Some notes for Casual Inference

Ref: https://www.stat.cmu.edu/~larry/=stat700/Lecture22.pdf

Counterfactual

$E[Y_1]=E[Y_1|X=1] Pr(X=1) + E[Y_1|X=0] Pr(X=0)$ and $Y=X Y_1 + (1-X) Y_0$ .

$Y_1$ is the potential outcome if an individual were treated. $Y_0$ is the potential outcome if an individual were not treated.

$Y$ is the observed outcome:

$Y = X Y_1 + (1 - X) Y_0$

which means:

If $X=1$ , then $Y = Y_1$ (we observe the treated outcome)
If $X=0$ , then $Y = Y_0$ (we observe the untreated outcome).

From $Y=X Y_1 + (1-X) Y_0$ , we have $E[Y_1|X=1]=E[Y|X=1]$ . Therefore

Note that $E[Y_1 | X=0]$ is an unobserved counterfactual. For example, if $X \in {0,1}$ represents whether an individual receives a treatment and $Y$ is the health outcome, then $E[Y_1 | X=0]$ represents the \textbf{expected outcome if the treatment were applied to individuals who did not receive it}. In contrast, $E[Y_1]$ represents the \textbf{expected outcome if the treatment were applied universally to everyone}.

If the treatment is effective, we expect $E[Y_1 | X=0] < E[Y_1 | X=1]$ , as individuals who voluntarily choose the treatment may have characteristics that lead to better outcomes. As a result, we also expect $E[Y | X=1] > E[Y_1]$ , since the observed group receiving treatment ( $X=1$ ) may include individuals who are more health-conscious and proactive about their well-being. On the other hand, $E[Y_1]$ accounts for individuals who might not be health-conscious but were hypothetically \textbf{forced} to take the treatment, potentially lowering the overall expected outcome.

Impossible to estimate treatment effect in general

Given treatment effect $\theta = E[Y_1]-E[Y_0]$ , it is generally impossible to estimate $\theta$ because we can only observe $Y_1$ or $Y_0$ at one time but not both.

Randomization to estimate treatment effect

It would be possible to estimate $\theta$ if $X \bot (Y_1,Y_0)$ . When the indepenence holds, we have $E[Y|X=1]\overset{(a)}{=}E[Y_1|X=1]\overset{(b)}{=}E[Y_1]$ , where $(a)$ always holds and $(b)$ is due to $Y_1 \bot X$ . Similarly, we have $E[Y|X=0]=E[Y_0|X=0]=E[Y_0]$ . Therefore, we have $E[Y_1]-E[Y_0] = E[Y|X=1]-E[Y|X=0]$ , where the latter can be readily observed.

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Counterfactual

Impossible to estimate treatment effect in general

Randomization to estimate treatment effect

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