Causal Inference Notation
ICS212 Fall 2022
simplified notation.pdf - 1
ICS212 Fall 2022
simplified notation.pdf - 2
What is: E [ Y | D = 1] ?
Is it observable?
The expected value of the outcome Y given that treatment 1 was received. If some of the units received treatment 0 then we cannot observe this.
ICS212 Fall 2022
simplified notation.pdf - 3
What is: E [ Y1 ] ?
Is it observable?
The expected value of the outcome Y given that treatment 1 was received. If some of the units received treatment 0 then we cannot observe this.
ICS212 Fall 2022
simplified notation.pdf - 4
What is: E [ Y | D = 0]?
Is it observable?
The expected value of the outcome Y given that treatment 0 was received. If some of the units received treatment 1 then we cannot observe this.
ICS212 Fall 2022
simplified notation.pdf - 5
What is: E [ Y0 ] ?
Is it observable?
The expected value of the outcome Y given that treatment 0 was received. If some of the units received treatment 1 then we cannot observe this.
ICS212 Fall 2022
simplified notation.pdf - 6
RECAP:
E [ Y0 ] E [ Y1 ] E [ Y | D = 0] E [ Y | D = 1]
ICS212 Fall 2022
simplified notation.pdf - 7
What is: E [ Y | D = 1] - E [ Y | D = 0] ?
Is it observable?
The expected value of the Y given that treatment 1 was received minus the expected value of the outcome given that treatment 0 was received. This is the true treatment effect. It is not observable.
ICS212 Fall 2022
simplified notation.pdf - 8
What is: E [ Y0 | D = 0]?
Is it observable?
The expected value of the outcome Y after treatment 0 given that treatment 0 was received. These are the control group cases and they can be observed.
ICS212 Fall 2022
simplified notation.pdf - 9
What is: E [ Y1 | D = 1]?
Is it observable?
The expected value of the outcome Y after treatment 1 given that treatment 1 was received. These are the treatment group cases and they can be observed.
ICS212 Fall 2022
simplified notation.pdf - 10
What is: E [ Y0 | D = 1]?
Is it observable?
The expected value of the outcome Y after treatment 0 given that treatment 1 was received. These are imaginary cases and they cannot be observed.
ICS212 Fall 2022
simplified notation.pdf - 11
What is: E [ Y1 | D = 0]?
Is it observable?
The expected value of the outcome Y after treatment 1 given that treatment 0 was received. These are imaginary cases and they cannot be observed.
ICS212 Fall 2022
simplified notation.pdf - 12
What is: E [ Y0 | D = 1] - E [ Y0 | D = 0]?
Is it observable?
The difference between the expected outcome for the treatment group had they not been treated and the expected outcomes of the control group. This is not observable
ICS212 Fall 2022