1

PH 240A: Chapter 8

Mark van der LaanUniversity of California Berkeley

(Slides by Nick Jewell)

2

Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions

Group Coffee (E )

No Coffee ( )

#

1 D D

2 D

3 D

4

E

D

DD

D

4Np

3Np

1Np

2Np

31

21

pp

ppRRcausal +

+=

)(1

)(1

31

31

21

21

pppppppp

ORcausal

+−++−

+

= )()( 3121 ppppERcausal +−+=

3

Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions

Group Coffee (E )

No Coffee ( )

#

1 D D

2 D

3 D

4

E

D

DD

D

4Np

3Np

1Np

2Np

Suppose Group 1 and 3 individuals are all coffee drinkersand Group 2 and 4 individuals abstain, then . . .

4

Population Data

Pancreatic Cancer

D Not D

Coffee Drinking

(cups/day)

(E)

0 (not E) 0

∞==RROR

1≥ 1Np

)( 42 ppN +

3Np

1Np )( 432 pppN ++

)( 31 ppN +

)( 42 ppN +

5

Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions

Group Coffee (E )

No Coffee ( )

#

1 D D

2 D

3 D

4

E

D

DD

D

4Np

3Np

1Np

2Np

Suppose Group 1 and 2 individuals are all abstainersand Group 3 and 4 individuals drink coffee, then . . .

6

Population Data

Pancreatic Cancer

D Not D

Coffee Drinking

(cups/day)

(E) 0

0 (not E)

! 0==RROR

1≥1Np

)( 43 ppN +

2Np

1Np )( 432 pppN ++

)( 21 ppN +

)( 43 ppN +

7

All is Lost?

Let’s quit—the first four weeks of the class have been a total waste of time . . .?

8

Counterfactual Pancreatic Cancer Responses to Coffee Drinking Conditions

Group Coffee (E )

No Coffee ( )

#

1 D D

2 D

3 D

4

E

D

DD

D

4Np

3Np

1Np

2Np

Suppose individuals choose whether to drink coffee or not at random, (say, toss a coin) then . . .

9

Population Data Under Random Counterfactual Observation

Pancreatic Cancer

D Not D

Coffee Drinking

(cups/day)

(E)

0 (not E)

causalOROR =

1≥ 2/N

2/)( 42 ppN +

2/)2( 432 pppN ++

2/)( 21 ppN + 2/)( 43 ppN +

2/)( 31 ppN + 2/N

2/)2( 321 pppN ++

causalRRRR =

causalERER =

10

Confounding Variables

Randomization assumption Probability of observing a specific exposure

condition (eg coffee drinking or not) must not depend on counterfactual outcome pattern (i.e. vary across groups)

Failure of randomization assumption Group 1 individuals are more likely to be

males than say Group 4 individuals If males are also more likely to drink coffee,

then we are more likely to observe the coffee drinking counterfactual in Group 1 than Group 4

11

Confounding Variables

For example, imagine a world where all males, and only males, drank coffee

Pancreatic Cancer

D Not D

Coffee Drinking

(cups/day)

males

0 females

1≥

Even if coffee had no effect we would observean association (due to sex)sex is a confounder

12

Confounding Variables

Conditions for confounding C must cause D C must cause E

C

E D?

13

Stratification to Control Confounding

Divide the population into strata defined by different levels of C Within a fixed stratum there can be

no confounding due to C New issue: causal effects of E may

vary across levels of C• Interaction or effect modification

When no interaction, need methods to combine common causal effects across C strata (Chapter 9)

14

Causal Graph Approach to Confounding

Possible causal effects of childhood vaccination on autism access to general medical care may affect autism

incidence and/or diagnosis Access to medical care increases vaccination Family SES influences access to medical care and

also ability to pay for vaccination Family medical history may affect risk for autism and

may also influence access to medical care Which of medical care access, SES, and family

history are confounders? Do we need to stratify on all three?

15

Directed Acyclic Graphs

Nodes, directed graphs (edges have direction)

Directed paths: B-A-DB-D-A, C-B-D

B

A D

C

16

Directed Acyclic Graphs

Acyclic No loops: A cannot cause itself Mother’s Smoking Status

Child’s Respiratory Condition

Mother’s Smoking Status (t=0) Mother’s Smoking Status (t=1)

Child’s Respiratory Condition (t=0) Child’s Respiratory Condition (t=0)

node A at the end of a directed path starting at B is a descendant of B(B is an ancestor of A)

17

Directed Acyclic Graphs

A node A can be a collider on a specific pathway if the path entering and leaving A both have arrows pointing into A. A path is blocked if it contains a collider.

D is a collider on the pathway C-D-A-F-B; this path is blocked

BF

D

C

A

18

Using Causal Graphs to Detect Confounding

Delete all arrows from E that point to any other node

Is there now any unblocked backdoor pathway from E to D? Yes—confounding exists No—no confounding

19

Vaccination & Autism Example

Medical Care Access

Vaccination

Autism

Family History

SES

20

Using Causal Graphs to Detect Confounding

FC

DE

FC

DE

FC

DE

FC

DE

21

Checking for Residual Confounding

After stratification on one or more factors, has confounding been removed? Cannot simply remove stratification

factors and relevant arrows and check residual DAG

Have to worry about colliders

22

Controlling for Colliders

Stratification on a collider can induce an association that did not exist previously

Rain

Sprinkler Wet Pavement

Diet sugar (B)

Fluoridation (A)

Tooth Decay (D)

23

Hypothetical Data on Water Flouridation,

High-Sugar Diet, and Tooth Decay Tooth Decay (D)

Flouridation D OR ER

A High-Sugar Diet

B 160 40

2.67 0.2

120 80

High-Sugar Diet

B 80 120

2.67 0.2

40 160

Tooth Decay (D)

High-Sugar Diet D

B Fluoridation A 160 40 6.00 0.4

80 120

Fluoridation A 120 80 6.00 0.4

40 160

Pooled table Fluoridation

A

High-Sugar Diet

B 200 200 1.00 0.00

200 200

D

A

BA

B

B

A

D

B

A

24

Hypothetical Data on Water Flouridation,

High-Sugar Diet, and Tooth Decay

Tooth Decay (D)

Fluoridation

A OR ER

B 160 80 0.67 -0.083

120 40

No Tooth Decay ( )

Fluoridation

A OR ER

B 40 120 0.67 -0.083

800 160

B

A

B

A

D

25

Checking for Residual Confounding

Delete all arrows from E that point to any other node

Add in new undirected edges for any pair of nodes that have a common descendant in the set of stratification factors S

Is there still any unblocked backdoor path from E to D that doesn’t pass through S ? If so there is still residual confounding, not accounted for by S .

26

Vaccination & Autism Example

Medical Care Access

Vaccination

Autism

Family History

SES

27

Vaccination & Autism Example: Stratification on Medical Care Access

Vaccination

Autism

Family History

SES

Still confounding: need to stratify additionally on SES or Family History, or both

28

Caution: Stratification Can Introduce Confounding!

C

E D

F

No Confounding

Stratification on C introduces confounding!

29

Collapsibility

No Confounding and No Interaction

500120280

400

14

86 100134

366

60340

400

7 93 100 67 433 500

00.2=CRR

CC

D

E

E E

E

00.2=CRR 00.2=RR

D D D

E

E

DD

Pooled

30

Collapsibility

Confounding and No Interaction

820216504

720

14

86 100230

590

12 68 80 7 93 100 19 161 1800

00.2=CRR

CC

D

E

E E

E

00.2=CRR 66.2=RR

D D D

E

E

DD

Pooled

31

Collapsibility and Confounding

With the assumption of the causal graph below, and in the absence of interaction, the conditions for collapsibility wrt RR (and ER) are the same as for no confounding (i.i either C & E are independent, or C & E are independent, given E, or both)

C

E D?

But note that collapsibilitycannot distinguish directions of the arrows

C

E D?

32

Collapsibility

No Confounding and Interaction

50060340

400

14

86 100 74426

120280

400

7 93 100127

373

500

50.0=CRR

CC

D

E

E E

E

00.2=CRR 58.0=RR

D D D

E

E

DD

Pooled

has causal interpretation

33

Collapsibility

Confounding and Interaction

820108612

720

14

86 100122

698

24 56 80 7 93 100 31149

180

50.0=CRR

CC

D

E

E E

E

00.2=CRR 86.0=RR

D D D

E

E

DD

Pooled

has no causal interpretation

34

Collapsibility and OR

Collapsibility and “No Confounding” not quite the same thing for OR