1. Hein Stigum http://folk.uio.no/heins/ courses
DAGs intro with exercises 8h (reordered ) DAG=Directed Acyclic Graph
MF 9570: Causal Inference:
Monday
:  Introduction to causal graphs (DAGs) h
:  Lunch break
:  Causal graphs (DAGs) continued. 3 h
Tuesday
: Analyzing DAGs h
: DAGitty h
Sum without DAGitty =6.75 h
Included If time Not included
1. day:
Concepts, Causal thinking, Paths, Analyzing, Drawing, Meth to remove conf. IPW, Outcome versus exposure models
Mendelian
2. day:
Selection bias strategies, DAGs and prop
Limitations, Problems and extensions
Norsk beskrivelse:
Introduksjon til kausale grafer (DAGs)
Kausale grafer (Directed Acyclic Graphs) er nyttige verktøy for å forstå grunnleggende begreper som konfundering, mediering og seleksjonsfeil. Grafene kan finne variable som må justeres for, og variable som ikke bør justeres for. Og grafene er en presis beskrivelse av antagelsene i analysen. Kurset vil gi en introduksjon til kausale grafer med mye eksempler og lite formalisme. Velkommen under mottoet «Draw your assumptions
before your conclusions»
Engelsk beskrivelse:
The causal graphs are useful tools to understand key concepts like confounding, mediation and colliding (selection bias). They help in the analysis by finding a group of variables that must be adjusted for (and variables that should not be adjusted for). And they give a clear statement of prior assumptions for the analysis.
Hein Stigum
courses
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2. Agenda Background DAG concepts Analyzing DAGs
Causal thinking, Paths
Analyzing DAGs
Examples
DAGs and stat/epi phenomena
Mediation, Matching, Mendelian randomization, Selection
More on DAGs
Limitations, problems
Exercises
DAG concepts
Define a few main concepts
Paths: Surprisingly few rules needed
Analyzing DAGs
Examples: conf, intermediate, collider
Selection bias, Information bias
Rand, Mend Rand
The two former: manual for use
More on DAGs
Deeper thoughts and problems
With exercises, difficult to guess time
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3. Background Potential outcomes: Neyman, 1923
Causal path diagrams: Wright, 1920
Causal DAGs: Pearl, 2000
Potential outcomes or counterfactual outcomes
Jerzy Neyman  (April 16, 1894 – August 5, 1981), born Jerzy Spława-Neyman, was a Polish mathematician and statistician who spent the first part of his professional career in various institutions in Warsaw, Poland, and the second part at the University of California, Berkeley. Neyman first introduced the modern concept of a confidence interval into statistical hypothesis testing[2] and co-devised null hypothesis testing (in collaboration with Egon Pearson).
Sewall Green Wright (December 16, 1889 – March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis.
Judea Pearl (born 1936) is an Israeli-born American computer scientist and philosopher, best known for championing the probabilistic approach to artificial intelligence and the development of Bayesian networks (see the article on belief propagation). He is also credited for developing a theory of causal and counterfactual inference based on structural models (see article on causality)
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4. Why causal graphs? Estimate effect of exposure on disease* Problem
Association measures are biased
Causal graphs help:
Understanding
Confounding, mediation, selection bias
Analysis
Adjust or not
Discussion
Precise statement of prior assumptions
*DAGs not applicable
for prediction models
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5. CONCEPTS Causal versus casual
(Rothman et al. 2008; Veieroed et al. 2012
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6. DAG=Directed Acyclic Graph
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
Arrows=lead to or causes
Time
E- exposure
D- disease
C, V - cofactor, variable
U- unmeasured
Directed= arrows
Acyclic = nothing can cause itself
Conditioning (Adjusting): E[C]UD
Time 
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7. Association and Cause Association 3 possible causal structure E D
(reverse cause) E D
Assume E precedes D in time
Association: observe
Cause: infer (extra knowledge)
Causal structure force on the data
Basic structures, may generalize with many more variables: use paths
+ more complicated structures
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8. Confounder idea + A common cause Adjust for smoking Smoking
Yellow fingers
Smoking
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)
“+” (assume monotonic effects)
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9. Two causes for selection to study
Collider idea
Two causes for selection to study
Selected subjects
Selected
Yellow fingers
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
“+” (assume monotonic effects)
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10. Mediator Have found a cause (E) How does it work? Mediator (M) M E D
indirect
effect
How does it work?
Mediator (M)
Paths E
direct effect D
𝑇𝑜𝑡𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡=𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡+𝑑𝑖𝑟𝑒𝑐𝑡
𝑀𝑒𝑑𝑖𝑎𝑡𝑒𝑑 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛= 𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 𝑡𝑜𝑡𝑎𝑙
Use ordinary regression methods if:
linear model and
no E-M interaction.
Otherwise, need new methods
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11. Causal thinking in analyses
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12. Regression before DAGs
Risk factors for D:
Use statistical criteria for variable selection
Variable OR
Comments E
2.0 C
1.2
Surprisingly low association
Report all variables in the model as equals
Association
Both can be misleading! C E D
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13. Statistical criteria for variable selection
- 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) C E D
The undirected graph above is compatible with three DAGs: C C C E D E D E D
Confounder
1. Adjust
Mediator
2. Direct: adjust
3. Total: not adjust
Collider
4. Not adjust
Hirotugu Akaike 赤池 弘次
Conclusion: 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
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14. Reporting variable as equals: Association versus causation
Risk factors for D:
Use statistical criteria for variable selection
Variable OR
Comments E
2.0 C
1.2
Surprisingly low association
Report all variables in the model as equals
Association
Causation
Base adjustments
on a DAG C C
Report only
the E-effect
or use different models
for each variable E D E D
Symmetrical
Directional
C is a confounder for E-D
C is a confounder for ED
E is a confounder for C-D
E is a mediator for CD
Westreich & Greenland 2013
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15. Exercise: report variables as equals?
Risk factors for Fractures
Interpret as effect of:
Variable OR
Comments (surprises)
Diabetes 2
2.0
Physical activity
1.2
Protective in other studies?
Obesity
1.0
No effect?
Bone density
0.8
Diabetes adjusted for all other vars.
Phy. act. adjusted for all other vars.
Obesity adjusted for all other vars.
Bone d. adjusted for all other vars.
physical activity P
P is a confounder for E→D, but is E a confounder for P→D?
Which effects are reported correctly in the table?
diabetes 2 E
fractures D
obesity O
bone density B
5 min
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16. Exercise: Stratify or not
Want the effect of action(A, exposure/treatment) on disease (D). Have stratified on C.
Make a guess at the population effect of A on D
Calculate the population effect of A on D
What is the correct analysis (and RR)? OBS several answers possible!
P=0.16
Re=2.0
Rd=5.0
Red=0.6
Low Bp=30%
C could be low/high blood pressure
Population = crude
Stratified = adjusted for C C A D
10 min
Hernan et al. 2011
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17. 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.
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18. Paths The Path of the Righteous Jan-19 Jan-19 Jan-19 H.S. H.S. H.S. 18
Ezekiel 25:17. "The Path of the Righteous Man Is Beset on All Sides by The inequities of the Selfish and the Tyranny of Evil Men."
(Pulp Fiction version)
Paths
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19. Path definitions
Path: any trail from E to D (without repeating itself)
Type: causal, non-causal
State: open, closed
Path 1
E®D 2
E®M®D 3
E¬C®D 4
E®K¬D
Four paths:
Notice: path with or against the arrows
Paths show potential association
Goal:
Keep causal paths of interest open
Close all non-causal paths
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20. Four rules 1. Causal path: ED 2. Closed path: K
(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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21. ANALYZING DAGs
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22. Confounding examples Jan-19 Jan-19 Jan-19 Jan-19 Jan-19 Jan-19 H.S.
Informal, no strict notation/def
Casual about the causal!
Confounding examples
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23. Vitamin and birth defects
Is there a bias in the crude E-D effect?
Should we adjust for C?
What happens if age also has a direct effect on D?
Unconditional
Path
Type
Status 1
E®D
Causal
Open 2
E¬C®U®D
Non-causal
Bias
This is an example
of confounding
Noncausal open=biasing path
Both C and U are confounders
Problem that we have ”forgotten” arrow C->D?
Conditioning on C
Path
Type
Status 1
E®D
Causal
Open 2
E¬[C]®U®D
Non-causal
Closed
Question:
Is U a confounder?
No bias 3
E¬[C] ®D
Non-causal
Closed
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24. Exercise: Physical activity and Coronary Heart Disease (CHD)
We want the total effect of Physical Activity on CHD.
Write down the paths.
Are they causal/non-causal, open/closed?
What should we adjust for?
Noncausal open=biasing path
5 minutes
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25. Intermediate variables
Direct and indirect effects
Intermediate variables
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26. Exercise: Tea and depression
Write down the paths.
You want the total effect of tea on depression. What would you adjust for?
You want the direct effect of tea on depression. What would you adjust for?
Is caffeine an intermediate variable or a variable on a confounder path?
Tea and depression: Finnish study
Caffeine reduces depression: Nurses Health Study
10 minutes
Hintikka et al. 2005
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27. Exercise: Statin and CHD
Write down the paths.
You want the total effect of statin on CHD. What would you adjust for?
If lifestyle is unmeasured, can we estimate the direct effect of statin on CHD (not mediated through cholesterol)?
Is cholesterol an intermediate variable or a collider? C
cholesterol U
lifestyle E
statin D
CHD
Statin: lipid (cholesterol) lowering drug
10 minutes
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28. Direct and indirect effects
So far: Controlled (in)direct effect
Causal interpretation: 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
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29. Mixed Confounder, collider and mediator Jan-19 Jan-19 Jan-19 Jan-19
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30. Diabetes and Fractures
We want the total effect of
Diabetes (type 2) on fractures
Conditional
Path
Type
Status 1
E→D
Causal
Open 2
E→F→D 3
E→B→D 4
E←[V]→B→D
Non-causal
Closed 5
E←[P]→B→D
Unconditional
Path
Type
Status 1
E→D
Causal
Open 2
E→F→D 3
E→B→D 4
E←V→B→D
Non-causal 5
E←P→B→D
Questions:
Paths ←→?
More paths?
B a collider?
V and P ind?
Diabetes->eye disease->fall,
could have ->eye disease->physical activity->
Diabetes II reduces bone density, BMI increases bone density
Questions: more paths (E-B-P-E-D)? Two (or three) arrows are colliding in B, is B a collider?
Mediators
Confounders
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31. Drawing DAGs
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32. Technical note on drawing DAGs
Drawing tools in Word (Add>Figure)
Use Dia
Use DAGitty
Hand-drawn figure.
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33. Direction of arrow C Does physical activity reduce smoking, or
does smoking reduce physical activity? ? E
Phys. Act. D
Diabetes 2 H
Health con. C
Smoking
Maybe an other variable
(health consciousness)
is causing both? E
Phys. Act. D
Diabetes 2
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34. Time C Does physical activity reduce smoking, or
does smoking reduce physical activity? ? E
Phys. Act. D
Diabetes 2 C
Smoking -5
Smoking measured 5 years ago
Physical activity measured 1 year ago E
Phys. Act. -1 D
Diabetes 2
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35. Drawing a causal DAG Start: E and D 1 exposure, 1 disease
add: [S] variables conditioned by design
add: C-s all common causes of 2 or more
variables in the DAG C
C must be included common cause
V may be excluded exogenous
M may be excluded mediator
K may be excluded unless we condition V E D M K
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36. Exercise: Drawing survivor bias
We what to study the effect of exposure early in life (E) on disease (D) later in life.
Exposure (E) decreases survival (S) in the period before D (deaths from other causes than D).
A risk factor (R) reduces survival (S) in the period before D.
The risk factor (R) increases disease (D).
Only survivors are available for analysis (look at Collider idea).
Draw and analyze the DAG
10 minutes
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37. Real world examples Jan-19 Jan-19 Jan-19 Jan-19 H.S. H.S. H.S. 37 37

38. Endurance training and Atrial fibrillation
Tobacco
Cardiovascular factors *
Alcohol consumption
Socioeconomic
Status **
BMI
Diabetes
Endurance training
Atrial fibrillation
Genetic disposition
Health *** consciousness
Hyperhyreosis
Height
Gender
Missing arrows: for example from Age and Gender to BMI, Tobacco, Cardio, …
Want direct effect of ET->AF, means close many paths, all missing paths are also closed.
Infinite chain of causality! Close factor distant factor
Age
Long-distance racing
Several arrows missing!
*Hypertension, heart disease, high cholesterol
** Socioeconomic status: Education, marital status
*** Unmeasured factors
(Blue: Mediators, red: confounders, violet: colliders)
Myrstad et al. 2014b

39. Methods to remove confounding
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40. Methods to remove confounding
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
Stratification: non-parametric adjustment
Regression: parametric adjustment
Matching in cohort (C=age): for every exposed person, find an unexposed of the same age.
Matching in CaseControl: for every case, find a control of the same age C
Case-Control matching?
Other methods?
Remove CD arrow E D
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41. Matching: Cohort vs Case-Control
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42. Matching in cohort, binary E
For every exposed person with a value of C, find an unexposed person with the same value of C S C E D
selected based on E and C
E independent of C after matching
 All open paths between C and E must C
C→E
C→[S]←E
sum to “null” E D
Cohort matching removes confounding
Unfaithful DAG
Stratification: non-parametric adjustment
Regression: parametric adjustment
Matching in cohort (C=age): for every exposed person, find an unexposed of the same age.
Matching in CaseControl: for every case, find a control of the same age
Cohort matching is not common,
except in propensity score matching
Mansournia et al. 2013; Shahar and Shahar 2012
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43. Matching in Case Control, binary D
For every case with a value of C, find a control with the same value of C C S E D
selected based on D and C
D independent of C after matching
 All open paths between C and D must C S
C→D
C→[S]←D
sum to “null” E D
C→E→D
Stratification: non-parametric adjustment
Regression: parametric adjustment
Matching in cohort (C=age): for every exposed person, find an unexposed of the same age.
Matching in CaseControl: for every case, find a control of the same age
Matched CC analysis:
1-1 matching: conditional logistic (condition on matched pair)
frequency matching: adjust for C
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
(Mansournia et al. 2013; Shahar and Shahar 2012
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44. Inverse probability weighting
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45. 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
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46. Inverse Probability WeightingC
Have observed data distribution:
Factorization:
𝑃(𝐶)∙𝑃(𝐸|𝐶)∙𝑃(𝐷|𝐸,𝐶) E D C
Want the RCT distribution:
Factorization:
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
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47. 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=1 𝐶 =1 if E=1
𝑃(𝐸=1|𝐶)/𝑃 𝐸 𝐶 =
𝑃(𝐸=1|𝐶)/𝑃 𝐸=0 𝐶 =OT if E=0
McCaffrey et al. 2004
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48. Marginal Structural Model
DAG for the reweighted pseudo data E D
MSM:
The expected value of a counterfactual outcome D
under a hypothetical exposure e:
𝐸 𝐷 𝑒 = 𝛼 0 + 𝛼 1 𝑒
effect =𝐸 𝐷 1 −𝐸 𝐷 0
Veieroed, Lydersen et al Daniel, Cousens et al Rothman, Greenland et al. 2008
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49. Outcome versus exposure modeling
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50. Methods to remove confounding
How can we remove confounding? A C B
What variables should be involved? C- A- B-
yes E D no
maybe
Close ECD by conditioning
(ordinary) regression model
outcome
modeling
E(D| E,C)
Remove the EC arrow, binary E
Propensity score matching
Inverse probability weighting
exposure
modeling
P(E|C)
Combine exposure and outcome modeling: doubly robust models

51. Outcome vs exposure modeling
Outcome model: E(D|E,C) (and possibly B)
Ex: Nano-particlesCardioVascularDisease
Know little about risk for nano-particles
Know a lot about risk factors for CVD A C B E D
Do outcome model
Exposure model: E(E|C)
Ex: SmokingBladder cancer
Know a lot about risk for smoking
Know little about risk factors for bladder cancer A C B E D
Do exposure model
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52. 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!
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53. Randomized experiments Mendelian randomization
Randomized experiments: 3 reasons
Interesting in the selves, understand ITT
Basis for conditions for causal estimation: exchangeability and positivity
Lead to IV methods, Mendelian rand
Randomized experiments Mendelian randomization
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54. Strength of arrow, randomizationC1
Not
deterministic E D C2
C1, C2, C3 exogenous C3 C
full
compliance
Full compliance
 no E-D confounding R E D
deterministic U
Not full compliance
weak E-D confounding
but R-D is unconfounded
U could be: a condition that gives side effects of drug E (therefore less compliant) and gives risk of D
not full
compliance R E D
Path
Type
Status 1
R®E®D
Causal
open 2
R®E¬U®D
non-causal
closed
Sub analysis conditioning on E
may lead to bias
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55. Randomized experiments
Observational study
Randomized experiment with full compliance
R= randomized treatment
E= actual treatment.
R=E
Randomized experiment with less than full compliance (c) c
IVe
No way of drawing that R=E or that R has strong effect on E (R-E arrow is determinisric)
If R has strong effect on E then U must have weak effect, that is little confounding.
ITT weaker than IV effect, combination of behavior (compliance) and biology
Per Protocol=as treated
If linear model: ITTe=c*IVe, c<1
ITTe
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
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56. RCT exercise
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.
Calculate the compliance (c) as a risk difference from the table.
Calculate the intention to treat effect (ITTe) as a risk difference.
Calculate the per-protocol effect (PP) as a risk difference.
Calculate the instrumental variable effect (IVe).
Explain the results in words.
P(U=1)=0.5 c
IVe
ITTe
-0.15
0.22
10 minutes
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57. Mendelian randomizationU
Observational study
Suffers from unmeasured confounding U
Randomized trial: 3 conditions
R affects E: balanced, strong effect
No direct R-D effect: R independent of D | E
R and D no common causes: R independent of U 3 R 1 E D 2
G->E Often E is a metabolite of common food or drink, G controls conversion into E, or out of E
G often rare allele - wild type N=100 in RCT, N= in Medelian
5% non-compliance gives RR=20 for R,E association
(If full compliance, can draw the do-operator)
What happens with R-D ass if compliance drops?
Discusiion paper: Sleiman and Grant, Clin Chem 2010
Examples of Mendelian rand from Sleiman and Grant:
Honkanen R, Kroger H, Alhava E, Turpeinen P,
Tuppurainen M, Saarikoski S. Lactose intolerance
associated with fractures of weight-bearing
bones in Finnish women aged 38–57 years. Bone
1997;21:473–7.
Brennan P, McKay J, Moore L, Zaridze D, Mukeria
A, Szeszenia-Dabrowska N, et al. Obesity and
cancer: mendelian randomization approach utiliz-
ing the FTO genotype. Int J Epidemiol 2009;38:971–5.
Freathy RM, Bennett AJ, Ring SM, Shields B,
Groves CJ, Timpson NJ, et al. Type 2 diabetes risk
alleles are associated with reduced size at birth.
Diabetes 2009;58:1428–33.
32. Zhao J, Li M, Bradfield JP, Wang K, Zhang H,
Sleiman P, et al. Examination of type 2 diabetes
loci implicates CDKAL1 as a birth weight gene.
Diabetes 2009;58:2414–8.
33. Thorgeirsson TE, Geller F, Sulem P, Rafnar T,
Wiste A, Magnusson KP, et al. A variant associ-
ated with nicotine dependence, lung cancer and
peripheral arterial disease. Nature 2008;452:
638–42. U
Medelian randomization: 3 conditions
G must affect E: unbalanced, weak  large N
No direct G-D effect: depends on gene function
G and D no common causes: Mendel’s 2. law 3 G 1 E D 2
Sheehan et al, 2008
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58. Ex: Alcohol and blood pressureBP U
Observational study
Alcohol use increases blood pressure
Many ”lifestyle” confounders
Gene: ALDH2, 2 alleles
2,2 type suffer nausea, headache after alcohol
 low alcohol regardless of lifestyle (U)
Medelian randomization
Gene ALDH2 is highly associated with alcohol
OK, gene function is known
Mendel’s 2. law, no ass. to obs. confounders U 3
John Maynard Smith
Chen: meta analysis, n=7,658, RR=6-7 for alcohol use, typical in SNP studies is RR=
Phenotypes: wildtype 1,1=57%, hetero 1,2=37%, risk homo 2,2=6%, that is fairly common risk type
ALDH2: aldehyde dehydrogenase 2
Medelian randomization
Gene common i Japan, RR=6-7 for alcohol use
Gene no associated with age, smoking, BMI, cholesterol. Gene could be ass with coffe
Gene function does not affect blood pressure
Result: Meta analysis by Chen at al. 2008, alcohol lead to higher bp (1,1 versus 2,2 =+7.4mmHg) G 1 A BP
Result:
1,1 type BP +7.4 mmHg
Alcohol increases blood pressure 2
Chen et al 2008
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59. DAGs and other causal models
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60. Greenland and Brumback, Causal modeling methods, Int J Epid 2002
DAGs and causal pies
SCC Sufficient Component Causes
background causes assumed!
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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61. Exercise: causal pies H E D hospital diabetes fractures 10 minutes
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.
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). H
hospital E
diabetes D
fractures
10 minutes
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62. Structural Equation Models, SEM
Causal assumptions + statistical model + data
SEM: parametric DAG
X is unmeasured (latent), x-I and y are measured variables
Legg in bilde fra AMOS som eksempel
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63. 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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64. Selection bias Two (3) concepts Jan-19 Jan-19 Jan-19 Jan-19 Jan-19
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65. 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
 Selection bias: effect of E overestimated
Selection bias often based on idea of difference: the selected are different from unselected. Must be different in what we are measuring.
Different in prevalence
Different in E-D effect
Weight by stratum size or inverse stratum variance
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66. Selection bias: concept 1 “Selected different from unselected”
Paths
Type
Status
smoke®CHD
Causal
Open S
age
smoke
CHD
Normally, selection variables unknown
Selection bias often based on idea of difference: the selected are different from unselected. Must be different in what we are measuring.
Different in prevalence
Different in E-D effect
Weight by stratum size or inverse stratum variance
Properties:
- Need smoke-age interaction
- Cannot be adjusted for, but stratum effects OK
True RR=weighted average of stratum effects
RR in “natural” range ( )
Scale dependent
Name:
interaction based?
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67. 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
Selection bias often based on idea of difference: the selected are different from unselected. Must be different in what we are measuring.
Different in prevalence
Different in E-D effect
Weight by stratum size or inverse stratum variance
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68. Selection bias: concept 2 “Distorted E-D distributions”
Paths
Type
Status
smoke®CHD
Causal
Open
smoke¬sex®[S]¬age®CHD
Non-causal
sex
age
smoke
CHD
Properties:
Open non-causal path (collider)
Does not need interaction
Can be adjusted for (sex or age)
Not in “natural” range (“surprising bias”)
Name:
Collider stratification bias
Common table of properties?
Both types of selection may operate in the named examples.
Ref to Pearl
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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69. 1) “Exclusive or” selection
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70. Exercise: Dust and COPD Chronic Obstructive Pulmonary Disease
Calculate the RR of dust on COPD in good and poor health groups.
Write down the paths for the effect of E on D. E0 and D0 are unknown (past) measures.
What would you adjust for?
Suppose the crude effect of dust on COPD is RR=0.7 and the true RR=2. What do you call this bias?
Could the concept 1 (interaction based) selection bias work here? S
cur. worker D0
diseases H
health E0
prior dust E
cur. dust D
COPD
COPD risks:
COPD: Chronic obstructive pulmonary disease
Risk factors: smoking, air pollution, genetics, workplace dust
10 minutes
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71. Convenience sample, homogenous sample
hospital
Convenience:
Conduct the study among
hospital patients? E
diabetes D
fractures
2. Homogeneous sample:
Population data,
exclude hospital patients?
Unconditional
Path
Type
Status 1
E→D
Causal
Open 2
E→H←D
Non-causal
Closed
Conditional
Path
Type
Status 1
E→D
Causal
Open 2
E→[H]←D
Non-Causal
Collider, selection bias
Collider stratification bias: at least on stratum is biased
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72. Outcome dependent selection
Selection into the study based on D.
Get bias among selected. E D U
Explanation:
Always have exogenous U.
D is a collider on E→D←U, S is a descendant of collider D.
Conditioning on (a descendant of) a collider opens the E→D←U path, and U becomes associated with E.
U now acts a confounder for E→D.
Selection depends on:
Strength of E→D. Strength of U→D
Example of non-causal confounding
Unmatched Case-Control
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73. Selection bias summing up
Concept 1
Concept 2
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
smoke
CHD
age S S
sex
age
smoke
CHD
Quite different concepts
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74. MORE ON DAGs
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75. Back door, front door, D-separated
Paths from E to D, all are “leaving” E
Paths open before conditioning:
back door
.
non-causal
open
need to close
front door
.
causal
open E
Plus paths closed at a collider
If all paths from A to B are closed  d-separated
Pearl 2000
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76. 3 strategies for estimating causal effectsD C
Back-door criterion
Condition to close all no-causal paths (between E and D) E D U M1 M2
Front-door criterion
Condition an all intermediate variables (between E and D)
Instrumental Variables
Use an IV to control the effect (of E on D)
IV criteria:
IV must affect E
No direct IV-D effect
IV and D no common causes U 3 IV 1 E D 2
Pearl 2009, Glymor and Greenland, 2008
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77. Example: front–door criterion
Weight and Coronary Heart Disease U
lifestyle
Assume:
adjusted for sex, age and smoke
lifestyle is unmeasured
no other mediators (between E and D) C
cholesterol E
weight D
CHD B
blood pressure
Can estimate effect of E on D
Path
Type
Status 1
E→C→D
Causal
Open 2
E→B→D 3
E←U→D
Non-causal
Difference = causal
Crude
Adjusted for B and C
Weight is not a good “action”
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78. 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 E D B A
Confounding:
Non-causal path
without colliders K E D B A
Selection bias:
Non-causal path open
due to conditioning
on a collider E D B A
Causal
Note: interaction based selection bias not included
Hernan et al, A structural approach to selection bias, Epidemiology 2004
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79. Testable implications
The DAG implies:
C is independent of D given E and O O
test
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
Textor et al. 2011
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80. Definitions
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81. Causal graphs: definitions
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: [(L,A),(A,Y)]
Parent → child
Ancestors → → descendants
Exogenous: variables with no parents U L A Y U
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82. Directed Acyclic Graphs
Ordinary DAG
Arrows = associations
Causal DAG
Arrows = cause
All common causes of any pair of variables in the DAG are included
Two types of variables
Immutable sex, age
Mutable exposure (actions), smoking
Mixing variables in a DAG is OK
All dependence/independence conclusions valid L A Y U
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83. 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)
DAGs are non-parametric
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84. DAG units DAG units individuals populations C C gene D gene D
(almost) No variable can influence a gene in an individual 
No confounding
A variable can influence gene frequency in a population
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85. D-separation, moralization
Directed graph-separation
two variables d-separated if no open path
otherwise d-connected
2 DAG analyses
Paths (Pearl)
Moralization (Lauritzen)
equivalent
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86. DAGs and probability theory
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87. DAGs rules and statistical independence
DAG correct?
Two assumptions
Compatibility: separated  independent
Faithfulness: separated  independent
= connected  dependent
Weak faithfulness: connected variables may be dependent B B A Y A Y
Pearl 2009, Glymor and Greenland, 2008
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88. Limitations, problems and extensions of DAGs
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89. 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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90. DAG focus: bias, not precision
Should we adjust for C?
DAG: no bias from C, need not adjust E D
May include C to improve precision, depends on model!
E->D=1 in linear regr
C->D=2 in linear regr
D2=10% in logistic, crude E->D=1.17, adjusted E->D=1.43,
no E-C interaction
Including C: better precision
Including C: worse precision
OR not collapsible
Robinson and Jewell 1991; Xing and Xing 2010
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91. Signed DAGs and direction of biasU  M
Positive or negative bias from confounding by U? + +
Neg
True
E→D
Pos E D
on average
Average monotonic effect
+ -
X  Y →
for all Y=y
Distributional monotonic effect 
To find direction of bias, multiply signs:
Need distributional monotonic effects except at each end
Positive bias from this confounding
VanderWeele, Hernan & Robins, 2008
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92. Size of bias from unmeasured UC 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
𝛼=𝑃 𝑢 𝑎 1 −𝑃 𝑢 𝑎 0
γ=𝐸 𝑌 𝑢 1 −𝐸 𝑌 𝑢 0 if linear model
γ=𝐸 𝑌 𝑢 1 /𝐸 𝑌 𝑢 if RR model
episens
Bias = 𝛼∗𝛾
if linear model
Bias = 1+ 𝛾−1 𝑃 𝑢 𝑎 𝛾−1 𝑃 𝑢 𝑎 0
if RR model
Stata: episens
VanderWeele & Arah 2011
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93. Interaction in DAGs + = DAG Causal pie Extended DAG C C C D E,C D E E
Mech-
anisms C C C C
E C + = D
E,C D E E E E
Red arrow =
interaction
Specify scale
VanderWeele and Robins 2007
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94. 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
HDL A1 A2
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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95. Most paths involving variables back in the chain (U)
Infinite causal chain U we
adjust
for
variables E D in
the
analysis
Most paths involving variables back in the chain (U)
will be closed
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96. DAGs are simplified DAGs are models of reality
must be large enough to be realistic,
small enough to be useful
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97. Better discussion based on DAGs before your conclusions
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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98. Recommended reading Books Papers
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
Hernan and Robins Causal inference (web)
Hernan a struct approach
Hernandez- From causal
Shahar
Rothman
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Jan-19
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99. References 1
Aalen OO, Roysland K, Gran JM, Ledergerber B Causality, mediation and time: A dynamic viewpoint. Journal of the Royal Statistical Society Series A 175:
Chen L, Davey SG, Harbord RM, Lewis SJ 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 Methods for dealing with time-dependent confounding. Statistics in Medicine 32:
Greenland S, Schlesselman JJ, Criqui MH Re: "The fallacy of employing standardized regression coefficients and correlations as measures of effect". AJE 125:
Greenland S, Robins JM, Pearl J Confounding and collapsibility in causal inference. Statistical Science 14:29-46.
Greenland S, Brumback B An overview of relations among causal modelling methods. Int J Epidemiol 31:
Greenland S, Mansournia MA 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 Standardized regression coefficients: A further critique and review of some alternatives. Epidemiology 2:6.
Hafeman DM, Schwartz S Opening the black box: A motivation for the assessment of mediation. International Journal of Epidemiology 38:
Hernan MA, Hernandez-Diaz S, Werler MM, Mitchell AA Causal knowledge as a prerequisite for confounding evaluation: An application to birth defects epidemiology. AJE 155:
Hernan MA, Hernandez-Diaz S, Robins JM A structural approach to selection bias. Epidemiology 15:
Hernan MA, Cole SR Causal diagrams and measurement bias. AJE 170:
Hernan MA, Clayton D, Keiding N The simpson's paradox unraveled. Int J Epidemiol.
Hintikka J, Tolmunen T, Honkalampi K, Haatainen K, Koivumaa-Honkanen H, Tanskanen A, et al Daily tea drinking is associated with a low level of depressive symptoms in the finnish general population. European Journal of Epidemiology 20:
Lange T, Hansen JV Direct and indirect effects in a survival context. Epidemiology 22:
Mansournia MA, Hernan MA, Greenland S Matched designs and causal diagrams. International Journal of Epidemiology 42:
McCaffrey DF, Ridgeway G, Morral AR Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychological Methods 9:
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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100. References 2
Pearl J Causality: Models, reasoning, and inference. Cambridge:Cambridge Univeristy Press.
Pearl J The causal mediation formula-a guide to the assessment of pathways and mechanisms. Prev Sci 13:
Robins JM, Greenland S Identifiability and exchangeability for direct and indirect effects. Epidemiology 3:
Robins JM Data, design, and background knowledge in etiologic inference. Epidemiology 12:
Robinson LD, Jewell NP Some surprising results about covariate adjustment in logistic-regression models. Int Stat Rev 59:
Rothman KJ, Greenland S, Lash TL Modern epidemiology. Philadelphia:Lippincott Williams & Wilkins.
Shahar E Causal diagrams for encoding and evaluation of information bias. Journal of evaluation in clinical practice 15:
Shahar E, Shahar DJ Causal diagrams and the logic of matched case-control studies. Clinical epidemiology 4:
Sheehan NA, Didelez V, Burton PR, Tobin MD Mendelian randomisation and causal inference in observational epidemiology. PLoS Med 5:e177.
Textor J, Hardt J, Knuppel S Dagitty a graphical tool for analyzing causal diagrams. Epidemiology 22:
VanderWeele TJ, Robins JM Directed acyclic graphs, sufficient causes, and the properties of conditioning on a common effect. AJE 166:
VanderWeele TJ, Hernan MA, Robins JM Causal directed acyclic graphs and the direction of unmeasured confounding bias. Epidemiology 19:
VanderWeele TJ Mediation and mechanism. Eur J Epidemiol 24:
VanderWeele TJ, Arah OA Bias formulas for sensitivity analysis of unmeasured confounding for general outcomes, treatments, and confounders. Epidemiology 22:42-52.
VanderWeele TJ, Hernan MA Results on differential and dependent measurement error of the exposure and the outcome using signed directed acyclic graphs. AJE 175:
VanderWeele TJ A unification of mediation and interaction: A 4-way decomposition. Epidemiology 25:
Veieroed M, Lydersen S, Laake P Medical statistics in clinical and epidemiological research. Oslo:Gyldendal Akademisk.
Westreich D, Greenland S The table 2 fallacy: Presenting and interpreting confounder and modifier coefficients. AJE 177:
Xing C, Xing GA Adjusting for covariates in logistic regression models. Genet Epidemiol 34:
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101. Jan-19
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102. Extra material
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103. Mediation Analysis Hafeman and Schwartz 2009; Lange and Hansen 2011;
Pearl 2012;
Robins and Greenland 1992;
VanderWeele 2009, 2014
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104. Why mediation analysis?
Have found a cause
How does it work? M
𝑇𝑜𝑡𝑎𝑙 𝑒𝑓𝑓𝑒𝑐𝑡=𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡+𝑑𝑖𝑟𝑒𝑐𝑡
indirect
effect
𝑀𝑒𝑑𝑖𝑎𝑡𝑒𝑑 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛= 𝑖𝑛𝑑𝑖𝑟𝑒𝑐𝑡 𝑡𝑜𝑡𝑎𝑙 A
direct effect Y
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105. Classic approach: controlling
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106. Controlled Direct effect
Effect of statin on CHD
“for the same cholesterol”
Fixed M m
Fixed M: controlled direct effect
CDE=E(Y|A=1,M=m) - E(Y|A=0,M=m) M
cholesterol
Problems
Conceptual: Can we fix cholesterol levels?
Technical: A*M Interaction?
(Technical: non-linear models?)
Statin: lipid (cholesterol) lowering drug A
statin Y
CHD
0/1
Solution?
Robins and Greenland 1992; VanderWeele 2009
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106

107. New approach: counterfactuals
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108. Counterfactual causal effect
Two possible outcome variables
Outcome if treated: Y1
Outcome if untreated: Y0
Counterfactuals
Potential outcomes
Causal effect
Individual: Y1i-Y0i
Average: E(Y1)-E(Y0)
or other effect measures
Fundamental problem: either Y1 or Y0 is missing
Hernan 2004
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109. Total causal effect, TCE
Potential (counterfactual) outcomes:
Y1 is the outcome if A is set to 1
Effect of statin on CHD
M1 is the mediator if A is set to 1 M1 M0
A set to 1
A set to 0 M1 M0 M
cholesterol Y1 Y0
Statin: lipid (cholesterol) lowering drug
𝑇𝐶𝐸=𝐸 𝑌 − 𝐸 𝑌 0 A
statin Y
CHD
=𝐸 𝑌 1, 𝑀 1 −𝐸 𝑌 0, 𝑀 0
0/1
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109
109

110. “for the same cholesterol”
Natural Direct effect
Direct effect:
Effect of statin on CHD
“for the same cholesterol” M1 M0
A set to 1
A set to 0 M0 M0 M
cholesterol
Natural Direct Effect: Keep M at M0
Statin: lipid (cholesterol) lowering drug
𝑁𝐷𝐸=𝐸 𝑌 1, 𝑀 0 −𝐸 𝑌 0, 𝑀 0 A
statin Y
CHD
Takes care of the 3 earlier problems:
Don’t need to fix M=m
OK for interactions
(OK for non-linear models) in 4 slides
0/1
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110
110

111. 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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112. Natural Indirect effect
Effect of statin via cholesterol on CHD
“for the same statin”
M set to M1
M set to M0 M1 M0
A=1
A=1 M
cholesterol
Natural Indirect Effect: Keep A at 1
𝑁𝐷𝐸=𝐸 𝑌 1, 𝑀 1 −𝐸 𝑌 1, 𝑀 0
Statin: lipid (cholesterol) lowering drug A
statin Y
CHD
Why keep A at 1?
0/1
then direct + indirect = total
Jan-19
Jan-19
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H.S.
112
112

113. Natural direct and indirect effects 1
Natural Direct Effect: 𝑁𝐷𝐸=𝐸 𝑌 1, 𝑀 0 −𝐸 𝑌 0, 𝑀 0 A=1 vs. 0 for M=M0 Natural Indirect Effect: 𝑁𝐼𝐸=𝐸 𝑌 1, 𝑀 1 −𝐸 𝑌 1, 𝑀 0 M=M1 vs. M0 for A=1 Total Causal Effect: 𝑇𝐶𝐸=𝐸 𝑌 1, 𝑀 1 −𝐸 𝑌 0, 𝑀 0 =𝑁𝐷𝐸+𝑁𝐼𝐸
Jan-19
H.S.

114. Binary outcome, RR effect measure
Natural Direct Effect: 𝑁𝐷𝐸 𝑅𝑅 = 𝑃 𝑌 1, 𝑀 0 =1 𝑃 𝑌 0, 𝑀 0 =1 A=1 vs. 0 for M=M0 Natural Indirect Effect: 𝑁𝐼𝐸 𝑅𝑅 = 𝑃 𝑌 1, 𝑀 1 =1 𝑃 𝑌 1, 𝑀 0 =1 M=M1 vs. M0 for A=1 Total Causal Effect: 𝑇𝐶𝐸 𝑅𝑅 = 𝑃 𝑌 1, 𝑀 1 =1 𝑃 𝑌 0, 𝑀 0 =1 = 𝑁𝐷𝐸 𝑅𝑅 ∙ 𝑁𝐼𝐸 𝑅𝑅
Jan-19
H.S.

115. Example: Labor marked discrimination Are Emily and Greg more employable than Lakisha and Jamal?
Names
White: Emely Walsh, Greg Baker
Af.-am.: Lakisha Washington, Jamal Jones
CV:
low/high quality
Job:
callback
Experiment:
5000 CV with random name type
Boston and Chicago, 2002?
Result:
White names: 10 CV-s before callback
African-am: 15 CV-s before callback CV
name
Job
Bertrand and Mullainathan 2004
Jan-19
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116. Exercise: Labor marked discrimination
Describe the experiment for estimating the controlled direct effect
Describe the experiment for estimating the natural direct effect CV
name
Job
5 minutes
Jan-19
H.S.

117. Assumptions U2 M U3 Classic method: Linear models No A-M interaction
Classic and new:
No unmeasured confounders (U1, U2, U3,)
No treatment dependent confounders (P) P A Y U1
VanderWeele 2009
Jan-19
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118. Summing up (so far) Classic method New approach
Linear models, no A-M interaction
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 A Y
Jan-19
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119. Mediation analysis
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120. 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
Jan-19
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121. Classic method Classic method (Baron & Kenny): Only OK for
Total effect: regress Y on A
Direct effect: regress Y on A and M
Indirect effect: Total – Direct
Only OK for
Linear models
No A*M interaction
Strong no-confounding assumptions M A Y
Jan-19
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122. New methods Estimate natural direct and indirect effects (Stata) D M
Multiple M
Sensitivity
Paramed
continuous
binary
count no
Idecomp
any
yes
Medeff
Gformula* ?
(Lange-2011)
Time-to-event
(Lange-2012)
* Only for treatment dependent confounding
Jan-19
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123. Mediation analysis: 4-way decomposition
Jan-19
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124. Notation Counterfactuals if A is set to a, M is set to m: M 𝑌 𝑎 𝑀 𝑎
𝑌 𝑎𝑚 A Y
For simplicity assume binary A and M:
(general results available)
𝑌 10
a m
Jan-19
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125. Interaction on additive scaleM
No additive interaction: 
𝑅𝐷 11 = 𝑅𝐷 𝑅𝐷 01 A Y 
𝑌 11 − 𝑌 00 = 𝑌 10 − 𝑌 00 + 𝑌 01 − 𝑌 00 
𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 =0
Measure of additive interaction:
𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00
Jan-19
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126. Interaction in DAGs + = DAG Causal pie Extended DAG M M M Y A,M Y A A
Mech-
anisms M M M M
A M + = Y
A,M Y A A A A
Red arrow =
interaction
Specify scale
VanderWeele and Robins 2007
Jan-19
H.S.

127. 4 way decomposition M
TE = total effect
CDE = controlled direct effect
INTref = interaction effect at reference
INTmed = interaction*mediator effect
PIE = pure indirect effect A Y
Can always decompose the total effect into 4:
Medi-
ation
Inter-
action
𝑌 1 − 𝑌 0 = TE
(𝑌 10 − 𝑌 00 )
CDE - -
+ 𝑀 0 ( 𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 )
INTref - +
+ ( 𝑀 1 −𝑀 0 )( 𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 )
INTmed + +
+ ( 𝑀 1 −𝑀 0 )( 𝑌 01 − 𝑌 00 )
PIE + -
Jan-19
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128. Individual 4 mechanisms
TE = total effect
CDE = controlled direct effect
INTref = interaction effect at reference
INTmed = interaction*mediator effect
PIE = pure indirect effect A Y
If AY for one subject then one of 4 mechanisms will work: M I
(𝑌 10 − 𝑌 00 )
AY|M=0 - -
𝑀 0 ( 𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 )
M0=1 & M*A inter. - +
( 𝑀 1 −𝑀 0 )( 𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 )
AM & M*A inter. + +
( 𝑀 1 −𝑀 0 )( 𝑌 01 − 𝑌 00 )
AM & MY|A=0 + -
Jan-19
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129. Interaction in DAGs Extended DAG M M A,M Y A A Mech- anisms
VanderWeele and Robins 2007
Jan-19
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130. Individual 4 mechanisms1 2 3 4 A Y M
A,M A Y M
A,M A Y M
A,M A Y M
A,M
If AY for one subject then one of 4 mechanisms will work: M I
(𝑌 10 − 𝑌 00 )
AY|M=0 - -
𝑀 0 ( 𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 )
M0=1 & M*A inter. - +
( 𝑀 1 −𝑀 0 )( 𝑌 11 − 𝑌 10 − 𝑌 01 + 𝑌 00 )
AM & M*A inter. + +
( 𝑀 1 −𝑀 0 )( 𝑌 01 − 𝑌 00 )
AM & MY|A=0 + -
Jan-19
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131. 4 and 2 way decompositions
Mediation:
TE = CDE + INTref + INTmed + PIE
TE =Natural Direct + Natural Indirect
Jan-19
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132. Identification (estimate from data)
Notation:
Counterfactuals if
A is set to a, M is set to m: M
𝑌 𝑎
𝑀 𝑎
𝑌 𝑎𝑚
For simplicity assume binary A and M:
(general results available)
𝑌 10 A Y
Consistency:
Consistency assumption:
𝐼𝑓 𝐴=𝑎 𝑎𝑛𝑑 𝑀=𝑚 𝑡ℎ𝑒𝑛 𝑌 𝑎𝑚 =𝑌
Composition assumption:
𝑌 𝑎 = 𝑌 𝑎 𝑀 𝑎
Confounding: U2 M U3
4 assumptions needed for estimation:
No unmeasured confounding (U1-U3)
No confounders affected by A (P) P A Y U1
Jan-19
H.S.

133. Y and M in linear regression models
Assume Y and M continuous and following: M
𝐸 𝑌 𝑎,𝑚,𝑐 = 𝜃 0 + 𝜃 1 𝑎+ 𝜃 2 𝑚+ 𝜃 3 𝑎𝑚+ 𝜃 4 ′ 𝑐
𝐸 𝑀 𝑎,𝑐 = 𝛽 0 + 𝛽 1 𝑎+ 𝛽 2 ′ 𝑐 A Y
then
𝐸 𝐶𝐷𝐸 𝑐 = 𝜃 1 (𝑎− 𝑎 ∗ )
𝐸 𝐼𝑁𝑇 𝑟𝑒𝑓 𝑐 = 𝜃 3 ( 𝛽 0 + 𝛽 1 𝑎 ∗ + 𝛽 2 ′ 𝑐)(𝑎− 𝑎 ∗ )
𝐸 𝐼𝑁𝑇 𝑚𝑒𝑑 𝑐 = 𝜃 3 𝛽 1 (𝑎− 𝑎 ∗ ) 2
𝐸 𝑃𝐼𝐸 𝑐 = (𝜃 2 𝛽 1 + 𝜃 3 𝛽 1 𝑎 ∗ ) (𝑎− 𝑎 ∗ )
a=1 and a*=0 would simplify further
Jan-19
H.S.

134. Binary outcomes and ratio scale
Similar results based on RR
(with a scaling factor) A Y
Example:
Smoking
Lung
Cancer
Gene
15q25.1 rs C alleles
Jan-19
H.S.

135. Non-collapsibility of the odds ratio
Jan-19
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136. Non-collapsibility of the ORD
Jan-19
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137. Non-collapsibility of the OR
Non-collapsibility depends on frequency of D E D
Not collapsible
Appr. collapsible
Collapsible
Jan-19
H.S.

138. Information bias Hernan and Cole 2009; Shahar 2009;
VanderWeele and Hernan 2012
Jan-19
Jan-19
Jan-19
Jan-19
Jan-19
Jan-19
H.S.
H.S.
H.S.
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138
138
138
138
138
138

139. Depicting measurement error
Error in E
E=true exposure
E*=measured exposure
UE=process giving error in E b)
Error in E and D
D=true disease
D*=diagnosed disease
UD=process giving error in D
E=fat intake, E* fat intake from questionnaire
D=infarctions, D*= diagnosed infarctions
a) and b) Can test H0
b) shows independent non-differential errors
Jan-19
H.S.

140. Dependent errors, differential errors
a) E and D measures both temp dependent
b) Alcohol in preg and malformations
c) Air pollution and asthma
No assumption of additive error or linear effect of E on D needed to explain concepts, but needed to estimate effect of errors
Dependent errors:
Temp. in lab
Differential error:
Recall bias in Case-Control study
Differential error:
Investigator bias in cohort study
Hernan and Cole, Causal Diagrams and Measurement Bias, AJE 2009
Jan-19
H.S.

141. 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.
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.
Show the paths for the effect of E on D. Will there be a bias?
Show the paths for the effect of E* on D. Will there be a bias?
Can E* be associated with D even if E→D is zero?
10 minutes
Jan-19
H.S.

142. Selection bias depicted
Jan-19
H.S.

143. Simplified example S E and D continuous, Z, normal
Selection S
E and D continuous, Z, normal
True effect of E on D: =0 E D
EMF IQ
Stratification  Selection
Selection in quadrants
(common understanding for continuous and binary variables)
Change scales?
Focus: selection types and bias patterns
(Clarity over realism)
Jan-19 HS

144. 1) “Exclusive or” selection
Jan-19
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145. 2) “Inclusive or” selection
Jan-19
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146. 3) “And” selection Jan-19 HS All: all states and all bp-s
Only Hypertensive will participate (just 5% participation among low bp)
Perform the study in a state with high fluoride in water (just 5% participation from low fluoride (private) water)
Jan-19 HS

147. 4) “Gradient” selection
Assume published studies show EMF->IQ effects.
Jan-19
H.S.

148. Z-scores
Jan-19
H.S.

149. Birth weight paradox M U E D Results: birth weight
Maternal smoking increases neonatal mortality overall
Maternal smoking decreases neonatal mortality among low birth weight
Possible explanation:
conditioning on M opens collider path via U
Some advocate standardizing birth weight with respect to smoking, i.e. Z-scores
birth weight M U E
smoke D
neonatal mort
Jan-19
H.S.
H.S.
149
149

150. Z-scores M U Z E D 𝑍 𝑖𝑗 = 𝑏𝑖𝑟𝑡ℎ 𝑤𝑒𝑖𝑔𝑡ℎ 𝑖𝑗 − 𝑚 𝑗 𝑠𝑑 𝑗 𝑗=0,1 (𝑠𝑚𝑜𝑘𝑒)
𝑍 𝑖𝑗 = 𝑏𝑖𝑟𝑡ℎ 𝑤𝑒𝑖𝑔𝑡ℎ 𝑖𝑗 − 𝑚 𝑗 𝑠𝑑 𝑗 𝑗=0,1 (𝑠𝑚𝑜𝑘𝑒)
E(smoke) independent of Z
 All open paths between E and Z must
E→Z
E→M→Z
sum to “null”
when we condition on Z
birth weight M U
Adjusting for Z estimates the total effect of E on D Z
No gain,
crude model also estimates the total effect of E on D E
smoke D
neonatal mort
“unfaithful”
Jan-19
H.S.
H.S.
150
150

151. Z-scores G U E D Should not adjust for gestational age: gest. age
removes part of the effect of exposure
opens up a collider path involving U
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 G
gest. age U E
toxicant D
birth weight
Jan-19
H.S.
H.S.
151
151