Difference between the counterfactual mean and average treatment effect

Question

I am new on the causality topic, I don't know the difference between the average treatment effect and counterfactual mean. Can anyone tell me?

Answer 1

Average treatment effect (ATE)

The average treatment effect (ATE) is a measure used to compare treatments (or interventions) in randomized experiments or observational studies. The ATE is the population mean of (hypothetical) outcomes should everyone receive treatment less the population mean of (hypothetical) outcomes should no-one receive treatment: $$ \text{ATE}=\mathbb{E}(\delta_i)=\mathbb{E}\!\left(Y_i^1-Y_i^0\right)=\mathbb{E}\!\left(Y_i^1\right)-\mathbb{E}\!\left(Y_i^0\right) $$ where $\delta_i$ is the treatment effect for individual $i$, $Y_i^1$ is the outcome for individual $i$ under treatment and $Y_i^0$ is the outcome for individual $i$ under no treatment. (Based on Cunningham (2018) p. 87 and Wikipedia.)

Counterfactual mean

According to Hernán & Robins (2015), counterfactural means are simply the two means defining the ATE, $\mathbb{E}\!\left(Y_i^1\right)$ and $\mathbb{E}\!\left(Y_i^0\right)$; i.e., the population mean of (hypothetical) outcomes should everyone receive treatment and the population mean of (hypothetical) outcomes should no-one receive treatment, respectively.

References

• Cunningham, S. (2018). Causal Inference: The Mixtape. Unpublished manuscript.
• Hernán, M. A., & Robins, J. M. (2015). Longitudinal causal inference. International Encyclopedia of the Social & Behavioral Sciences (Second Edition), p. 340-344.
• Wikipedia contributors. (2019, December 11). Average treatment effect. In Wikipedia, The Free Encyclopedia. Retrieved 10:09, May 23, 2020, from https://en.wikipedia.org/w/index.php?title=Average_treatment_effect&oldid=930240228