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Fix a probability space $(\Omega,\mathcal{F},P)$ and, for $i\in\{0,...,n\}$, random variables $X_i$, $Y_i$, $Y^0_i$, $Y^1_i$ which are all measurable functions $\Omega\to \{0,1\}$. Call $X_i$ "treatment of unit $i$", $Y_i$ "the outcome of unit $i$" and $Y^1_i$, $Y^0_i$ "the potential outcomes of unit $i$".

Rubin, 2005 says that the SUTVA assumption is:

In any case, the depiction in Figure 1 requires assumptions for it to be adequate—in particular, SUTVA (stable unit treatment value assumption) (Rubin 1980), which comprises two subassumptions. First, there is no interference between units (Cox 1958); that is, neither $Y^0_i$ for $Y^1_i$ is affected by what action any other unit received. Second, it assumes there are no hidden versions of the treatments; no matter how unit $i$ received treatment $1$, the outcome that would be observed would be $Y^1_i$ and similarly for treatment 0.

I have tried interpreting this as saying:

  • Firstly, $\{Y_i^1,Y_i^0\}\perp \{X_j:j\neq i\}$
  • Secondly, for all $\omega\in \Omega$, $X_i(\omega)=x\implies Y^x_i(\omega) = Y_i(\omega)$

The latter being closely related (if not identical) to the assumption of consistency.

I am not confident that this is a correct interpretation of SUTVA.

In particular, it's not obvious how violating either of these would lead to Figure 1 being inadequate. It certainly does not preclude $Y^0_i$ or $Y^1_i$ or functions thereof from being well-defined random variables. The objection seems to be more along the lines that violating SUTVA will mean that we cannot necessarily regard $Y^1_i-Y^0_i$ as a "causal effect". My concern then is that if SUTVA is intended to ensure that this can be regarded as a causal effect then it is perhaps invoking nonprobabilistic assumptions that are not captured by the pair I have supplied above (for example, "unaffected by" is not synonymous with "independent from").

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These are great questions. Your interpretation of "no hidden versions of the treatment" is perfect, and this is indeed equal to consistency.

Your interpretation of no interference is not exactly right, and you already guessed why,

for example, "unaffected by" is not synonymous with "independent from").

Thus, it should read (I will use parenthesis for potential outcomes instead of your superscripts),

$$ Y_{i}(x_1,\dots, x_n) = Y_{i}(x_i) $$

Which is basically an exclusion restriction: the potential response of unit $i$ responds only to the treatment assigned to unit $i$ and not the treatment assigned to any other unit.

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    $\begingroup$ Thanks, I wasn't thinking potential-outcomey enough $\endgroup$ – David J Sep 28 '19 at 6:02

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