I have a question regarding the notation in causal meditation.
The conventional potential outcomes is generally written as follow. For the observed (potential) outcome under treatment we have $Y(1) \mid D = 1$ and for the observed (potential) outcome under no treatment we have $Y(0) \mid D = 0$.
We then have the two unobserved quantities, $Y(0) \mid D = 1$, and $Y(1) \mid D = 0$.
Now, in causal mediation (ex Robins and Greenland 1991), we have $Y(1), M(1)$ denoting the potential outcome under treatment $Y(D=1)$ and the Mediator $M$, and we also have $Y(0), M(0)$.
I do not understand what is unobserved in causal mediation. Is $Y(1), M(0)$ or $Y(0), M(1)$ unobserved?
My sense is that it is not unobserved because we could observe people getting the treatment but not deciding to take up the mediator in $Y(1), M(0)$ and vice versa.
Does $M(1)$ means $M(D=1)$ (mediator under treatment) or $M(M=1)$ the mediator under the mediator condition?
This confuses me because it seems that a fundamental estimand in causal mediation is
\begin{equation}
\mathbb{E}[Y(1), M(1)] – \mathbb{E}[Y(0), M(0)]
\end{equation}
Does this means that for instance the so-called "pure direct effect" is unobservable?
\begin{equation}
\mathbb{E}[Y(1), M(0)] – \mathbb{E}[Y(0), M(0)]
\end{equation}
In summary, what are the observed and unobserved quantities in causal meditation and does $M(1)$ means $M(D=1)$ or $M(M=1)$?
Best Answer
Going with the notation of VanderWeele and Vansteelandt (2009), with treatment indicator A, we can never observe $Y(A=1,M=0)$, because this is the outcome for treated individuals who are exposed to the mediator as if they were in the control group. Conversely, we cannot observe $Y(A=0,M=1)$. Therefore, these are quantities that require to be predicted in causal mediation analysis.
In summary, M(1) means M(A=1).
I add the following slide by Stijn Vansteelandt, which illustrates the notation quite nicely:
Edit: Some further clarifications as response to comments.
Sure you can have people also taking up the mediator in the control group, which is good, you can use that information to predict take up of the mediator under the counterfactual.
Let's say you are interested in the natural direct effect in an RCT. This effect is conditional on some baseline covariates C, which are sufficient to adjust for confounding of the mediator-outcome relationship (which is potentially confounded even if treatment assignment was random):
$\mathbb{E}(Y(A=1,M=0) - Y(A=0,M=0)|C)$.
Then you will need to "impute" counterfactual outcomes $Y(A=1,M=0 | C)$, which are unobserved. A common approach to do so is to use an auxiliary model on the way, in which you estimate the effects of $A$, $M$ and $C$ on $Y$. Using that model, you can predict the missing outcomes $Y(A=1,M=0 | C)$ and - given all structural and statistical assumptions hold - arrive at an unbiased estimate for the natural direct (and similarly the indirect) effect.
So those individuals taking up the mediator in the control group will give you valuable information for fitting this auxiliary model, making it easier (from a model-fitting perspective) to separate the effects of $A$ and $M$.