Concepts of Interaction

Matthew FoxAdvanced Epi

What is interaction?

Interaction?

Covar + Covar -E+ E- E+ E-

D+ 600 300 40 20D- 400 700 960 980Total 1000 1000 1000 1000Risk 0.4 0.2 0.04 0.02

RR 2 2

OR 3.5 2.04

Interaction?

Smokers Non-smokersAsbestos + Asbestos - Asbestos + Asbestos -

LC 20 10 3 1No LC 980 990 997 999Total 1000 1000 1000 1000Risk 0.02 0.01 0.003 0.001

RR 2 3

RD 0.010 0.002

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:0.2 + 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

EMM on Relative Scale?Smokers Non-smokers

Asbestos + Asbestos - Asbestos + Asbestos -

LC 20 10 3 1No LC 980 990 997 999Total 1000 1000 1000 1000Risk 0.02 0.01 0.002 0.001RR 2 Ref 2 Ref

S+, A+ vs S-,A-

S+ vs S- among A-

A+ vs A- among S-

RR 0.02/0.001 0.01/0.001 0.002/0.001RR 20 10 2

20 = 10 * 2

EMM on Difference Scale?Smokers Non-smokers

Asbestos + Asbestos - Asbestos + Asbestos -LC 20 10 3 1No LC 980 990 997 999Total 1000 1000 1000 1000Risk 0.02 0.01 0.002 0.001RD 0.01 Ref 0.001 Ref

S+, A+ vs S-,A-

S+ vs S- among A-

A+ vs A- among S-

RD 0.02-0.001 0.01-0.001 0.002-0.001RD 0.019 0.009 0.001

0.019 ≠ 0.009 + 0.001 = 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)

LCIS - AllCD4 <200 CD4 ≥200

Death 175 89Person-years 25915 25949Rate 68/10000 34/10000rate difference 33/10000 0relative rate 1.97

Example 1 (2)Untreated Treated LCIS + or -

CD4 <200 CD4 ≥200 CD4 <200 CD4 ≥200Death 18 8 157 81Person-years 1384 1404 24531 24545Rate 130/10000 57/10000 64/10000 33/10000Rate Difference 73/10000 0 31/10000 0Relative Rate 2.28 1 1.94 1

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+ 600 300 40 20D- 400 700 960 980Total 1000 1000 1000 1000Risk 0.4 0.2 0.04 0.02

RR 2.0 2.0

OR 3.5 2.04

A simple test for homogeneity

2

22

1

21

22

21

21

)ˆ(ln)ˆ(ln

ˆlnˆln

)ˆ()ˆ(

ˆˆ

RRSERRSE

RRRRz

DRSEDRSEDRDRz

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

)1()1()(

)(

22

22

EEEE

EE

NNdb

NNcaRDSE

PTb

PTaIRDSE

SE for relative measures

dcbaORSE

NbNaRRSE

baIRRSE

EE

1111)(

1111)(

11)(

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 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)

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)

Baseline riskEffect of E1Effect of E2Anything else

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

4

8

6

2

100

8 6

2 2

2

100 100 10020/100 10/100 8/100 2/100

2

TotalRisk

RRRD

4

0.1 0.06

B+ B-A+

B+ B-A-

0

8

6

2

100

8 6

2 2

2

100 100 10016/100 10/100 8/100 2/100

1.6

TotalRisk

RRRD

4

0.06 0.06

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

Untreated Treated LCIS + or -CD4 <200 CD4 ≥200 CD4 <200 CD4 ≥200

Death 18 8 157 81Person-years 1384 1404 24531 24545Rate 130/10000 57/10000 64/10000 33/10000Rate difference 24/10000 31/10000 0

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

Untreated Treated LCIS + or -CD4 <200 CD4 ≥200 CD4 <200 CD4 ≥200

Death 18 8 157 81Person-years 1384 1404 24531 24545Rate 130/10000 57/10000 64/10000 33/10000Rate difference 24/10000 31/10000 0

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

Untreated Treated LCIS + or -CD4 <200 CD4 ≥200 CD4 <200 CD4 ≥200

Death 18 8 157 81Person-years 1384 1404 24531 24545Rate 130/10000 57/10000 64/10000 33/10000Rate difference 24/10000 31/10000 0

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

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24000,1031

000,1033

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IR

IRCRDERDCERCER

Interdependence (8) Calculate the rate due to interdependence two ways:

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CERCERCERCERIR

orCER

CERCERCERCER

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Component 1

Component 2

Component 3

Interdependence (9)

000,1042

10,00031

000,1073)(

000,1042

000,1033

000,1031

000,1024

000,10130)(

IR

or

IR

Calculate the rate due to interdependence two ways:

Untreated Treated LCIS + or -CD4 <200 CD4 ≥200 CD4 <200 CD4 ≥200

Death 18 8 157 81Person-years 1384 1404 24531 24545Rate 130/10000 57/10000 64/10000 33/10000Rate difference 24/10000 31/10000 0Rate Difference 73/10000 31 / 10000

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

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)

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

The four possible combinations of factors X and Y

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Strata of Y

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Indicates whether or not the outcome was experiencedFor a particular type of subject with that combination of X and Y

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Example of one susceptibility type

Interdependence under the CST model: the non-interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 1 (doomed) 1 1 1 1 0 0 0

4 1 1 0 0 0 0 0

6 1 0 1 0 1 1 0

11 0 1 0 1 -1 -1 0

13 0 0 1 1 0 0 0

16 (immune) 0 0 0 0 0 0 0

Interdependence under the CST model: the positive interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of Y

y=A y=B Difference 3 1 1 0 1 0 -1 1

5 1 0 1 1 1 0 1

7 1 0 0 1 1 -1 2

8 (causal synergism) 1 0 0 0 1 0 1

15 0 0 0 1 0 -1 1

Interdependence under the CST model: the negative interdependence class

Type (description) Risks x=I x=R x=I x=R y=A y=A y=B y=B

Effect of x=I within strata of y

y=A y=B Difference 2 (single plus joint

causation by x=1 and y=1)

1 1 1 0 0 1 -1

9 0 1 1 1 -1 0 -1

10 0 1 1 0 -1 1 -2

12 0 1 0 0 -1 0 -1

14 0 0 1 0 0 1 -1

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 TypesR(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?

Rate (per 10,000 PY)

Proportion of total

Proportion of total

Baseline 33 25.4%Low CD4 24 18.5%Untreated 31 23.8%R(I) 42 32.3%Total 130 100%

Attributable Proportions

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

Rate (per 10,000 PY)

Proportion of total

Proportion of total

Baseline 33 25.4% 25.4%Low CD4 24 18.5% 50.8%Untreated 31 23.8% 56.2%R(I) 42 32.3%Total 130 100% 132%

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)

covariateexposureb covariateb exposureb

intercept outcome

3

2

1

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

covariateexposure10,000

42

covariate10,000

24

exposure10,000

31

10,000

33 rate

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)

covariateexposureln(1.17) covariateln(1.73) exposureln(1.94)

ln(1)ln(RR)

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