Rubin Causal Model - An Extended Example

An Extended Example

Rubin defines a causal effect:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from to is the difference between what would have happened at time if the unit had been exposed to E initiated at and what would have happened at if the unit had been exposed to C initiated at : 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning.

According to the RCM, the causal effect of your taking or not taking aspirin one hour ago is the difference between how your head would have felt in case 1 (taking the aspirin) and case 2 (not taking the aspirin). If your headache would remain without aspirin but disappear if you took aspirin, then the causal effect of taking aspirin is headache relief.

Suppose that Joe is participating in an FDA test for a new hypertension drug. If we were omniscient, we would know the outcomes for Joe under both the treatments and therefore know the treatment effect.

is the change in Joe's blood pressure if he takes the pill. In general, this notation expresses the effect of a treatment, t, on a unit, u. Similarly, is the effect of a different treatment, c or control, on a unit, u. In this case, is the change in Joe's blood pressure if he doesn't take the pill. is the causal effect of taking the drug.

From this table we only know the causal effect on Joe. Everyone else in the study might have an increase in blood pressure. However, regardless of what the causal effect is for the other subjects, the causal effect for Joe is a decrease in blood pressure.

Consider a larger sample of patients:

The causal effect is different for every subject, but the drug works for everyone because everyone's blood pressure decreases.