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Concepts of Interaction

Concepts of Interaction. Matthew Fox Advanced Epi. What is interaction?. Interaction?. Interaction?. 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

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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

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