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I'm reading this slides.

At slide 10 there is written that in "Single Experiment Design" we assume "Randomization of treatment", that is:

$ \{ Y_i(t,m),M_i(t') \} \perp T_i \lvert X $

I don't understand how the outcome and the treatment can be conditionally independent given X, as long as the treatment has an effect on the outcome.

Here there is written that:

The rule here is that after you've drawn out the graph, two events are conditionally independent if you can't traverse from one node to the other without going through a "blocked" node, where a "blocked" node is an event that has has already happened.

But it looks like, if the treatment has an effect on the outcome, I can go from treatment to outcome without going through the X.

What am I not understanding?

I think he wants to say that the treatment is random, meaning there are no unobserved covariates that determine both treatment and outcome.

But this seems not to imply that treatment and outcome are conditionally independent of each other given X.

Or maybe he is abusing notation?

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    $\begingroup$ Are you mixing up potential outcomes and observed outcomes? $\endgroup$
    – dimitriy
    Commented Jan 29, 2023 at 21:53
  • $\begingroup$ Related question here. $\endgroup$
    – dimitriy
    Commented Jan 29, 2023 at 21:59
  • $\begingroup$ @dimitry I think the point I'm missing is that T is treatment assignment and not actually treated? But still, treatment assignment is correlated with actually treated, and actually treated is correlated with the potential outcome (if the treatment effect is non-zero). So I still not understand why treatment assignment can be conditionally independent on potential outcome. I guess I need clear definitions and maybe an example? $\endgroup$
    – robertspierre
    Commented Jan 30, 2023 at 10:19
  • $\begingroup$ I am not sure you need to worry about compliance here. Suppose the treatment effect is positive, and I ranked people by $Y_1-Y_0$ in descending order and gave treatment to the top of the list. Giving treatment to the people with the biggest effect would violate independence with potential outcomes. The actual/observed outcome is $Y=Y_1 \cdot T + (1-T) \cdot Y_0$, so it's always going to be correlated with treatment, assuming treatment does something. $\endgroup$
    – dimitriy
    Commented Jan 30, 2023 at 17:43

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This is what I think I was missing:

If the treatment assignment is randomized, as in the Brader, Valentino, and Suhay (2008) study, then the treatment is jointly independent of the potential outcomes because the probability of receiving the treatment is identical regardless of the values of the potential outcomes. We can write this as {Yi(1), Yi(0)}⊥⊥ Ti with the standard symbol of statistical independence.

So it's not treatment that is not correlated with outcome, it's the probability of receiving treatment that does not depend on the value of potential outcome.

Reference:

Imai, K., Keele, L., Tingley, D., Yamamoto, T., 2011. Unpacking the Black Box of Causality: Learning about Causal Mechanisms from Experimental and Observational Studies. Am Polit Sci Rev 105, 765–789. https://doi.org/10.1017/S0003055411000414

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