DAGs intro
without exercises
1h
Directed Acyclic Graph
Hein Stigum
http://folk.uio.no/heins/
courses
May-16
H.S.
1
Motivating example
Want the effect of E on D
C
E
(E precedes D)
Observe the two associations
C-E and C-D
Not enough information!
D
Different analyses for:
C
E
Confounder
C
D
E
D
Causal information
Mediator
Need causal information from outside the data to do a proper analysis!
May-16
H.S.
2
Agenda
• Background
• DAG concepts
– Association, Cause
– Confounder, Collider
– Paths
• Analyzing DAGs
– Examples
May-16
H.S.
3
Why causal graphs?
• Problem
– Association measures are biased
• Causal graphs help:
– Understanding
• Confounding, mediation, selection bias
– Analysis
• Adjust or not
– Discussion
• Precise statement of prior assumptions
May-16
H.S.
4
Causal versus casual
CONCEPTS
May-16
H.S.
5
god-DAG
Causal Graph:
Node = variable
Arrow = cause
E=exposure, D=disease
DAG=Directed Acyclic Graph
Read of the DAG:
Causality
= arrows
Associations = paths
Independencies = no paths
Estimations:
E-D association has two parts:
ED
causal effect
keep open
ECUD bias
try to close
E[C]UD Condition (adjust) to close
May-16
H.S.
6
Association
and
Cause
Association
3
possible
causal
Association
3 possible causal structures
structures
3 possible causal structure
Association
1
1
Yellow
Yellow
fingers
fingers
Lung
Lung
cancer
cancer
Cause
Cause
(Reversed cause)
E
Yellow
Yellow
fingers
fingers
Smoke
Smoke
D
Lung
Lung
cancer
cancer
2
2
Yellow
Yellow
fingers
fingers
Confounder
Confounder
Lung
Lung
cancer
cancer
U
U
3
3
Yellow
Yellow
fingers
fingers
Collider
Collider
Lung
Lung
cancer
cancer
+ more complicated structures
May-16
H.S.
7
Confounder idea
A common cause
Smoking
+
Adjust for smoking
Smoking
+
Yellow fingers
Lung cancer
+
Yellow fingers
+
Lung cancer
+
• A confounder induces an association between its effects
• Conditioning on a confounder removes the association
• Condition = (restrict, stratify, adjust)
• Simplest form
May-16
H.S.
8
Collider idea
Two causes for selection to study
Selected
+
Yellow fingers
Selected subjects
Selected
+
Lung cancer
+
+
Yellow fingers
Lung cancer
- or
+ and
• Conditioning on a collider induces an association
between its causes
• “And” and “or” selection leads to different bias
• Simplest form
May-16
H.S.
9
Data driven analysis
C
E
D
- Want the effect of E on D (E precedes D)
- Observe the two associations C-E and C-D
- Assume statistical criteria dictates adjusting for C
(likelihood ratio, Akaike (赤池 弘次) or 10% change in estimate)
The undirected graph above is compatible with three DAGs:
C
C
E
D
Confounder
1. Adjust
Conclusion:
May-16
E
C
D
Mediator
2. Direct: adjust
3. Total: not adjust
E
D
Collider
4. Not adjust
The data driven method is correct in 2 out of 4 situations
Need information from outside the data to do a proper analysis
H.S.
10
The Path of the Righteous
Paths
May-16
H.S.
11
Path definitions
Path: any trail from E to D (without repeating itself)
Type: causal, non-causal
State: open, closed
1
2
3
4
Four paths:
Path
ED
EMD
ECD
EKD
Goal:
Keep causal paths of interest open
Close all non-causal paths
May-16
H.S.
12
Four rules
1. Causal path: ED
(all arrows in the same direction) otherwise non-causal
Before conditioning:
2. Closed path: K
(closed at a collider, otherwise open)
Conditioning on:
3. a non-collider closes: [M] or [C]
4. a collider opens:
[K]
(or a descendant of a collider)
May-16
H.S.
13
ANALYZING DAGs
May-16
H.S.
14
Confounding examples
May-16
H.S.
15
Physical activity and
Coronary Heart Disease (CHD)
1. We want the total effect of
Physical Activity on CHD. What
would we adjust for?
Unconditional
Path
1 ED
2 EC1D
3 EC2D
Type
Causal
Non-causal
Non-causal
Status
Open
Open
Open
Conditioning on C1
and C2
Path
1 ED
2 EC1]D
3 EC2]D
Type
Causal
Non-causal
Non-causal
Status
Open
Closed
Closed
May-16
May-16
Bias
H.S.
This is an example
of confounding
No bias
16
Intermediate variables
May-16
H.S.
17
Tea and depression
O
coffee
caffeine
E
tea
Path
1 E→D
2 E→C→D
3 E←O→C→D
May-16
C
Total effect:
adjust for O
Direct effect:
adjust for C (and O)
Caffeine
intermediate or
confounder?
Caffeine is both
intermediate and part
of a confounder path.
D
depression
Type
Status
Causal
Open
Causal
Open
Non-causal Open
direct
indirect
H.S.
total
18
Statin and CHD
We want the total effect of statin
on CHD. What would you adjust
for?
Nothing
Can we estimate the direct effect
of statin on CHD (not mediated
through cholesterol)?
No, U is
unmeasured
Unconditional
Path
1 ED
2 ECD
3 ECUD
Type
Causal
Causal
Non-causal
Status
Open
Open
Closed
Conditioning on C
Path
1 ED
2 EC]D
3 EC]UD
Type
Causal
Causal
Non-causal
Status
Open
Closed
Open
May-16
H.S.
No adjustments gives
the total effect
Adjusting for C will close path 2
but will open path 3 and give bias!
C is a collider on path 3
19
Two concepts
Selection bias
May-16
H.S.
20
Convenience sample, homogenous sample
H
1. Convenience sample:
Conduct the study among
hospital patients?
hospital
E
D
2. Homogeneous sample:
Population data,
exclude hospital patients?
fractures
diabetes
Conditional
Unconditional
Path
1 E→D
2 E→H←D
E→[H]←D
Type
Causal
Non-causal
Non-Causal
Status
Open
Closed
Open
Collider, selection bias
Collider stratification bias: at least on stratum is biased
This type of selection bias can be adjusted for!
May-16
H.S.
21
Summing up
• Data driven analyses do not work. Need (causal) information
from outside the data.
• DAGs are intuitive and accurate tools to display that
information.
• Paths show the flow of causality and of bias and guide the
analysis.
• DAGs clarify concepts like confounding and selection bias,
and show that we can adjust for both.
Better discussion based on DAGs
Draw your assumptions
before your conclusions
May-16
H.S.
22
Recommended reading
• Books
–
–
–
–
–
Hernan, M. A. and J. Robins. Causal Inference. Web:
Rothman, K. J., S. Greenland, and T. L. Lash. Modern Epidemiology, 2008.
Morgan and Winship, Counterfactuals and Causal Inference, 2009
Pearl J, Causality – Models, Reasoning and Inference, 2009
Veierød, M.B., Lydersen, S. Laake,P. Medical Statistics. 2012
• Papers
– Greenland, S., J. Pearl, and J. M. Robins. Causal diagrams for epidemiologic
research, Epidemiology 1999
– Hernandez-Diaz, S., E. F. Schisterman, and M. A. Hernan. The birth weight
"paradox" uncovered? Am J Epidemiol 2006
– Hernan, M. A., S. Hernandez-Diaz, and J. M. Robins. A structural approach to
selection bias, Epidemiology 2004
– Berk, R.A. An introduction to selection bias in sociological data, Am Soc R 1983
– Greenland, S. and B. Brumback. An overview of relations among causal modeling
methods, Int J Epidemiol 2002
– Weinberg, C. R. Can DAGs clarify effect modification? Epidemiology 2007
May-16
H.S.
23