So how do we deal with confounders and estimate the ATE when the treatment assignment is not randomized? Let’s return to the therapy animal example once more.
Suppose that instead of randomizing patients to receive therapy animal services, we allow the twelve hospital patients to CHOOSE whether or not they want therapy. Imagine that we also have data for a new confounding variable X that represents whether an individual does (X = 1) or does not (X = 0) have a diagnosis of anxiety disorder. Here, X is a confounder and impacts both the treatment and the outcome: patients who have anxiety might be more likely to choose to receive therapy animal services AND have higher cortisol levels generally.
This confounding is problematic because it means there may be more people with anxiety in the treatment group than in the control group. More people with anxiety means the treatment group may have a higher average cortisol level compared to that of the control group before therapy animal services even occur!
When the treatment groups are unbalanced with respect to confounders, the treatment groups are not exchangeable: we would observe different outcomes if the treatment groups swapped treatment conditions.
To avoid making poor comparisons between potentially imbalanced treatment groups, we have to be able to assume conditional exchangeablility:
- Conditional exchangeability means that the treatment groups are exchangeable if we take into account confounding variables.
- This is also called ignorability or unconfoundedness.
By taking anxiety diagnosis (variable X) into account, we avoid getting a biased estimate of the cortisol levels produced by those receiving therapy animal services in comparison to those who do not.
Take a look at the slides in the learning environment to see how the anxiety confounder might bias our estimate of the treatment effect in our therapy animal analysis.
Slide 1In our table, we have the data for our twelve hospital patients who chose whether or not to receive therapy animal services. The variables include anxiety diagnosis (X=1 for anxiety and X=0 for no anxiety) and treatment assignment (Z=1 for therapy and Z=0 for no therapy). Because this is reality, we cannot see both potential outcomes but rather just the observed outcome Y (observed cortisol level).
If we compute the estimated ATE as though treatment was randomized and don’t take anxiety (X) into account, we get an estimated ATE of -2.9. We conclude that therapy animal services reduce cortisol levels by 2.9 units on average.
In order to account for the anxiety variable, we first compute the ATE for just those patients diagnosed with anxiety disorder (X=1). We do this as we always do: by subtracting the average cortisol level for the control patients from the average for the treated patients. Because we are only comparing patients with anxiety, the groups should theoretically have similar cortisol levels before receiving therapy. We find an estimated ATE of -5.5 for individuals with anxiety—much greater in magnitude than our initial estimate of -2.9.
Now we compute the ATE for just those patients NOT diagnosed with anxiety disorder (X=0). We find an estimated ATE of -5.5 for individuals WITHOUT anxiety.
Averaging the ATEs of our two groups gives us an estimated ATE of -5.5 when we take X into account. If we recall from Exercise 5, the true ATE was -5.8: therapy animal services reduced cortisol levels by 5.8 units on average. Our estimate when we took confounding into account was much closer to the true treatment effect.
Why did this happen? Anxiety correlated with higher cortisol, and more people with anxiety were in the treatment group. This made the treatment group’s cortisol levels tend toward higher levels, making it look as though therapy animal services were not bringing down cortisol levels by much.