DAGs intro
with exercises 8h
(reordered 4.5+3.5)
DAG=Directed Acyclic Graph
Hein Stigum
http://folk.uio.no/heins/
courses
May-16
H.S.
1
Agenda
• Background
• DAG concepts
– Causal thinking, Paths
• Analyzing DAGs
– Examples
• DAGs and stat/epi phenomena
Exercises
– Mediation, Matching, Mendelian
randomization, Selection
• More on DAGs
•
Limitations, problems
May-16
H.S.
2
Background
• Potential outcomes:
Neyman, 1923
• Causal path diagrams:
Wright,
1920
• Causal DAGs:
Pearl,
2000
May-16
H.S.
3
Why causal graphs?
• Estimate effect of exposure on disease*
*DAGs not applicable
for prediction models
• 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
(Rothman et al. 2008; Veieroed et al. 2012
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
= arrow
Association
= path
Independency = no path
Estimations:
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 
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
(reverse 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
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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)
• Paths
• Simplest form
•
Causal confounding, (exception: see outcome dependent selection)
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
•
Paths
• Simplest form
May-16
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9
Mediator
M
• Have found a cause (E)
• How does it work?
– Mediator (M)
E
direct effect
D
– Paths
𝑇𝑜𝑡𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡 = 𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 + 𝑑𝑖𝑟𝑒𝑐𝑡
𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡
𝑀𝑒𝑑𝑖𝑎𝑡𝑒𝑑 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 =
𝑡𝑜𝑡𝑎𝑙
Use ordinary regression methods if:
linear model and
no E-M interaction.
Otherwise, need new methods
May-16
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10
Causal thinking in analyses
May-16
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11
Regression before DAGs
Use statistical criteria
for variable selection
Risk factors for D:
Variable
OR
E
2.0
C
1.2
Comments
Surprisingly low association
Association
Both can be
misleading!
C
E
May-16
Report all variables in
the model as equals
D
H.S.
12
Statistical criteria for variable selection
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:
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
DAGs variable selection: close all non-causal paths
May-16
H.S.
13
Reporting variable as equals:
Association versus causation
Use statistical criteria
for variable selection
Risk factors for D:
Variable
OR
E
2.0
C
1.2
Comments
Surprisingly low association
Association
Causation
C
C
E
D
Symmetrical
C is a confounder for E-D
E is a confounder for C-D
E
Report all variables in
the model as equals
Base adjustments
on a DAG
Report only
the E-effect
D
or use different models
for each variable
Directional
C is a confounder for ED
E is a mediator for CD
Westreich & Greenland 2013
May-16
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14
Exercise: report variables as equals?
Risk factors for Fractures
Comments (surprises)
Interpret as effect of:
Physical activity 1.2
Protective in other studies?
Obesity
1.0
No effect?
Diabetes adjusted for all other vars.
Phy. act. adjusted for all other vars.
Obesity adjusted for all other vars.
Bone density
0.8
Variable
OR
Diabetes 2
2.0
Bone d.
adjusted for all other vars.
P
physical activity
D
E
fractures
diabetes 2
O
B
obesity bone density
May-16
1. P is a confounder for E→D, but is E
a confounder for P→D?
2. Which effects are reported correctly
in the table?
5 min
H.S.
15
Exercise: Stratify or not
Want the effect of action(A, exposure/treatment) on disease (D). Have stratified on C.
1. Make a guess at the population effect of A on D
2. Calculate the population effect of A on D
3. What is the correct analysis (and RR)? OBS several answers possible!
Population
D
1
A
1
0
C=0
D
sum
risk
210
A
0
1
0
RR=
C=1
D
1
0
sum
risk
10
70
90
330
100
400
500
0.10
0.18
RR=
0.6
40% less disease if treated
Population = crude
A
May-16
1
A0
1
0
sum
risk
200
80
200
20
400
100
500
0.50
0.80
RR=
0.6
40% less disease if treated
Stratified = adjusted for C
C
D
10 min
H.S.
Hernan et al. 2011
16
Summing up so far
• Associations visible in data. Causation from outside the data.
• Data driven analyses do not work. Need causal information
from outside the data.
• Reporting table of adjusted associations is misleading.
• Simpson’s paradox: causal information resolves the paradox.
May-16
H.S.
17
The Path of the Righteous
Paths
May-16
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18
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
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19
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
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20
ANALYZING DAGs
May-16
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21
Confounding examples
May-16
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22
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
1 ED
2 EC]UD
3 EC] D
Type
Causal
Non-causal
Non-causal
May-16
May-16
Status
Open
Closed
Closed
Bias
No bias
H.S.
This is an example
of confounding
Question:
Is U a confounder?
23
Exercise: Physical activity and
Coronary Heart Disease (CHD)
We want the total effect of Physical
Activity on CHD.
1. Write down the paths.
2. Are they causal/non-causal,
open/closed?
3. What should we adjust for?
5 minutes
May-16
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24
Direct and indirect effects
Intermediate variables
May-16
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25
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 variable on a
confounder path?
10 minutes
Hintikka et al. 2005
May-16
H.S.
26
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
H.S.
27
Direct and indirect effects
So far:
Causal interpretation:
Controlled (in)direct effect
linear model and no E-M interaction
New concept:
Natural (in)direct effect
Causal interpretation also for: non-linear model , E-M interaction
Controlled effect = Natural effect if
linear model and no E-M interaction
Hafeman and Schwartz 2009;
Lange and Hansen 2011;
Pearl 2012;
Robins and Greenland 1992;
VanderWeele 2009, 2014
May-16
H.S.
28
Confounder, collider and mediator
Mixed
May-16
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29
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
30
Drawing DAGs
May-16
H.S.
31
Technical note on drawing DAGs
• Drawing tools in Word (Add>Figure)
• Use Dia
• Use DAGitty
• Hand-drawn figure.
May-16
H.S.
32
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.
33
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.
34
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
H.S.
35
Exercise: Drawing survivor bias
1. We what to study the effect of exposure early in life (E)
on disease (D) later in life.
2. Exposure (E) decreases survival (S) in the period
before D (deaths from other causes than D).
3. A risk factor (R) reduces survival (S) in the period before D.
4. The risk factor (R) increases disease (D).
5. Only survivors are available for analysis (look at Collider
idea).
Draw and analyze the DAG
10 minutes
May-16
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36
Real world examples
May-16
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37
Endurance training and Atrial fibrillation
Tobacco
Socioeconomic
Status **
Endurance
training
Cardiovascular
factors *
Alcohol
consumption
BMI
Diabetes
Genetic disposition
Atrial
fibrillation
Hyperhyreosis
Health ***
consciousness
Height
Gender
Long-distance
racing
Several arrows missing!
Age
*Hypertension, heart disease, high cholesterol
** Socioeconomic status: Education, marital status
*** Unmeasured factors
(Blue: Mediators, red: confounders, violet: colliders)
Myrstad et al. 2014b
METHODS TO REMOVE
CONFOUNDING
May-16
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39
Methods to remove confounding
Method
Action
DAG effect
C
Condition: Restrict, Stratify, Adjust
Close path
E
D
C
Cohort matching, Propensity Score
Inverse Probability Treatment Weighting
Remove CE arrow
E
D
C
Case-Control matching?
Other methods?
May-16
Remove CD arrow
H.S.
E
D
40
Matching:
Cohort vs Case-Control
May-16
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41
Matching in cohort, binary E
Matching:
For every exposed person with a value of C,
find an unexposed person with the same value of C
S
E
 selected based on E and C
 E independent of C after matching
 All open paths between C and E must
C→E
C→[S]←E
C
D
C
sum to “null”
E
Cohort matching removes confounding
D
Unfaithful DAG
Cohort matching is not common,
except in propensity score matching
May-16
H.S.
Mansournia et al. 2013; Shahar and Shahar 2012
42
Matching in Case Control, binary D
C
Matching:
For every case with a value of C,
find a control with the same value of C
E
 selected based on D and C
 D independent of C after matching
 All open paths between C and D must
C→[S]←D
D
C
C→D
sum to “null”
C→E→D
S
E
S
D
Case-Control matching does not removes confounding,
unless E→D=0 (or C→E=0)
 must adjust for C in all analyses
Case-Control matching common, may improve precision
May-16
H.S.
(Mansournia et al. 2013; Shahar and Shahar 2012
43
Inverse probability weighting
May-16
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44
Marcov decomposition
• DAG implies:
Joint probability can be factorized into
the product of conditional distributions
of each variable given its parents
C
𝑃 𝐶, 𝐸, 𝐷 =
𝑃(𝐶) ∙ 𝑃(𝐸|𝐶) ∙ 𝑃(𝐷|𝐸, 𝐶)
E
D
Pearl 2000
May-16
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45
Inverse Probability Weighting
C
Have observed data distribution:
Factorization:
𝑃(𝐶) ∙ 𝑃(𝐸|𝐶) ∙ 𝑃(𝐷|𝐸, 𝐶)
E
C
Want the RCT distribution:
Factorization:
D
𝑃(𝐶) ∙ 𝑃(𝐸) ∙ 𝑃(𝐷|𝐸, 𝐶)
E
D
Can reweight the observed data with
weights:
𝑃(𝐸)/𝑃(𝐸|𝐶)
to obtain the RCT distribution
IPW knocks out arrows in the DAG
May-16
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46
Odds of Treatment weighting (OT)
Previous weighting targeted the Average Causal Effect
Want now the effect among the exposed (treated) instead
𝑃(𝐸 = 1|𝐶) is the probability of treatment
Can reweight the observed data with
weights:
𝑃(𝐸 = 1|𝐶)/𝑃(𝐸|𝐶)
to obtain the RCT distribution among the exposed
𝑃(𝐸 = 1|𝐶)/𝑃 𝐸 𝐶 =
𝑃(𝐸 = 1|𝐶)/𝑃 𝐸 = 1 𝐶 =1
𝑃(𝐸 = 1|𝐶)/𝑃 𝐸 = 0 𝐶 =OT
if E=1
if E=0
McCaffrey et al. 2004
May-16
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47
Marginal Structural Model
C
DAG for the reweighted pseudo data
E
D
MSM:
The expected value of a counterfactual outcome D
under a hypothetical exposure e:
𝐸 𝐷𝑒 = 𝛼0 + 𝛼1 𝑒
effect = 𝐸 𝐷1 − 𝐸 𝐷0
May-16
Veieroed, Lydersen et al. 2012. Daniel, Cousens et al. 2013. Rothman, Greenland et al. 2008
HS
48
Outcome versus exposure
modeling
May-16
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49
Methods to remove confounding
A
E
C
B
D
How can we remove confounding?
What variables should be involved?
C- yes
A- no
B- maybe
• Close ECD by conditioning
– (ordinary) regression model
E(D| E,C)
outcome
modeling
• Remove the EC arrow, binary E
–
–
Propensity score matching
Inverse probability weighting
P(E|C)
exposure
modeling
Combine exposure and outcome modeling:
doubly robust models
Outcome vs exposure modeling
• Outcome model: E(D|E,C) (and possibly B)
A
• Ex: Nano-particlesCardioVascularDisease
E
– Know little about risk for nano-particles
– Know a lot about risk factors for CVD
A
• Ex: SmokingBladder cancer
E
May-16
H.S.
B
D
Do outcome model
• Exposure model: E(E|C)
– Know a lot about risk for smoking
– Know little about risk factors for bladder
cancer
C
C
B
D
Do exposure model
51
Doubly robust methods
• Combine the outcome and the exposure
models:
Do the regression E(D|E,C)
with inverse probability weighting (OT)
• Will be unbiased if the outcome- or the
exposure-model is correct
• Doubly robust methods: Twice as right!
May-16
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52
Randomized experiments
Mendelian randomization
May-16
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53
Strength of arrow, randomization
E
Not
deterministic
C1
D
C2
C1, C2, C3 exogenous
C3
C
R
full
compliance
deterministic
E
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
54
Randomized experiments
C
E
Observational study
D
C
R
E
D
U
R c E IVe
ITTe
D
Randomized experiment with full compliance
R= randomized treatment
E= actual treatment.
R=E
Randomized experiment with less than full
compliance (c)
If linear model: ITTe=c*IVe, c<1
IntentionToTreat effect:
effect of R on D (unconfounded)
population
PerProtocol:
crude
effect of E on D (confounded by U)
InstrumentalVariable effect: adjusted effect of E on D (if c is known, 2SLS) individual
May-16
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55
RCT exercise
E
R
+
-
+
85
0
85
15
100
115
D
R
+
-
+
43
63
106
57
37
94
D
E
+
-
+
32
74
106
53
41
94
N
Risk
100
100
200
0.85
0.00
N
Risk
1.
100
100
0.43
0.63
Calculate the compliance (c) as a
risk difference from the table.
2.
Calculate the intention to treat effect
(ITTe) as a risk difference.
N
Risk
3.
85
115
0.38
0.64
Calculate the per-protocol effect (PP)
as a risk difference.
4.
Calculate the instrumental variable
effect (IVe).
5.
Explain the results in words.
R+ means randomized to treatment,
E+ means treated and D+ means getting disease.
0.85 is the risk of treatment for R+ subjects,
0.00 is the risk for R- subjects,
the risk difference is the difference between these.
U
0.22
-0.15
R c E
IVe
D
10 minutes
ITTe
May-16
H.S.
56
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
E
R independent of U
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
D
U
3
G
1
E
D
2
Sheehan et al, 2008
May-16
May-16
H.S.
H.S
57
57
Ex: Alcohol and blood pressure
•
U
Observational study
–
–
Alcohol use increases blood pressure
Many ”lifestyle” confounders
A
BP
•
Gene: ALDH2, 2 alleles
–
–
•
2,2 type suffer nausea, headache after alcohol
 low alcohol regardless of lifestyle (U)
40
30
20
10
0
1,1
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
•
alcohol ml/day
Alcohol use
Result:
–
–
1,2
Genotype
U
3
G
1
2,2
A
BP
2
1,1 type BP +7.4 mmHg
Alcohol increases blood pressure
Chen et al 2008
May-16
May-16
H.S.
H.S
58
58
DAGs and other causal models
May-16
H.S.
59
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
May-16
H.S.
60
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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61
Structural Equation Models, SEM
• Causal assumptions + statistical model + data
• SEM: parametric DAG
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Causal models compared
• DAGs
– qualitative population assumptions
– sources of bias (not easily seen with other approaches)
• Causal Pies (SCC)
– specific hypotheses about mechanisms of action
• SEM
– quantitative analysis of effects
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Two (3) concepts
Selection bias
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Selection bias: concept 1
Simple version
• “Selected different from unselected”
• Prevalence (D)
Old have lower prevalence than young
Old respond less to survey
 Selection bias: prevalence overestimated
• Effect (E→D)
Old have lower effect of E than young
Old respond less to survey
 Selection bias: effect of E overestimated
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Selection bias: concept 1
“Selected different from unselected”
Paths
smokeCHD
S
age
smoke
CHD
Age
Young
Old
All
Type
Causal
Status
Open
Population RRsmoke Selected RRsmoke
50 %
4.0
75 %
4.0
50 %
2.0
25 %
2.0
3.0
3.5
Normally, selection variables unknown
Name:
interaction based?
May-16
• Properties:
- Need smoke-age interaction
- Cannot be adjusted for, but stratum effects OK
- True RR=weighted average of stratum effects
- RR in “natural” range (2.0-4.0)
- Scale dependent
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Selection bias: concept 2
Simple version
• “Distorted E-D distributions”
• DAG
Collider bias
• Words
Selection by sex and/or age
Distorted sex-age distribution
Old have more disease
Men are more exposed
 Distorted E - D distribution
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Selection bias: concept 2
“Distorted E-D distributions”
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
Open non-causal path (collider)
Does not need interaction
Can be adjusted for (sex or age)
Not in “natural” range (“surprising bias”)
Selection bias types:
Berkson’s, loss to follow up, nonresponse, self-selection, healthy worker
Hernan et al, A structural approach to selection bias, Epidemiology 2004
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22
1) “Exclusive or” selection
S=5%
-0.5
0.5
0.0
-1
00
IQ
11
S=95%
S=95%
-2
S=5%
-2
-2
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-1-1
0 0
EMF
H.S.
1
1
2
2
69
Exercise: Dust and COPD
Chronic Obstructive Pulmonary Disease
D0
S
cur. worker
H
health
diseases
E
E0
D
COPD
prior dust cur. dust
COPD risks:
Dust
low
Health
good
poor
high
5 % 10 %
10 % 20 %
1. Calculate the RR of dust on COPD
in good and poor health groups.
2. Write down the paths for the effect
of E on D. E0 and D0 are unknown
(past) measures.
3. What would you adjust for?
4. Suppose the crude effect of dust on
COPD is RR=0.7 and the true
RR=2. What do you call this bias?
5. Could the concept 1 (interaction
based) selection bias work here?
10 minutes
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Convenience sample, homogenous sample
H
1. Convenience:
Conduct the study among
hospital patients?
hospital
E
diabetes
Conditional
Unconditional
Path
1 E→D
2 E→H←D
E→[H]←D
D
2. Homogeneous sample:
Population data,
exclude hospital patients?
fractures
Type
Causal
Non-causal
Non-Causal
Status
Open
Closed
Open
Collider, selection bias
Collider stratification bias: at least on stratum is biased
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Outcome dependent selection
Selection into the study based on D.
Get bias among selected.
S
D
E
U
4
5
Explanation:
• Always have exogenous U.
0.6
2
D
3
• D is a collider on E→D←U, S is a
descendant of collider D.
1
1.0
0
• Conditioning on (a descendant of) a
collider opens the E→D←U path,
and U becomes associated with E.
0.6
0
2
3
E
• U now acts a confounder for E→D.
Selected if D<= 2.5 + 0.0*E
Selection depends on:
Strength of E→D. Strength of U→D
Unmatched Case-Control
Example of non-causal confounding
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Selection bias summing up
Concept 1
Concept 2
S
S
smoke
age
sex
age
CHD
smoke
CHD
Selected differ from
unselected in E-D effects
Selected differ from
unselected in E-D distributions
Interaction
Collider bias
“natural” effects
“surprising” effects
Report stratum effects
Adjust
Quite different concepts
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MORE ON DAGs
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Back door, front door, D-separated
Paths from E to D, all are “leaving” E
Paths open before conditioning:
E
back door
.
non-causal open
front door
.
causal open
need to close
Plus paths closed at a collider
If all paths from A to B are closed  d-separated
Pearl 2000
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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)
• Instrumental Variables
–
–
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
D
E
D
2
Pearl 2009, Glymor and Greenland, 2008
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76
Example: front–door criterion
• Weight and Coronary Heart Disease
U
lifestyle
• Assume:
C
– adjusted for sex, age and smoke
E
– lifestyle is unmeasured
– no other mediators (between E and D)
cholesterol
weight
B
D
CHD
blood pressure
• Can estimate effect of E on D
Path
1 E→C→D
2 E→B→D
3 E←U→D
•
Type
Causal
Causal
Non-causal
Status
Open
Open
Open
Crude
Difference = causal
Adjusted for B and C
Weight is not a good “action”
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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
K
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
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Testable implications
The DAG implies:
C is independent of D given E and O
O
Regress D on E, C and O,
if the C coefficient is different from zero
we reject the DAG or rather add the arrow.
DAGitty gives a list of testable implications
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Textor et al. 2011
79
DEFINITIONS
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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
L
A
Y
U
• Path
– Sequence of connected edges:
– Parent → child
– Ancestors → → descendants
• Exogenous: variables with no parents
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[(L,A),(A,Y)]
U
81
D
irected
A
cyclic
G
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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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 non-parametric)
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DAG units
DAG units
individuals
populations
C
gene
(almost) No
C
D
variable can
influence a gene in an
individual

No confounding
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gene
D
A variable can influence
gene frequency in a
population
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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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DAGs and probability theory
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DAGs rules and statistical independence
DAG correct?
• Two assumptions
1. Compatibility:
– Faithfulness:
– =
separated  independent
separated  independent
connected  dependent
2. Weak faithfulness:
connected variables may
be dependent
B
A
B
Y
A
Y
Pearl 2009, Glymor and Greenland, 2008
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LIMITATIONS, PROBLEMS AND
EXTENSIONS OF DAGS
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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
direction and magnitude
• Interaction
scale dependent
• Static
may include time varying variables
• Simplified
infinite causal chain
• Simplified
do not capture reality
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DAG focus: bias, not precision
C
Should we adjust for C?
DAG: no bias from C, need not adjust
E
D
May include C to improve precision, depends on model!
Linear regression
Logistic regression
crude
crude
adjusted
adjusted
.9
1
1.1
Effect of E on D with 95% CI
1.2
1.4
1.6
Effect of E on D with 95% CI
Including C: better precision
Including C: worse precision
OR not collapsible
Robinson and Jewell 1991; Xing and Xing 2010
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Signed DAGs and direction of bias
M
+
+
E
D
X Y
Positive or negative bias from confounding by U?
Neg
True
Pos
E→D
on average
Average monotonic effect
+-
for all Y=y
Distributional monotonic effect


U
To find direction of bias, multiply signs:
Need distributional monotonic effects except at each end
Positive bias from this confounding
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91
Size of bias
from unmeasured U
C
A
Y
U
Assume: Difference in the distribution
of U for two levels for A: a1 ,a0 ,
does not vary with C
Assume: Difference in expected
value of Y for two levels of U : u1 ,u0 ,
does not vary with A and C
γ = 𝐸 𝑌 𝑢1 − 𝐸 𝑌 𝑢0 if linear model
𝛼 = 𝑃 𝑢 𝑎1 − 𝑃 𝑢 𝑎0
γ = 𝐸 𝑌 𝑢1 /𝐸 𝑌 𝑢0
Bias = 𝛼 ∗ 𝛾
Bias =
1+ 𝛾−1 𝑃
1+ 𝛾−1 𝑃
if linear model
𝑢 𝑎1
𝑢 𝑎0
Stata: episens
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if RR model
H.S.
if RR model
VanderWeele & Arah 2011
92
Interaction in DAGs
DAG
Causal pie
Extended DAG
Mechanisms
C
C
D
+
C
=
E C
E
E,C
E
E
C
D
E
Red arrow
=
interaction
Specify scale
VanderWeele and Robins 2007
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DAGs and time processes
DAGs often static,
but may have time varying A1, A2,…
Want total effect of A-s, Time Dependent Confounding
DAG
Process graph
HDL
A1
A2
HDL
CHD
Alcohol
CHD
The process graph is simpler but less specific
The process graph allows feedback loops
and has a clear time component
Aalen et al. 2012
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94
Infinite causal chain
U
E
D
the
Most paths involving variables back in the chain (U)
will be closed
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DAGs are simplified
DAGs are models of reality
must be large enough to be realistic,
small enough to be useful
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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
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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.
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
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References 1
•
Aalen OO, Roysland K, Gran JM, Ledergerber B. 2012. Causality, mediation and time: A dynamic viewpoint. Journal of the Royal Statistical
Society Series A 175:831-861.
•
Chen L, Davey SG, Harbord RM, Lewis SJ. 2008. Alcohol intake and blood pressure: A systematic review implementing a mendelian
randomization approach. PLoS Med 5:e52.
•
Daniel RM, Cousens SN, De Stavola BL, Kenward MG, Sterne JAC. 2013. Methods for dealing with time-dependent confounding. Statistics in
Medicine 32:1584-1618.
•
Greenland S, Schlesselman JJ, Criqui MH. 1987. Re: "The fallacy of employing standardized regression coefficients and correlations as measures
of effect". AJE 125:349-350.
•
Greenland S, Robins JM, Pearl J. 1999. Confounding and collapsibility in causal inference. Statistical Science 14:29-46.
•
Greenland S, Brumback B. 2002. An overview of relations among causal modelling methods. Int J Epidemiol 31:1030-1037.
•
Greenland S, Mansournia MA. 2015. Limitations of individual causal models, causal graphs, and ignorability assumptions, as illustrated by random
confounding and design unfaithfulness. Eur J Epidemiol.
•
Greenland SM, Malcolm; Schlesselman, James J.; Poole, Charles; Morgenstern, Hal. 1991. Standardized regression coefficients: A further critique
and review of some alternatives. Epidemiology 2:6.
•
Hafeman DM, Schwartz S. 2009. Opening the black box: A motivation for the assessment of mediation. International Journal of Epidemiology
38:838-845.
•
Hernan MA, Hernandez-Diaz S, Werler MM, Mitchell AA. 2002. Causal knowledge as a prerequisite for confounding evaluation: An application to
birth defects epidemiology. AJE 155:176-184.
•
Hernan MA, Hernandez-Diaz S, Robins JM. 2004. A structural approach to selection bias. Epidemiology 15:615-625.
•
Hernan MA, Cole SR. 2009. Causal diagrams and measurement bias. AJE 170:959-962.
•
Hernan MA, Clayton D, Keiding N. 2011. The simpson's paradox unraveled. Int J Epidemiol.
•
Hintikka J, Tolmunen T, Honkalampi K, Haatainen K, Koivumaa-Honkanen H, Tanskanen A, et al. 2005. Daily tea drinking is associated with a low
level of depressive symptoms in the finnish general population. European Journal of Epidemiology 20:359-363.
•
Lange T, Hansen JV. 2011. Direct and indirect effects in a survival context. Epidemiology 22:575-581.
•
Mansournia MA, Hernan MA, Greenland S. 2013. Matched designs and causal diagrams. International Journal of Epidemiology 42:860-869.
•
McCaffrey DF, Ridgeway G, Morral AR. 2004. Propensity score estimation with boosted regression for evaluating causal effects in observational
studies. Psychological Methods 9:403-425.
•
Myrstad M, Lochen ML, Graff-Iversen S, Gulsvik AK, Thelle DS, Stigum H, et al. 2014a. Increased risk of atrial fibrillation among elderly norwegian
men with a history of long- term endurance sport practice. Scand J Med Sci Spor 24:E238-E244.
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References 2
•
Pearl J. 2000. Causality: Models, reasoning, and inference. Cambridge:Cambridge Univeristy Press.
•
Pearl J. 2012. The causal mediation formula-a guide to the assessment of pathways and mechanisms. Prev Sci 13:426-436.
•
Robins JM, Greenland S. 1992. Identifiability and exchangeability for direct and indirect effects. Epidemiology 3:143-155.
•
Robins JM. 2001. Data, design, and background knowledge in etiologic inference. Epidemiology 12:313-320.
•
Robinson LD, Jewell NP. 1991. Some surprising results about covariate adjustment in logistic-regression models. Int Stat Rev
59:227-240.
•
Rothman KJ, Greenland S, Lash TL. 2008. Modern epidemiology. Philadelphia:Lippincott Williams & Wilkins.
•
Shahar E. 2009. Causal diagrams for encoding and evaluation of information bias. Journal of evaluation in clinical practice
15:436-440.
•
Shahar E, Shahar DJ. 2012. Causal diagrams and the logic of matched case-control studies. Clinical epidemiology 4:137-144.
•
Sheehan NA, Didelez V, Burton PR, Tobin MD. 2008. Mendelian randomisation and causal inference in observational
epidemiology. PLoS Med 5:e177.
•
Textor J, Hardt J, Knuppel S. 2011. Dagitty a graphical tool for analyzing causal diagrams. Epidemiology 22:745-745.
•
VanderWeele TJ, Robins JM. 2007. Directed acyclic graphs, sufficient causes, and the properties of conditioning on a common
effect. AJE 166:1096-1104.
•
VanderWeele TJ, Hernan MA, Robins JM. 2008. Causal directed acyclic graphs and the direction of unmeasured confounding
bias. Epidemiology 19:720-728.
•
VanderWeele TJ. 2009. Mediation and mechanism. Eur J Epidemiol 24:217-224.
•
VanderWeele TJ, Arah OA. 2011. Bias formulas for sensitivity analysis of unmeasured confounding for general outcomes,
treatments, and confounders. Epidemiology 22:42-52.
•
VanderWeele TJ, Hernan MA. 2012. Results on differential and dependent measurement error of the exposure and the outcome
using signed directed acyclic graphs. AJE 175:1303-1310.
•
VanderWeele TJ. 2014. A unification of mediation and interaction: A 4-way decomposition. Epidemiology 25:749-761.
•
Veieroed M, Lydersen S, Laake P. 2012. Medical statistics in clinical and epidemiological research. Oslo:Gyldendal Akademisk.
•
Westreich D, Greenland S. 2013. The table 2 fallacy: Presenting and interpreting confounder and modifier coefficients. AJE
177:292-298.
•
Xing C, Xing GA. 2010. Adjusting for covariates in logistic regression models. Genet Epidemiol 34:937-937.
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Extra material
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MEDIATION ANALYSIS
Hafeman and Schwartz 2009;
Lange and Hansen 2011;
Pearl 2012;
Robins and Greenland 1992;
VanderWeele 2009, 2014
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Why mediation analysis?
• Have found a cause
• How does it work?
M
A
May-16
direct effect
𝑇𝑜𝑡𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡 = 𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 + 𝑑𝑖𝑟𝑒𝑐𝑡
Y
𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡
𝑀𝑒𝑑𝑖𝑎𝑡𝑒𝑑 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 =
𝑡𝑜𝑡𝑎𝑙
H.S.
104
Classic approach: controlling
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Controlled Direct effect
Direct effect:
Effect of statin on CHD
“for the same cholesterol”
Fixed M
Fixed M: controlled direct effect
CDE=E(Y|A=1,M=m) - E(Y|A=0,M=m)
m
M
cholesterol
A
statin
Y
CHD
Problems
1. Conceptual: Can we fix cholesterol levels?
2. Technical: A*M Interaction?
3. (Technical: non-linear models?)
0/1
Solution?
Robins and Greenland 1992; VanderWeele 2009
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New approach: counterfactuals
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Counterfactual causal effect
• Two possible outcome variables
– Outcome if treated:
– Outcome if untreated:
Y1
Y0
Counterfactuals
Potential outcomes
• Causal effect
– Individual:
– Average:
Y1i-Y0i
E(Y1)-E(Y0)
or other effect measures
Fundamental problem: either Y1 or Y0 is missing
Hernan 2004
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Total causal effect, TCE
Potential (counterfactual) outcomes:
Effect of statin on CHD
Y1 is the outcome if A is set to 1
M1 is the mediator if A is set to 1
M1
M0
M
cholesterol
A
statin
A set to 0
M0
Y1
Y0
𝑇𝐶𝐸 = 𝐸 𝑌1
Y
CHD
− 𝐸 𝑌0
= 𝐸 𝑌1,𝑀1 − 𝐸 𝑌0,𝑀0
0/1
May-16
A set to 1
M1
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109
Natural Direct effect
Direct effect:
Effect of statin on CHD
“for the same cholesterol”
M1
M0
M
statin
0/1
May-16
A set to 0
M0
Natural Direct Effect: Keep M at M0
cholesterol
A
A set to 1
M0
Y
CHD
𝑁𝐷𝐸 = 𝐸 𝑌1,𝑀0 − 𝐸 𝑌0,𝑀0
Takes care of the 3 earlier problems:
1. Don’t need to fix M=m
2. OK for interactions
3. (OK for non-linear models) in 4 slides
H.S.
110
Exercise: nested counterfactuals
• Write counterfactual outcomes for:
– All are treated (A set to 1), the mediator is fixed at m
– All are untreated (A set to 0), the mediator is fixed at m
– All are treated, the mediator is at its natural
distribution if all are untreated
– All are untreated, the mediator is at its natural
distribution if all are untreated
5 minutes
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Natural Indirect effect
Indirect effect:
Effect of statin via cholesterol on CHD
“for the same statin”
M1
M set to M1
A=1
M0
Natural Indirect Effect: Keep A at 1
M
cholesterol
A
statin
0/1
M set to M0
A=1
𝑁𝐷𝐸 = 𝐸 𝑌1,𝑀1 − 𝐸 𝑌1,𝑀0
Y
CHD
Why keep A at 1?
then direct + indirect = total
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Natural direct and indirect effects 1
Natural Direct Effect:
𝑁𝐷𝐸 = 𝐸 𝑌1,𝑀0 − 𝐸 𝑌0,𝑀0 A=1 vs. 0 for M=M0
Natural Indirect Effect:
𝑁𝐼𝐸 = 𝐸 𝑌1,𝑀1 − 𝐸 𝑌1,𝑀0
M=M1 vs. M0 for A=1
Total Causal Effect:
𝑇𝐶𝐸 = 𝐸 𝑌1,𝑀1 − 𝐸 𝑌0,𝑀0
= 𝑁𝐷𝐸 + 𝑁𝐼𝐸
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Binary outcome, RR effect measure
Natural Direct Effect:
𝑁𝐷𝐸𝑅𝑅 =
𝑃 𝑌1,𝑀0 =1
𝑃 𝑌0,𝑀0 =1
A=1 vs. 0 for M=M0
Natural Indirect Effect:
𝑁𝐼𝐸𝑅𝑅 =
𝑃 𝑌1,𝑀1 =1
𝑃 𝑌1,𝑀0 =1
M=M1 vs. M0 for A=1
Total Causal Effect:
𝑃 𝑌1,𝑀1 = 1
𝑇𝐶𝐸𝑅𝑅 =
= 𝑁𝐷𝐸𝑅𝑅 ∙ 𝑁𝐼𝐸𝑅𝑅
𝑃 𝑌0,𝑀0 = 1
May-16
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Example: Labor marked discrimination
Are Emily and Greg more employable than Lakisha and Jamal?
Names
CV
White: Emely Walsh, Greg Baker
Af.-am.: Lakisha Washington, Jamal Jones
CV:
Job
name
low/high quality
Job:
callback
Experiment:
5000 CV with random name type
Boston and Chicago, 2002?
Result:
White names:
African-am:
10 CV-s before callback
15 CV-s before callback
Bertrand and Mullainathan 2004
May-16
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Exercise: Labor marked discrimination
1. Describe the experiment for estimating
the controlled direct effect
2. Describe the experiment for estimating
the natural direct effect
CV
Job
name
5 minutes
May-16
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116
Assumptions
U2
Classic method:
Linear models
No A-M interaction
M
U3
P
Y
A
Classic and new:
No unmeasured confounders (U1, U2, U3,)
No treatment dependent confounders (P)
U1
VanderWeele 2009
May-16
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117
Summing up (so far)
M
• Classic method
– Linear models, no A-M interaction
Y
A
• New approach
– Linear/non-linear, with/without A-M interaction
– Natural Direct Effect
• effect of A keeping M at its natural distribution
– More assumptions
• Mediation > Total
• Natural
> Controlled
May-16
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Mediation analysis
May-16
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119
Highlights
• Classic decomposing into direct and indirect effects fail:
– Do not account for interaction
– Not meaningful decomposition in non-linear models
• New methods define natural direct and indirect effects
– Need special (often limited) software for estimation
• A 4-way decomposition of mediation and interaction
– Theoretical insight
– Estimate with standard regression models*
– * use delta or bootstrap for confidence intervals
– All types of exposures and mediators
May-16
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Classic method
Classic method (Baron & Kenny):
Total effect:
regress Y on A
Direct effect: regress Y on A and M
Indirect effect: Total – Direct
M
Y
A
• Only OK for
– Linear models
– No A*M interaction
– Strong no-confounding assumptions
May-16
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121
New methods
Estimate natural direct and indirect effects (Stata)
D
M
Multiple M
Sensitivity
Paramed
continuous
binary
count
continuous
binary
no
no
Idecomp
binary
any
yes
no
continuous
binary
continuous
binary
no
yes
any
any
?
?
(Lange-2011)
Time-to-event
continuous
no
no
(Lange-2012)
any
any
no
no
Medeff
Gformula*
* Only for treatment dependent confounding
May-16
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Mediation analysis:
4-WAY DECOMPOSITION
May-16
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Notation
Counterfactuals if
A is set to a, M is set to m: 𝑌𝑎
M
𝑀𝑎
𝑌𝑎𝑚
A
Y
For simplicity assume binary A and M: 𝑌10
(general results available)
a
May-16
H.S.
m
124
Interaction on additive scale
M
No additive interaction:



𝑅𝐷11 =
𝑅𝐷10
+ 𝑅𝐷01
Y
A
𝑌11 − 𝑌00 = 𝑌10 − 𝑌00 + 𝑌01 − 𝑌00
𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 = 0
Measure of additive interaction:
𝑌11 − 𝑌10 − 𝑌01 + 𝑌00
May-16
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125
Interaction in DAGs
DAG
Causal pie
Extended DAG
Mechanisms
M
M
Y
+
M
=
A M
A
A,M
A
A
M
Y
A
Red arrow
=
interaction
Specify scale
VanderWeele and Robins 2007
May-16
H.S.
126
4 way decomposition
TE
CDE
INTref
INTmed
PIE
M
= total effect
= controlled direct effect
= interaction effect at reference
= interaction*mediator effect
= pure indirect effect
Y
A
Can always decompose the total effect into 4:
𝑌1 − 𝑌0 =
Mediation
Interaction
TE
(𝑌10 −𝑌00 )
CDE
-
-
+𝑀0 (𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 )
INTref
-
+
+(𝑀1 − 𝑀0 )(𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 )
INTmed
+
+
+(𝑀1 − 𝑀0 )(𝑌01 − 𝑌00 )
PIE
+
-
May-16
H.S.
127
Individual 4 mechanisms
TE
CDE
INTref
INTmed
PIE
M
= total effect
= controlled direct effect
= interaction effect at reference
= interaction*mediator effect
= pure indirect effect
Y
A
If AY for one subject then one of 4 mechanisms will work:
M I
(𝑌10 −𝑌00 )
AY|M=0
-
𝑀0 (𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 )
M0=1 & M*A inter.
- +
(𝑀1 − 𝑀0 )(𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 )
AM & M*A inter.
+ +
(𝑀1 − 𝑀0 )(𝑌01 − 𝑌00 )
AM & MY|A=0
+ -
May-16
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128
-
Interaction in DAGs
Extended DAG
Mechanisms
M
M
A,M
A
Y
A
VanderWeele and Robins 2007
May-16
H.S.
129
Individual 4 mechanisms
1
M
2
M
A,M
A
A
M
Y
3
M
A,M
A
M
Y
A
4
M
A,M
A
M
Y
A
M
A,M
A
Y
A
If AY for one subject then one of 4 mechanisms will work:
M I
(𝑌10 −𝑌00 )
AY|M=0
-
𝑀0 (𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 )
M0=1 & M*A inter.
- +
(𝑀1 − 𝑀0 )(𝑌11 − 𝑌10 − 𝑌01 + 𝑌00 )
AM & M*A inter.
+ +
(𝑀1 − 𝑀0 )(𝑌01 − 𝑌00 )
AM & MY|A=0
+ -
May-16
H.S.
130
-
4 and 2 way decompositions
Mediation:
-
-
+
+
TE = CDE + INTref + INTmed + PIE
TE =Natural Direct + Natural Indirect
May-16
H.S.
131
Notation:
Identification (estimate from data)
Counterfactuals if
A is set to a, M is set to m:
M
𝑌𝑎
𝑀𝑎
For simplicity assume binary A and M:
(general results available)
𝑌𝑎𝑚
𝑌10
Y
A
Consistency:
Consistency assumption:
Composition assumption:
𝐼𝑓 𝐴 = 𝑎 𝑎𝑛𝑑 𝑀 = 𝑚 𝑡ℎ𝑒𝑛 𝑌𝑎𝑚 = 𝑌
𝑌𝑎 = 𝑌𝑎𝑀𝑎
U2
Confounding:
4 assumptions needed for estimation:
No unmeasured confounding (U1-U3)
No confounders affected by A (P)
M
U3
P
Y
A
U1
May-16
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132
Y and M in linear regression models
Assume Y and M continuous and following:
M
𝐸 𝑌 𝑎, 𝑚, 𝑐 = 𝜃0 + 𝜃1 𝑎 + 𝜃2 𝑚 + 𝜃3 𝑎𝑚 + 𝜃4′ 𝑐
𝐸 𝑀 𝑎, 𝑐 = 𝛽0 + 𝛽1 𝑎 + 𝛽2′ 𝑐
Y
A
then
𝐸 𝐶𝐷𝐸 𝑐
= 𝜃1 (𝑎 − 𝑎∗ )
𝐸 𝐼𝑁𝑇𝑟𝑒𝑓 𝑐 = 𝜃3 (𝛽0 + 𝛽1 𝑎∗ + 𝛽2′ 𝑐)(𝑎 − 𝑎∗ )
𝐸 𝐼𝑁𝑇𝑚𝑒𝑑 𝑐 = 𝜃3 𝛽1 (𝑎 − 𝑎∗ )2
𝐸 𝑃𝐼𝐸 𝑐
= (𝜃2 𝛽1 + 𝜃3 𝛽1 𝑎∗ ) (𝑎 − 𝑎∗ )
a=1 and a*=0 would simplify further
May-16
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133
Binary outcomes and ratio scale
M
Similar results based on RR
(with a scaling factor)
Y
A
Example:
Smoking
Gene
Lung
Cancer
15q25.1 rs8034191 C alleles
May-16
H.S.
134
Non-collapsibility of the odds
ratio
May-16
H.S.
135
Non-collapsibility of the OR
Population
D
C
OR=10
E
A
D
1
0
1
0
sum
470
1 775
530
7 225
1 000
9 000
10 000
OR=
3.6
C=0
A
0
D
1
0
sum
105
225
395
4 275
500
4 500
5 000
OR=
5.1
May-16
0.89
0.25
C=1
D
1
odds
odds
0.27
0.05
1
A0
H.S.
1
0
sum
365
1 550
135
2 950
500
4 500
5 000
OR=
5.1
odds
2.70
0.53
136
Non-collapsibility of the OR
C
OR=10
E
D
Non-collapsibility depends on frequency of D
Logistic regression
crude
adjusted
D=22% :
Not collapsible
crude
adjusted
D=6% :
Appr. collapsible
crude
adjusted
D=1% :
Collapsible
2
May-16
3
4
5
OR for E on D
H.S.
6 7
137
Information bias
Hernan and Cole 2009;
Shahar 2009;
VanderWeele and Hernan 2012
May-16
H.S.
138
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
May-16
H.S.
139
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
H.S.
140
Exercise: Hair dye and
congenital
malformations
We study the effect of hair dye (E) during pregnancy on malformations
(D) in the baby in a traditional case–control study. Mothers are asked
after birth how often they dyed their hair during pregnancy.
1. Draw a DAG of the situation were mothers do not recall exactly
how often they dyed their hair, and were the recall is different for
mothers with malformed babies. Use E for the correct amount of hair dye,
and E* for the reported. Malformations are assumed to be without
misclassification.
2. Show the paths for the effect of E on D. Will there be a bias?
3. Show the paths for the effect of E* on D. Will there be a bias?
4. Can E* be associated with D even if E→D is zero?
10 minutes
May-16
H.S.
141
Selection bias depicted
May-16
H.S.
142
Simplified example
Selection
S
IQ
• (common understanding
for continuous and binary
variables)
• Focus: selection types
and bias patterns
IQ
• Selection in quadrants
1
• Stratification  Selection
0
EMF
-1
D
-2
E
2
E and D continuous, Z, normal
True effect of E on D: =0
-2
0
EMF
2
• (Clarity over realism)
May-16
HS
143
22
1) “Exclusive or” selection
S=5%
-0.5
0.5
0.0
-1
00
IQ
11
S=95%
S=95%
-2
S=5%
-2
-2
May-16
-1-1
0 0
EMF
H.S.
1
1
2
2
144
2
2) “Inclusive or” selection
S=95%
1
S=95%
0.0
0.1
-1
0
IQ
-0.2
S=95%
-2
S=5%
-2
May-16
-1
0
Parent income
H.S.
1
2
145
2
3) “And” selection
S=95%
0.2
0.0
-0.2
-1
0
1
S=5%
S=4%
-2
S=5%
-2
May-16
-1
0
Fluoride in water
HS
1
2
146
2
4) “Gradient” selection
S=95%
1
S=50%
0.0
0
IQ
-0.1
-1
-0.1
S=50%
-2
S=4%
-2
May-16
-1
0
EMF
H.S.
1
2
147
Z-scores
May-16
H.S.
148
Birth weight paradox
•
Results:
–
–
•
Possible explanation:
–
•
Maternal smoking increases neonatal
mortality overall
Maternal smoking decreases neonatal
mortality among low birth weight
M
birth weight
E
smoke
U
D
neonatal mort
conditioning on M opens collider path
via U
Some advocate standardizing birth
weight with respect to smoking,
i.e. Z-scores
May-16
H.S.
149
Z-scores
𝑍𝑖𝑗 =
𝑏𝑖𝑟𝑡ℎ 𝑤𝑒𝑖𝑔𝑡ℎ𝑖𝑗 −𝑚𝑗
𝑠𝑑𝑗
𝑗 = 0,1 (𝑠𝑚𝑜𝑘𝑒)
0
 E(smoke) independent of Z
 All open paths between E and Z must
E→Z
E→M→Z
birth weight
-4
-2
0
2
4
Z-scores of birth weight by smoke
U
Adjusting for Z estimates the
total effect of E on D
Z
E
smoke
No gain,
crude model also estimates the
total effect of E on D
D
neonatal mort
“unfaithful”
May-16
6000
sum to “null”
when we condition on Z
M
2000
4000
Birth weight by smoke
H.S.
150
Z-scores
•
Should not adjust for
gestational age:
–
–
•
G
gest. age
removes part of the effect of
exposure
opens up a collider path
involving U
E
toxicant
U
D
birth weight
Some advocate standardizing
birth weight with respect to
gestational age, i.e. Z-scores
–
but this also represents some
type of adjustment for
gestational age
May-16
H.S.
151