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I am reading the book Counterfactuals and Causal Inference: Methods and Principles for Social Research. I have a question related to Section 2.5, and 2.6.

Suppose $d$ is an $N \times 1$ vector of treatment indicator variables, the treatment effect for each individual $i$ is $$\delta_i(d) = y_{i}^{1}(d) - y_{i}^{0}(d)$$

If SUTVA is valid, $\delta_i(d) = y_{i}^{1} - y_{i}^{0}$. I think it means the potential outcome $Y$ is independent of the treatment assignment pattern.

For Ignorability, the book defined it as "treatment status is independent of the potential outcomes." $(Y^0,Y^1)$ independent of $D$.

What is the difference between the two?

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  • $\begingroup$ One way to describe the difference is that ignorability is an assumption about the treatment assignment process, and SUTVA is an assumption about the number of potential outcomes each unit has. $\endgroup$
    – Noah
    Nov 17 '21 at 15:14
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SUTVA says that $\delta_i(d)$ doesn't depend on treatment assignment for individuals other than $i$. Ignorability says it doesn't depend on treatment assignment for individual $i$.

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  • $\begingroup$ How can I see that $d$ is the treatment assignment for individuals other than $i$. By looking at $d$, it seems to me that it means the treatment assignment for all individuals. $\endgroup$
    – JOHN
    Nov 17 '21 at 14:02
  • $\begingroup$ At the start of section 2.5, in block quote: "SUTVA is simply the a priori assumption that the value of Y for unit u when exposed to treatment t will be the same no matter what mechanism is used to assign treatment t to unit u and no matter what treatments the other units receive." $\endgroup$ Nov 17 '21 at 23:48
  • $\begingroup$ "the value of Y for unit u when exposed to treatment t will be the same no matter what mechanism is used to assign treatment t to unit u" Is it the same as ignorability? $\endgroup$
    – JOHN
    Nov 18 '21 at 0:50
  • $\begingroup$ No, it's different. SUTVA says that $Y_i(1)$ and $Y_i(0)$ aren't causally affected by the assignment mechanism and so can be regarded as well-defined attributes of unit $i$. That's much weaker than ignorability, which rules out the random variables $D$ and $(Y(1), Y(0))$ being correlated or otherwise statistically dependent in some way. $\endgroup$ Nov 18 '21 at 2:09

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