How can the potential outcomes $(Y_0,Y_1)$ be independent of the treatment variable $T$? - Causal Inference

Question

currently I am studying the model of potential outcomes and am still confused after reading the other answers on the other threads but also books treating this model.

How can the potential outcomes $(Y_{0i},Y_{1i})$ be independent of the treatment variable $T$?

Formally: $(Y_ {0i},Y_{1i}) \perp\!\!\!\perp T$ whereby $i$ is the $i$-th unit.

I know that we make this assumption to be able to eliminate the selection bias: $E[Y_{0i} | T_i=1] - E[Y_{0i} | T_i=0]$.
But how can the potential outcomes be independent of the treatment variable if we want to check for a causal relationship between $Y$ and $T$?

Looking at the switching equation: $Y_i=Y_{0i}+(Y_{1i}-Y_{0i})\cdot T_i$ with $T_i \in \{0,1\}$

$Y_i=Y_{0i}$ for $T_i=0$,
$Y_i=Y_{1i}$ for $T_i=1$

So I can not see why we can assume that the potential outcomes are independent of $T$ if the observed outcome we get is $Y_{0i}$ or $Y_{1i}$ depending on the treatment variable $T$.

The switching equations says that the potential outcome that will be assigned to our observed result $Y_i$ is dependent on the treatment variable $T$.
But assuming $(Y_ {0i},Y_{1i}) \perp\!\!\!\perp T$ while looking at the switching equation seems like a contradiction to me.

It looks like I am missing something here.
Would be happy if someone could explain me where my misconception lies.

Answer 1

Here is an example of an effective treatment assignment that satisfies PO orthogonality:

$$ \begin{array}{|l|c|c|c|c|} \hline \text{Person} & Y_0 & Y_1 & T & Y \\ \hline\hline \text{Alice} & 3 & 5 & 1 & 5 \\ \hline \text{Bob} & 3 & 5 & 0 & 3 \\ \hline \text{Cathy} & 3 & 5 & 1 & 5 \\ \hline \text{David} & 3 & 5 & 0 & 3 \\ \hline \end{array} $$

Above, the effect is 2 and $(Y_ {0},Y_{1}) \perp\!\!\!\perp T$ since the potential outcomes $(3,5)$ are the same when $T=0$ and $T=1$.

Below is an example where $(Y_ {0},Y_{1}) \not\perp\!\!\!\!\!\perp T$. Suppose the cost of treatment is $\$2.25$, and people get to choose if they are treated. Alice and Cathy opt in since $(5-2.25) = 2.75 > 0$. Bob and David do not since $(5-2.25)=2.75 < 3$. The potential outcomes are $(0,5)$ when $T=1$ and $(3,5)$ when $T=0$, so they are not orthogonal.

$$ \begin{array}{|l|c|c|c|c|} \hline \text{Person} & Y_0 & Y_1 & T & Y \\ \hline\hline \text{Alice} & 0 & 5 & 1 & 5 \\ \hline \text{Bob} & 3 & 5 & 0 & 3 \\ \hline \text{Cathy} & 0 & 5 & 1 & 5 \\ \hline \text{David} & 3 & 5 & 0 & 3 \\ \hline \end{array} $$

Here the actual effect is 3.5, and the measured effect is only 2 because you have self-selection out of treatment by folks with high $Y_0$s and in from people with low $Y_0$s. This biases the measured effect from the naive T bs C comparison down.

In both examples, $Y\not\perp\!\!\!\!\!\perp T$ since the treatment is effective.