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 exposed Hypothetical disease experience, if unexposed The Counterfactual 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 compared Call the counterfactual outcomes: – Y a=1 vs Y a=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: – Y a=1 ≠Y a=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)Y a=1 Y a=0 Y Person A1101 Person B1111 Person C1010 Person D1000 Person E0000 Person F0100 Person G0111 Person H0011 Effect : [Pr(Y a=1 =1) - Pr(Y a=0 =1)] = Assume each person represents 100,000 people

A (E)Y a=1 Y a=0 Y Person A1101 Person B1111 Person C1010 Person D1000 Person E0000 Person F0100 Person G0111 Person H0011 Effect : [Pr(Y a=1 =1) - Pr(Y a=0 =1)] = Assume each person represents 100,000 people [4/8 – 4/8] = 0 Association : [Pr(Y=1|A=1) - Pr(Y=1|A=0)] =

A (E)Y a=1 Y a=0 Y Person A1101 Person B1111 Person C1010 Person D1000 Person E0000 Person F0100 Person G0111 Person H0011 Effect : [Pr(Y a=1 =1) - Pr(Y a=0 =1)] = Assume each person represents 100,000 people Association : [Pr(Y=1|A=1) - Pr(Y=1|A=0)] = [4/8 – 4/8] = 0 [2/4 – 2/4] = 0

The counterfactual model Susceptibility types Envision 4 responses to exposure, relative to unexposed – Type 1 - Doomed – Type 2 - E causal – Type 3 - E preventive – Type 4 - Immune CST: Counterfactual susceptibility type

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 1 – 1 = 0 1 / 1 = 1 1 – 0 = 11 / 0 = undef 0 – 1 = -10 / 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 populations An average causal effect of treatment A on outcome Y occurs when: – Pr(Y a=1 = 1) ≠ Pr(Y a=0 = 1) – Or more generally, E(Y a=1 ) ≠ E(Y a=0 ) Note makes no reference to relative vs. absolute

Effects vs. Associations Effects measures – RD: Pr(Y a=1 = 1) - Pr(Y a=0 = 1) – RR: Pr(Y a=1 = 1) / Pr(Y a=0 = 1) – OR: Pr(Y a=1 = 1)/Pr(Y a=1 = 0)/ Pr(Y a=0 = 1)/Pr(Y a=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)

Traditional Approaches to Confounding and Confounders

Extend the CST model of causation to populations 1 1

What is the risk of disease in exposed? 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? 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? 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 exposed Pr(Y a=1 =1|A=1) = Pr(Y a=1 =1|A=0) – Risk unexposed had = risk exposed would have had if they were unexposed Pr(Y a=0 =1|A=1) = Pr(Y a=0 =1|A=0) Observed 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(Y a=0 =1|A=1) = Pr(Y a=0 =1|A=0) Observed Counterfactual

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 p 1 = q 1 under this definition. The risk difference [(p 1 + p 2 ) - (q 1 + q 3 )] equals (p2 - q3), assuming partial exchangeability. p 1 = q 1 p1p1 p 2 – q 3 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 [(p 1 + p 2 ) - (q 1 + q 3 )] equals (p2 - p3 ), assuming partial exchangeability. p 1 + p 3 = q 1 + q 3 p 1 + p 3 p 2 – p 3 NOTE that RD related to balance of p 2 and p 3

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: I AE+ - I AE- We observe: I AE+ - I BE- If we add I AE- - I AE- to this we get:  (I AE+ - I AE- ) + (I AE- - I BE- )  (I AE+ - I AE- ) is the causal RD  (I AE- - I BE- ) 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? – p 2 – p 3 – 5/100 – 10/100 = -5/100 Observed RD? – (p 1 +p 2 ) – (q 1 +q 3 ) – 15/100 – 15/100 = 0 Confounding? – Does (p 1 +p 3 ) = (q 1 +q 3 ) ? – 20/100 ≠ 15/100, Yes Causal = Observed? – No 100

Causal RD vs. Observed Causal RD? – p 2 – p 3 – 5/100 – 5/100 = 0 Observed RD? – (p 1 +p 2 ) – (q 1 +q 3 ) – 15/100 – 15/100 = 0 Confounding? – Does (p 1 +p 3 ) = (q 1 +q 3 ) ? – 15/100 = 15/100, No Causal = Observed? – Yes 100

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 Variable Stratify 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 (1-2) (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 misleading as this is NECESSARY but not SUFFICIENT

Example (1-3) (CST balance within strata) Does p1 + p3 = q1 + q3? (30+20)/1000 <> (35+35)/1000

Example (1-4) (CST balance within strata) Does p1 + p3 = q1 + q3 within strata?

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

Example: MH Pooling 95% CI: 1.4, 2.7 p for heterogeneity 0.09 CrudeC1C1 C0C0 E+E-E+E-E+E- D D+3204D D D D Total Total152049Total RR1.3RR2.6RR1.9

Example: MH Pooling Weight is N 1 *N 0 /N which weights towards the strata with highest total N and most evenly distributed exposure distribution CrudeC1C1 C0C0 E+E-E+E-E+E- D D+3204D D D D Total Total152049Total RR1.3RR2.6RR1.9

Mantel Haenszel Weights The weight, (N 1 *N 0 )/ N is at its minimum if N 1 =1 so N 0 = (N-1). Weight is then (N-1)/N which is about 1 The weight, (N 1 *N 0 )/ N is at its maximum if N 1 = N 0 = 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 Exposed (Index) Unexposed (reference) Cases agag bgbg Controls cgcg dgdg

MH estimates for 3 designs Exposed (Index) Unexposed (reference) Cases agag bgbg Undiseased cgcg dgdg Totaln 1g n 0g

MH estimates for 3 designs Exposed (Index) Unexposed (reference) Cases agag bgbg Person-time L 1g L 0g

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, , 2.7 CrudeC1C1 C0C0 E+E-E+E-E+E- D D+3204D D D D Total Total152049Total RR1.3RR2.6RR1.9

Example: Standardization When we standardize, we can use whatever distribution we want. If we use the distribution of the exposed group, we call this an SMR. CrudeC1C1 C0C0 E+E-E+E-E+E- D D+3204D D D D Total Total152049Total RR1.3RR2.6RR1.9

Example: Standardization We could also ask what would happen if everyone was both exposed and unexposed: corresponds to PO model CrudeC1C1 C0C0 E+E-E+E-E+E- D D+3204D D D D Total Total152049Total RR1.3RR2.6RR1.9

MH estimates for 3 designs Exposed (Index) Unexposed (reference) Cases agag bgbg Controls cgcg dgdg

MH estimates for 3 designs Exposed (Index) Unexposed (reference) Cases agag bgbg Undiseased cgcg dgdg Totaln 1g n 0g

MH estimates for 3 designs Exposed (Index) Unexposed (reference) Cases agag bgbg Person-time L 1g L 0g

Practical summary Use the RR c to measure the direction and magnitude of confounding: cRR = SMR*RR c RR c = 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

Practical summary Use the RR c to measure the direction and magnitude of confounding: cRR = SMR*RR c RR c = 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: 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