3 Causal Models Part II: Counterfactual Theory and Traditional Approaches to Confounding (Bias?)...

3 Causal Models Part II:Counterfactual Theory and Traditional Approaches to

Confounding (Bias?)

Confounding, Identifiability, Collapsibility and Causal inference

Thursday reception at lunch time at SACEMA

Review Yesterday

Causes – definition Sufficient causes model

– Component causes– Attributes– Causal complements

Lessons– Disease causation is poorly understood– Diseases don’t have induction periods– Strength of effects determined by prevalence of complements– Only need to prevent one component to prevent disease

This Morning

Counterfactual model– Susceptibility types– Potential outcomes

Confounding under the counterfactual susceptibility model of causation

Stratification– Identifying confounders– Standardization versus pooling

What is Confounding?

Give me the definition you were taught or describe how you understand it

What is an “adjusted” measure of effect?

Is red wine cardio-protective?

In an adjusted model to remove confounding of the E-D relationship, is it reasonable to remove variables that are not statistically significant

and include those that are?

Counterfactual Theory

Potential Outcomes,

Susceptibility types

Poor Clare

Doctor prescribes antibiotics

3 days later she is cured

Did the antibiotic cure her?

Cinema d’Counterfactual

The counterfactual model:The counterfactual ideal

Disease experience, given exposedHypothetical disease experience, if unexposed

TheCounterfactual

Ideal

Counterfactual theory

Only one can actually be observed– The other is “counterfactual” in that it is counter to

what is actually observed

Ask, what would have happened had things been different, all other things being equal?– Leads to the causal contrast

Exposure must be changeable to have effect– We will come back to this

The counterfactual model:The counterfactual ideal

Disease experience, given exposed

Substitute disease experience of truly unexposed

Approximation to

The Counterfactual

Ideal

Take home message 1:We’re often interested in what happens

to index (exposed). Reference (unexposed) are useful only insofar as

they tell us about index group.

Must Specify a Causal Contrast

Events are not causes themselves– Only causes as part of a causal contrast

What is the effect of oral contraceptives on risk of death?– The question, as defined, has no meaning

Compared to condoms, increased risk– Through stroke and heart attack

Compared to no contraceptive, maybe decreased risk

– Some places childbirth may be a greater risk

Take home message 2:“Effects” of exposures only have meaning when defined in contrast

to an alternative

If ethics were not a concern, how would you design an RCT of smoking and lung cancer?

Think about dose, duration

What about gender and cancer?

What about obesity and MI?

Effects Must be Amenable to Action

To have an effect, must be changeable– What is effect of sex on heart disease?– How would you change sex?

Defining the action helps define the causal contrast well– What is the effect of obesity on death?– How would you change obesity?

Each has a different effect, some good, some bad

To remind us, use A for Action, not E

Think of the action, inclusion criteria, the placebo, etc.

Take Home Message 3:For etiologic observational studies,

think of RCT you would do first. Develop your observational study

with the RCT in mind.

To identify a causal effect in an individual

Need three things:– Outcome, actions compared, person whose

2+ counterfactual outcomes comparedCall the counterfactual outcomes:

– Ya=1 vs Ya=0, read: Y that would occur if A=aNote counterfactuals different from:

– Y|A=1 (or just Y), read: Y given A=1Effect can be precisely defined as:

– Ya=1 ≠Ya=0

All examples, assume each person represents 1,000,000 people exactly the same as them so no random error problem

Assume infinite population with no information or selection bias, a

dichotomous A and Y

A (E) Ya=1 Ya=0 Y

Person A 1 1 0 1

Person B 1 1 1 1

Person C 1 0 1 0

Person D 1 0 0 0

Person E 0 0 0 0

Person F 0 1 0 0

Person G 0 1 1 1

Person H 0 0 1 1

Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =

Assume each person represents 100,000 people

A (E) Ya=1 Ya=0 Y

Person A 1 1 0 1

Person B 1 1 1 1

Person C 1 0 1 0

Person D 1 0 0 0

Person E 0 0 0 0

Person F 0 1 0 0

Person G 0 1 1 1

Person H 0 0 1 1

Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =


[4/8 – 4/8] = 0

Association : [Pr(Y=1|A=1) - Pr(Y=1|A=0)] =

A (E) Ya=1 Ya=0 Y

Person A 1 1 0 1

Person B 1 1 1 1

Person C 1 0 1 0

Person D 1 0 0 0

Person E 0 0 0 0

Person F 0 1 0 0

Person G 0 1 1 1

Person H 0 0 1 1

Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =


Association : [Pr(Y=1|A=1) - Pr(Y=1|A=0)] =

[4/8 – 4/8] = 0

[2/4 – 2/4] = 0

The counterfactual modelSusceptibility types

Envision 4 responses to exposure, relative to unexposed– Type 1 - Doomed– Type 2 - E causal– Type 3 - E preventive– Type 4 - Immune

Susceptibility Type

Exposed Outcome

(Ya=1)

Unexposed Outcome

(Ya=0)

Type 1 – doomed

Type 2 – E causal

Type 3 – E preventive

Type 4 – immune

CST: Counterfactual susceptibility type

1 1

1 0

0 1

0 0

The counterfactual model

The index condition, relative to the reference condition, affects only susceptibility types 2 and 3– Types 2 get the disease, but would not get disease

had they had the reference condition– Types 3 do not get the disease, but would have got

the disease had they had the reference condition

Individual Susceptibility under the CST model

Individual Susceptibility

Risk Difference Ya=1 – Ya=0

Risk Ratio Ya=1 / Ya=0

Type 1

Type 2

Type 3

Type 4

1 – 1 = 0 1 / 1 = 1

1 – 0 = 1 1 / 0 = undef

0 – 1 = -1 0 / 1 = 0

0 – 0 = 0 0 / 0 = undef

Can type 2 and 3 co-exist?

Are there exposures that can both prevent and causes disease? – Vaccination and polio– Exercise and heart attack– Seat belts and death in a motor vehicle accident– Heart transplant and mortality

So what does RD = 0 or RR=1 mean?– Could mean no effect– Could be balance of causal/preventive mechanisms– We call no effect “sharp null” but it is not identifiable

Take home message 4:If exposures can be causal and

preventive, estimates of effect only tell us about the balance of causal and

preventive effects

Average causal effects

Individual effects rarely identifiable because we don’t have both conditions– But average causal effects may be

identifiable in populationsAn average causal effect of treatment A

on outcome Y occurs when:– Pr(Ya=1 = 1) ≠ Pr(Ya=0 = 1) – Or more generally, E(Ya=1) ≠ E(Ya=0)

Note makes no reference to relative vs. absolute

Traditional Approaches to Confounding and Confounders

Extend the CST model of causation to populations

Susceptibility Type

Index Outcome

Reference Outcome

Proportion in

Index Pop

Proportion in

Reference Pop

Type 1 – doomed

1 1 p1 q1

Type 2 – index causal

1 0 p2 q2

Type 3 – index preventive

0 1 p3 q3

Type 4 – immune

0 0 p4 q4

1 1

What is the risk of disease in exposed?

Susceptibility Type

Index Outcome

Reference Outcome

Proportion in

Index Pop

Proportion in

Reference Pop

Type 1 – doomed

1 1 p1 q1


1 0 p2 q2


0 1 p3 q3

Type 4 – immune

0 0 p4 q4

1 1

Observed risk in exposed is p1 + p2, but we cannot tell how many of each

What would the risk of disease be in the exposed had they been unexposed?

Susceptibility Type

Index Outcome

Reference Outcome

Proportion in

Index Pop

Proportion in

Reference Pop

Type 1 – doomed

1 1 p1 q1


1 0 p2 q2


0 1 p3 q3

Type 4 – immune

0 0 p4 q4

1 1

Counterfactual risk is the risk the

exposed would have had had they

been exposed: p1+p3

When can reference group stand in for the exposed had they been unexposed?

Susceptibility Type

Index Outcome

Reference Outcome

Proportion in

Index Pop

Proportion in

Reference Pop

Type 1 – doomed

1 1 p1 q1


1 0 p2 q2


0 1 p3 q3

Type 4 – immune

0 0 p4 q4

1 1

To have a valid comparison, we require the disease experience of reference

group be able to stand in for the counterfactual risk. This

is partial exchangeability

Exchangeability

Full exchangeability means the two groups can stand in for each other– Risk exposed had = risk unexposed would

have had if they were exposedPr(Ya=1=1|A=1) = Pr(Ya=1=1|A=0)

– Risk unexposed had = risk exposed would have had if they were unexposed

Pr(Ya=0=1|A=1) = Pr(Ya=0=1|A=0)

ObservedCounterfactual

Exchangeability

Partial exchangeability means the E- can stand in for what would have happened to the E+ had they been unexposed– Risk unexposed had = risk exposed would

have had if they were unexposedPr(Ya=0=1|A=1) = Pr(Ya=0=1|A=0)

ObservedCounterfactual

Take Home Message 5:The unexposed have to be able to

stand in for the exposed had they been unexposed. Not vice versa.

Partial exchangeability

Two possible definitions of no confounding (1)

Definition One — the risk of disease due to background causes is equal in the index and reference populations

So p1 = q1 under this definition.

The risk difference [(p1 + p2) - (q1 + q3)] equals

(p2 - q3), assuming partial exchangeability.

p1 = q1

p1

p2 – q3

But effect should be based only on exposed

Two possible definitions of no confounding (2)

Definition Two -- the risk of disease in the reference population equals the risk the index population would have had, if they had been unexposed

So p1 + p3 = q1 + qunder this definition.

The risk difference [(p1 + p2) - (q1 + q3)]

equals (p2 - p3 ), assuming partial

exchangeability.

p1 + p3 = q1+ q3

p1 + p3

p2 – p3

NOTE that RD related to balance of p2 and p3

We choose the second definition

First forces inclusion of effect of absence of exposure in reference group

Second measures effect of exposure only in index group – Holds under randomization– However, it is counterfactual

If exposure is never preventive, they are same

We choose the second definition

A measure of association is unconfounded if:– Experience of the reference group = the

disease occurrence the index population would have had, had they been unexposed

Risk difference tells about balance of causal/preventive action in index– Effect, not an estimate

To put it mathematically

Suppose we have two populations A and B We want to observe: IAE+ - IAE-

We observe: IAE+ - IBE-

If we add IAE- - IAE- to this we get: (IAE+ - IAE-) + (IAE- - IBE-) (IAE+ - IAE-) is the causal RD (IAE- - IBE-) is a bias factor (i.e. confounding)

Bias is difference between counterfactual unexposed experience of exposed and experience of truly unexposed

Causal RD vs. Observed

Susceptibility Type

Index Frequency

Reference Frequency

Type 1 – doomed

10 10


5 10


10 5

Type 4 – immune

75 75

Causal RD?– p2 – p3

– 5/100 – 10/100 = -5/100

Observed RD?– (p1+p2) – (q1+q3)

– 15/100 – 15/100 = 0 Confounding?

– Does (p1+p3) = (q1+q3) ?

– 20/100 ≠ 15/100, Yes Causal = Observed?

– No100 100

Causal RD vs. Observed

Causal RD?– p2 – p3

– 5/100 – 5/100 = 0

Observed RD?– (p1+p2) – (q1+q3)

– 15/100 – 15/100 = 0 Confounding?

– Does (p1+p3) = (q1+q3) ?

– 15/100 = 15/100, No Causal = Observed?

– Yes100 100

Susceptibility Type

Index Frequency

Reference Frequency

Type 1 – doomed

10 5


5 10


5 10

Type 4 – immune

80 75

Take Home Message 6:Lack of confounding doesn’t mean

perfect balance of CST types which we would expect under randomization

Take Home Message 7:If there is no confounding, the causal risk difference (i.e. the true effect) is

the observed effect

Assuming no other bias and random error

Getting the observed contrast close to the counterfactual ideal

Design– Randomization– Creating similar

populations Matching Restriction

Analysis– Stratification based

methods Stratification, Mantel-

Haenszel, Regression

– Standardization based methods

Standardization, G-estimation, IPTW, Marginal structural models

Confounders

Note we have defined confounding with no reference to imbalances in covariates– Separate confounder from confounding– Confounder is a factor that explains discrepancy

between observed risk in reference and desired counterfactual risk

Must be imbalanced in index/reference groups, a cause of disease and not on causal pathway– Use data as guide only

Non-identifiability and collapsibility:Identifying confounding in practice

Because we can’t identify individuals’ CST types, can’t use comparability definition in practice – Call this “ the non-identifiability problem” – Except thoughtfully

Instead a traditional approach uses the collapsibility criterion – If crude measure equals adjusted for potential

confounder, no confounding by that variable

What adjusted measure of effect?

Take Home Message 8: Confounding is when the

unexposed can’t stand in for the exposed had they been unexposed.

Confounders are variables that explain confounding.

Stratified Analysis: Introduction

One method for control of extraneous variables in the analysis– Analysis of disease-exposure association within

categories of confounder / modifier prevents external influence of that variable

Advantages/disadvantages– Straight-forward, few statistical assumptions– Data become thin with many categories/ variables

Candidate variables– Confounders, Modifiers, Matched factors

Stratified Analysis 1 VariableStratify then ask:

Are measures of effect within each stratum heterogeneous? – Yes = Interaction, stratified analysis?– No = No Interaction, assess confounding

Does summary measure of effect across strata equal crude? – Yes = No confounding, collapse– No = Confounding, use summary measure

Note, this is about change in estimate of effect, nothing about p-values

Example (1-1) (CST balance within strata)

All agesMen Women

cases 150 70non-cases 850 930total 1000 1000risk 0.15 0.07risk ratio 2.1

Old Young All agesMen Women Men Women

cases 90 60 60 10non-cases 243 607 607 323total 333 667 667 333risk 0.27 0.09 0.09 0.03risk ratio 3.0 3.0

Example (1-2) (CST balance within strata)

Take home message 9:In practice, confounding USUALLY presents as – within levels of the

confounder, uneven distribution of the exposure and different risk of outcome

among unexposed

But be careful, as this can be misleadingas this is NECESSARY but not SUFFICIENT

Example (1-3) (CST balance within strata)All ages

Men Womencases 150 70non-cases 850 930total 1000 1000risk 0.15 0.07risk ratio 2.1

type1 30 35type2 120 40type3 20 35type4 830 890total 1000 1000confounding -0.020

Does p1 + p3 = q1 + q3?

(30+20)/1000 <>(35+35)/1000

Example (1-4) (CST balance within strata)Old Young All ages

Men Women Men Womencases 90 60 60 10non-cases 243 607 607 323total 333 667 667 333risk 0.27 0.09 0.09 0.03risk ratio 3.0 3.0

type1 20 30 10 5type2 70 30 50 10type3 10 30 10 5type4 233 577 597 313total 333 667 667 333confounding 0.000 0.000

Does p1 + p3 = q1 + q3

within strata?

Example (2-1) (choice of effect measure) E+ E-

cases 100 50 non-cases 900 950

total 1000 1000 risk 0.1 0.05 RR 2

odds 0.111 0.053 OR 2.11

males females E+ E- E+ E-

cases 90 45 cases 10 5 non-cases 110 155 non-cases 790 795

total 200 200 total 800 800 risk 0.45 0.225 0.0125 0.00625 RR 2 2

MHRR 2 odds 0.818 0.290 0.013 0.006 OR 2.82 2.01

MHOR 2.68

Collapsible? Does crude = adjusted?Collapsible? Does crude = adjusted?

Outcome needs to be rare in all levels of the

exposure/confounder

Take Home Message 10:The odds ratio is not strictly

collapsible. Change in estimate of effect after adjustment can be just an artifact of the data. Outcome must be

rare in ALL strata.

But this can go wrong

Counfounding?

Does p1+p3=q1+q3?

Is the exposure distribution different across strata?

Is the risk in the unexposed different?

Take Home Message 11:Statistical Criteria Are Not Sufficient

to Determine What to Keep in a Model to Observe Causal Effects

Pooled adjusted estimate

Assumes uniform RR/RD across strata – Precision enhancing

Pooled estimates are weighted averages of effects in strata– Pooled estimate are between stratum estimates– Weights measure information in strata (inverse

variance) but can be computed differently

Ex: Mantel-Haenzel, Logistic/Cox Reg– So long as there are no interaction terms– Regression models are analogous to stratification

Review of weighting

Pooling means we average the stratum specific estimates to get one estimate– Thus the pooled estimate must be between the two

stratum specific estimates

We can choose the weights however we like– Different weighting schemes have different

properties and logics

wwR

wwR

0

1

RR weighted

Example: MH Pooling

95% CI:

1.4, 2.79.1

2422290

*39215691520

*4

24222132

*1001569

49*320

ˆR MH1

0

NbNNaN

R

3.12181/396

1810/420ˆ RCrudeR

p for heterogeneity 0.09

Crude C1 C0

E+ E- E+ E- E+ E-

D+ 420 396 D+ 320 4 D+ 100 392

D- 1390 1785 D- 1200 45 D- 190 1740

Total 1810 2181 Total 1520 49 Total 290 2132

RR 1.3 RR 2.6 RR 1.9

Example: MH Pooling

9.1

2422290

*39215691520

*4

24222132

*1001569

49*320

*

*

ˆR MH0

1

10

0

10

10

1

10

NbNNaN

NNN

Nc

NNN

NNN

Na

NNN

R

Weight is N1*N0/N which weights towards the strata with highest total N and most evenly distributed exposure distribution

Crude C1 C0

E+ E- E+ E- E+ E-D+ 420 396 D+ 320 4 D+ 100 392D- 1390 1785 D- 1200 45 D- 190 1740Total 1810 2181 Total 1520 49 Total 290 2132RR 1.3 RR 2.6 RR 1.9

Mantel Haenszel Weights

The weight, (N1*N0 )/ N is at its minimum if N1=1 so N0 = (N-1). Weight is then (N-1)/N which is about 1

The weight, (N1*N0 )/ N is at its maximum if N1= N0 = N/2. Weight is then (N/2)2/N which is N/4

So a larger sample size will increase the weight, as will an even distribution of exposed an unexposed subjects

Summary estimates: Mantel-Haenszel

A pooled summary estimate: – Weighted average of estimates of effect from each

stratum– Weight is highest for stratum with most information

(subjects)

Precision optimizing Calculation depends on design

MH estimates for 3 designs

g g

gg

g g

gg

MH

n

cb

n

da

OR design control-Case

Exposed

(Index)

Unexposed

(reference)

Casesag bg

Controlscg dg


g g

gg

g g

gg

MH

n

nb

n

na

RR1

0

design Risk

Exposed

(Index)

Unexposed

(reference)

Casesag bg

Undiseasedcg dg

Total n1g n0g


g g

gg

g g

gg

MH

L

Lb

L

La

IRR1

0

design Rate

Exposed

(Index)

Unexposed

(reference)

Casesag bg

Person-timeL1g L0g

Summary estimates:Standardized RR (SMR)

Standardize the risk or rate– Weighted average of risk or rate in strata, using the

index group’s experience as the weight

Choose index group because:– Want reference group to reflect the rate we would

have seen in the exposed had they been unexposed

No assumption of homogeneity across strata

Example: Standardization

95% CI:

0.9, 6.4

3.12181/396

1810/420ˆ RCrudeR

4.2

2132392

*290494

*1520

100320

*

ˆ1

oNN

b

ORSM

1.7, 2.7

0.7, 2.50.9, 6.4

1.4, 2.7

Crude C1 C0



When we standardize, we can use whatever distribution we want. If we use the distribution of the exposed group, we call this an SMR.

Crude C1 C0


4.2

2132392

*290494

*1520

290100

*2901520320

*1520

**

*

*

*

ˆ

01

01

11

01

11

Nb

N

a

Nb

N

Na

N

WNb

N

WNa

N

RSM


0.2

2132392

*2422494

*1569

290100

*24221520320

*1569

*

*

*

*

ˆ

0

1

0

1

Nb

N

Na

N

WNb

W

WNa

W

RSM

We could also ask what would happen if everyone was both exposed and unexposed: corresponds to PO model

Crude C1 C0



g g

gg

gg

d

bc

a

SMR design control-Case

Exposed

(Index)

Unexposed

(reference)

Casesag bg

Controlscg dg


g g

gg

gg

n

bn

a

SMR

01

design Risk

Exposed

(Index)

Unexposed

(reference)

Casesag bg

Undiseasedcg dg

Total n1g n0g


g g

gg

gg

L

bL

a

SMR

01

design Rate

Exposed

(Index)

Unexposed

(reference)

Casesag bg

Person-timeL1g L0g

Practical summary

Use the RRc to measure the direction and magnitude of confounding:

cRR = SMR*RRc

RRc = cRR/SMR

Use pooled estimates to maximize precision when effects are homogeneous within strata.

Use the SMR as an unconfounded summary estimate when effects are heterogeneous

1.3 1.31

Practical summary

Use the RRc to measure the direction and magnitude of confounding:

cRR = SMR*RRc

RRc = cRR/SMR

Use pooled estimates to maximize precision when effects are homogeneous within strata.

Use the SMR as an unconfounded summary estimate when effects are heterogeneous

1.3 2.10.6

Take Home Message 12:Mantel-Haenszel is only appropriate when no interaction. Standardization can be used with interaction but isn’t

precision optimizing.

Conclusion

Counterfactual model Causal contrast is between disease experience

of exposed and counterfactual experience they would have had had they been unexposed

Use unexposed group to stand in for counterfactual ideal

Confounding occurs when the unexposed can’t stand in for exposed had they been unexposed

3 Causal Models Part II: Counterfactual Theory and Traditional Approaches to Confounding (Bias?)...

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Transcript of 3 Causal Models Part II: Counterfactual Theory and Traditional Approaches to Confounding (Bias?)...