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Consider some experiment with $n$ units along outcomes $Y=(Y_1,\dots, Y_n)$, covariates $X=(X_1,\dots, X_n)$ and treatment vector $W=(W_1,\dots, W_n)$ where $Y_i=(Y_i^1,Y_i^0)$ for treatment and control. $W_i=1$ for treatment and $W_i=0$ for control.

"Assignment mechanism is called unconfounded assignment if $P(W|X,Y)=P(W|X)$." This is definition of unfounded assignment in Rubin's Causal Inference.

Why do we call confounded assignment here? Confound with what? This is just a statement on MAR assumption to me if I think $1-W$ contains missing information as $X$ is fully observed here.

user45765
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The word "unconfounded" as used in this context is a bit confusing because when we think of unconfoundedness we often think of unconditional unconfoundedness, e.g., in a randomized experiment where treatment does not depend on any covariates and compliance is perfect. Rubin's use of the word "unconfounded" can be more simply understood as "conditionally unconfounded", i.e., that the treatment assignment is unconfounded conditional on the set of measured covariates.

What does conditional unconfoundedness entail? At its most basic, it is that $P(W|X, Y^1, Y^0) = P(W|X)$, which is a bit different from what you wrote. $Y$ is not involved in a statement about unconfoundedness because it is not observed until after treatment has been received. In contrast, the potential outcomes $Y^1$ and $Y^0$ exist (conceptually) prior to treatment receipt.

When would conditional unconfoundedness be violated? Any time the potential outcomes are associated with the treatment after conditioning on the measured covariates. This occurs whenever there is an open backdoor path between the treatment and outcome. If conditioning on the measured covariates closes all backdoor paths, then treatment assignment is (conditionally) unconfounded.

Note that I talk about treatment assignment being unconfounded; unconfoundedness is a description of the treatment assignment mechanism; it says that whatever mechanism is used to assign treatment is independent of the potential outcomes (and therefore, the outcomes) given the covariates.

If you are used to thinking about missingness, (conditional) unconfoundedness corresponds to missing at random (MAR) for the potential outcomes $Y^1$ and $Y^0$. $Y^1$ is missing for all units with $W \ne 1$, and $Y^0$ is missing for all units with $W \ne 0$. Since our estimand is (often) $E[Y^1]-E[Y^0]$, we need to make an assumption about the missingness of the potential outcomes. When unconfoundedness is true, covariate adjustment methods (e.g., regression, propensity scores, etc.) can be used to recover the treatment effect. Other methods may be available under other assumptions when unconfoundedness is not met.

Noah
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  • I just checked. I think conditional unconfoundedness implies MAR but I do not think reverse is true as assignment mechanism is unrelated to outcome at all. I am not sure what is formal definition of unconfoundedness. Is it just $P(W|Y)=P(W)$ with conditioning on $X$ dropped? If that is the case, I would not think this will imply $P(W|X,Y)=P(W|X)$ then. – user45765 Jan 24 '22 at 20:24
  • You keep writing $Y$ in your statements, but $Y$ is unrelated to unconfoundedness. The formal definition of (conditional) unconfoundedness is $\{Y^1, Y^0\} \perp W|X$. If $X$ is the empty set, as in a randomized experiment, then unconditional unconfoundedness means $\{Y^1, Y^0\} \perp W$. When talking about missing data, you still have potential outcomes, which are the outcomes under non-missingness (i.e., missingness corresponds to treatment). See my answers [here](https://stats.stackexchange.com/a/474669/116195) and [here](https://stats.stackexchange.com/a/404939/116195). – Noah Jan 24 '22 at 20:37