Concepts of Interaction
Matthew Fox
Advanced Epi
What is interaction?
Interaction?
Covar +
Covar -
D+
D-
E+
600
400
E300
700
E+
40
960
E20
980
Total
Risk
1000
0.4
1000
0.2
1000
0.04
1000
0.02
RR
2
2
OR
3.5
2.04
Interaction?
Smokers
Non-smokers
Asbestos +
Asbestos -
Asbestos +
Asbestos -
LC
20
10
3
1
No LC
980
990
997
999
Total
1000
1000
1000
1000
Risk
0.02
0.01
0.003
0.001
RR
2
3
RD
0.010
0.002
Last Session


New approaches to confounding
Instrumental variables
–

Propensity scores
–
–

Variable strongly predictive of exposure, no direct
link to outcome, no common causes with outcome
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
–

Interdependence
–

Measure of effect is different in the strata of the
modifying variable
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
1
No LC
980
990
997
999
Total
1000
1000
1000
1000
Risk
0.02
0.01
0.002
0.001
2
Ref
2
Ref
S+, A+
vs S-,A-
S+ vs Samong A-
A+ vs Aamong S-
RR
0.02/0.001
0.01/0.001
0.002/0.001
RR
20
10
2
RR
20 = 10 * 2
EMM on Difference Scale?
Smokers
Non-smokers
Asbestos +
Asbestos -
Asbestos +
Asbestos -
LC
20
10
3
1
No LC
980
990
997
999
Total
1000
1000
1000
1000
Risk
0.02
0.01
0.002
0.001
RD
0.01
Ref
0.001
Ref
S+, A+
vs S-,A-
S+ vs Samong A-
A+ vs Aamong S-
RD
0.02-0.001
0.01-0.001
0.002-0.001
RD
0.019
0.009
0.001
0.019 ≠ 0.009 + 0.001 = 0.010
Effect measure modification (2)

If:
–
–

Then:
–
–

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)
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)
All
Death
Person-years
Rate
rate difference
relative rate
CD4 <200
175
25915
68/10000
33/10000
1.97
CD4 ≥200
89
25949
34/10000
0
Example 1 (2)
Death
Person-years
Rate
Rate Difference
Relative Rate

–
–

Treated
CD4 <200
CD4 ≥200
157
81
24531
24545
64/10000
33/10000
31/10000
0
1.94
1
Is there confounding?
–

Untreated
CD4 <200
CD4 ≥200
18
8
1384
1404
130/10000
57/10000
73/10000
0
2.28
1
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
Total
Risk
Covar +
E+
600
400
1000
0.4
RR
2.0
2.0
OR
3.5
2.04
D+
D-
E300
700
1000
0.2
Covar E+
40
960
1000
0.04
E20
980
1000
0.02
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
ˆ  RDˆ 

R
D
z
1
2

ln RRˆ
z
1
SE (RDˆ1 )2  SE (RDˆ 2 )2
 ln RRˆ 2

SE (ln RRˆ 1 )2  SE (ln RRˆ 2 )2
SE for difference measures
a
b
SE (IRD) 

2
2
PTE  PTE 
a c
bd
SE (RD) 
 2
2
NE  (NE   1) NE  (NE   1)
SE for relative measures
SE (IRR ) 
1 1

a b
SE (RR ) 
1
1
1
1

 
a NE  b NE 
SE (OR) 
1 1 1 1
  
a b c d
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)

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?
Baseline risk
Effect of E1
Effect of E2
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
A+
A-
B+
B-
B+
B-
4
8
8
6
2
6
2
2
100
10/100
100
8/100
4
0.06
2
Total
Risk
RR
RD
100
20/100
2
0.1
100
2/100
A+
A-
B+
B-
B+
B-
0
8
8
6
2
6
2
2
100
10/100
100
8/100
4
0.06
2
Total
Risk
RR
RD
100
16/100
1.6
0.06
100
2/100
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)
Death
Person-years
Rate
Rate difference

Treated
CD4 <200
CD4 ≥200
157
81
24531
24545
64/10000
33/10000
31/10000
0
Component 1:
–

Untreated
CD4 <200
CD4 ≥200
18
8
1384
1404
130/10000
57/10000
24/10000
The baseline rate in the doubly unexposed
The doubly unexposed = high CD4/treated
–
Their mortality rate is 33/10,000
Interdependence (4)
Death
Person-years
Rate
Rate difference

Treated
CD4 <200
CD4 ≥200
157
81
24531
24545
64/10000
33/10000
31/10000
0
Component 2:
–

Untreated
CD4 <200
CD4 ≥200
18
8
1384
1404
130/10000
57/10000
24/10000
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)
Death
Person-years
Rate
Rate difference

Treated
CD4 <200
CD4 ≥200
157
81
24531
24545
64/10000
33/10000
31/10000
0
Component 3:
–

Untreated
CD4 <200
CD4 ≥200
18
8
1384
1404
130/10000
57/10000
24/10000
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?
–

Rate in doubly exposed is 130/10,000
–
–
–

Do components add to rate in doubly exposed (low
CD4 count, untreated)?
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
R(E ,C  )  R(E ,C )  RD(E  )  RD(C  )  R(I )
130
33
31
24



 ?[R(I )]
10,000 10,000 10,000 10,000
Interdependence (8)

Calculate the rate due to interdependence two ways:
R(I )  R (E ,C  )
 [R (E ,C  )  R (E ,C )]
 [R (E ,C )  R (E ,C )]
 R (E ,C  )
or
R(I )  [R (E ,C  )  R(E ,C  )] 
[R (E ,C  )  R(E ,C  )]
Component 3
Component 2
Component 1
Interdependence (9)
Death
Person-years
Rate
Rate difference
Rate Difference

Untreated
CD4 <200
CD4 ≥200
18
8
1384
1404
130/10000
57/10000
24/10000
73/10000
Treated
CD4 <200
CD4 ≥200
157
81
24531
24545
64/10000
33/10000
31/10000
0
31 / 10000
Calculate the rate due to interdependence two ways:
130
24
31
33
42
R(I ) 




10,000 10,000 10,000 10,000 10,000
or
73
31
42
R(I ) 


10,000 10,000 10,000
Perspective of interdependence

With interdependence we care about the
joint effect of two actions
–
–

Now we care about:
–

Action is A+B+, A+B-, A-B+, A-BLeads to four potential outcomes per person
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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 experienced
For a particular type of subject with that combination of X and Y
Interdependence under the CST model:
the non-interdependence class
Type (description)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
1 (doomed)
Risks
x=I x=R x=I x=R
y=A y=A y=B y=B
1
1
1
1
Effect of x=I within
strata of y
y=A y=B Difference
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)
3
Risks
Effect of x=I within
x=I x=R x=I x=R
strata of Y
y=A y=A y=B y=B y=A y=B Difference
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)
2 (single plus joint
causation by x=1
and y=1)
9
Risks
Effect of x=I within
x=I x=R x=I x=R
strata of y
y=A y=A y=B y=B y=A y=B Difference
1
1
1
0
0
1
-1
0
1
1
1
-1
10
0
1
1
0
-1
12
0
1
0
0
-1
14
0
0
1
0
0
0
1
0
1
-1
-2
-1
-1
Assessing biologic interdependence: Are there any
cases due to joint occurrence of component causes?
CST Types
R(E+,C+) = R(I,A) =
[p1+p4+p6+p3+p5+p7+p8+p2]
The risk in the doubly exposed
[R(I,A)] equals:
the effect of x=I in y=B
R(E+,C-)=R(I,B) =
[r1+r6+r13+r5+r2+r9+r10+r14]
[R(I,B) - R(R,B]
+ the effect of y=A in x=R
R(E-,C+)=R(R,A) =
[q1+q4+q11+q3+q2+q9+q10+q12]
[R(R,A) - R(R,B]
+ the baseline risk
R(E-,C-)=R(R,B) =
[s1+s11+s13+s3+s5+s7+s15+s9]
[R(x=R,y=B)]
+ the interaction contrast [IC]
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?
Baseline
Low CD4
Untreated
R(I)
Total
Rate
(per 10,000 PY)
33
24
31
42
130
Proportion of
total
25.4%
18.5%
23.8%
32.3%
100%
Proportion of
total
Attributable Proportions

What proportion of the risk in the doubly exposed
can be attributed to each exposure?
Baseline
Low CD4
Untreated
R(I)
Total
Rate
(per 10,000 PY)
33
24
31
42
130
Proportion of
total
25.4%
18.5%
23.8%
32.3%
100%
Proportion of
total
25.4%
50.8%
56.2%
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)
outcome  intercept 
b1  exposure
b2  covariate 
b3  exposure covariate
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
33
rate 

10,000
31
 exposure
10,000
24
 covariate 
10,000
42
 exposure covariate
10,000
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)
ln(RR)  ln(1) 
ln(1.94)  exposure
ln(1.73)  covariate 
ln(1.17)  exposure covariate
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
