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Apr 21, 2023
H.S. 1
Causal inference
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/talks
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Apr 21, 2023
H.S. 2
Contents
• Background– Error
– Bias
• Define causal effect– Individual
– Average
• Identify causal effect– Exchangeability
– Positivity
– Consistency
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Background
Apr 21, 2023
H.S. 3
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04/21/23 H.S. 4
Error
Random error
• Source: sampling
• Expressed as:– p-values
– Confidence intervals (precision)
• Affect– All measures
Systematic error
• Source: design
• Expressed as bias:1. Selection bias
2. Information bias
3. Confounding
• Affect:– Frequency measure
– Association measure
Causality field: Strong focus on bias at the expense of precision
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04/21/23 H.S. 5
Precision and Bias
True value
Estimate
Precision
Bias
Causaleffect
Association
Precision
BiasBias:
association causal effect
Objective:find effects
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Define Causal Effects
Apr 21, 2023
H.S. 6
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Individual causal effect
• Counterfactual outcome
• Important– Clear definition
– Notation mathematical proofs
– Notation new methods
• Estimate individual effect?– No, but Crossover design
Apr 21, 2023
H.S. 7
Treated Not treated Individual causal effect
Zeus Died Lived Yes
Hera Lived Lived No
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Apr 21, 2023
H.S. 8
Individual causal effects
20 subjects 12 individual causal effects
No YesRheia 0 1Kronos 1 0Demeter 0 0Hades 0 0Hestia 0 0Poseidon 1 0Hera 0 0Zeus 0 1Artemis 1 1Apollo 1 0Leto 0 1Ares 1 1Athena 1 1Hephaestus 0 1Aphrodite 0 1Cyclope 0 1Persephone 1 1Hermes 1 0Hebe 1 0Dionysus 1 0
Treatment
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Average causal effect
• Counterfactual outcome
• Estimate average effect?– Yes, Randomized controlled trial
Apr 21, 2023
H.S. 9
Treated Not treated Average causal effect
Population 10/20=0.5 10/20=0.5 No
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Identify Causal Effects
Apr 21, 2023
H.S. 10
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Ideal Randomized Trial
• Trial– Randomize, Treat, Compare outcomes
• Features– Exchangeability
• Comparable groups
– Positivity• Both treated and untreated
– Consistency• Well defined intervention and contrast
Apr 21, 2023
H.S. 11
E D
C
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Conditional Randomized Trial
• Conditional Trial– By sex: Randomize, Treat,
Compare outcomes
• Features– Conditional Exchangeability
• Comparable groups by sex
– Conditional Positivity• Both treated and untreated by sex
– Consistency• Well defined intervention and
contrast
Apr 21, 2023
H.S. 12
E D
C
sex
E D
C
E D
C
Males
Females
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Observational study
• Observational studyMake = conditionally randomized trial
• Need Features– Conditional Exchangeability
• Comparable groups by all values of C
– Conditional Positivity• Both treated and untreated by all values of C
– Consistency• Well defined intervention and contrast
Apr 21, 2023
H.S. 13
E D
C
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Conditional exchangeability
Need to measure all relevant factors
Apr 21, 2023
H.S. 14
Conditional exchangeability=
No unmeasured confounding
E D
C
E D
CTwo ways to remove confounding:
Adjust:
Balance:
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• Weights– Estimate probability of exposed by C = pi
• Balance– Weight exposed by 1/ pi , for plot 100/pi
– Weight unexposed by 1/(1- pi) , for plot 100/(1-pi)
• Effect
Balance by Inverse Probability Weights
Apr 21, 2023
H.S. 15
E D
C
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IPW and plots
Apr 21, 2023
H.S. 16
50 100 150 200Blood pressure mmHg
Weight:NormalOverweight
Crude distributions
Eoverweight
DBlood pressure
Csmoke
- +
Effect of E on D:Crude: 0 biased Adjusted: 4 true
50 100 150 200Blood pressure mmHg
Weight:Normal(mean=113)Overweight(mean=117)
Inverse probability weighted distributions
Balance the data using IPWResult: all plots of D versus E are adjustedProblem: N gets large
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Conditional positivity example• Prior knowledge
– Dose response is linear
• Positivity problem– Estimate dose response
for each sex?
010
2030
40R
esp
onse
low highDose
All
010
2030
40R
espo
nse
low highDose
Males
010
2030
40R
espo
nse
low highDose
Females
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Conditional positivity
Apr 21, 2023
H.S. 18
Conditional positivity=
exposed and unexposed for all
values of C
Parametric assumption:linear “dose response”
E D
Cpositivity
E=0 E=1
30 40 55 70Confounder, C
C<40
E=0 E=1
150
200
250
300
350
Dis
eas
e
70 90 110 130 150 170Exposure
C=40 to 55
E=0 E=1
150
200
250
300
350
70 90 110 130 150 170Exposure
C>55
E=0 E=1
150
200
250
300
350
70 90 110 130 150 170Exposure
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Consistency
Consistency
=
Well defined intervention and contrast
Apr 21, 2023
H.S. 19
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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
Apr 21, 2023
H.S. 20
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Body Mass Index
Excess mortality from obesity?Standard method: estimate attributable fractionImplicit contrast: 30 versus <25
ExerciseImplicit intervention: Diet Mortality
Smoking
Apr 21, 2023
H.S. 21
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Poorly defined intervention may affect exchangeability
• Adjust for lung disease?
Apr 21, 2023
H.S. 22
Eexercise
Dmortality
Clung disease
Adjust
Ediet
Dmortality
Clung disease
Need not adjust
Esmoking
Dmortality
Clung disease
Should not adjust
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Poorly defined intervention may affect positivity
• Confounder status unknown– Can not asses positivity
Apr 21, 2023
H.S. 23
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Summing up
• Defined bias– Objective: find effects
• Conditions to find effects– Exchangeability: comparable E+ and E-
– Positivity: E+ and E- in all strata
– Consistency: well defined intervention and contrast
Apr 21, 2023
H.S. 24
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Apr 21, 2023
H.S. 25
Litterature
• Hernan and Robins, Causal Inference