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In observational studies, one can use the Rubin Causal Model to retrieve unbiased estimates, which usually there is a statement that is usually required which states that:

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

where $(Y(1), Y(0))$ are the potential outcomes, $T$ the treatment variable, and $X$ being the covariates. The theory behind this assumption usually takes the potential outcomes to be fixed and known ahead of time. The Rubin book mentions that fact explicitly here. That is, they are non-random or non-stochastic. Given this, how would one interpret the independence statement above? If $(Y(1), Y(0))$ are non-random and, say, scalar, how does the stochastic relation above even make sense?

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  • $\begingroup$ Check the paper with title "A new approach to causal inference in mortality studies with a sustained exposure period --- ...." Try to understand formula (2.4) on page 1402. Maybe you can get the answer by yourself. But I have no answer to your question. $\endgroup$
    – user158565
    Oct 22, 2018 at 17:07
  • $\begingroup$ Where do you take the assumption from that potential outcomes are "known ahead of time"? $\endgroup$ Oct 23, 2018 at 7:17
  • $\begingroup$ @JulianSchuessler This is an assumption from pretty much every Rubin paper, such as here in his books.google.com/… $\endgroup$
    – user321627
    Oct 23, 2018 at 16:02

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Potential outcomes are features of units $U$, so it helps in our case to index them:

$$Y_u(1), Y_u(0)$$

The usual statement is that potential outcomes are fixed for a specific unit $u$, but may of course vary across $U$. The conditional independence assumptions then becomes

$$Y_u(1), Y_u(0) \perp T_u | X_u$$

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  • $\begingroup$ Even if the statement holds for a unit $u$, the values are still fixed. I fail to see how a stochastic statement like the conditional independence statement can be made. Is it then an abuse of notation in the usual theory and how its stated? $\endgroup$
    – user321627
    Oct 23, 2018 at 16:04
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Because this is a property that holds for the population, not for the individual. The potential outcome of any individual is fixed. The conditional independence claim holds in the population.

Say we have 6 individuals with potential outcomes, $Y(1) = \{1, 2, 3,4,5,6\}$ and $Y(0) = \{1,2,3,4,5,6\}$. Note the potential outcomes are fixed. Now say you randomly sample 3 individuals for the treatment group and 3 individuals for the control group. Let the treatment indicator be $T$. Clearly, $\{Y(1),Y(0)\} \perp T$, since you treated them at random we have that $P(T = 1|Y(1), Y(0)) = P(T = 1) = 1/2$. This independence is a property of the population.

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  • $\begingroup$ So you are implying that the randomness comes from the assignment mechanism? Is there a relation here to super/finite sample population frameworks? $\endgroup$
    – user321627
    Oct 23, 2018 at 18:05
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    $\begingroup$ @user321627 I answered this here stats.stackexchange.com/a/310902/39630 $\endgroup$ Oct 23, 2018 at 18:23
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You cited the Imbens & Rubin book in a comment...

From the introduction of chapter 7:

"The most important difference between the methods discussed in Chapters 5 and 6 and the ones discussed here is that they rely on different sampling perspectives. Both the Fisher approach discussed in Chapter 5 and the Neyman methods discussed in Chapter 6 view the potential outcomes as fixed and the treatment assignments as the sole source of randomness. In the regression analysis discussed in this chapter, the starting point is an infinite super-population of units. Properties of the estimators are assessed by resampling from that population, sometimes conditional on the predictor variables including the treatment indicator. From that perspective, the potential outcomes in the sample are random..."

From the introduction of chapter 8:

"As discussed in Chapters 5 and 6, both Fisher’s and Neyman’s approaches for assessing treatment effects in completely randomized experiments viewed the potential outcomes as fixed quantities, some observed and some missing. The randomness in the observed outcomes was generated primarily through the assignment mechanism, and sometimes also through random sampling from a population. In this chapter, as in the preceding chapter on regression methods, we consider a different approach to inference, where the potential outcomes themselves are also viewed as random variables, even in the finite sample."

So it seems this is just a misunderstanding. Or where else did you read Rubin making this assumption of fixed potential outcomes (other than in explaining Fisher's and Neyman's work)?

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