A Brief Introduction to
Causal Inference
Brady Nealcausalcourse.com

What is causal inference?
Inferring the effects of any treatment/policy/intervention/etc.

Brady Neal

2 / 28What is causal inference?
Inferring the effects of any treatment/policy/intervention/etc.
Examples:• Effect of treatment on a disease

Brady Neal

2 / 28What is causal inference?
Inferring the effects of any treatment/policy/intervention/etc.
Examples:• Effect of treatment on a disease• Effect of climate change policy on emissions

Brady Neal

2 / 28What is causal inference?
Inferring the effects of any treatment/policy/intervention/etc.
Examples:• Effect of treatment on a disease• Effect of climate change policy on emissions• Effect of social media on mental health

Brady Neal

2 / 28What is causal inference?
Inferring the effects of any treatment/policy/intervention/etc.
Examples:• Effect of treatment on a disease• Effect of climate change policy on emissions• Effect of social media on mental health• Many more (effect of X on Y)

Brady Neal

2 / 28

Motivating example: Simpson’s paradox
Correlation does not imply causation
Then, what does imply causation?
Causation in observational studies

Brady Neal

3 / 28

Motivating example: Simpson’s paradox
Correlation does not imply causation
Then, what does imply causation?
Causation in observational studies

Brady Neal

Motivating example: Simpson’s paradox

4 / 28

Simpson’s paradox: COVID-27
New disease: COVID-27

Brady Neal

Motivating example: Simpson’s paradox

5 / 28

Simpson’s paradox: COVID-27
New disease: COVID-27
Treatment T: A (0) and B (1)Brady Neal

Motivating example: Simpson’s paradox

5 / 28

Simpson’s paradox: COVID-27
New disease: COVID-27

YOU

Treatment T: A (0) and B (1)Brady Neal

Motivating example: Simpson’s paradox

5 / 28

Simpson’s paradox: COVID-27
New disease: COVID-27

YOU

Treatment T: A (0) and B (1)Condition C: mild (0) or severe (1)Brady Neal

Motivating example: Simpson’s paradox

5 / 28

Simpson’s paradox: COVID-27
New disease: COVID-27

YOU

Treatment T: A (0) and B (1)Condition C: mild (0) or severe (1)Outcome Y: alive (0) or dead (1)Brady Neal

Motivating example: Simpson’s paradox

5 / 28

Simpson’s paradox: mortality rate table

Tr
ea
tm
en
t

Total
A
B

16%
(240/1500)

19%(105/550)
E[Y |T ]Brady Neal

Motivating example: Simpson’s paradox

6 / 28

Simpson’s paradox: mortality rate table

Tr
ea
tm
en
t

Condition

Brady Neal

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |T, C = 0]

E[Y |T, C = 1]

E[Y |T ]Motivating example: Simpson’s paradox

6 / 28

Simpson’s paradox: mortality rate table

Tr
ea
tm
en
t

Condition

Brady Neal

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |T, C = 0]

E[Y |T, C = 1]

E[Y |T ]Motivating example: Simpson’s paradox

1400
100
0.15 +
0.30 = 0.16
1500
1500
50
0.10
550

+

500
0.20
550

= 0.196 / 28

Simpson’s paradox: mortality rate table

Tr
ea
tm
en
t

Condition

Brady Neal

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |T, C = 0]

E[Y |T, C = 1]

E[Y |T ]Motivating example: Simpson’s paradox

1400
100
0.15 +
0.30 = 0.16
1500
1500
50
0.10
550

+

500
0.20
550

= 0.196 / 28

Simpson’s paradox: mortality rate table

Tr
ea
tm
en
t

Condition

Brady Neal

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |T, C = 0]

E[Y |T, C = 1]

E[Y |T ]Motivating example: Simpson’s paradox

1400
100
0.15 +
0.30 = 0.16
1500
1500
50
0.10
550

+

500
0.20
550

= 0.196 / 28

Simpson’s paradox: mortality rate table

Tr
ea
tm
en
t

Condition
A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |T, C = 0]

E[Y |T, C = 1]

E[Y |T ]1400
100
0.15 +
0.30 = 0.16
1500
1500
50
0.10
550

+

500
0.20
550

= 0.19Which treatment should you choose?
Brady Neal

Motivating example: Simpson’s paradox

6 / 28

Simpson’s paradox: scenario 1 (treatment B)

A

Tr
e

at
m

en

t

Condition

Brady Neal

B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

Motivating example: Simpson’s paradox

7 / 28

Simpson’s paradox: scenario 1 (treatment B)

A

Tr
e

at
m

en

t

Condition

B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

Causal Graph
T

Brady NealC

Y

Motivating example: Simpson’s paradox

7 / 28

Simpson’s paradox: scenario 1 (treatment B)

A

Tr
e

at
m

en

t

Condition

B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

Causal Graph

Mild

C

Treatment A
T

Brady NealY

Motivating example: Simpson’s paradox

7 / 28

Simpson’s paradox: scenario 1 (treatment B)

A

Tr
e

at
m

en

t

Condition

B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

Causal Graph
T

Brady Neal

Mild

SevereTreatment A

Treatment B

CY

Motivating example: Simpson’s paradox

7 / 28

Simpson’s paradox: scenario 2 (treatment A)

A

Tr
e

at
m

en

t

Condition

Brady Neal

B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

Motivating example: Simpson’s paradox

8 / 28

Simpson’s paradox: scenario 2 (treatment A)
Condition

Tr
e

at
m

en

t

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

T

Causal Graph
Brady NealC

Y

Motivating example: Simpson’s paradox

8 / 28

Simpson’s paradox: scenario 2 (treatment A)
Condition

Tr
e

at
m

en

t

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

T

Causal Graph
Brady NealCTreatment B

SevereY

Motivating example: Simpson’s paradox

8 / 28

Simpson’s paradox: scenario 2 (treatment A)
Condition

Tr
e

at
m

en

t

A
B

Mild

Severe

Total

15%

30%

16%

(210/1400)(30/100)(240/1500)

10%

20%

19%

(5/50)(100/500)(105/550)

T

Causal Graph
Brady NealCTreatment A

Treatment B

Mild

SevereY

Motivating example: Simpson’s paradox

8 / 28

Motivating example: Simpson’s paradox
Correlation does not imply causation
Then, what does imply causation?
Causation in observational studies

Brady Neal

Correlation does not imply causation

9 / 28

Correlation does not
imply causation

Brady Neal

Correlation does not imply causation

10 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache

Brady Neal

Correlation does not imply causation

11 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache
Common cause: drinking the night before

Brady Neal

Correlation does not imply causation

11 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache
Common cause: drinking the night before
1. Shoe-sleepers differ from non-shoesleepers in a key way

Brady Neal

Correlation does not imply causation

11 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache
Common cause: drinking the night before
1. Shoe-sleepers differ from non-shoesleepers in a key way
2. Confounding

Brady Neal

Correlation does not imply causation

11 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache
Common cause: drinking the night before
1. Shoe-sleepers differ from non-shoesleepers in a key way
2. Confounding

Brady Neal

Correlation does not imply causation

11 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache
Common cause: drinking the night before
1. Shoe-sleepers differ from non-shoesleepers in a key way
2. Confounding

Causal association

Brady Neal

Correlation does not imply causation

11 / 28

Correlation does not imply causation
Sleeping with shoes on is strongly correlated with waking up with a headache
Common cause: drinking the night before
1. Shoe-sleepers differ from non-shoesleepers in a key way
2. Confounding
Total association (e.g. correlation):
mixture of causal andconfounding association
Causal association

Brady Neal

Correlation does not imply causation

11 / 28

“Correlation = Causation” is a cognitive bias1

Causal association
1The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018)

Brady Neal

Correlation does not imply causation

12 / 28

“Correlation = Causation” is a cognitive bias1

confounderCausal association
1The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018)

Brady Neal

Correlation does not imply causation

12 / 28

“Correlation = Causation” is a cognitive bias1
Availability heuristic (another cognitive bias) gives us

confounderCausal association
1The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018)

Brady Neal

Correlation does not imply causation

12 / 28

“Correlation = Causation” is a cognitive bias1
Availability heuristic (another cognitive bias) gives us
Why?
confounderCausal association
1The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018)

Brady Neal

Correlation does not imply causation

12 / 28

“Correlation = Causation” is a cognitive bias1
Availability heuristic (another cognitive bias) gives us
Motivated reasoning (another cognitive bias)
Why?
confounderCausal association
1The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018)

Brady Neal

Correlation does not imply causation

12 / 28

“Correlation = Causation” is a cognitive bias1
Availability heuristic (another cognitive bias) gives us
Motivated reasoning (another cognitive bias)
Why?

explains

confoundercorrelation = causation

Causal association
1The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018)

Brady Neal

Correlation does not imply causation

12 / 28

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"

https://www.tylervigen.com/spurious-correlations

Brady Neal

Correlation does not imply causation

13 / 28

Correlation does not
imply causation

Brady Neal

Correlation does not imply causation

14 / 28

Then, what does imply causation?

Motivating example: Simpson’s paradox
Correlation does not imply causation
Then, what does imply causation?
Causation in observational studies

Brady Neal

Then, what does imply causation?

16 / 28

Potential outcomes: intuition
Inferring the effect of treatment/policy on some outcome

Brady Neal

Then, what does imply causation?

17 / 28

Potential outcomes: intuition
Inferring the effect of treatment/policy on some outcome
Take pill

Brady Neal

Then, what does imply causation?

17 / 28

Potential outcomes: intuition
Inferring the effect of treatment/policy on some outcome
Take pill

Don’t take pill

Brady Neal

Then, what does imply causation?

17 / 28

Potential outcomes: intuition
Inferring the effect of treatment/policy on some outcome
Take pillcausal effectDon’t take pill

Brady Neal

Then, what does imply causation?

17 / 28

Potential outcomes: intuition
Inferring the effect of treatment/policy on some outcome
Take pillno causal effectDon’t take pill

Brady Neal

Then, what does imply causation?

17 / 28

Potential outcomes: notation
TY

do(T = 1): observed treatment: observed outcomedo(T = 0)

Brady Neal

Then, what does imply causation?

18 / 28

Potential outcomes: notationdo(T = 1)

Yi |do(T =1)

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

do(T = 0)

Brady Neal

Then, what does imply causation?

18 / 28

Potential outcomes: notationdo(T = 1)

Yi |do(T =1)

do(T = 0)

Yi |do(T =0)

Brady Neal

Then, what does imply causation?

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

18 / 28

Potential outcomes: notationdo(T = 1)

Yi |do(T =1) , Yi (1)do(T = 0)

Yi |do(T =0)

Brady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment

Then, what does imply causation?

TYi

18 / 28

Potential outcomes: notationdo(T = 1)

Yi |do(T =1) , Yi (1)do(T = 0)

Yi |do(T =0) , Yi (0)Brady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

Then, what does imply causation?

TYi

18 / 28

Potential outcomes: notationdo(T = 1)

Yi |do(T =1) , Yi (1)do(T = 0)

Yi |do(T =0) , Yi (0)Brady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

Then, what does imply causation?

T
Yi

Causal effectYi (1) Yi (0)18 / 28

Fundamental problem of causal inferencedo(T = 1)

do(T = 0)

Brady NealYi (1) = 1: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

TYi

Yi (0) = 0

Then, what does imply causation?

Causal effectYi (1) Yi (0) = 1

19 / 28

Fundamental problem of causal inferencedo(T = 1)

Yi (1) = 1

Counterfactualdo(T = 0): observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

Yi (0) = 0

FactualBrady NealT
Y
iThen, what does imply causation?

Causal effectYi (1) Yi (0) = 1

19 / 28

Fundamental problem of causal inferencedo(T = 1)

Yi (1) = 1

Factualdo(T = 0): observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

Yi (0) = 0

Counterfactual

Brady NealT
Y
iThen, what does imply causation?

Causal effectYi (1) Yi (0) = 1

19 / 28

Average treatment effect (ATE)
Individual treatment effect (ITE): Yi (1)Brady Neal

Yi (0): observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

Then, what does imply causation?

TYi

20 / 28

Average treatment effect (ATE)
Individual treatment effect (ITE): Yi (1)Yi (0)

Average treatment effect (ATE):
E[Yi (1) Yi (0)] = E[Y (1)] E[Y (0)]6= E[Y |T = 1] E[Y |T = 0]

Brady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

Then, what does imply causation?

TYi

20 / 28

Average treatment effect (ATE)
Individual treatment effect (ITE): Yi (1)Yi (0)

Average treatment effect (ATE):
E[Yi (1) Yi (0)] = E[Y (1)] E[Y (0)]6= E[Y |T = 1] E[Y |T = 0]

Brady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

Then, what does imply causation?

TYi

20 / 28

Average treatment effect (ATE)
Individual treatment effect (ITE): Yi (1)Yi (0)

Average treatment effect (ATE):
E[Yi (1) Yi (0)] = E[Y (1)] E[Y (0)]6= E[Y |T = 1] E[Y |T = 0]CT

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcomeTYi

Recall: correlation does not imply causation
Y

Causal association

Brady Neal

Then, what does imply causation?

20 / 28

Average treatment effect (ATE)
Individual treatment effect (ITE): Yi (1)Yi (0)

Average treatment effect (ATE):
E[Yi (1) Yi (0)] = E[Y (1)] E[Y (0)]6= E[Y |T = 1] E[Y |T = 0]CT

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcomeTYi

Recall: correlation does not imply causation
Y

Causal association

Brady Neal

Then, what does imply causation?

20 / 28

Average treatment effect (ATE)
Individual treatment effect (ITE): Yi (1)Yi (0)

Average treatment effect (ATE):
E[Yi (1) Yi (0)] = E[Y (1)] E[Y (0)]6= E[Y |T = 1] E[Y |T = 0]CT

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcomeTYi

Recall: correlation does not imply causation
Y

Causal association

Brady Neal

Then, what does imply causation?

20 / 28

Randomized control trials (RCTs)
ATE when there is confounding:E[Y (1)] E[Y (0)] 6= E[Y |T = 1]
CT

E[Y |T = 0]

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

T
Y
i

Y
Causal association

Brady Neal

Then, what does imply causation?

21 / 28

Randomized control trials (RCTs)
ATE when there is confounding:E[Y (1)] E[Y (0)] 6= E[Y |T = 1]
C

T

C

Y
Causal association

Brady NealE[Y |T = 0]T

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

T
Y
i

Y
Causal association

Then, what does imply causation?

21 / 28

Randomized control trials (RCTs)
ATE when there is confounding:E[Y (1)] E[Y (0)] 6= E[Y |T = 1]
C

T

C

Y
Causal association

Brady NealE[Y |T = 0]T

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

Y
Causal association

Then, what does imply causation?

T
Yi

RCTs: experimenter randomizes
subjects into treatment groupor control group
1. T cannot have any causal
parents
21 / 28

Randomized control trials (RCTs)
ATE when there is confounding:E[Y (1)] E[Y (0)] 6= E[Y |T = 1]
C

T

C

Y
Causal association

Brady NealE[Y |T = 0]T

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

Y
Causal association

Then, what does imply causation?

T
Yi

RCTs: experimenter randomizes
subjects into treatment groupor control group
1. T cannot have any causal
parents
2. Groups are comparable
21 / 28

Randomized control trials (RCTs)
ATE when there is confounding:E[Y (1)] E[Y (0)] 6= E[Y |T = 1]
C

T

C

Y
Causal association

E[Y |T = 0]T

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment
Y (t) : population-level potential outcome

Y
Causal association

ATE when there is no confounding (e.g. RCTs):
E[Y (1)] E[Y (0)] = E[Y |T = 1] E[Y |T = 0]
Brady Neal

Then, what does imply causation?

T
Yi

RCTs: experimenter randomizes
subjects into treatment groupor control group
1. T cannot have any causal
parents
2. Groups are comparable
21 / 28

Motivating example: Simpson’s paradox
Correlation does not imply causation
Then, what does imply causation?
Causation in observational studies

Brady Neal

Causation in observational studies

22 / 28

Observational studies
Ideal

C

T

Y
Causal association

Observational
studiesC

T

Y
Causal association

Brady Neal

Causation in observational studies

23 / 28

Observational studies
Ideal

Can’t always randomize treatment

CT

Y
Causal association

Observational
studiesC

T

Y
Causal association

Brady Neal

Causation in observational studies

23 / 28

Observational studies
Ideal

Can’t always randomize treatment• Ethical reasons (e.g. unethical torandomize people to smoke formeasuring effect on lung cancer)

C

TY
Causal association

Observational
studiesC

T

Y
Causal association

Brady Neal

Causation in observational studies

23 / 28

Observational studies
Can’t always randomize treatment• Ethical reasons (e.g. unethical torandomize people to smoke formeasuring effect on lung cancer)• Infeasibility (e.g. can’t randomize
countries into communist/capitalist
systems to measure effect on GDP)

Ideal

C

T

Y
Causal association

Observational
studiesC

T

Y
Causal association

Brady Neal

Causation in observational studies

23 / 28

Observational studies
C
Ideal
Can’t always randomize treatment• Ethical reasons (e.g. unethical torandomize people to smoke forTYmeasuring effect on lung cancer)
Causal association• Infeasibility (e.g. can’t randomize
Observationalcountries into communist/capitalist
studiessystems to measure effect on GDP)C
• Impossibility (e.g. can’t change a living
person’s DNA at birth for measuringTYeffect on breast cancer)
Causal association

Brady Neal

Causation in observational studies

23 / 28

How do we measure causal
effects in observational studies?

Solution: adjust/control for confounders

C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

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Solution: adjust/control for confounders
Adjust/control for the right variables W.

C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

25 / 28

Solution: adjust/control for confounders
Adjust/control for the right variables W.
If W is a sufficient adjustment set, we haveE[Y (t)|W = w] , E[Y |do(T = t), W = w] = E[Y |t, w]
C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

25 / 28

Solution: adjust/control for confounders
Adjust/control for the right variables W.
If W is a sufficient adjustment set, we haveE[Y (t)|W = w] , E[Y |do(T = t), W = w] = E[Y |t, w]
C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

25 / 28

Solution: adjust/control for confounders
Adjust/control for the right variables W.
If W is a sufficient adjustment set, we haveE[Y (t)|W = w] , E[Y |do(T = t), W = w] = E[Y |t, w]
E[Y (t)] , E[Y |do(T = t)] = EW E[Y |t, W ]

C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

25 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]

C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]

C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]Shaded nodes are examples of sufficient adjustment sets W

C

T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]Shaded nodes are examples of sufficient adjustment sets W
W2
C
W1
T

M
Causal association

C

W3

Y
T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]Shaded nodes are examples of sufficient adjustment sets W
W2
C
W1
T

M
Causal association

C

W3

Y
T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]Shaded nodes are examples of sufficient adjustment sets W
W2
C
W1
T

M
Causal association

C

W3

Y
T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]Shaded nodes are examples of sufficient adjustment sets W
W2
C
W1
T

M
Causal association

C

W3

Y
T

M

Y

Causal association

Brady Neal

Causation in observational studies

26 / 28

Solution: backdoor adjustment
E[Y |do(T = t)] = EW E[Y |t, W ]Shaded nodes are examples of sufficient adjustment sets W
W2

Z1

Z3

C
W1
T

M
Causal association

C

W3

Y
T

M

Y

Causal association

Brady Neal

Z2

Causation in observational studies

T

M

Y

Causal association

26 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
c

E[Y |t, c]P (c)

A

Tr
e

at
m

en

t

Condition

B

Brady Neal

C

T

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |t, C = 0] E[Y |t, C = 1]

Causal Graph

E[Y |t]Y

12.9%
E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
c

E[Y |t, c]P (c)

A

Tr
e

at
m

en

t

Condition

B

Brady Neal

C

T

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |t, C = 0] E[Y |t, C = 1]

Causal Graph

E[Y |t]Y

12.9%
E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
c

E[Y |t, c]P (c)

A

Tr
e

at
m

en

t

Condition

B

Brady Neal

C

T

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |t, C = 0] E[Y |t, C = 1]

Causal Graph

E[Y |t]Y

12.9%
E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
c

E[Y |t, c]P (c)

A

Tr
e

at
m

en

t

Condition

B

Brady Neal

C

T

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

10%

20%

19%

(5/50)(100/500)(105/550)

E[Y |t, C = 0] E[Y |t, C = 1]

Causal Graph

E[Y |t]Y

12.9%
E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
cCausal Graph

E[Y |t, c]P (c)

C

A

Tr
e

at
m

en

t

Condition

B

Y

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

1450
600
0.15 +
0.30 ≈ 0.194
2050
2050

10%

20%

19%

(5/50)(100/500)(105/550)

12.9%

1450
600
0.10 +
0.20 ≈ 0.129
2050
2050

E[Y |t, C = 0] E[Y |t, C = 1]

Brady NealT

E[Y |t]E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
cCausal Graph

E[Y |t, c]P (c)

C

A

Tr
e

at
m

en

t

Condition

B

Y

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

1450
600
0.15 +
0.30 ≈ 0.194
2050
2050

10%

20%

19%

(5/50)(100/500)(105/550)

12.9%

1450
600
0.10 +
0.20 ≈ 0.129
2050
2050

E[Y |t, C = 0] E[Y |t, C = 1]

Brady NealT

E[Y |t]E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
cCausal Graph

E[Y |t, c]P (c)

C

A

Tr
e

at
m

en

t

Condition

B

Y

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

1450
600
0.15 +
0.30 ≈ 0.194
2050
2050

10%

20%

19%

(5/50)(100/500)(105/550)

12.9%

1450
600
0.10 +
0.20 ≈ 0.129
2050
2050

E[Y |t, C = 0] E[Y |t, C = 1]

Brady NealT

E[Y |t]E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
cCausal Graph

E[Y |t, c]P (c)

C

A

Tr
e

at
m

en

t

Condition

B

Y

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

1450
600
0.15 +
0.30 ≈ 0.194
2050
2050

10%

20%

19%

(5/50)(100/500)(105/550)

12.9%

1450
600
0.10 +
0.20 ≈ 0.129
2050
2050

E[Y |t, C = 0] E[Y |t, C = 1]

Brady NealT

E[Y |t]E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
E[Y |do(T = t)] = EC E[Y |t, C] =X
cCausal Graph

E[Y |t, c]P (c)

C

A

Tr
e

at
m

en

t

Condition

B

Y

Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

1450
600
0.15 +
0.30 ≈ 0.194
2050
2050

10%

20%

19%

(5/50)(100/500)(105/550)

12.9%

1450
600
0.10 +
0.20 ≈ 0.129
2050
2050

E[Y |t, C = 0] E[Y |t, C = 1]

Brady NealT

E[Y |t]E[Y |do(t)]Causation in observational studies

27 / 28

Application to the COVID-27 example
Naive

50
0.10
550

Tr
e

at
m

en

t

Condition

A
B

500
+
0.20
550

= 0.19Mild

SevereTotal

Causal

15%

30%

16%

(210/1400)(30/100)(240/1500)

19.4%

1450
600
0.15 +
0.30 ≈ 0.194
2050
2050

10%

20%

19%

(5/50)(100/500)(105/550)

12.9%

1450
600
0.10 +
0.20 ≈ 0.129
2050
2050

E[Y |t, C = 0] E[Y |t, C = 1]

Brady Neal

1400
100
0.15 +
0.30 = 0.16
1500
1500

E[Y |t]E[Y |do(t)]Causation in observational studies

27 / 28

Brady Neal

28 / 28