Summary of Causality
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/courses
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Contents
• Concepts
– Statistics and Causality
– Counterfactuals, Actions
• Causal models
– DAGs, Pies, SEM, (MSM)
• Causal inference
– Exchangeability, Positivity, Consistency
• Methods of adjustment
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Statistics and Causality (J. Pearl)
Concepts
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Traditional Statistics
Data
P
Joint
distribution
Q(P)
(Aspects of P)
• Inference
– Infer if customers who by product A will also by product B
– Q=P(B|A)
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From Statistical to Causal analysis
P
Joint
distribution
Data
P’
Joint
distribution
change
Q(P’)
(Aspects of P’)
• Intervention: P changes to P’
– Infer if customers who by product A will also by product B when
we double the price
– Statistics deals with static relations, P does not tell us how it
ought to change: P’(v)P(v|price=2)
– Need assumptions about aspects of P that stay invariant under
intervention (change)
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Statistical and Causal concepts
• Statistical and causal concepts do not mix
Statistical
Causal
Association
Randomization / intervention
Controlling for / Conditioning
Confounding
Independence
Instrumental variable
Collapsibility
Endogeneity
• No causes in – no causes out
Causal assumptions +
Statistical assumptions + Data
= causal conclusions
• Standard mathematics
– Causal assumptions cannot be expressed
• Non-standard mathematics
– Structural Equation Models (Wright 1920, Simon 1969)
– Counterfactuals (Neyman-Rubin)
– Do-operator (Pearl)
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Potential outcome, Counterfactual
Concepts
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Individual causal effect
• Individual causal effect
– Outcome if exposed
– Outcome if unexposed
– Causal effect if
Y1
Y0
Y1  Y0
Potential outcomes
Counterfactual
• Important
– Clear definition
– Notation  mathematical proofs
– Notation  new methods
• Estimate individual effect?
– No, (but Crossover design)
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Population causal effect
• Average causal effect
– Expected outcome if all exposed
– Expected outcome if all unexposed
– Causal effect if
E(Y1)
E(Y0)
E(Y1)  E(Y0)
• Causal effect measures
– RDcausal= E(Y1) - E(Y0)
– RRcausal= E(Y1) / E(Y0)
• Estimate average effect?
– Yes, Randomized Controlled Trial
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Actions
Modifiable risk factors
• Examples
Not modifiable risk factors
• Examples
– Smoking, Radon
• Actions
– Sex, Age
• Actions
– Reduce prevalence of
smoking from 15% to 10%
– ?
Causal effects are strictly speaking only defined for actions
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DAGs, Pies and SEM
Causal models
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Causal models
• Four models
–
–
–
–
Causal graphs (DAGs)
Causal Pies, Sufficient Component Cause (SCC)
Structural Equation Models (SEM)
Potential outcome models
• Marginal Structural Models (MSM)
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Causal graphs, DAGs
C
• Causal assumptions
• Units: individuals (also other units)
E
D
• E->D reads E causes D, any definition of cause
• Qualitative (non-parametric): the E->D may be linear,
threshold, U-shaped, …
• Simple, only 4 rules needed for analysis
• No estimation, only qualitative results: confounding yes/no
• Non-action (immutable) variables as exogenous, all rules
apply
• New understanding: collider
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Causal Pies (SCC)
• Causal assumptions
U
• Units: causal mechanisms
A
B
• Any definition of cause
• No estimation, only qualitative results: interaction yes/no
• Logically finer than DAGs
1 DAG
B
A
D
U
A
U
B
U
A
5 SCCs
B
• Additive scale (interaction)
• New understanding: sufficient-,necessary cause, interaction
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Structural Equation Models, SEM
• Causal assumptions + statistical model + data
• Units: individuals (also other units)
• Any definition of cause
• Quantitative (parametric): linear
• Estimation: direct and indirect effects
• Ordinary regression: association of actual covariates with
actual outcomes
• SEM: effect of actions on potential outcomes
• SEM: parametric DAG
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Causal Inference
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Causal inference question
• Counterfactual definition of cause
– Cannot estimate individual causal effects
– Can estimate average causal effects from RCTs
• Can we estimate average causal effects from
observational data?
– Find conditions needed for causal inference
• Examine RCTs for conditions
• Apply to observational studies
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Randomized Controlled Trial, RCT
•
U
Observational study
–
Suffers from unmeasured confounders
E
D
U
•
Randomized trial
–
If full compliance:
•
•
•
R
R=E
No arrow from U to E
E
D
Three (trivial) conditions in RCTs :
–
–
–
Exchangeability:
Positivity:
Consistency:
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exposed and unexposed may be switched
have both exposed and unexposed
well defined treatment
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RCTs versus Observational studies
C
R
E
C
D
E
D
RCT get
Observational need
strength
test
Exchangeability
Conditional exchangeability
weaker
untestable
Positivity
Conditional positivity
stronger
testable
Consistency
Consistency
-
-
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Exchangeability, Positivity and Consistency
U1
K
K
K
non-causal
C
C
EE
Sufficient causes for E
D
D
D
M
M
M
C
causal
C U1
Conditions for estimating causal effect:
1. Cond. Exchangeability
No open non-causal paths
2. Cond. Positivity
Arrows into E not deterministic
3. Consistency
Causal paths well defined
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Conditional positivity example
All
– Estimate dose response
for each sex?
20
10
• Positivity problem
0
Response
– Dose response is linear
30
40
• Prior knowledge
low
high
Dose
Females
30
20
Response
30
20
10
10
0
0
Response
40
40
Males
low
high
Dose
low
high
Dose
Conditional positivity,
Common support
E=0
Conditional positivity
=
exposed and
unexposed for all
values of C
E=1
C
positivity
30
40
55
70
E=1
E=0
110
130
Exposure
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150
170
250
C>55
150
C=40 to 55
150
90
Parametric assumption:
linear “dose response”
200
250
200
250
200
150
C<40
70
D
E=1
300
E=0
300
E=1
300
E=0
E
350
350
350
Confounder, C
70
90
110
130
Exposure
150
170
70
90
110
130
Exposure
H.S.
150
170
22
Consistency
Consistency
=
Well defined intervention and contrast
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Air pollution
Excess mortality from air pollution?
Standard method:
estimate attributable fraction
Implicit contrast:
current levels versus zero
Implicit intervention: not existent
 no consistency
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Body Mass Index
Excess mortality from obesity?
Standard method:
Implicit contrast:
Implicit intervention:
estimate attributable fraction
30 versus <25
Exercise
Diet
 Mortality
Smoking
 no consistency
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G-methods versus Stratification based methods
Methods of adjustment
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Adjusting for confounding
G-methods
Stratification-methods
C
E
C
D
E
Simulated population
with exchangeability
D
Sub population
with C constant
Causal effect valid for
entire population
Causal effect valid for sub
population
IPW, standarization
←Non-parametric→
Stratification, matching
MSM, NSM
←parametric→
regression
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Stratification based adjustment
H
We want the direct effect of
tea on depression
chocolate
O
coffee
E
tea
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C
caffeine
U
low carb
Try stratification based
adjustment
D
Fails: one non-causal path is
left open
depression
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Inverse probability weighting
H
We want the direct effect of
tea on depression
chocolate
O
coffee
E
tea
C
caffeine
U
Try adjustment by IPW:
D
Choose a variable V
and
weight by the inverse of
P(V| direct causes)
Try C
low carb
depression
Works: all non-causal paths
are closed,
only direct effect left
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Summing up
• Concepts
– Causal definition: counterfactual (potential outcome)
– Causal conclusion requires causal assumptions
• Models
– DAGs, Pies
– SEM, MSM
causal assumptions
statistical + causal assumptions
• Causal inference
– Exchangeability: comparable E+ and E– Positivity:
E+ and E- in all strata
– Consistency:
well defined intervention and contrast
• Adjustment
– Stratification based
– G-methods
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stratification, matching, regression
IPW, MSM
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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.
• Papers
– Greenland, S., J. Pearl, and J. M. Robins. "Causal diagrams for epidemiologic
research." Epidemiology 1999
– Hernandez-Diaz, S., E. F. Schisterman, and M. A. Hernan. "The birth weight
"paradox" uncovered?" Am J Epidemiol 2006
– Hernan, M. A., S. Hernandez-Diaz, and J. M. Robins. "A structural approach to
selection bias." Epidemiology 2004
– Greenland, S. and B. Brumback. "An overview of relations among causal modeling
methods." Int J Epidemiol 2002
– Weinberg, C. R. "Can DAGs clarify effect modification?" Epidemiology 2007
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References
•
Chen, L., et al. "Alcohol intake and blood pressure: a systematic review implementing
a Mendelian randomization approach." PLoS Med 2008
•
Greenland, S. and B. Brumback. "An overview of relations among causal modelling
methods." Int J Epidemiol 2002
•
Hernan, M. A., S. Hernandez-Diaz, and J. M. Robins. "A structural approach to
selection bias." Epidemiology 2004
•
Hernan, M. A. and S. R. Cole. "Causal diagrams and measurement bias." Am J
Epidemiol 2009
•
Sheehan, N. A., et al. "Mendelian randomisation and causal inference in observational
epidemiology." PLoS Med 2008
•
VanderWeele, T. J. and J. M. Robins. "Directed acyclic graphs, sufficient causes, and
the properties of conditioning on a common effect." Am J Epidemiol 2007
•
VanderWeele, T. J., M. A. Hernan, and J. M. Robins. "Causal directed acyclic graphs
and the direction of unmeasured confounding bias." Epidemiology 2008
•
VanderWeele, T. J. "The sign of the bias of unmeasured confounding." Biometrics
2008
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