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

1. 3 Causal Models Part II:Counterfactual Theory and Traditional Approaches to Confounding (Bias?) Confounding, Identifiability, Collapsibility and Causal inference

2. Thursday reception at lunch time at SACEMA

3. 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

4. This Morning • Counterfactual model • Susceptibility types • Potential outcomes • Confounding under the counterfactual susceptibility model of causation • Stratification • Identifying confounders • Standardization versus pooling

5. What is Confounding? Give me the definition you were taught or describe how you understand it

6. What is an “adjusted” measure of effect?

7. Is red wine cardio-protective?

8. 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?

9. Counterfactual Theory Potential Outcomes, Susceptibility types

10. Poor Clare • Doctor prescribes antibiotics • 3 days later she is cured • Did the antibiotic cure her?

11. Cinema d’Counterfactual

12. The counterfactual model:The counterfactual ideal Disease experience, given exposed Hypothetical disease experience, if unexposed The Counterfactual Ideal

13. 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

14. Approximation to The Counterfactual Ideal The counterfactual model:The counterfactual ideal Disease experience, given exposed Substitute disease experience of truly unexposed

15. 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.

16. 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

17. Take home message 2:“Effects” of exposures only have meaning when defined in contrast to an alternative

18. If ethics were not a concern, how would you design an RCT of smoking and lung cancer? Think about dose, duration

19. What about obesity and MI? What about gender and cancer?

20. 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

22. Take Home Message 3: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.

23. To identify a causal effect in an individual • 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

24. 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

25. Assume each person represents 100,000 people Effect : [Pr(Ya=1=1) - Pr(Ya=0=1)] =

26. Assume each person represents 100,000 people [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)] =

27. Assume each person represents 100,000 people [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

28. The counterfactual modelSusceptibility 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

29. 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

30. Individual Susceptibility under the CST model 1 – 1 = 0 1 / 1 = 1 1 – 0 = 1 1 / 0 = undef 0 – 1 = -1 0 / 1 = 0 0 – 0 = 0 0 / 0 = undef

31. 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

32. 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

33. 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(Ya=1 = 1) ≠ Pr(Ya=0 = 1) • Or more generally, E(Ya=1) ≠ E(Ya=0) • Note makes no reference to relative vs. absolute

34. Effects vs. Associations • 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)

35. Traditional Approaches to Confounding and Confounders

36. Extend the CST model of causation to populations 1 1

37. What is the risk of disease in exposed? Observed risk in exposed is p1 + p2, but we cannot tell how many of each 1 1

38. 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

39. 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

40. Observed 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)

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

42. 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

43. 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 p1 = q1 p2 – q3 But effect should be based only on exposed

44. 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 p1 + p3 = q1+ q3 p2 – p3 NOTE that RD related to balance of p2 and p3

45. 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

46. 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

47. 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

48. Causal RD vs. Observed • 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

49. 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? • Yes 100 100

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