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I'm new to causality and have a basic understanding of Average Treatment Effect (ATE). Recently, I came across Judea Pearl's definition of Total Effect/Total Causal Effect. Could anyone tell me what is its relationship to ATE? Do they measure same thing?

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The ATE and total effect do indeed refer to the same quantity, $E[Y_1]-E[Y_0]$, where $Y_1$ and $Y_0$ are the potential outcomes under treatment and control, respectively. The difference is in the context: the total effect is used to distinguish among other causal contrasts in the context of causal mediation analysis, where an intermediate variable $M$ lies along the causal path between the treatment and outcome.

$Y_1$ can be written as $Y_{1, M_1}$, where $M_a$ is the potential mediator value under treatment $a$, emphasizing that the potential outcome under treatment is equivalent to the potential outcome under treatment and under the value the mediator takes under treatment (and analogously for $Y_0$). This identity (i.e., $Y_a = Y_{a, M_a}$ for all $a$) is the key link between the ATE and total effect, as the total effect is often written as $E[Y_{1, M_1}] - E[Y_{0, M_0}]$, which is equivalent to $E[Y_1]-E[Y_0]$.

One may ask why we need two different terms for the same quantity. The phrase "total effect" emphasizes that is the sum of other effects. The total effect can be decomposed as follows: \begin{align} E[Y_{1, M_1}] - E[Y_{0, M_0}] &= E[Y_{1, M_1}] - E[Y_{1, M_0}] + E[Y_{1, M_0}] - E[Y_{0, M_0}] \\ &= (E[Y_{1, M_1}] - E[Y_{1, M_0}]) + (E[Y_{1, M_0}] - E[Y_{0, M_0}]) \\ &= \text{NIE} + \text{NDE} \end{align} where NIE is the natural indirect effect (the effect of the treatment on the outcome solely through its effect on the mediator) and the NDE is the natural direct effect (the effect of the treatment on the outcome that does not pass through the mediator). Again, the use of "total effect" emphasizes that the quantity in question is the sum (i.e., total) of the natural indirect and natural direct effects.

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  • $\begingroup$ Thank you for the very clear explanation! $\endgroup$ – matais Jun 3 at 8:17

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