# For the ATT (Average Treatment Effect on the Treated), why is it usually talked about in the population perspective when it is defined w.r.t. samples?

The $$ATT$$ or the Average Treatment Effect on the Treated, is defined as:

$$ATT = E[Y(1) - Y(0) | T=1]$$

for potential outcomes $$Y(1), Y(0)$$ and treatment indicator $$T \in \{0,1\}$$. It is my understanding that the above is an estimand and in observational studies, the $$ATT$$ is not equal to the $$ATE$$, or the average treatment effect. It is however, equal under randomized studies.

The $$ATT$$ seems to be a measure of the average effect from a treatment to a treated individual randomly drawn from the treated population, rather than to any member of the population. Therefore, it seems to be dependent on the sample itself. Why is it then talked about as an estimand, which hints at the population level?

It seems to me that the reason the $$ATT$$ is not equal to the $$ATE$$ because there is not sufficient randomization for the treatment assignment. This in turn has nothing to do with the original groups themselves, but rather what constituted an individual being put into "treatment". This selection seems to be a property that goes into the sample. Why then is the $$ATT$$ a population estimand parameter?

You are correct in saying $$ATT$$ is not necessarily equal to the $$ATE$$ in observational studies due to randomization. Consider a variable $$M$$ that I will call an effect measure modifier. Consider the following relationship, $$M$$ changes the effect as such $$E[Y(1) - Y(0)|M=0] > E[Y(1) - Y(0)|M=1]$$.
For randomized trials, $$M$$ is distributed evenly between the treated and the untreated in expectation. Therefore, the proportion with $$M$$ will not differ between the treated and untreated. Because of this, the $$ATE$$ and $$ATT$$ will be the same in expectation in randomized trials. In your particular trial, the $$ATE$$ and $$ATT$$ may not necessarily be equal.
For observational studies, there is no general expectation that $$M$$ would be evenly distributed among groups (it may be a confounder in this setting). If it is unevenly distributed, the $$ATE$$ and $$ATT$$ won't be the same. Using the above relation, if the treated group has more $$M=1$$ individuals, then the $$ATT$$ will be less than the $$ATE$$. However, the $$ATE$$ and $$ATT$$ can be equal in observational studies as long as all $$M$$ are evenly distributed between the treated and untreated (a strong assumption).
However, the estimand is still a population parameter. Like the $$ATE$$, the $$ATT$$ is similarly a average effect and not an individual causal effect. The population is only the population of treated individuals for the $$ATT$$.
Why the $$ATT$$ may be preferred: the identifiability requirements for the $$ATT$$ are slightly weaker than $$ATE$$ since we observe all the potential outcomes under $$Y(1)$$ for the population of treated individuals. Therefore, we can rewrite the $$ATT$$ as $$E[Y - Y(0)|A=1]$$ where $$A$$ is the treatment.
• Can $M$ (effect measure modifier) also be thought of as a covariate? Is your statement of "...$M$ are evenly distributed between the treated and untreated..." also known as an unconfoundedness assumption? Aug 29, 2019 at 13:16
• So $M$ does not need to be a confounder in either context, but it can be. The only requirement for $M$ is that it is a cause (or proxy for a cause) of the outcome $Y$. Regarding the evenly distributed, this is also unconfoundedness assumption (since $M$ must always at least be a cause of $Y$) Aug 29, 2019 at 14:41