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Suppose we have that $Z\in \{0,1\}$ is the treatment, $(Y(1),Y(0))$ the potential outcomes, and $X$ the covariates. Suppose we have know that unconfoundedness holds on $X$, such that:

$$ (Y(1),Y(0)) \perp Z \mid X $$

Now suppose that there is a new set of covariates $X'$ we may want to conduct matching with. I am wondering if the specific condition,

$$ X' \perp Z \mid (Y(1),Y(0)), X $$

always holds. That is, will it always be the case that conditioning on the potential outcomes directly implies knowledge of the treatment assignment $Z$, such that it will be conditionally independent of ANY possible variable? Thank you in advance for any insights.

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  • $\begingroup$ What is $(Y(1),Y(0))$ as a probabilistic event? Is it just the probability frequency of event or is it the observed event for an individual. If you told me someone had lung cancer, I couldn't say whether they smoked or not regardless of how old they are. If you told me someone had an 80% chance of lung cancer, and gave me the risk model and their age, I could backwards-derive their smoking status. If I didn't actually know the model, I'd be stuck. $\endgroup$
    – AdamO
    Commented May 14, 2019 at 15:03

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Here's one way that the second condition could fail to hold.

First, consider that $X'$ is an instrument, that is, it is independent from the potential outcomes but causes selection into treatment, and, for simplicity, it is independent from $X$. It is not necessary to condition on $X'$ to arrive at unconfoundedness, so whatever association $X'$ has with $Z$ will remain after conditioning on $X$. Because $X'$ is unrelated to the potential outcomes given $X$, conditioning on the potential outcomes will not change the association between $X'$ and $Z$, so they will not be conditionally independent.

Consider the following example. $X'$ is the order of one's last name in the alphabet. $Z$ is whether the patient gets the medicine or not; it is entirely dependent on whether one's last name is near the beginning of the alphabet (i.e., on $X'$). $Y$ is whether the patient survives or not. In this experiment, there is unconfoundedness already; $X$ is empty. Let's condition on $(Y(0), Y(1))$ by considering one stratum: the doomed (people who will die no matter what; $Y(0)=Y(1)=0$. Among the doomed, there is still an association between a patient's last name and whether the patient received treatment. This is true in all strata of the potential outcomes. Regardless of what variables you control for, there will always be an association between $X'$ and $Z$.

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