Concepts of Interaction Matthew Fox Advanced Epi

What is interaction?

Interaction? Covar +Covar - E+E-E+E- D D Total1000 Risk RR22 OR

Interaction? SmokersNon-smokers Asbestos +Asbestos -Asbestos +Asbestos - LC No LC Total1000 Risk RR23 RD

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

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

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

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

Concept 1: Effect Measure Modification

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)

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: = 0.3 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

EMM on Relative Scale? SmokersNon-smokers Asbestos +Asbestos -Asbestos +Asbestos - LC No LC Total1000 Risk RR2Ref2 S+, A+ vs S-,A- S+ vs S- among A- A+ vs A- among S- RR0.02/ / /0.001 RR = 10 * 2

EMM on Difference Scale? SmokersNon-smokers Asbestos +Asbestos -Asbestos +Asbestos - LC No LC Total1000 Risk RD0.01Ref0.001Ref S+, A+ vs S-,A- S+ vs S- among A- A+ vs A- among S- RD RD ≠ = 0.010

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”

Example 1 (1)

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?

But EMM of OR can be misleading Covar +Covar - E+E-E+E- D D Total1000 Risk RR2.0 OR

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

SE for difference measures

SE for relative measures

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

Perspective With modification, concerned only with the outcome of one variable within levels of 2 nd – 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(Y a=1 =1|M=1) - Pr(Y a=0 =1|M=1) = Pr(Y a=1 =1|M=0) - Pr(Y a=0 =1|M=0) or – Pr(Y a=1 =1|M=1) / Pr(Y a=0 =1|M=1) = Pr(Y a=1 =1|M=0) / Pr(Y a=0 =1|M=0)

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

Concept 2: Interdependence

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?

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

Four ways to get disease

Cases of D in doubly unexposed

Cases of D in those exposed to A

Cases of D in those exposed to B

Cases of D in double exposed

/10010/1008/1002/100 2 Total Risk RR RD B+B- A+ B+B- A-

/10010/1008/1002/ Total Risk RR RD B+B- A+ B+B- A-

So how to get at interaction?

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

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

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)

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

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

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

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

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

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

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

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(Y a=1,b=1 =1) - Pr(Y a=0,b=1 =1) = Pr(Y a=1,b=0 =1) - Pr(Y a=0,b=0 =1) Both actions need to have an effect to have interdependence – Surrogates are not possible

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 (2 4 ) Three classes of susceptibility types: – Non-interdependence (like doomed & immune) – Positive interdependence (like causal CST) – Negative interdependence (like preventive CST)

Interdependence under the CST model: the non-interdependence class

The four possible combinations of factors X and Y

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

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

Interdependence under the CST model: the non-interdependence class Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class

Interdependence under the CST model: the positive interdependence class

Interdependence under the CST model: the negative interdependence class

Assessing biologic interdependence: Are there any cases due to joint occurrence of component causes? The risk in the doubly exposed [R(I,A)] equals: the effect of x=I in y=B [R(I,B) - R(R,B] + the effect of y=A in x=R [R(R,A) - R(R,B] + the baseline risk [R(x=R,y=B)] + the interaction contrast [IC] CST Types R(E+,C+) = R(I,A) = [p1+p4+p6+p3+p5+p7+p8+p2] R(E+,C-)=R(I,B) = [r1+r6+r13+r5+r2+r9+r10+r14] R(E-,C+)=R(R,A) = [q1+q4+q11+q3+q2+q9+q10+q12] R(E-,C-)=R(R,B) = [s1+s11+s13+s3+s5+s7+s15+s9]

Solve for IC, assess direction and magnitude IC 0 IC = R(I,A) - [R(I,B) - R(R,B)]- [R(R,A) - R(R,B)]- [R(R,B)] Require both have 3-way partial exchangeability: IC = (p3+p5+2p7+p8+p15) - (p2+p9+2p10+p12+p14) IC is the difference in the sum of positive interdependence CSTs and sum of negative interdependence CSTs

Point 5: Remember, lack of additive EMM usually means multiplicative EMM. So interaction in a logistic regression model cannot tell us about additive interaction!

Attributable Proportions What proportion of the risk in the doubly exposed can be attributed to each exposure?

Attributable Proportions What proportion of the risk in the doubly exposed can be attributed to each exposure?

Back to Rothman’s Model: Attributable %s don’t need to add to 100%

Concept 3: Statistical Interaction

Statistical Interaction (1) Most easily understood in regression modeling Write model as effect = baseline + effects of predictor variables (exposure, covariates, and their interaction)

Statistical Interaction (2) Where: – Exposure: 1 = exposed, 0 = unexposed – Covariate: 1 = covariate+, 0 = covariate- And: – Intercept is baseline risk or rate – b1 is effect of exposure – b2 is effect of covariate – b3 is the statistical interaction

Statistical Interaction (3) Rate or risk model (e.g., linear regression) – Effects are the risk differences b3 is risk due to interdependence

Statistical Interaction (4) Relative risk model (e.g., logistic regression) – Effects are the log of the relative effects – On log scale division becomes addition b3 is departure from MULTIPLICATIVE interaction – Because deviation from additivity on log scale =relative – NO correspondence to R(I)

Summary: Difference Scale Effect measure modification on the difference scale implies:  A non-zero risk due to interdependence [R(I)], because risk due to interdependence equals difference in risk differences  A non-zero cross-product term in linear regression models of the risk or rate  Nothing about departure from multiplicativity

Summary: Relative Scale Effect measure modification on the relative scale implies:  A non-zero cross-product term in logistic regression models  Nothing about departure from additivity