DAGs intro
with exercises
6h
DAGs = Directed Acyclic Graphs
Hein Stigum
http://folk.uio.no/heins/
courses
May-16
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1
Agenda
• Background
• DAG concepts
– Association, Cause
– Confounder, Collider
– Paths
Exercises
• Analyzing DAGs
– Examples
• More on DAGs
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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
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Causal versus casual
CONCEPTS
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Precision and validity
• Measures of populations
– precision - random error - statistics
– validity - systematic error - epidemiology
Precision
Bias
True
value
Estimate
DAGs only consider bias
29.05.2016
H.S.
5
god-DAG
Causal Graph:
Node = variable
Arrow = cause
E=exposure, D=disease
DAG=Directed Acyclic Graph
Read of the DAG:
Estimations:
Causality
= arrows
Associations = paths
Independencies = no paths
E-D association has two parts:
ED
causal effect
keep open
ECUD bias
try to close
Conditioning (Adjusting): E[C]UD
Time 
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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
Smoke
Smoke
Yellow
Yellow
fingers
fingers
Lung
Lung
cancer
cancer
2
2
Yellow
Yellow
fingers
fingers
Confounder
Confounder
Lung
Lung
cancer
cancer
U
U
3
3
Yellow
Yellow
fingers
fingers
Cause
Cause
Collider
Collider
Lung
Lung
cancer
cancer
+ more complicated structures
May-16
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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
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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
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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 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
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The Path of the Righteous
Paths
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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
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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)
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ANALYZING DAGs
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Confounding examples
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Vitamin and birth defects
1. Is there a bias in the crude E-D effect?
2. Should we adjust for C?
3. What happens if age also has a direct effect on D?
Unconditional
Path
1 ED
2 ECUD
Type
Status
Causal
Open
Non-causal Open
Conditioning on C
Path
Type
Status
1 ED
Causal
Open
2 EC]UD Non-causal Closed
May-16
May-16
Bias
This is an example
of confounding
No bias
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16
Exercise: Physical activity and
Coronary Heart Disease (CHD)
1. We want the total effect of
Physical Activity on CHD. What
should we adjust for?
5 minutes
May-16
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Direct and indirect effects
Intermediate variables
May-16
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Exercise: Tea and depression
1. Write down the paths.
O
coffee
E
tea
2. You want the total effect of
tea on depression. What
would you adjust for?
C
caffeine
D
depression
3. You want the direct effect
of tea on depression. What
would you adjust for?
4. Is caffeine an intermediate
variable or a confounder?
10 minutes
May-16
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Exercise: Statin and CHD
C
cholesterol
E
U
lifestyle
D
CHD
statin
1. Write down the paths.
2. You want the total effect of
statin on CHD. What would
you adjust for?
3. If lifestyle is unmeasured, can
we estimate the direct effect of
statin on CHD (not mediated
through cholesterol)?
4. Is cholesterol an intermediate
variable or a collider?
10 minutes
May-16
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Direct and indirect effects
U2
M
E
U3
• Total effect:
– no unmeasured U1
– no unmeasured U2
+
D
• Direct and indirect effects
– (linear model)
– no unmeasured U3
U1
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Confounder, collider and mediator
Mixed
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Diabetes and Fractures
We want the total effect of
Diabetes (type 2) on fractures
Conditional
Unconditional
Path
Path
11 E→D
E→D
22 E→F→D
E→F→D
33 E→B→D
E→B→D
44 E←[V]→B→D
E←V→B→D
55 E←[P]→B→D
E←P→B→D
May-16
Type
Type
Causal
Causal
Causal
Causal
Causal
Causal
Non-causal
Non-causal
Non-causal
Non-causal
Status
Status
Open
Open
Open
Open
Open
Open
Closed
Open
Closed
Open
H.S.
Questions:
Mediators
Paths ←→?
More paths?
B a collider?
V and P ind?
Confounders
23
Selection bias
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Selection bias: concept 1
S
age
smoke
CHD
Age
Young
Old
All
Population RRsmoke Selected RRsmoke
50 %
4.0
75 %
4.0
50 %
2.0
25 %
2.0
3.0
3.5
• Properties:
Name:
interaction based?
- Need heterogeneity of smoke effect
- Can not be adjusted for
- True RR=weighted average of stratum effects
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Selection bias: concept 2
S
sex
age
smoke
CHD
Paths
Type
Status
smokeCHD
Causal
Open
smokesexSageCHD Non-causal Open
Properties:
Name:
Collider stratification bias
Does not need heterogeneity
Can be adjusted for (sex and age)
(Some counterintuitive results)
Hernan et al, A structural approach to selection bias, Epidemiology 2004
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Diabetes and Fractures
1. Population based:
Path
1 E→D
2 E→H←D
Type
Causal
Non-causal
Status
Open
Closed
OK
2. Convenience:
Conduct the study among
hospital patients?
3. Homogeneous sample:
Path
1 E→D
2 E→[H]←D
Type
Status
Causal
Open
Non-Causal Open
Exclude hospital patients
Collider stratification bias:
at least on stratum is biased
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Selection examples (collider stratification)
“Or”-selection:
E
H hospital
2.0
← sufficient causes for hospital
Population
RR=
2.0
3.0
Hospital
RR= 1.6
Not-hospital
RR= 1.5
D
E
diabetes
D
2.0
fractures
Bias
“And”-selection:
E
D ← sufficient causes for hospital
H hospital
1.8
1.4
May-16
Hospital
RR=
4.1
Not-hospital
RR=
1.3
D
E
diabetes
Population
RR=
2.0
2.0
fractures
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Exercise: Survivior bias
• Study exposure early in life (E) on later
disease (D) among survivors (S)
• Early exposure decreases survival
• A risk factor (R) increases later disease
(D) and reduces survival (S)
Draw and analyze the DAG
10 minutes
May-16
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Milk intake and bone density
• Study
S
calcium supp.
E
D
milk
bone density
S
C
calcium supp.
family history
E
D
milk
bone density
Structure:
Collider stratification
May-16
– Milk on bone density
– Exclude Calcium supplements?
Path
1 ED
2 E[S]D
Type
Status
Causal
Open
Non-causal Open
2 E[S][C]D Non-causal Closed
Lessons learned:
Biological effect not protected
May adjust for selection bias
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Examples
S
C
respond
education
E
D
alcohol
CHD
S
C
loss to follow up
smoking
E
D
drug
disease
• Differential response
– Survey: Alcohol and CHD
• Differential loss to follow up
– Randomized trial: drug and disease
Note: no confounding
Hernan et al, A structural approach to selection bias, Epidemiology 2004
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Exercise: Dust and lung disease
• Background
– We are interested in the effect of dust and heat exposure on
lung disease, and do a cross-sectional study among workers
in aluminum melt halls.
• Assumptions
– Workers who are sensitive to dust and heat are more likely
leave melt hall work. Subjects with general good health
(genes) are more likely to keep the job, and less likely to
develop lung disease.
• Draw and analyze the DAG
– What type of bias do you get?
– Could the ”interaction based” selection bias also operate
here?
15 minutes
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Information bias
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Depicting measurement error
a)
•
UE
– E=true exposure
– E*=measured exposure
– UE=process giving error in E
E*
b)
Error in E
E
D
UE
UD
E*
D*
E
D
•
Error in E and D
– D=true disease
– D*=diagnosed disease
– UD=process giving error in D
a) and b) Can test H0
b) shows independent non-differential errors
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Dependent errors, differential errors
a)a)a)
b)b)b)
c)c)c)
UED
UED
UED
UEUEUE
UDUDUD
UEUEUE
UDUDUD
UEUEUE
UDUDUD
E*E*E* D*D*
D*
E*E*E* D*D*
D*
E*E*E* D*D*
D*
EEE
EEE
EEE
DDD
Dependent errors:
Temp. in lab
DDD
Differential error:
Recall bias in
Case-Control study
DDD
Differential error:
Investigator bias
in cohort study
Hernan and Cole, Causal Diagrams and Measurement Bias, AJE 2009
May-16
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Randomized experiments
Mendelian randomization
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Strength of arrow
E
Not
deterministic
C1
D
C2
C1, C2, C3 exogenous
C3
C
R
full
compliance
deterministic
E
Randomization:
Full compliance
 no E-D confounding
D
U
R
not full
compliance
E
D
Sub analysis conditioning on E
may lead to bias
May-16
Not full compliance
 weak E-D confounding
but R-D is unconfounded
Path
1 RED
2 REUD
H.S.
Type
Status
Causal
open
non-causal closed
37
Randomized experiments
C
Observational study
E
D
C
R
E
D
U
Rc E
b
Randomized experiment with full compliance
R= randomized treatment
E= actual treatment, R=E
Randomized experiment with less than full
compliance (c)
D
If linear model: a=c*b, c<1
a
Intention To Treat (a):
effect of R on D (unconfounded)
population
Per Protocol:
crude effect of E on D (confounded by U)
Instrumental Variable (b): adjusted effect of E on D (if c is known, 2SLS) individual
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R
C
andomized
T example
ontrolled
rial
E
R
+
-
+
93
0
U
7
100
RD
0.93
0
57
38
RD
-0.20
0
ITT
59
37
RD
-0.22
0
PP
-0.12
c
R
D
R
+
-
+
43
62
c
IV
E
0.09
D
ITT
𝐼𝑉 =
𝐼𝑇𝑇
0.20
= −
= −0.21
𝑐
0.93
D
+
E
-
+
41
63
Stata command for IV:
ivregress (2SLS)
True value
Negative bias
from confounding
May-16
ITT always weaker than IV
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Mendelian randomization
•
U
Observational study
–
•
Suffers from unmeasured confounding
Randomized trial:
3 conditions
1. R affects E:
2. No direct R-D effect:
3. R and D no common causes:
•
E
D
U
3
balanced, strong effect
R independent of D | E
R
1
R independent of U
D
2
Medelian randomization: 3 conditions
1. G must affect E:
unbalanced, weak  large N
2. No direct G-D effect:
depends on gene function
3. G and D no common causes: Mendel’s 2. law
E
U
3
G
1
E
D
2
Sheehan et al, Mendelian Randomisation and Causal Inference in Observational Epidemiology, PLoS Med 2008
May-16
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40
Ex: Alcohol and blood pressure
•
U
Observational study
–
–
Alcohol use increases blood pressure
Many ”lifestyle” confounders
A
BP
Alcohol use
Gene: ALDH2, 2 alleles
–
–
alcohol ml/day
•
2,2 type suffer nausea, headache after alcohol
 low alcohol regardless of lifestyle (U)
40
30
20
10
0
1,1
•
1,2
Genotype
Medelian randomization
1. Gene ALDH2 is highly associated with alcohol
2. OK, gene function is known
3. Mendel’s 2. law, no ass. to obs. confounders
–
–
Result: 1,1 type BP +7.4 mmHg
Alcohol increases blood pressure
U
3
G
1
A
BP
2
Chen et al, Alcohol Intake and Blood Pressure, PLoS Med 2008
May-16
May-16
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H.S
2,2
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MORE ON DAGs
May-16
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Methods of adjusting
Method
Action
Effect
C
Conditioning, Stratification
Close path
E
D
C
Matching in cohort
Remove arrow
E
Matching in Case-Control
May-16
C
Remove arrow
E
InverseProbabilityWeighting ~ matching
H.S.
D
D
43
Confounding versus selection bias
Path:
Any trail from E to D (without repeating itself)
Open non-causal path = biasing path
Confounding and selection bias not always distinct
May use DAG to give distinct definitions:
C
A
E
A B D
Causal
C
B
A
B
E
D
Confounding:
E
D
Selection bias:
Non-causal path
without colliders
Non-causal path open
due to conditioning
on a collider
Note: interaction based selection bias not included
Hernan et al, A structural approach to selection bias, Epidemiology 2004
May-16
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Exercise: M-structure
A
B
C
E
D
1. Show the paths
2. Should we adjust for C?
3. If the design implies a selection on C, what would you
call the resulting bias: selection bias or confounding?
5 minutes
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Variables and arrows
• Variable
at least two values
• 
cause, almost any causal definition will work
• ED
usually on the individual level, “at least
one subject with an effect of the exposure”
• ?  age
only possible on group level
• ED
the dose response can be linear, threshold,
U-shaped or any other (DAGs are nonparametric)
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Causal graphs: definitions
• Causal graph
– Graph showing causal relations and conditional
independencies between variables
– G={V,E}
• Vertices=random variables
• Edges=associations or cause
– Edges
 undirected or → directed
– Path
• Sequence of connected edges:
• Parent → child
• Ancestors → descendants
– Exogenous: variables with no parents
May-16
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L
A
Y
U
[(L,A),(A,Y)]
L,U
47
D
irected
A G
cyclic
raphs
• Ordinary DAG
– Arrows = associations
L
• Causal DAG
A
Y
U
1. Arrows = cause
2. All common causes of any pair of variables in the
DAG are included
• Two types of variables
– Immutable
– Mutable
sex, age
exposure (actions), smoking
• Mixing variables in a DAG is OK
– All dependence/independence conclusions valid
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DAGs and probability theory
• Two assumptions
– Causal Markov assumption
• Factorize distributions
– Faithfulness assumption
• No perfect cancellations of effects
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D-separation, moralization
• Directed graph-separation
– two variables d-separated
– otherwise d-connected
if no open path
• 2 DAG analyses
– Paths
(Pearl)
– Moralization (Lauritzen)
– equivalent
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3 strategies for estimating causal effects
C
• Back-door criterion
–
Condition to close all no-causal paths
E
(between E and D)
D
U
• Front-door criterion
–
Condition an all intermediate variables
M1
M2
E
(between E and D)
• InstrumentalVariables
–
–
1.
2.
3.
IV
IV must affect E
No direct IV-D effect
IV and D no common causes
U
3
Use an IV to control the effect (of E on D)
IV criteria:
1
E
D
2
Pearl 2009, Glymor and Greenland, 2008
May-16
May-16
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H.S
D
51
51
Example: front–door criterion
• Weight and CoronaryHeartDisease
• Assume:
– adjusted for sex, age and smoke
– lifestyle is unmeasured
– no other mediators (between E and D)
U
lifestyle
C
E
cholesterol
weight
B
D
CHD
blood pressure
• Can estimate effect of E on D
•
Weight is not a good “action”
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DAGs and causal pies
DAG
Sufficient causes
E1
E1
1. E1 or E2
E2
D
E1 E2 2. E1 and E2
E2
E1
E2
E1 E2 3. both
DAGs are less specific than causal pies
DAGs are scale free, interaction is scale dependent
Greenland and Brumback, Causal modeling methods, Int J Epid 2002
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Exercise: causal pies
1. Write down the causal pies for getting
into hospital based on the DAG.
Show that the DAG is compatible with
at least 3 different combinations of
sufficient causes.
H
hospital
E
diabetes
D
fractures
2. Selection bias: Discuss how the
different combinations of sufficient
causes for getting into hospital might
affect the estimate of E on D among
hospital patients (perhaps difficult).
10 minutes
May-16
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Limitations and problems of DAGs
• New tool
relevance debated, focus on causality
• Focus on bias
precision also important
• Bias or not
magnitude and direction
• Interaction
scale dependent
• Drawing
capture reality, large enough to be realistic
small enough to be useful
VanderWeele, The sign of the bias of unmeasured confounding. Biometrics 2008
VanderWeele, Causal directed acyclic graphs and the direction of unmeasured confounding bias" Epidemiology 2008
VanderWeele, Directed acyclic graphs, sufficient causes, and the properties of conditioning on a common effect." Am J Epid 2007
May-16
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Drawing DAGs
May-16
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Technical note on drawing DAGs
• Drawing tools in Word (Add>Figure)
• Use Dia
• Use DAGitty
• Hand-drawn figure.
May-16
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Direction of arrow
C
Smoking
?
E
D
Phys. Act.
Diabetes 2
H
C
Health con.
Smoking
E
D
Phys. Act.
Diabetes 2
May-16
Does physical activity reduce smoking,
or
does smoking reduce physical activity?
Maybe an other variable
(health consciousness)
is causing both?
H.S.
58
Time
C
Smoking
?
E
D
Phys. Act.
Diabetes 2
Does physical activity reduce smoking,
or
does smoking reduce physical activity?
C
Smoking -5
E
D
Phys. Act. -1
Diabetes 2
May-16
Smoking measured 5 years ago
Physical activity measured 1 year ago
H.S.
59
Drawing a causal DAG
Start: E and D
add: [S]
add: C-s
1 exposure, 1 disease
variables conditioned by design
all common causes of 2 or more
variables in the DAG
C
V
E
D
C must be included
V may be excluded
M may be excluded
K may be excluded
common cause
exogenous
mediator
unless we condition
M
K
May-16
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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
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Recommended reading
• Books
– Hernan, M. A. and J. Robins. Causal Inference. Web:
– Rothman, K. J., S. Greenland, and T. L. Lash. Modern Epidemiology. 2008.
– 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
– 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
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References
•
Chen, L., et al. "Alcohol intake and blood pressure: a systematic review implementing
a Mendelian randomization approach." PLoS Med 2008
•
Greenland, S. and B. Brumback. "An overview of relations among causal modelling
methods." Int J Epidemiol 2002
•
Hernan, M. A., S. Hernandez-Diaz, and J. M. Robins. "A structural approach to
selection bias." Epidemiology 2004
•
Hernan, M. A. and S. R. Cole. "Causal diagrams and measurement bias." Am J
Epidemiol 2009
•
Sheehan, N. A., et al. "Mendelian randomisation and causal inference in observational
epidemiology." PLoS Med 2008
•
VanderWeele, T. J. and J. M. Robins. "Directed acyclic graphs, sufficient causes, and
the properties of conditioning on a common effect." Am J Epidemiol 2007
•
VanderWeele, T. J., M. A. Hernan, and J. M. Robins. "Causal directed acyclic graphs
and the direction of unmeasured confounding bias." Epidemiology 2008
•
VanderWeele, T. J. "The sign of the bias of unmeasured confounding." Biometrics
2008
May-16
H.S.
63
Exercise: Collider stratification
H
Hospital risk:
D
E
1
0
1
0
0.6
0.3
0.2
0.1
Response=
16 %
E
0
1
0
sum
36
164
64
736
100
900
1000
May-16
E
3.0
E
Hospital
D
Population
D
1
2.0
1
0
D
2.0
No hospital
D
1
0
sum
22
49
13
74
35
123
157
H.S.
E
1
0
1
0
sum
15
115
51
663
65
777
843
64
Exercise: Collider stratification
• Selection:
1. Confirm that exposure doubles the risk of
hospitalization (hint RREH=0.6/0.3=2)
2. Confirm that disease triples the risk of hospitalization
3. Confirm (roughly) that the hospital 2*2 table is correct
(hint 35*0.6=22,…)
• Relative risk of exposure on disease:
4. What is the RR (of E on D) in the hospital group?
5. What is the RR (of E on D) in the non-hospital group?
6. Are the RRs biased?
10 minutes
May-16
H.S.
65