How can I interpret the conditional expectation mean $E[Y(0)\mid Z=1]$ where $Y(0)$ is the potential outcome on control and $Z$ the treatment assignment?
I believe it is the "expected outcome if the individuals who received the treatment had not been treated." Does this imply it isn't actually observable from the data? In that we are asking, among the units that were treated, what was their actual control potential outcome?
Is there a cleaner way to think about this? thanks.
Yes, the way to interpret this quantity is the expected outcome under control for those who receive treatment. This is a fundamental quantity in most causal inference applications. The average treatment effect in the treated (ATT) is defined as $E[Y(1)|Z=1] - E[Y(0)|Z=1]$, that is, the difference between the expect value of the outcome under treatment and the expected value of the outcome under control for those who received treatment. This is an important estimand for understanding the effect of harmful exposures on those potentially exposed or the effect of experimental programs or procedures on individuals likely to receive them.
Due to the assumption of causal consistency, $E[Y(1)|Z=1] = E[Y|Z=1]$, so the left quantity in the ATT is observed in the data (assuming you have data from the full population). $E[Y(0)|Z=1]$ is not observed, which is why statistical methods for causal inference exist; they are designed to estimate this quantity (or similar unobserved quantities involving potential outcomes that do not correspond to the observed treatment status of the corresponding units).
There are several ways to estimate $E[Y(0)|Z=1]$ depending on the assumptions one wants to endorse. For example, if pretreatment values of $Y$ are measured, the assumption of parallel trends allows one to estimate $E[Y(0)|Z=1]$ using difference-in-differences methodology. If instead there are no unmeasured confounders, then regression, (propensity score) matching, or weighting can be used to estimate it.
