Thanks, I now have a clearer idea of what these expressions mean and why they matter. You write on page 15:
Defining the influence of A on Y for a particular unit u as Y(1,M(0,u),u) involved a seemingly impossible hypothetical situation, where the treatment given to u was 0 for the purposes of the mediator M, and 1 for the purposes of the outcome Y.
For the A/M/Y = smoking/tar/cancer situation I can imagine a simple way of creating this situation: have someone smoke cigarettes with filters that remove all of the tar but nothing else. There may be practical engineering problems in creating such a filter, and ethical considerations in having experimental subjects smoke, but it does not seem impossible in principle. This intervention sets A to 1 and M to M(0,u), allowing the measurement of Y(1,M(0,u),u).
As with the case of the word “untestable”, I am wondering if “impossible” is here being understood to mean, not impossible in an absolute sense, but “impossible within some context of available means, assumed as part of the background”. For example, “impossible without specific domain knowledge”, or “impossible given only the causal diagram and some limited repertoire of feasible interventions and observations”. The tar filter scenario goes outside those bounds by using domain knowledge to devise a way of physically erasing the arrow from A to M.
I have the same question about page 18, where you say that equation (15):
Y(1,m) _||_ M(0)
is untestable (this is the example you expressed in words upthread), even though you have shown that it mathematically follows from any SEM of a certain form relating the variables, and could be violated if it has certain different forms. The true causal relationships, whatever they are, are observable physical processes. If we could observe them all, we would observe whether Y(1,m) _||_ M(0).
Again, by “untestable” do you here mean untestable within certain limits on what experiments can be done?
This paper is about an argument the authors are having with Judea Pearl about whether assumptions like the one we are talking about are sensible to make. Of particular relevance for us is section 5.1. If I understood the point the authors are making, whenever Judea justifies such an assumption, he tells a story that is effectively interventional (very similar to your story about a filter). That is, what really is happening is we are replacing the graph:
A → M → Y, A → Y
by another graph:
A → A1 → Y, A → A2 → M → Y
where A1 is the “non tar-producing part” of smoking, and A2 is the “tar-producing part” of smoking (the example in 5.1 was talking about nicotine instead). As long as we can tell such a story, the relevant counterfactual is implemented via interventions, and all is well. That is, Y(A=1,M(A=0)) in graph 1 is the same thing as Y(A1=1,A2=0) in graph 2.
The true causal relationships, whatever they are, are observable physical processes. If we could observe
them all, we would observe whether Y(1,m) || M(0).
The point of doing mediation analysis in the first place is because we are being empiricist—using data for scientific discovery. In particular, we are trying to learn a fairly crude fact about cause-effect relationships of A, M and Y. If, as you say, we were able to observe the entire relevant DAG, and all biochemical events involved in the A → M → Y chain, then we would already be done, and would not need to do our analysis in the first place.
“Testability” (the concept I am confused about) comes up in the process of scientific work, which is crudely about expanding a lit circle of the known via sensible procedures. So intuitively, “testability” has to involve the resources of the lit circle itself, not of things in the darkness. This is because there is a danger of circularity otherwise.
Thanks, I now have a clearer idea of what these expressions mean and why they matter. You write on page 15:
For the A/M/Y = smoking/tar/cancer situation I can imagine a simple way of creating this situation: have someone smoke cigarettes with filters that remove all of the tar but nothing else. There may be practical engineering problems in creating such a filter, and ethical considerations in having experimental subjects smoke, but it does not seem impossible in principle. This intervention sets A to 1 and M to M(0,u), allowing the measurement of Y(1,M(0,u),u).
As with the case of the word “untestable”, I am wondering if “impossible” is here being understood to mean, not impossible in an absolute sense, but “impossible within some context of available means, assumed as part of the background”. For example, “impossible without specific domain knowledge”, or “impossible given only the causal diagram and some limited repertoire of feasible interventions and observations”. The tar filter scenario goes outside those bounds by using domain knowledge to devise a way of physically erasing the arrow from A to M.
I have the same question about page 18, where you say that equation (15):
is untestable (this is the example you expressed in words upthread), even though you have shown that it mathematically follows from any SEM of a certain form relating the variables, and could be violated if it has certain different forms. The true causal relationships, whatever they are, are observable physical processes. If we could observe them all, we would observe whether Y(1,m) _||_ M(0).
Again, by “untestable” do you here mean untestable within certain limits on what experiments can be done?
Richard, thanks for your message, and for reading my paper.
At the risk of giving you more homework, I thought I would point you to the following paper, which you might find interesting:
http://www.hsph.harvard.edu/james-robins/files/2013/03/wp100.pdf
This paper is about an argument the authors are having with Judea Pearl about whether assumptions like the one we are talking about are sensible to make. Of particular relevance for us is section 5.1. If I understood the point the authors are making, whenever Judea justifies such an assumption, he tells a story that is effectively interventional (very similar to your story about a filter). That is, what really is happening is we are replacing the graph:
A → M → Y, A → Y
by another graph:
A → A1 → Y, A → A2 → M → Y
where A1 is the “non tar-producing part” of smoking, and A2 is the “tar-producing part” of smoking (the example in 5.1 was talking about nicotine instead). As long as we can tell such a story, the relevant counterfactual is implemented via interventions, and all is well. That is, Y(A=1,M(A=0)) in graph 1 is the same thing as Y(A1=1,A2=0) in graph 2.
The point of doing mediation analysis in the first place is because we are being empiricist—using data for scientific discovery. In particular, we are trying to learn a fairly crude fact about cause-effect relationships of A, M and Y. If, as you say, we were able to observe the entire relevant DAG, and all biochemical events involved in the A → M → Y chain, then we would already be done, and would not need to do our analysis in the first place.
“Testability” (the concept I am confused about) comes up in the process of scientific work, which is crudely about expanding a lit circle of the known via sensible procedures. So intuitively, “testability” has to involve the resources of the lit circle itself, not of things in the darkness. This is because there is a danger of circularity otherwise.