When implementing Rubin’s causal model, one of the (untestable) assumptions that we need is unconfoundedness, which means
Where the LHS are the counterfactuals, the T is the treatment, and X are the covariates that we control for.
I am wondering how to describe this to a person who doesn’t know much about the Rubin Causal Model. I understand why theoretically we need this assumption, but I am not sure about conceptually why this is important. Specifically, if T is the treatment, shouldn’t the potential outcome be very dependent on it? As well, if we have a randomized controlled trial, then automatically, (Y(0),Y(1))⊥T. Why does this hold true?
How would you describe the uncoundedness/ignorability assumption to somebody who has not studied the RCM?
I think you are getting hung up on the difference between potential outcomes (Y0,Y1) and the observed outcome Y. The latter is very much influenced by treatment, but we hope the former pair is not.
Here’s the intuition (putting aside conditioning on X for simplicity) about the observed outcome. For each observation, the realized outcome can be expressed as
This means that Y and T are dependent because the average value of T⋅Y1 will not equal the average (1−T)⋅Y0 (as long as the treatment effect is nonzero and treatment is randomized/ignorable).
Here’s the intuition for the second part. If we are going to learn about the causal effect of T, we will be comparing treated and untreated observations, while taking differences in X into account. We are assuming that the control group is the counterfactual for the treatment group had they not received treatment. But if people choose their own treatment based on their potential outcomes (or expectations about the potential outcomes), this comparison is apples to orangutans. This is like a medical trial where only the healthier patients opt for the painful surgery because it is worth the cost for them. Our comparison will be contaminated if the choice to opt for treatment is not random after conditioning on X (variables that measure current health status which should be observable to the doctor and the patients). One example of an unobservable variable might be having a spouse who loves you very much, so she urges you to get the surgery, but also makes sure you stick to the doctor’s instructions post-op, thereby improving Y1 outcome. The measured effect is now some combination of surgery and loving help, which is not what we want to measure. A better example is an X that is affected by the treatment, either ex post or ex ante in anticipation of treatment.