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# Concepts of Interaction

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1. Concepts of Interaction Matthew Fox Advanced Epi

2. What is interaction?

3. Interaction?

4. Interaction?

5. Last Session • New approaches to confounding • Instrumental variables • Variable strongly predictive of exposure, no direct link to outcome, no common causes with outcome • Propensity scores • Summarize confounding with a single variable • Useful when have lots of potential comparisons • Marginal structural models • Use weighting rather than stratification to adjust • Useful when we have time dependent confounding

6. This session • Concepts of interaction • Very poorly understood concept • Often not clear what a person means when they suggest it exists • Often confused with bias • Define each concept • Distinguish between them • Which is the most useful

7. 3 Concepts of Interaction • Effect Measure Modification • Measure of effect is different in the strata of the modifying variable • Interdependence • Risk in the doubly exposed can’t be explained by the independent effects of two single exposures • Statistical Interaction • Cross-product term in a regression model not = 0

8. Point 1: Confounding is a threat to validity. Interaction is a threat to interpretation.

9. Concept 1:Effect Measure Modification

10. Effect measure modification (1) • Measures of effect can be either: • Difference scale (e.g., risk difference) • Relative scale (e.g., relative risk) • To assess effect measure modification: • Stratify on the potential effect measure modifier • Calculate measure of effect in all strata • Decide whether measures of effect are different • Can use statistical tests to help (only)

11. No EMM corresponds to • Difference scale: • If RD comparing A+ vs A- among B- = 0.2 and • RD comparing B+ vs B- among A- = 0.1, then • RD comparing A+,B+ to A-,B- (doubly exposed to doubly unexposed) should be: • 0.2 + 0.1 = 0.3 • Relative scale: • If RR comparing A+ vs A- among B- = 2 and • RR comparing B+ vs B- among A- = 3, then • RR comparing A+,B+ to A-,B- should be: • 2 * 3 = 6

12. EMM on Relative Scale? 20 = 10 * 2

13. EMM on Difference Scale? 0.019 ≠ 0.009 + 0.001 = 0.010

14. Effect measure modification (2) • If: • Exposure has an effect in all strata of the modifier • Risk is different in unexposed group of each stratum of the modifier (i.e., modifier affects disease) • Then: • There will always be some effect measure modification on one scale or other (or both) • you must to decide if it is important • Therefore: • More appropriate to use the terms “effect measure modification on the difference or relative scale”

15. Example 1 (1)

16. Example 1 (2) • Is there confounding? • Does the disease rate depend on treatment in unexposed? • Does exposure prevalence depend on treatment in pop? • Is the relative rate collapsible? • Effect measure modification — difference scale? • Effect measure modification — relative scale?

17. But EMM of OR can be misleading

18. A simple test for homogeneity • Large sample test • More sophisticated tests exist (e.g., Breslow-Day) • Assumes homogeneity, must show heterogeneous • Tests provide guidance, not the answer

19. SE for difference measures

20. SE for relative measures

21. Point 3: Effect measure modification often exists on one scale by definition. Doesn’t imply any interaction between variables.

22. Perspective • With modification, concerned only with the outcome of one variable within levels of 2nd • The second may have no causal interpretation • Sex, race, can’t have causal effects, can be modifiers • Want to know effect of smoking A by sex M: • Pr(Ya=1=1|M=1) - Pr(Ya=0=1|M=1) = Pr(Ya=1=1|M=0) - Pr(Ya=0=1|M=0) or • Pr(Ya=1=1|M=1) / Pr(Ya=0=1|M=1) = Pr(Ya=1=1|M=0) / Pr(Ya=0=1|M=0)

23. Surrogate modifiers • Just because stratification shows different effects doesn’t mean intervening on the modifier will cause a change in outcome • Cost of surgery may modify the effect of heart transplant on mortality • More expensive shows a bigger effect • Likely a marker of level of proficiency of the surgeon • Changing price will have no impact on the size of the effect

24. Concept 2:Interdependence

25. Interdependence (1) • Think of the risk of disease in the doubly exposed as having four components: • Baseline risk (risk in doubly unexposed) • Effect of the first exposure (risk difference 1) • Effect of the second exposure (risk difference 2) • Anything else?

26. Think again about multiplicative scale • Additive scale: • Risk difference • Implies population risk is general risk in the population PLUS risk due to the exposure • Assumes no relationship between the two • Multiplicative scale: • Risk ratio • Implies population risk is general risk in the population PLUS risk due to the exposure • Further assumes the effect of the exposure is some multiple of the baseline risk

27. Four ways to get disease

28. Cases of D in doubly unexposed

29. Cases of D in those exposed to A

30. Cases of D in those exposed to B

31. Cases of D in double exposed

32. A+ A- B+ B- B+ B- 4 8 6 8 2 6 2 2 2 Total 100 100 100 100 Risk 20/100 10/100 8/100 2/100 4 RR 2 0.06 RD 0.1

33. A+ A- B+ B- B+ B- 0 8 6 8 2 6 2 2 2 Total 100 100 100 100 Risk 16/100 10/100 8/100 2/100 4 RR 1.6 0.06 RD 0.06

34. So how to get at interaction?

35. Point 4: It is the absolute scale that tells us about biologic interaction (biologic doesn’t need to be read literally)

36. Point 4a: Since Rothman’s model shows us interdependence is ubiquitous, there is no such thing as “the effect” as it will always depend on the distribution of the complement

37. Interdependence (2) • In example, doubly exposed group are low CD4 count who were untreated • Their mortality rate is 130/10,000 • Separate this rate into components: • Baseline mortality rate in doubly unexposed (high CD4 count, treated) • Effect of low CD4 count instead of high • Effect of no treatment instead of treatment • Anything else (rate due to interdependence)

38. Interdependence (3) • Component 1: • The baseline rate in the doubly unexposed • The doubly unexposed = high CD4/treated • Their mortality rate is 33/10,000

39. Interdependence (4) • Component 2: • The effect of exposure 1 (low CD4 vs. high) • Calculate as rate difference • (low - high) in treated stratum • Rate difference is 31/10,000

40. Interdependence (5) • Component 3: • Effect of exposure 2 (untreated vs treated) • Calculate as rate difference • (untreated - treated), in unexposed (high CD4) • Rate difference is 24/10,000

41. Interdependence (6) • Anything else left over? • Do components add to rate in doubly exposed (low CD4 count, untreated)? • Rate in doubly exposed is 130/10,000 • component 1 (rate in doubly unexposed): 33/10,000 • component 2 (effect of low CD4 vs high): 31/10,000 • component 3 (effect of not vs treated): 24/10,000 • These 3 components add to 88/10,000 • There must be something else to get to 130/10,000

42. Interdependence (7) • The something else is the “risk (or rate) due to interdependence” between CD4 count and treatment

43. Interdependence (8) Component 3 Component 2 • Calculate the rate due to interdependence two ways: Component 1

44. Interdependence (9) • Calculate the rate due to interdependence two ways:

45. Perspective of interdependence • With interdependence we care about the joint effect of two actions • Action is A+B+, A+B-, A-B+, A-B- • Leads to four potential outcomes per person • Now we care about: • Pr(Ya=1,b=1=1) - Pr(Ya=0,b=1=1) = Pr(Ya=1,b=0=1) - Pr(Ya=0,b=0=1) • Both actions need to have an effect to have interdependence • Surrogates are not possible

46. Biologic interaction under the CST model: general • A study with two binary factors (X & Y), producing four possible combinations: • x=I, y=A; x=R, y=A; x=I, y=B; x=R, y=B • Binary outcome (D=1 or 0) • 16 possible susceptibility types (24) • Three classes of susceptibility types: • Non-interdependence (like doomed & immune) • Positive interdependence (like causal CST) • Negative interdependence (like preventive CST)

47. Interdependence under the CST model: the non-interdependence class

48. Interdependence under the CST model: the non-interdependence class The four possible combinations of factors X and Y

49. Interdependence under the CST model: the non-interdependence class Strata of Y

50. Interdependence under the CST model: the non-interdependence class Indicates whether or not the outcome was experienced For a particular type of subject with that combination of X and Y