Issue on page /01-Introduction-To-Causality.html: "association IS CAUSATION!" is precipitate.

[chep0k]

Hello!
Before all, thanks to the authors for the book!

Below is some math proving that $\mathbb{E}[Y_0 \mid T=0] = \mathbb{E}[Y_0 \mid T=1]$ is not yet enough to equate association $\mathbb{E}[Y \mid T=1] - \mathbb{E}[Y \mid T=0]$ and causation $\mathbb{E}[Y_1 - Y_0]$ as they are defined above in the chapter.

When it does not hold

Drawing on the numerical example from the document, here's a counterexample.

i	$\text{Y}_0$	$\text{Y}_1$	T	Y	TE
1	500	700	0	500	200
2	600	800	0	600	200
3	500	600	1	600	100
4	600	700	1	700	100

Bias = $\mathbb{E}[Y_0 \mid T=0] - \mathbb{E}[Y_0 \mid T=1] = 550 - 550 = 0$.
At the same time, $\mathbb{E}[Y_1 \mid T=1] = \mathbb{E}[Y_1 \mid T=0] \ne 0$.

Association = $\mathbb{E}[Y \mid T=1] - \mathbb{E}[Y \mid T=0] = 650 - 550 = 100$.
ATT = $\mathbb{E}[Y_1 - Y_0 \mid T=1] = \frac{100 + 100}{2} = 100$.
Causation = ATE = $$\mathbb{E}[Y_1 - Y_0] = \frac{200 + 200 + 100 + 100}{4} = 150$$.

Association = ATT $\ne$ ATE = Causation.

Suggested improvement

Instead of
"If $\mathbb{E}[Y_0 \mid T=0] = \mathbb{E}[Y_0 \mid T=1]$, then, association IS CAUSATION!"
Write e.g.
"If $\mathbb{E}[Y_0 \mid T=0] = \mathbb{E}[Y_0 \mid T=1]$, then, association is the causal effect on the treated, i.e. ATT. If, additionally, $\mathbb{E}[Y_1 \mid T=1] = \mathbb{E}[Y_1 \mid T=0]$, then, association IS CAUSATION!"