as causes of interest Theory #1 A B A/B Disease If the ratio has no effect, any association between the ratio and D is fully confounded by A and B. According to this theory, we should not model the ratio. It has no effect on D.

5. Consider ratios as causes of interest Theory #2 A B A/B Disease But this diagram is identical to the following causal structures! (and don’t call it interaction: interactions are not meaningfully depicted in DAG) A A/B Disease B A/B Disease These structures say the following: if A/B has an effect on D, then it is a mediator of the effects of A and B (and of nothing else). The effect of A/B is fully due to the effects of A on D and of B on D. There is no additional “unique effect” of their ratio. In other words, D will change based on the changes in A and B. Variables A and B (not A/B!) contain all of the causal information about what will happen to the value of D (assuming no other causes of D). A/B would have been “interesting” on its own merit only if there were other causes of it, besides A and B. But there aren’t!

6. Consider ratios as causes of interest Theory #3 A B A/B Disease Here, the ratio cannot be modeled alone. A and B are confounders for the effect of A/B on D. Still, there is no “unique effect” of their ratio. In other words, D will change based on the changes in A and B. Variables A and B (not A/B!) contain all of the causal information about what will happen to the value of D. In fact, they contain additional information that is not mediated by A/B at all (direct arrows from A to D and from B to D).

7. Consider ratios as causes of interest Theory #4 A B A/B Disease Again, the ratio cannot be modeled alone. A and B are confounders for the effect of A/B on D. Still, there is no “unique effect” of their ratio. In other words, D will change based on the changes in A and B. Variables A and B (not A/B!) contain all of the causal information about what will happen to the value of D. In fact, they contain additional information that is not mediated by A/B at all (direct arrows from A to D and from B to D). Notice that A is a confounder for the effect of B, but B is a mediator (not a confounder) for the effect of A

8. Where do ratios come from? Why are ratios popular? Why do they appear intuitively “right”? What problem do they try to fix? Loosely speaking, they serve a desire for “standardization” Example: I weigh 170 pounds. Am I over weight? Well, it depends on my height. It is “normal” for taller people” to be heavier. But what does “normal weight” and “abnormal weight” mean, if not the causal effect of weight? We usually call something “abnormal” when the contrast between “abnormal” and “normal” has some interesting causal effect (say, on diabetes or on death). And what role is height playing here, if not the role of a confounder or an effect modifier? (There are no other possible roles in causal thinking.) The ratio seems to be a primitive (and poor) method to deal with confounding. It is a kind of “standardization” that is worse than classical standardization (“direct” or “indirect”).

9. Summary From the viewpoint of DAG, I cannot find any reason to study ratios, either as causes or as effects. Typical arguments say things like, “but weight alone does not matter; only weight relative to height”. That’s not a clear causal statement. What exactly does it mean “weight alone does not matter”? All of the diagrams show a causal pathway from weight to D. It does matter! It is a cause of D. Is this a claim of confounding? Then, what causal structure is assumed? And why does the ratio help to deconfound? Is ratio a method to adjust for confounding? Where is the formal theory that justifies ratios as a method to deconfound? It does not appear in any textbook. Does it mean “the effect of gaining 1 kg differs according to height”? If so, the claim is about effect modification. Is ratio a method to model effect modification? I don’t think so. On a philosophical note: we sometimes use things that seem “intuitively right”, but that’s not a good practice in science. We need to have a clear understanding of the rationale.

10. Corrections and references are welcome!