If there is really both reverse causation and regular causation between Xr and Y, you have a cycle, and you have to explain what the semantics of that cycle are (not a deal breaker, but not so simple to do. For example if you think the cycle really represents mutual causation over time, what you really should do is unroll your causal diagram so it’s a DAG over time, and redo the problem there).
You might be interested in this paper (https://arxiv.org/pdf/1611.09414.pdf) that splits the outcome rather than the treatment (although I don’t really endorse that paper).
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The real question is, why should Xc be unconfounded with Y? In an RCT you get lack of confounding by study design (but then we don’t need to split the treatment at all). But this is not really realistic in general—can you think of some practical examples where you would get lucky in this way?
If there is really both reverse causation and regular causation between Xr and Y, you have a cycle, and you have to explain what the semantics of that cycle are (not a deal breaker, but not so simple to do. For example if you think the cycle really represents mutual causation over time, what you really should do is unroll your causal diagram so it’s a DAG over time, and redo the problem there).
I agree, but I think this is much more dependent on the actual problem that one is trying to solve. There’s tons of assumptions and technical details that different approaches use, but I’m trying to sketch out some overview that abstracts over these and gets at the heart of the matter.
(There might also be cases where there is believed to be a unidirectional causal relationship, but the direction isn’t know.)
The real question is, why should Xc be unconfounded with Y? In an RCT you get lack of confounding by study design (but then we don’t need to split the treatment at all). But this is not really realistic in general—can you think of some practical examples where you would get lucky in this way?
Indeed that is the big difficulty. Considering how often people use these methods in social science, it seems like there is some general belief that one can have Xc be unconfounded with Y, but this is rarely proven and seems often barely even justified. It seems to me that the general approach is to appeal to parsimony and assume that if you can’t think of any major confounders, then they probably don’t exist.
This obviously doesn’t work well. I think people find it hard to get an intuition for how poorly it works, and I personally found that it made much more sense to me when I framed it in terms of the “Know your Xc!” point; the goal shouldn’t be to think of possible confounders, but instead to think of possible nonconfounded variance. I also have an additional blog post in the works arguing that parsimony is empirically testable and usually wrong, but it will be some time before I post this.
If there is really both reverse causation and regular causation between Xr and Y, you have a cycle, and you have to explain what the semantics of that cycle are (not a deal breaker, but not so simple to do. For example if you think the cycle really represents mutual causation over time, what you really should do is unroll your causal diagram so it’s a DAG over time, and redo the problem there).
You might be interested in this paper (https://arxiv.org/pdf/1611.09414.pdf) that splits the outcome rather than the treatment (although I don’t really endorse that paper).
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The real question is, why should Xc be unconfounded with Y? In an RCT you get lack of confounding by study design (but then we don’t need to split the treatment at all). But this is not really realistic in general—can you think of some practical examples where you would get lucky in this way?
I agree, but I think this is much more dependent on the actual problem that one is trying to solve. There’s tons of assumptions and technical details that different approaches use, but I’m trying to sketch out some overview that abstracts over these and gets at the heart of the matter.
(There might also be cases where there is believed to be a unidirectional causal relationship, but the direction isn’t know.)
Indeed that is the big difficulty. Considering how often people use these methods in social science, it seems like there is some general belief that one can have Xc be unconfounded with Y, but this is rarely proven and seems often barely even justified. It seems to me that the general approach is to appeal to parsimony and assume that if you can’t think of any major confounders, then they probably don’t exist.
This obviously doesn’t work well. I think people find it hard to get an intuition for how poorly it works, and I personally found that it made much more sense to me when I framed it in terms of the “Know your Xc!” point; the goal shouldn’t be to think of possible confounders, but instead to think of possible nonconfounded variance. I also have an additional blog post in the works arguing that parsimony is empirically testable and usually wrong, but it will be some time before I post this.