3 causal models part ii counterfactual theory and traditional approaches to confounding bias
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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

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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



Review yesterday
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
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

What is Confounding?

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


What is an adjusted measure of effect

What is an “adjusted” measure of effect?



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

Counterfactual Theory relationship, is it reasonable to remove variables that are not statistically significant and include those that are?

Potential Outcomes,

Susceptibility types


Poor clare
Poor Clare relationship, is it reasonable to remove variables that are not statistically significant and include those that are?

  • Doctor prescribes antibiotics

  • 3 days later she is cured

  • Did the antibiotic cure her?


Cinema d counterfactual
Cinema d’Counterfactual relationship, is it reasonable to remove variables that are not statistically significant and include those that are?


The counterfactual model the counterfactual ideal
The counterfactual model: relationship, is it reasonable to remove variables that are not statistically significant and include those that are?The counterfactual ideal

Disease experience, given exposed

Hypothetical disease experience, if unexposed

The

Counterfactual

Ideal


Counterfactual theory1
Counterfactual theory relationship, is it reasonable to remove variables that are not statistically significant and include those that are?

  • 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 ideal1

Approximation to relationship, is it reasonable to remove variables that are not statistically significant and include those that are?

The Counterfactual

Ideal

The counterfactual model:The counterfactual ideal

Disease experience, given exposed

Substitute disease experience of truly unexposed


Take home message 1: relationship, is it reasonable to remove variables that are not statistically significant and include those that are?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
Must Specify a Causal Contrast relationship, is it reasonable to remove variables that are not statistically significant and include those that are?

  • 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: relationship, is it reasonable to remove variables that are not statistically significant and include those that are?“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

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

Think about dose, duration


What about obesity and mi

What about obesity and MI? smoking and lung cancer?

What about gender and cancer?


Effects must be amenable to action
Effects Must be Amenable to Action smoking and lung cancer?

  • 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: smoking and lung cancer?For etiologic observational studies, think of RCT you would do first. Develop your observational study with the RCT in mind.

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


To identify a causal effect in an individual
To identify a causal effect in an individual smoking and lung cancer?

  • Need three things:

    • Outcome, actions compared, person whose 2+ counterfactual outcomes compared

  • Call the counterfactual outcomes:

    • Ya=1 vs Ya=0, read: Y that would occur if A=a

  • Note counterfactuals different from:

    • Y|A=1 (or just Y), read: Y given A=1

  • Effect can be precisely defined as:

    • Ya=1 ≠Ya=0


Assume infinite population with no information or selection bias a dichotomous a and y

Assume infinite population with no information or selection bias, a dichotomous A and Y

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


Assume each person represents 100,000 people bias, a dichotomous A and Y

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


Assume each person represents 100,000 people bias, a dichotomous A and Y

[4/8 – 4/8] = 0

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

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


Assume each person represents 100,000 people bias, a dichotomous A and Y

[4/8 – 4/8] = 0

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

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

[2/4 – 2/4] = 0


The counterfactual model bias, a dichotomous A and YSusceptibility types

CST: Counterfactual susceptibility type

  • Envision 4 responses to exposure, relative to unexposed

    • Type 1 - Doomed

    • Type 2 - E causal

    • Type 3 - E preventive

    • Type 4 - Immune

1

1

1

0

0

1

0

0


The counterfactual model
The counterfactual model bias, a dichotomous A and Y

  • 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 under the CST model bias, a dichotomous A and Y

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
Can type 2 and 3 co-exist? bias, a dichotomous A and Y

  • 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: bias, a dichotomous A and YIf exposures can be causal and preventive, estimates of effect only tell us about the balance of causal and preventive effects


Average causal effects
Average causal effects bias, a dichotomous A and Y

  • Individual effects rarely identifiable because we don’t have both conditions

    • But average causal effects may be identifiable in populations

  • An 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


Effects vs associations
Effects vs. Associations bias, a dichotomous A and Y

  • Effects measures

    • RD: Pr(Ya=1 = 1) - Pr(Ya=0 = 1)

    • RR: Pr(Ya=1 = 1) / Pr(Ya=0 = 1)

    • OR: Pr(Ya=1 = 1)/Pr(Ya=1 = 0)/ Pr(Ya=0 = 1)/Pr(Ya=0 = 0)

  • Associational measures

    • RD: Pr(Y = 1|A=1) - Pr(Y = 1|A=0)

    • RR: Pr(Y = 1|A=1) / Pr(Y = 1|A=0)

    • OR: Pr(Y = 1|A=1) / Pr(Y = 0|A=1) / Pr(Y = 1|A=0) / Pr(Y = 0|A=0)



Extend the cst model of causation to populations
Extend the CST model of causation to populations bias, a dichotomous A and Y

1

1


What is the risk of disease in exposed
What is the risk of disease in exposed? bias, a dichotomous A and Y

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

1

1


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

Counterfactual risk is the risk the exposed would have had had they been exposed: p1+p3

1

1


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

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

1

1


Exchangeability

Observed been unexposed?

Counterfactual

Exchangeability

  • Full exchangeability means the two groups can stand in for each other

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

      • Pr(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)


Exchangeability1

Observed been unexposed?

Counterfactual

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 unexposed

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


Take Home Message 5: been unexposed?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
Two possible definitions of no confounding (1) been unexposed?

  • 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

p1 = q1

p2 – q3

But effect should be based only on exposed


Two possible definitions of no confounding 2
Two possible definitions of no confounding (2) been unexposed?

  • 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

p1 + p3 = q1+ q3

p2 – p3

NOTE that RD related to balance of p2 and p3


We choose the second definition
We choose the second definition been unexposed?

  • 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 definition1
We choose the second definition been unexposed?

  • 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
To put it mathematically been unexposed?

  • 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
Causal RD vs. Observed been unexposed?

  • 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?

    • No

100

100


Causal rd vs observed1
Causal RD vs. Observed been unexposed?

  • 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?

    • Yes

100

100


Take Home Message 6: been unexposed?Lack of confounding doesn’t mean perfect balance of CST types which we would expect under randomization


Take Home Message 7: been unexposed?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
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
Confounders ideal

  • 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
Non-identifiability and collapsibility: idealIdentifying 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: ideal 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
Stratified Analysis: Introduction ideal

  • 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 variable stratify then ask
Stratified Analysis 1 Variable idealStratify 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
Example (1-1) ideal (CST balance within strata)


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


Take home message 9: idealIn 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
Example (1-3) ideal(CST balance within strata)

Does

p1 + p3 = q1 + q3?

(30+20)/1000 <>

(35+35)/1000


Example 1 4 cst balance within strata
Example (1-4) ideal(CST balance within strata)

Does

p1 + p3 = q1 + q3

within strata?


Example 2 1 choice of effect measure
Example (2-1) ideal(choice of effect measure)

Collapsible?

Does crude = adjusted?

Outcome needs to be rare in all levels of the exposure/confounder

Collapsible?

Does crude = adjusted?


Take Home Message 10: idealThe 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
But this can go wrong ideal

Counfounding?

Does p1+p3=q1+q3?

Is the exposure distribution different across strata?

Is the risk in the unexposed different?


Take Home Message 11: idealStatistical Criteria Are Not Sufficient to Determine What to Keep in a Model to Observe Causal Effects


Pooled adjusted estimate
Pooled adjusted estimate ideal

  • 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
Review of weighting ideal

  • 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


p for heterogeneity 0.09 ideal

Example: MH Pooling

95% CI:

1.4, 2.7


Example: MH Pooling ideal

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


Mantel haenszel weights
Mantel Haenszel Weights ideal

  • 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
Summary estimates: Mantel-Haenszel ideal

  • 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





Summary estimates standardized rr smr
Summary estimates: idealStandardized 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


1.4, 2.7 ideal

Example: Standardization

0.9, 6.4

95% CI:

0.9, 6.4


Example: Standardization ideal

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


Example: Standardization ideal

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





Practical summary

1 ideal

1.3

1.3

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


Practical summary1

0.6 ideal

1.3

2.1

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


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


Conclusion
Conclusion ideal

  • 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


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