Potential Outcomes Brady Neal causalcourse.com What are potential - - PowerPoint PPT Presentation

Potential Outcomes

Brady Neal

causalcourse.com

Brady Neal / 41 2

What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation

Brady Neal / 41 3

What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation

Brady Neal / 41

Potential outcomes: intuition

Inferring the effect of treatment/policy on some outcome

4

Brady Neal / 41

Potential outcomes: intuition

Inferring the effect of treatment/policy on some outcome

4

Take pill

Brady Neal / 41

Potential outcomes: intuition

Inferring the effect of treatment/policy on some outcome

4

Take pill Don’t take pill

Brady Neal / 41

Potential outcomes: intuition

Inferring the effect of treatment/policy on some outcome

4

Take pill Don’t take pill causal effect

Brady Neal / 41

Potential outcomes: intuition

Inferring the effect of treatment/policy on some outcome

4

Take pill Don’t take pill causal effect?

Brady Neal / 41

Potential outcomes: intuition

Inferring the effect of treatment/policy on some outcome

4

Take pill Don’t take pill no causal effect

Brady Neal / 41

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome

T

Y

Brady Neal / 41

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual

Yi|do(T =1)

T

Y

i

Brady Neal / 41

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual

Yi|do(T =1)

Yi|do(T =0)

T

Y

i

Brady Neal / 41 ) , Yi(1)

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment

Yi|do(T =1)

Yi|do(T =0)

T

Y

i

Yi(1)

Brady Neal / 41 ) , Yi(1)

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment

Yi|do(T =1)

Yi|do(T =0)

T

Y

i

Yi(1)

Brady Neal / 41 ) , Yi(1)

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

Yi|do(T =1)

Yi|do(T =0)

) , Yi(0)

T

Y

i

Yi(1)

Yi(0)

Brady Neal / 41 ) , Yi(1)

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

Yi|do(T =1)

Yi|do(T =0)

) , Yi(0)

T

Y

i

Yi(1)

Yi(0)

Brady Neal / 41 ) , Yi(1)

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

Yi|do(T =1)

Yi|do(T =0)

) , Yi(0)

Causal effect

T

Y

i

Yi(1)

Yi(0)

Yi(1) − Yi(0)

Brady Neal / 41 ) , Yi(1)

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

Yi|do(T =1)

Causal effect

T

Y

i

Yi(1)

Yi(0)

Yi(1) − Yi(0)

Yi(0) = 0

Brady Neal / 41

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

Causal effect

T

Y

i

Yi(1)

Yi(0)

Yi(1) − Yi(0)

Yi(1) = 1

Yi(0) = 0

Brady Neal / 41

Yi(1) − Yi(0) = 1

Potential outcomes: notation

5

do(T = 1)

do(T = 0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

Causal effect

T

Y

i

Yi(1)

Yi(0)

Yi(1) = 1

Yi(0) = 0

Brady Neal / 41 6

What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation

Brady Neal / 41

Yi(1) − Yi(0) = 1

Fundamental problem of causal inference

7

do(T = 1)

do(T = 0)

Causal effect Yi(0) = 0

Yi(1) = 1

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Then, what does imply causation?

Brady Neal / 41

Yi(1) − Yi(0) = 1

Fundamental problem of causal inference

7

do(T = 1)

do(T = 0)

Causal effect Yi(0) = 0

Yi(1) = 1

Factual Counterfactual

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Then, what does imply causation?

Brady Neal / 41

Yi(1) − Yi(0) = 1

Counterfactual

Fundamental problem of causal inference

7

do(T = 1)

do(T = 0)

Causal effect Yi(0) = 0

Yi(1) = 1

Factual

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Then, what does imply causation?

Brady Neal / 41

Missing data interpretation

8

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Question: What is the fundamental problem of causal inference?

Brady Neal / 41 10

What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

Average treatment effect (ATE)

11

Yi(1) − Yi(0)

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Brady Neal / 41

E[Yi(1) − Yi(0)]

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

Average treatment effect (ATE)

11

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

2/ 3

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

2/ 3

− 1/

3

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

= E[Y (1)] − E[Y (0)]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Average treatment effect (ATE)

11

E[Y (1) − Y (0)]

= E[Y | T = 1] − E[Y | T = 0]

: observed treatment : observed outcome : used in subscript to denote a specific unit/individual : potential outcome under treatment : potential outcome under no treatment

T

Y

i

Yi(1)

Yi(0)

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

associational difference = E[Y (1)] − E[Y (0)]

Brady Neal / 41

Association is not causation

Sleeping with shoes on is strongly correlated with waking up with a headache

12

Brady Neal / 41

Association is not causation

Sleeping with shoes on is strongly correlated with waking up with a headache

12

Common cause: drinking the night before

Brady Neal / 41

Association is not causation

Sleeping with shoes on is strongly correlated with waking up with a headache

12

Common cause: drinking the night before 1. Confounding

Brady Neal / 41

Association is not causation

Sleeping with shoes on is strongly correlated with waking up with a headache

12

Common cause: drinking the night before 1. Confounding X T Y

Brady Neal / 41

Association is not causation

Sleeping with shoes on is strongly correlated with waking up with a headache

12

Common cause: drinking the night before 1. Confounding 2. Shoe-sleepers differ from non-shoe- sleepers in a key way X T Y

Brady Neal / 41

Why? Because the groups are not comparable

13

Went to sleep with shoes on Went to sleep without shoes on (T = 1)

(T = 0)

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

Why? Because the groups are not comparable

13

Went to sleep with shoes on Went to sleep without shoes on

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

(T = 1)

(T = 0)

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

Why? Because the groups are not comparable

13

Went to sleep with shoes on Went to sleep without shoes on

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

(T = 1)

(T = 0)

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

Why? Because the groups are not comparable

13

Went to sleep with shoes on Went to sleep without shoes on

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

(T = 1)

(T = 0)

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

What would comparable groups look like?

14

Went to sleep with shoes on Went to sleep without shoes on (T = 1)

(T = 0)

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

Brady Neal / 41

What would comparable groups look like?

14

Went to sleep with shoes on Went to sleep without shoes on (T = 1)

(T = 0)

sober drunk sober sober sober sober drunk drunk drunk drunk sober drunk sober sober sober sober drunk drunk drunk drunk

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

What would comparable groups look like?

14

Went to sleep with shoes on Went to sleep without shoes on (T = 1)

(T = 0)

sober drunk sober sober sober sober drunk drunk drunk drunk sober drunk sober sober sober sober drunk drunk drunk drunk

E[Y (1)] − E[Y (0)] = E[Y | T = 1] − E[Y | T = 0]

Question: Why is association not causation?

What assumptions would make the ATE equal to the associational difference?

Brady Neal / 41

Ignorability:

17

(Y (1), Y (0)) ⊥ ⊥ T

Brady Neal / 41

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

(Y (1), Y (0)) ⊥ ⊥ T

Brady Neal / 41

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? ? 2 1 1 1 ? ? 3 1 ? ? 4 ? ? 5 1 ? 1 ? 6 1 1 1 ? ?

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

− 1/

3

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

i T Y Y (1) Y (0) Y (1) − Y (0) 1 ? 2 1 1 1 ? 3 1 ? 4 ? 5 1 1 ? 6 1 1 1 ?

Ignorability:

17

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

2/ 3

− 1/

3

/

3 =

/

3 −

− 1/

3

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

Group B Group A Group A

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

Group B Group A Group A

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

Group B Group A Group A

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

Group B E[Y (1) | T = 1] = E[Y (1) | T = 0]

Before switch After switch

Group A Group A

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

Group B E[Y (1) | T = 1] = E[Y (1) | T = 0]

Before switch After switch

0] = E[Y (1)]

Group A Group A

Brady Neal / 41

Another perspective: exchangeability

18

T = 1

T = 0

E[Y | T = 0] = y0

E[Y | T = 1] = y1

E[Y (0) | T = 0] = E[Y (0) | T = 1]

Group B E[Y (1) | T = 1] = E[Y (1) | T = 0]

Before switch After switch

0] = E[Y (1)]

1] = E[Y (0)]

Group A Group A

Brady Neal / 41

Aside: identifiability

19

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

− | − | = E[Y | T = 1] − E[Y | T = 0]

Brady Neal / 41

Aside: identifiability

19

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

− | − | = E[Y | T = 1] − E[Y | T = 0]

Causal quantities

Brady Neal / 41

Aside: identifiability

19

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

− | − | = E[Y | T = 1] − E[Y | T = 0]

Causal quantities Statistical quantities (accessible, since we have )

P(x, t, y)

Brady Neal / 41

Aside: identifiability

19

(ignorability)

E[Y (1)] − E[Y (0)] = E[Y (1) | T = 1] − E[Y (0) | T = 0]

− | − | = E[Y | T = 1] − E[Y | T = 0]

Causal quantities Statistical quantities (accessible, since we have )

P(x, t, y)

A causal quantity (e.g. ) is identifiable if we can compute it from a purely statistical quantity (e.g. )

E[Y (t)]

E[Y | t]

Brady Neal / 41

Randomized control trial (RCT)

20

T

Brady Neal / 41

Randomized control trial (RCT)

20

Went to sleep with shoes on Went to sleep without shoes on (T = 1)

(T = 0)

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

T

Brady Neal / 41

Randomized control trial (RCT)

20

Went to sleep without shoes on (T = 1)

(T = 0)

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

Slept with shoes on

T

Brady Neal / 41

Randomized control trial (RCT)

20

(T = 1)

(T = 0)

sober drunk drunk sober drunk drunk drunk drunk drunk drunk drunk drunk drunk drunk sober sober sober sober sober sober sober sober sober sober sober sober sober sober

Slept with shoes on Slept without shoes on

T

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Randomized control trial (RCT)

20

(T = 1)

(T = 0)

sober drunk sober sober sober sober drunk drunk drunk drunk sober drunk sober sober sober sober drunk drunk drunk drunk

Slept with shoes on Slept without shoes on

T

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Graphical interpretation of RCT

21

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Graphical interpretation of RCT

21

X T Y

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Graphical interpretation of RCT

21

T

X T Y

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Graphical interpretation of RCT

21

T

X T Y

Question: What important property does an RCT give us?

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Conditional exchangeability

23

(Y (1), Y (0)) ⊥ ⊥ T

Exchangeability:

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Conditional exchangeability

23

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

Exchangeability:

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Conditional exchangeability

23

X T Y

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

Exchangeability:

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Conditional exchangeability

23

X T Y

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

(Y (1), Y (0)) ⊥ ⊥ T | X

Exchangeability: Conditional exchangeability:

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Conditional exchangeability

23

X T Y

X T Y

(Y (1), Y (0)) ⊥ ⊥ T

(Y (1), Y (0)) ⊥ ⊥ T | X

Exchangeability: Conditional exchangeability:

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Identification of conditional average treatment effect

24

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

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Identification of conditional average treatment effect

24

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

E[Y (1) − Y (0) | X] = E[Y (1) | X] − E[Y (0) | X]

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Identification of conditional average treatment effect

24

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

E[Y (1) − Y (0) | X] = E[Y (1) | X] − E[Y (0) | X]

− | | − | = E[Y (1) | T = 1, X] − E[Y (0) | T = 0, X]

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Identification of conditional average treatment effect

24

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

E[Y (1) − Y (0) | X] = E[Y (1) | X] − E[Y (0) | X]

− | | − | = E[Y (1) | T = 1, X] − E[Y (0) | T = 0, X]

| − | = E[Y | T = 1, X] − E[Y | T = 0, X]

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Identification of conditional average treatment effect

24

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

E[Y (1) − Y (0) | X] = E[Y (1) | X] − E[Y (0) | X]

E[Y (1) − Y (0)]

What about the ATE?

− | | − | = E[Y (1) | T = 1, X] − E[Y (0) | T = 0, X]

| − | = E[Y | T = 1, X] − E[Y | T = 0, X]

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The Adjustment Formula (identification of ATE)

25

E[Y (1) − Y (0)] = EXE[Y (1) − Y (0) | X] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

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The Adjustment Formula (identification of ATE)

25

E[Y (1) − Y (0)] = EXE[Y (1) − Y (0) | X] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X T Y

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The Adjustment Formula (identification of ATE)

25

E[Y (1) − Y (0)] = EXE[Y (1) − Y (0) | X] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X T Y

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The Adjustment Formula (identification of ATE)

25

E[Y (1) − Y (0)] = EXE[Y (1) − Y (0) | X] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X T Y

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Unconfoundedness is an untestable assumption

26

unconfoundedness = conditional ignorability = conditional exchangeability

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Unconfoundedness is an untestable assumption

26

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

X T Y

unconfoundedness = conditional ignorability = conditional exchangeability

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X W T Y

Unconfoundedness is an untestable assumption

26

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

unconfoundedness = conditional ignorability = conditional exchangeability

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X W T Y

Unconfoundedness is an untestable assumption

26

(Y (1), Y (0)) ⊥ ⊥ T | X

Conditional exchangeability:

unconfoundedness = conditional ignorability = conditional exchangeability

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Positivity

For all values of covariates present in the population of interest (i.e. such that ),

27

P(X = x) > 0

x

x

0 < P(T = 1 | X = x) < 1

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Positivity

For all values of covariates present in the population of interest (i.e. such that ),

27

P(X = x) > 0

x

x

0 < P(T = 1 | X = x) < 1

Why? Recall the adjustment formula:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

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Positivity

For all values of covariates present in the population of interest (i.e. such that ),

27

P(X = x) > 0

x

x

0 < P(T = 1 | X = x) < 1

Why? Recall the adjustment formula:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X

x

X

y

y P(Y = y | T = 1, X = x) − X

y

y P(Y = y | T = 0, X = x) !

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Positivity

For all values of covariates present in the population of interest (i.e. such that ),

27

P(X = x) > 0

x

x

0 < P(T = 1 | X = x) < 1

Why? Recall the adjustment formula:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X

x

X

y

y P(Y = y | T = 1, X = x) − X

y

y P(Y = y | T = 0, X = x) !

X

x

X

y

y P(Y = y, T = 1, X = x) P(T = 1 | X = x)P(X = x) − X

y

y P(Y = y, T = 0, X = x) P(T = 0 | X = x)P(X = x) !

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Positivity

For all values of covariates present in the population of interest (i.e. such that ),

27

P(X = x) > 0

x

x

0 < P(T = 1 | X = x) < 1

Why? Recall the adjustment formula:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X

x

X

y

y P(Y = y | T = 1, X = x) − X

y

y P(Y = y | T = 0, X = x) !

X

x

X

y

y P(Y = y, T = 1, X = x) P(T = 1 | X = x)P(X = x) − X

y

y P(Y = y, T = 0, X = x) P(T = 0 | X = x)P(X = x) !

Brady Neal / 41

Positivity

For all values of covariates present in the population of interest (i.e. such that ),

27

P(X = x) > 0

x

x

0 < P(T = 1 | X = x) < 1

Why? Recall the adjustment formula:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

X

x

X

y

y P(Y = y | T = 1, X = x) − X

y

y P(Y = y | T = 0, X = x) !

X

x

X

y

y P(Y = y, T = 1, X = x) P(T = 1 | X = x)P(X = x) − X

y

y P(Y = y, T = 0, X = x) P(T = 0 | X = x)P(X = x) !

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Positivity: intuition

28

Total population

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Positivity: intuition

28

Total population

X = x

Brady Neal / 41

T = 0

T = 0

T = 0

T = 0

T = 0

T = 0

Positivity: intuition

28

Total population

X = x

No one treated

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T = 1

T = 1

T = 1

T = 1

T = 1

T = 1

Positivity: intuition

28

Total population

X = x

Everyone treated

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Another perspective: overlap

29

P(X | T = 1)

P(X | T = 0)

Overlap between and

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Another perspective: overlap

29

x

P(x | t)

P(X | T = 1)

P(X | T = 0)

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Another perspective: overlap

29

x

P(x | t)

P(X | T = 1)

P(X | T = 0)

No overlap means severe positivity violation

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Another perspective: overlap

29

x

P(x | t)

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Another perspective: overlap

29

x

P(x | t)

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Another perspective: overlap

29

x

P(x | t)

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Another perspective: overlap

29

x

P(x | t)

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Another perspective: overlap

29

x

P(x | t)

Complete overlap means no positivity violation

Question: What goes wrong if we don’t have positivity?

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The Positivity-Unconfoundedness Tradeoff

31

P(X | T = 1)

P(X | T = 0)

x

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The Positivity-Unconfoundedness Tradeoff

31

x

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The Positivity-Unconfoundedness Tradeoff

31

50% overlap

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The Positivity-Unconfoundedness Tradeoff

31

50% overlap 1-dimensional

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The Positivity-Unconfoundedness Tradeoff

31

50% overlap

x1

x2

1-dimensional 2-dimensional

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The Positivity-Unconfoundedness Tradeoff

31

50% overlap 25% overlap

x1

x2

1-dimensional 2-dimensional

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The Positivity-Unconfoundedness Tradeoff

31

50% overlap 25% overlap

x1

x2

1-dimensional 2-dimensional 3-dimensional – 12.5% overlap

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The Positivity-Unconfoundedness Tradeoff

31

50% overlap 25% overlap

x1

x2

1-dimensional 2-dimensional 3-dimensional – 12.5% overlap … and so on (curse of dimensionality)

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Extrapolation

32

x

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

Adjustment formula:

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f1(x)

Adjustment formula: Model with

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f1(x)

and

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f0(x)

f1(x)

and

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f1(x)

f0(x)

f1(x)

and

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f1(x)

f0(x)

f1(x)

and

T = 1

T = 0

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Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f1(x)

f0(x)

f1(x)

and

T = 1

T = 0

f1(x)

?

Brady Neal / 41

Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f1(x)

f0(x)

f1(x)

and

T = 1

T = 0

f1(x)

?

f0(x)

?

Brady Neal / 41

Extrapolation

32

x

X

x

(E[Y | T = 1, x] − E[Y | T = 0, x])

f0(x)

f1(x)

Adjustment formula: Model with Model with

f0(x)

f1(x)

f0(x)

f1(x)

and

T = 1

T = 0

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No interference

33

Yi(t1, . . . , ti−1, ti, ti+1, . . . , tn) = Yi(ti)

Brady Neal / 41

Ti Ti−1 . . . T1 Ti+1 . . . Tn Yi

No interference

33

Yi(t1, . . . , ti−1, ti, ti+1, . . . , tn) = Yi(ti)

Brady Neal / 41

Ti Ti−1 . . . T1 Ti+1 . . . Tn Yi

No interference

33

Yi(t1, . . . , ti−1, ti, ti+1, . . . , tn) = Yi(ti)

My happiness

Brady Neal / 41

Ti Ti−1 . . . T1 Ti+1 . . . Tn Yi

No interference

33

Yi(t1, . . . , ti−1, ti, ti+1, . . . , tn) = Yi(ti)

Whether friends get dogs Whether friends get dogs My happiness

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No interference

33

Yi(t1, . . . , ti−1, ti, ti+1, . . . , tn) = Yi(ti)

My happiness

Ti Yi

Brady Neal / 41

Consistency:

34

T = t = ⇒ Y = Y (t)

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Consistency:

34

T = t = ⇒ Y = Y (t)

T = 1

“I get a dog”

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Consistency:

34

T = t = ⇒ Y = Y (t)

T = 1

“I get a dog”

T = 0

“I don’t get a dog”

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Consistency:

34

T = t = ⇒ Y = Y (t)

T = 1

“I get a dog”

T = 0

“I don’t get a dog”

(T = 1)

= ⇒ Y = 1

(I’m happy)

Brady Neal / 41

Consistency:

34

T = t = ⇒ Y = Y (t)

T = 1

“I get a dog”

T = 0

“I don’t get a dog”

(T = 1)

= ⇒ Y = 1

= ⇒ Y = 0

(T = 1)

(I’m happy) (I’m not happy)

Brady Neal / 41

Consistency:

34

T = t = ⇒ Y = Y (t)

T = 1

“I get a dog”

T = 0

“I don’t get a dog”

(T = 1)

= ⇒ Y = 1

= ⇒ Y = 0

(T = 1)

(I’m happy) (I’m not happy) Consistency assumption violated

Recall:

• 1. What were the four main assumptions?
• 2. Why do positivity violations require extrapolation?
• 3. Can you test if unconfoundedness is satisfied?
• 4. What is identifiability?

Brady Neal / 41

Tying it all together

36

E[Y (1) − Y (0)]

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Tying it all together

36

No interference

E[Y (1) − Y (0)]

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Tying it all together

36

No interference

E[Y (1) − Y (0)]

0)] = E[Y (1)] − E[Y (0)] (linearity of expectation)

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Tying it all together

36

− − = EX [E[Y (1) | X] − E[Y (0) | X]] (law of iterated expectations)

No interference

E[Y (1) − Y (0)]

0)] = E[Y (1)] − E[Y (0)] (linearity of expectation)

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Tying it all together

36

− − = EX [E[Y (1) | X] − E[Y (0) | X]] (law of iterated expectations)

No interference

E[Y (1) − Y (0)]

0)] = E[Y (1)] − E[Y (0)] (linearity of expectation)

= EX [E[Y (1) | T = 1, X] − E[Y (0) | T = 0, X]] (unconfoundedness and positivity)

Brady Neal / 41

Tying it all together

36

− − = EX [E[Y (1) | X] − E[Y (0) | X]] (law of iterated expectations)

No interference

E[Y (1) − Y (0)]

0)] = E[Y (1)] − E[Y (0)] (linearity of expectation)

= EX [E[Y (1) | T = 1, X] − E[Y (0) | T = 0, X]] (unconfoundedness and positivity)

= EX [E[Y | T = 1, X] − E[Y | T = 0, X]] (consistency)

Brady Neal / 41 37

What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation

Brady Neal / 41

Estimands, estimates, and the Identification-Estimation Flowchart

38

Brady Neal / 41

Estimands, estimates, and the Identification-Estimation Flowchart

38

• Estimand - any quantity we want to estimate

Brady Neal / 41

Estimands, estimates, and the Identification-Estimation Flowchart

38

• Estimand - any quantity we want to estimate
• Causal estimand (e.g. )

E[Y (1) − Y (0)]

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Estimands, estimates, and the Identification-Estimation Flowchart

38

• Estimand - any quantity we want to estimate
• Causal estimand (e.g. )
• Statistical estimand (e.g. )

E[Y (1) − Y (0)]

EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

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Estimands, estimates, and the Identification-Estimation Flowchart

38

• Estimand - any quantity we want to estimate
• Causal estimand (e.g. )
• Statistical estimand (e.g. )
• Estimate: approximation of some estimand, using data

E[Y (1) − Y (0)]

EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

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Estimands, estimates, and the Identification-Estimation Flowchart

38

• Estimand - any quantity we want to estimate
• Causal estimand (e.g. )
• Statistical estimand (e.g. )
• Estimate: approximation of some estimand, using data
• Estimation: process for getting from data + estimand to estimate

E[Y (1) − Y (0)]

EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

Brady Neal / 41

Estimands, estimates, and the Identification-Estimation Flowchart

38

Causal Estimand Statistical Estimand Estimate Identification Estimation

• Estimand - any quantity we want to estimate
• Causal estimand (e.g. )
• Statistical estimand (e.g. )
• Estimate: approximation of some estimand, using data
• Estimation: process for getting from data + estimand to estimate

The Identification-Estimation Flowchart

E[Y (1) − Y (0)]

EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

39

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

Motivation: 46% of Americans have high blood pressure and high blood pressure is associated with increased risk of mortality

39

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

Motivation: 46% of Americans have high blood pressure and high blood pressure is associated with increased risk of mortality Data:

• Epidemiological example taken from Luque-Fernandez et al. (2018)

39

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

Motivation: 46% of Americans have high blood pressure and high blood pressure is associated with increased risk of mortality Data:

• Epidemiological example taken from Luque-Fernandez et al. (2018)
• Outcome Y: (systolic) blood pressure (continuous)

39

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

Motivation: 46% of Americans have high blood pressure and high blood pressure is associated with increased risk of mortality Data:

• Epidemiological example taken from Luque-Fernandez et al. (2018)
• Outcome Y: (systolic) blood pressure (continuous)
• Treatment T: sodium intake (1 if above 3.5 mg and 0 if below)

39

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

Motivation: 46% of Americans have high blood pressure and high blood pressure is associated with increased risk of mortality Data:

• Epidemiological example taken from Luque-Fernandez et al. (2018)
• Outcome Y: (systolic) blood pressure (continuous)
• Treatment T: sodium intake (1 if above 3.5 mg and 0 if below)
• Covariates X: age and amount of protein excreted in urine

39

Brady Neal / 41

Problem: effect of sodium intake on blood pressure

Motivation: 46% of Americans have high blood pressure and high blood pressure is associated with increased risk of mortality Data:

• Epidemiological example taken from Luque-Fernandez et al. (2018)
• Outcome Y: (systolic) blood pressure (continuous)
• Treatment T: sodium intake (1 if above 3.5 mg and 0 if below)
• Covariates X: age and amount of protein excreted in urine
• Simulation: so we know the “true” ATE is 1.05

39

Brady Neal / 41

Estimation of ATE

40

True ATE: E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: True ATE:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: True ATE:

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: True ATE:

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: True ATE:

Model (linear regression)

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: True ATE:

Model (linear regression)

Estimate: 0.85

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

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Estimation of ATE

40

Identification: Estimation: Naive: True ATE:

Model (linear regression)

Estimate: 0.85

E[Y | T = 1] − E[Y | T = 0]

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: Naive: True ATE:

Model (linear regression)

Estimate: 0.85 Naive estimate: 5.33

E[Y | T = 1] − E[Y | T = 0]

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: Naive: True ATE:

Model (linear regression)

Estimate: 0.85 Naive estimate: 5.33

E[Y | T = 1] − E[Y | T = 0]

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

|5.33 − 1.05| 1.05 × 100% = 407%

Brady Neal / 41

Estimation of ATE

40

Identification: Estimation: Naive: True ATE:

Model (linear regression)

Estimate: 0.85 Naive estimate: 5.33

E[Y | T = 1] − E[Y | T = 0]

1 n X

x

[E[Y | T = 1, x] − E[Y | T = 0, x]]

E[Y (1) − Y (0)] = EX [E[Y | T = 1, X] − E[Y | T = 0, X]]

E[Y (1) − Y (0)] = 1.05

|5.33 − 1.05| 1.05 × 100% = 407%

|0.85 − 1.05| 1.05 × 100% = 19%

Brady Neal / 41

Using coefficient of linear regression

41

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Y = αT + βX

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression:

Y = αT + βX

Y = ˆ αT + ˆ βX

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression:

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Brady Neal / 41 E[Y (1) − Y (0)]

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment:

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Severe limitations:

Brady Neal / 41

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Severe limitations: the causal effect is the same for all individuals

Brady Neal / 41 Yi(t) = αt + βxi

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Severe limitations: the causal effect is the same for all individuals

Brady Neal / 41 −α · 0 − βxi

Yi(1) − Yi(0) = α · 1 + βxi

Yi(t) = αt + βxi

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Severe limitations: the causal effect is the same for all individuals

Brady Neal / 41 −α · 0 − βxi

Yi(1) − Yi(0) = α · 1 + βxi

Yi(t) = αt + βxi

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Severe limitations: the causal effect is the same for all individuals

= α

Brady Neal / 41 −α · 0 − βxi

Yi(1) − Yi(0) = α · 1 + βxi

Yi(t) = αt + βxi

Using coefficient of linear regression

41

Assume linear parametric form: Run linear regression: Continuous treatment: E[Y (t)]

ˆ α = 0.85

Y = αT + βX

Y = ˆ αT + ˆ βX

ˆ α = 0.85

Severe limitations: the causal effect is the same for all individuals See Sections 6.2 and 6.3 of Morgan & Winship (2014) for more complete critique

= α