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Suppose that a set of covariates, $X_i$ follows a distribution that is conditional on another variable, $A_i$, for $i \in \{1, \ldots N\}$ individuals. For example, $X_i$ can be income, and $A_i$ can be age, defined as young, middle-age, and elderly. Then, suppose that we have:

$$ X_i \mid A_i \overset{iid}{\sim} F $$

Then, we say that $X_i$ is only i.i.d. within subsets defined by the age variable. Now let $Y(1),Y(0)$ denote the potential outcomes and $T$ the treatment indicator. Suppose that the joint distribution of the potential outcomes, treatment, and covariates are only i.i.d on subsets defined by $A_i$, such that

$$ (Y_i(1),Y_i(0),T_i, X_i)\mid A_i \overset{iid}{\sim} F $$

This scenario appears to hint at a heterogeneous treatment effect, since the distributions of the variables are potentially different given which age is it conditioned on. In example,

$$ \tau_i = E[Y_i(1)-Y_i(0)] $$

where $\tau_i$ may be different than a single global $\tau$.

In such an example, I am wondering how the unconfoundedness assumption needs to be modified given an observational study. For example, if we assume that the following holds,

$$ (Y_i(1),Y_i(0)) \perp T_i \mid X_i $$

will it be enough to identify $\tau_i$? In other words, if we have the joint distribution specification above, how will identification and estimation of the average treatment effect be impacted?

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  • $\begingroup$ What is being averaged in the expectation $\tau_i$, why does it differ across subjects? And how do you define the global $\tau$? $\endgroup$ – CloseToC Aug 23 '19 at 20:45
  • $\begingroup$ It differs because each unit may have a different distribution for $Y_i(1)-Y_i(0)$ based on the distribution specification. The global $\tau$ can be obtained by $\tau = E[\tau_i(A_i)]$, which is through averaging over the $\tau_i$. $\endgroup$ – user321627 Aug 24 '19 at 20:09
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If, as your notation suggests, that $\tau_i$ is an individual causal effect, then it still remains un-identifiable. It is clearer why if we frame the potential outcomes and our observations as a type of missing data.

Consider individual $i=1$, we observed $T=1$ and $Y=1$. Via causal consistency ($Y=Y(t)$), we can know that $Y(1)=1$. However, $Y(0)$ is still missing. We can't know $i=1$'s $Y(0)$ unless we were to have an exact copy of them aside from $T$. This means we would need all the causes of $Y$ in our model. Overcoming this is near impossible. This is why both randomized trials and observational studies only require mean exchangeability (an alternative phrase to unconfoundedness) $Y(t) \bot T$ or conditional mean exchangeability $Y(t) \bot T | X$. Notice that these are for the full population, not an individual.

Regarding identifiability, as long as exchangeability (unconfoundedness holds), then nothing needs to change. We are still able to identify the average causal effect (all-treated versus none-treated). However, the problem of generalizability (also sometimes referred to as transportability) becomes apparent. When there are heterogeneous effects in the observed sample, the results will only apply to populations with a similar distribution of the effect measure modifiers. Using your example, the $ATE$ would only be generalizabile to populations with a similar distribution of $A$.

So to summarize, nothing needs to change in an observational study when we are estimating the $ATE$ as long as the observed study sample is a random sample of the population we want to generalize to. If it is not, then special methods need to be used. This applies to both randomized and observational studies. I highly recommend Lesko 2017 et al. for some further reading on this problem.

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