Apr 21, 2023

H.S. 1

Causal inference

Hein Stigum

Presentation, data and programs at:

http://folk.uio.no/heins/talks

Apr 21, 2023

H.S. 2

Contents

• Background– Error

– Bias

• Define causal effect– Individual

– Average

• Identify causal effect– Exchangeability

– Positivity

– Consistency

Background

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H.S. 3

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

04/21/23 H.S. 5

Precision and Bias

True value

Estimate

Precision

Bias

Causaleffect

Association

Precision

BiasBias:

association causal effect

Objective:find effects

Define Causal Effects

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H.S. 6

Individual causal effect

• Counterfactual outcome

• Important– Clear definition

– Notation mathematical proofs

– Notation new methods

• Estimate individual effect?– No, but Crossover design

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H.S. 7

Treated Not treated Individual causal effect

Zeus Died Lived Yes

Hera Lived Lived No

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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

Average causal effect

• Counterfactual outcome

• Estimate average effect?– Yes, Randomized controlled trial

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H.S. 9

Treated Not treated Average causal effect

Population 10/20=0.5 10/20=0.5 No

Identify Causal Effects

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H.S. 10

Ideal Randomized Trial

• Trial– Randomize, Treat, Compare outcomes

• Features– Exchangeability

• Comparable groups

– Positivity• Both treated and untreated

– Consistency• Well defined intervention and contrast

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H.S. 11

E D

C

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

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H.S. 12

E D

C

sex

E D

C

E D

C

Males

Females

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

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H.S. 13

E D

C

Conditional exchangeability

Need to measure all relevant factors

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H.S. 14

Conditional exchangeability=

No unmeasured confounding

E D

C

E D

CTwo ways to remove confounding:

Adjust:

Balance:

• 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

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H.S. 15

E D

C

IPW and plots

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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

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

Conditional positivity

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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

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

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H.S. 20

Body Mass Index

Excess mortality from obesity?Standard method: estimate attributable fractionImplicit contrast: 30 versus <25

ExerciseImplicit intervention: Diet Mortality

Smoking

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H.S. 21

Poorly defined intervention may affect exchangeability

• Adjust for lung disease?

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H.S. 22

Eexercise

Dmortality

Clung disease

Adjust

Ediet

Dmortality

Clung disease

Need not adjust

Esmoking

Dmortality

Clung disease

Should not adjust

Poorly defined intervention may affect positivity

• Confounder status unknown– Can not asses positivity

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H.S. 23

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

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H.S. 25

Litterature

• Hernan and Robins, Causal Inference