4.18 ATE: Decomposition

• ATE can be decomposed as a function of 5 quantities (e.g. Keele 2015b, 4)²⁰
• \(\pi\) proportion of sample assigned to treatment (2/6 = 0.33..)
• E[Y_i1|D_i = 1]: Average outcome under treatment (Y_i1) for units | in treatment condition (D_i = 1)²¹
• E[Y_i1 - Y_i0] = \(\pi\) (E[Y_i1|D_i = 1] - E[Y_i0|D_i = 1]) + (1 - \(\pi\))(E[Y_i1|D_i = 0] - E[Y_i0|D_i = 0])
• Q: Which parts do we have and which parts don’t we have?

Unit	Di (Aspirin: Yes/No)	Yi1 (Pain | YesAspirin)	Yi0 (Pain | NoAspirin)
Simone	1	0	?
Julia	1	1	?
Paul	0	?	1
Trump	0	?	0
Fabrizio	0	?	0
Diego	0	?	0

References

Keele, Luke. 2015b. “The Statistics of Causal Inference: A View from Political Methodology.” Polit. Anal. 23 (3): 313–35.

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20. Start again by explaining columns of table.↩

21. Red: Average outcome under treatment for units (Y_i1) | in treatment condition (D_i = 1); Blue: Average outcome under control for units (Y_i0) | in treatment condition (D_i = 1)↩