Minimally Causal

In Rubin's classic 1974 paper, he introduces the potential outcomes model and uses that model to discuss causality. Following that idea, I will say that a policy requiring or encouraging a unit to receive treatment A instead of treatment B, would improve that unit's outcome if for that unit, their potential outcome for treatment A is higher than their potential outcome for treatment B. So note the first part. A policy of interest may not lead all units to receive the treatment. For example, if the FDA approves a new drug, then only some proportion of the patient population may end up taking the new drug. The second part states that there exists a unit who is better off under treatment A relative to treatment B. The first part means that the policy may not actually impact the unit that is made better off, but at least it is possible.

A minimally causal effect of treatment A relative to treatment B is thus one in which at least one unit is made better off from a policy moving that unit from treatment A to treatment B. Note that this definition does not rule out the possibility that treatment A has no effect for most units or that treatment B is actually better for some units. Nor does it rule out the possibility that the policy itself has no impact even though some units would be better off under the treatment.

To prove that treatment A is minimally causal relative to treatment B, then we must show that at least some strictly positive proportion of the population of interest is strictly better off with treatment A relative to treatment B. If we have access to perfect randomized controlled trials, then we can use the Frechet-Hoeffding bounds to prove minimal causality. Note that we don't need perfect randomization, in order for the data to imply minimal causality. We do need to know that for some positive proportion of the population, their assignment to treatment is "essentially at random". If for some subset of the population we can use the observed distribution outcomes as a reasonable estimate of the distribution of potential outcomes, then we can use the Frechet-Hoeffding bounds to show minimal causality. What if we don't have access to such data? What if the data is observational or there was a problem in the trial such that we are unwilling to claim that for any part of the population the observed distribution of outcomes is a reasonable estimate of the potential outcomes distribution?

One approach is to make no restrictions on the unobserved counter-factual outcomes, except for the restrictions implied by mathematics. Unfortunately, it is straightforward to show that these natural bounds are not enough to allow the data to imply minimal causality.

In Bounds Math, I go through the implications of various assumptions that could be made with observational data. Two of the assumptions, revealed preference and monotonic treatment response, imply minimal causality. To see this note that we can limit the population of interest to those people assigned to one of the treatments. Then the Frechet-Hoeffding Bounds and the assumption imply minimal causality.

The two other assumptions, monotonic treatment selection and monotonic treatment matching, do not immediately imply minimal causality. However, the analysis shows that the monotonic treatment selection assumption can never imply minimal causality. The matching assumption, on the other hand, can. The assumption states that those assigned to the treatment have "better" outcomes given the treatment than the potential outcomes of those not assigned to the treatment.

The assumption of monotonic treatment matching, which is generally a weaker assumption than random assignment, allows for the possibility that the data will show minimal causality. It is thus possible to make causal claims without experimentation. In some cases, correlation is indeed causation.