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I'm reading up on Rubin causal inference and haven't been able to find a clear distinction between a potential outcome and the outcome conditioned on treatment, although this distinction is supposed to be crucial.

Writing subject $i$'s response, $Y_i$, in terms of potential outcomes and treatment $X_i$ I see this definition in many sources: $$Y_i = \begin{cases} Y_i(1), X_i = 1 \\ Y_i(0), X_i = 0 \end{cases} = Y_i(1)X_i + Y_i(0)(1 - X_i)$$

By definition, potential outcome $Y_i(1)$ is the outcome when subject $i$ receives treatment. So how exactly is this different from $Y_i | X_i = 1$, which would be subject $i$'s outcome given they received treatment?

BBrooklyn
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    You've stated the so-called consistency assumption. Assuming consistency, it's true that $[Y \mid X=1] = [Y(1) \mid X=1]$, but it's not necessarily true $[Y(1) \mid X=1] = [Y(1)]$. It could be that the treatment assignment is informed by an individual's potential outcomes. – user551504 Mar 20 '22 at 01:11
  • Why would the second equality not be true? Isn't it the case that $Y(1)$ only exists when $X = 1$? – BBrooklyn Mar 20 '22 at 01:16
  • The third sentence in my previous message explains why. – user551504 Mar 20 '22 at 01:17
  • I'm under the impression that $[Y(1) | X = 1] = [Y(1)]$ by definition. If you have a different understanding or can point me to a reliable source I'd appreciate it. – BBrooklyn Mar 20 '22 at 01:24
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    That's asserting that $Y(1)$ is independent of $X$. That's sometimes true, such as in a perfectly conducted RCT. However imagine that $X$ is a type of medication; a doctor will do their best to choose to give you the medication type that leads you to have the best outcome. Therefore in this case $Y(1)$ and $X$ are not independent. – user551504 Mar 20 '22 at 01:54

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See my post about this here. The gist is that potential outcomes are pretreatment variables that exist prior to treatment assignment. Receiving the treatment reveals one of the potential outcomes and leaves the other hidden. It is true that after assigning treatment and measuring the outcome, the outcome conditional on the treatment is equal to the potential outcome corresponding to the received treatment level, but you can talk about potential outcomes without even considering the treatment as a realized random variable, such as in defining the causal effect $E[Y(1)] - E[Y(0)]$, for which the treatment does not need to be realized or measured.

It is true that $[Y|X=x] = [Y(x)|X=x]$ for treatment $X$ (under consistency). This fact is used in the critical proofs for identifying causal effects under strong ignorability, i.e., the proof that $E_W[E[Y|X=1, W]] = E[Y(1)]$ for a sufficient adjustment set $W$ relies on the first step $$ E_W[E[Y|X=1, W]] = E_W[E[Y(1)|X=1, W]] $$ which is a direct result of consistency. But we can also talk about potential outcomes without considering them as the outcome conditional on treatment, like in the statement of strong ignorability, $$ Y(x) \perp X|W $$ which does not involve the outcomes at all and is a statement about the assignment mechanism. To say instead $Y \perp X|W,X$, that is, replacing $Y(x)$ with $Y|X$, makes the statement trivial when in fact it is a very strong assumption.

Noah
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  • You write "It is true that $[Y|X=x] = Y(x)$ for treatment $X$ (under consistency)," however, it isn't. You must also assume $Y(x) \perp !!! \perp X$. Also, strong ignorability evaluates $Y(x)$ rather than $Y(X)$. – user551504 Mar 20 '22 at 14:09
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    You're right about it being $Y(x)$ rather than $Y(X)$, I edited that. I also clarified that consistency is $[Y|X=x] = [Y(x)|X=x]$. Thanks for the comment. – Noah Mar 20 '22 at 17:09