04/19/23 H.S. 104/19/23 H.S. 104/19/23 H.S. 1

Directed Acyclic GraphsDAGs

Hein Stigum

http://folk.uio.no/heins/

04/19/23 H.S. 204/19/23 H.S. 204/19/23 H.S. 2

Graphical Models

• Concepts– Confounder, Collider

– Conditional independence

– Undirected graphs, separation

• Directed Acyclic Graphs (DAGs)– Graphic tools: d-separation

– Examples

– Selection bias examples

04/19/23 H.S. 304/19/23 H.S. 304/19/23 H.S. 3

Concepts

“Close to the edge”

04/19/23 H.S. 404/19/23 H.S. 404/19/23 H.S. 4

Confounder, Collider

– C is a common cause of E and D

– C is a common effect of E and D (bias if we condition on C)

04/19/23 H.S. 4

E D

C

E D

C

Confounder

Collider

Condition on: adjust stratify

Bias: Selection Information Confounding Confounder

Collider

Hernan et al, A structural approach to selection bias, Epidemiology 2004

04/19/23 H.S. 504/19/23 H.S. 5

Collider idea

• Conditioning on a collider induces an association between the causes

• Condition = (restrict, stratify, adjust)• Bias direction?

Hip arthritis

Limp

Knee injury

Two causes for limping

-

++

Hip arthritis

Limp

Knee injury

Select limping subjects

++

04/19/23 H.S. 604/19/23 H.S. 604/19/23 H.S. 6

Marginal and conditional dependence

AllE- 10 %E+ 14 %

DiseaseAll C- C+

E- 10 % 3 % 21 %E+ 14 % 3 % 21 %

Disease

E and D are:Marginallydependent

ConditionallyIndependent | C

04/19/23 H.S. 704/19/23 H.S. 704/19/23 H.S. 7

Undirected Graph, separation

1

2 3

4 5Are 1 and 5 separated by 2 ?Are 1 and 5 separated by 3 ?Are 1 and 5 separated by 3 and 4 ?

Yes

If 1 and 5 are separated by 2, then 1 and 5 are conditionally independent given 2

Definition: “1” and “5” are separated by “2” if all paths from “1” to “5” pass thru “2”

If 1 and 5 are separated by 2,path from 1 and 5 is blocked by 2

NoYes

04/19/23 H.S. 804/19/23 H.S. 804/19/23 H.S. 8

Graphic tools:d-separation

“We have the moral edge!”

04/19/23 H.S. 904/19/23 H.S. 904/19/23 H.S. 9

No bias

No E-D bias

if

E and D are conditionally independent under H0, given the variables we adjust for (C)

E D

C

E D

C

OK Not OK

Need graphic tools:“d-separation”

04/19/23 H.S. 1004/19/23 H.S. 1004/19/23 H.S. 10

D-separation

1

2 4

3 5

Are 1 and 4 separated by 2 ? No

Steps:1. Take ancestral graph of {1,2,4}2. Moralize the graph3. Look for separation

04/19/23 H.S. 1104/19/23 H.S. 1104/19/23 H.S. 11

D-separation, cont

1

2 4

3 5 Are 1 and 4 separated by 2 ? No

Take ancestral graph of {1,2,4}1

2 4

3

Moralize the graph

1

2 4

3

Are 1 and 4 separated by 2 ? No

04/19/23 H.S. 1204/19/23 H.S. 1204/19/23 H.S. 12

Examples

04/19/23 H.S. 13

1. Ancestral graph of E,D,C2. Moralize3. Separated?

1. Ancestral graph of E,D2. Moralize3. Separated?

Vitamine example

Bias in E-D:

Evitamine

Dbirth defects

Cage

Uobesity

E ╨ Dno

Evitamine

Dbirth defects

Cage

Uobesity

Bias in E-D?Adjust for C?

Adjust for C :

E ╨ D|C yesE

vitamineD

birth defects

Cage

Uobesity

04/19/23 H.S. 14

Conditions giving confounding

Evitamine

Dbirth defects

Cage

Uobesity

Evitamine

Dbirth defects

Cage

Uobesity

Direct C-D effect?Path still blocked undirected graph!

Unmeasured U1 leads to age and vitamineNot logical (age), path still blocked!

Unmeasured U1 leads to obesity and vitamineNew path via U1 and U, Bias!

U1

U1

04/19/23 H.S. 1504/19/23 H.S. 1504/19/23 H.S. 15

Selection bias examples

04/19/23 H.S. 16H.S.

HCB and SGA

SGA HCB

SBreast

SGA HCB

SBreast

Directed Acyclic Graph

Moralized undirected graph

Conclusion:Selection bias: HCB and SGA are not independent under selection.Adjusting for breastfeeding blocks the bias.

Breastfeeding leads to higher participationSGA leads to lower participationSelection bias?Adjust for breastfeeding?

04/19/23 H.S. 17H.S.

Conditions giving selection bias

SGA HCB

SBreastU1

SGA HCB

SBreast

HCB leads to higher participation?Unlikely!

U1

U2

U2

Unmeasured U1 leads to HCB and SNo new path in UDG!

Unmeasured U2 leads to SGA and SNew path via U1 and U2, Bias!

04/19/23 H.S. 18

Cohort: Differential loss to follow up

• Loss to follow up:Severe symptoms

Side effects of therapy

• Questions:Bias in the E-D effect?

Adjust for symptoms?

Etherapy

DAIDS

Sfollow up

Uimmuno sup

Vsympt

1. Take ancestral graph of {E,D,S}2. Moralize3. Separation by S?4. Separation by S,V?

No

Yes

Etherapy

DAIDS

Sfollow up

Uimmuno sup

Vsympt

04/19/23 H.S. 19

Self-selection bias

• Self-selection:Awareness of disease

Awareness of smoke effect

• Bias:Biased in the sample

May adjust for awareness

Esmoke

DCHD

SSelf selection

Ufamily hist

Vawareness

04/19/23 H.S. 20

Berkson’s bias

• Selection:Cases from hospital

Controls also from hospital

• Bias:Any cause of D2 will be

associated with D in the sample

Eexposure

Dcases

Shospital

D2controls

E D

S

not D

• Proper control selection:Controls sampled independent of

exposure

• Problem:E-D bias for linear models???

04/19/23 H.S. 21

Healthy worker effect

• Working:Good health (proven by test)

Exposed with symp. leave work

• Bias:Negative bias in the sample

Echemical

Dmortality

Sworking

Uhealth

Vtest

Echemical

Dmortality

Sworking

Uhealth

Vtest

+ -

-

++

biased true

Negative bias

0

04/19/23 H.S. 22

Exercise, Firefighters

• Show:No E-D bias in the pop. E ╨ D under H0

E-D bias in the sample. E ╨ D|S under H0

Adj for V does help. E ╨ D|S,V under H0

• Draw DAG:Study exercise on CHD among firefighters

High socioeconomic status have less CHD and less firefighters

Unmeasured personality have more exercise and more firefighters

04/19/23 H.S. 23

DAG

No E-D bias in the pop. E ╨ D under H0

E-D bias in the sample. E ╨ D|S under H0

Adj for V does help.

E ╨ D|S,V under H0

Eexercise

DCHD

Sfirefighter

Vsocio stat.

Upersonality

Eexercise

DCHD

Vsocio stat.

Upersonality

Eexercise

DCHD

Sfirefighter

Vsocio stat.

Upersonality

Eexercise

DCHD

Sfirefighter

Vsocio stat.

Upersonality

04/19/23 H.S. 24

Inverse probability weighting

• Idea:Each subject not lost should weigh

up for those lost (with the same E,V values)

• Method:In (E=0, V=1), 4 subject, 3 lost,

1 left: s=1/4, w=4

4 org. pop. 4 copies pseudo pop.

• ResultPseudo pop. measure is unbiased

given the DAG, (w|V,E)

More general than adjusting for V

Etherapy

DAIDS

Sfollow up

Uimmuno sup

Vsympt

04/19/23 H.S. 25

Exercise, AIDS

• Show:No E-D bias in the pop. E ╨ D

E-D bias in the sample. E ╨ D|S

Adj for V does not help. E ╨ D|S,V

Etherapy

DAIDS

Sfollow up

Uimmuno sup

Vsympt

Inverse probability weighting would still work here(w|V)

04/19/23 H.S. 26H.S.

Lifestyle, diet

SGA HCB

SLifestyle

DietDirected Acyclic Graph

Moralized undirected graph

SGA HCB

SLifestyle

Diet

Conclusion:HCB and SGA are not independent under selectionunless we condition (adjust) on lifestyle or diet