# Causal Mediation and Potential Outcomes Notations

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

$$$$\mathbb{E}[Y(1), M(1)] - \mathbb{E}[Y(0), M(0)]$$$$

Does this means that for instance the so-called "pure direct effect" is unobservable?

$$$$\mathbb{E}[Y(1), M(0)] - \mathbb{E}[Y(0), M(0)]$$$$

In summary, what are the observed and unobserved quantities in causal meditation and does $$M(1)$$ means $$M(D=1)$$ or $$M(M=1)$$?

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 $$M$$.

• Ok thanks that makes sense. But what about the fact that we can observed people not being treated $Y(A=0)$ but taking up the mediator $M(A=1)$? In this example, some untreated use condoms?
– giac
Commented Jan 31, 2022 at 15:27
• Where could I find these slides? Thank you.
– giac
Commented Jan 31, 2022 at 15:35
• I have extended above answer. The slides are from a conference workshop and I am not sure if the presenter are fine with sharing them publicly. Here is a tutotial-like open access paper by partly the same authors with similar content: e-epih.org/journal/view.php?doi=10.4178/epih.e2017035 Commented Jan 31, 2022 at 16:53

I'll use the notation of VanderWeele (2014).

There are many relevant quantities in mediation.

$$Y_{am}$$ is the outcome had someone received treatment level $$a$$ (i.e., 0 or 1 for a binary treatment) and mediator level $$m$$ (i.e., 0 or 1 for a binary mediator). For example, $$Y_{01}$$ refers to the value had someone not taken the treatment but had taken the mediator. From this quantity, we can compute the control direct effect, $$E[Y_{am}]-E[Y_{a'm}]$$, between two treatment levels $$a$$ and $$a'$$. For example, this could be $$E[Y_{11}]-E[Y_{01}]$$: the direct effect of the treatment when the mediator is set to the value 1, or $$E[Y_{10}]-E[Y_{00}]$$: the direct effect of the treatment when the mediator is set to the value 0.

Sometimes, though, we are less interested in selecting a specific value for the mediator to take but are more interested in its natural value under some treatment. For this, we have $$M_a$$, the value of the mediator under treatment level $$a$$. $$M_a$$ can be 0 or 1 for any value of $$a$$, and will differ across individuals; it is a random variable. For example, for some units $$M_1 = 1$$, and for other units $$M_1=0$$; that is, under treatment, some units can take the mediator and other units can not.

We can now reconsider the potential outcomes under different treatment and mediator values without specifying a specific value for the mediator, but instead letting take whatever value it would take under the corresponding treatment. We can represent this as $$Y_{aM_a}$$. For example, $$Y_{1M_1}$$ is the potential outcome under treatment where the mediator is set to its value under treatment. We don't immediately know what that value is and it differs across units. For some units, $$Y_{1M_1} = Y_{11}$$, and for other units, $$Y_{1M_1} = Y_{10}$$. It depends on what value $$M_1$$ takes for each unit.

An important definition is that $$Y_{aM_a} = Y_{a}$$; that is, when the treatment is set to level $$a$$, and the mediator is set to whatever it happens to be when the treatment is set to level $$a$$, the potential outcome is equal to the potential outcome under treatment level $$a$$, ignoring the mediator. In this sense, the treatment and mediator are compatible because the mediator takes the value corresponding to the treatment value set for the potential outcome.

In mediation analysis, we are often interested in the "incompatible" potential outcome $$Y_{aM_{a'}}$$, the potential outcome when treatment is set to level $$a$$ but the mediator is set to whatever it would be under the other treatment level $$a'$$. For example, $$Y_{1M_{0}}$$ is the potential outcome under treatment but with the mediator set to whatever value it would take under control. The reason this is incompatible is because if a unit takes treatment, we can't observe the mediator value it would have received under control, but we need to have both in order to consider the potential outcome.

However, even though we can't observe this potential outcome for anyone, we can use it to define specific mediation contrasts. The natural indirect effect is $$E[Y_{aM_{a}}] - E[Y_{aM_{a'}}]$$; that is, if treatment is fixed at level $$a$$, what is the difference in the outcome if the mediator had taken its value under $$a$$ vs. if the mediator had taken its value under $$a'$$? Again, this is not comparing $$M=1$$ and $$M=0$$; for some units, $$M_{a}$$ is 1 and for other units it is 0, and the same for $$M_{a'}$$.

We can also talk about the natural direct effect $$E[Y_{aM_{a}}] - E[Y_{a'M_{a}}]$$; that is, if the mediator is fixed to whatever value it would take under treatment level $$a$$, what is the difference in the outcome if treatment were set to level $$a$$ vs. $$a'$$?

Because the "incompatible" potential outcomes cannot be observed, we need to rely on assumptions to estimate them. This is the crux of causal mediation analysis.

• That makes sense, especially $Y_{aM_a} = Y_{a}$. Thanks
– giac
Commented Feb 1, 2022 at 10:03