Transcript Lecture 7- Interaction 030905

Interaction
Aspirin
Note: to assess interaction, a
minimum of 3 variables were
needed in this study:
•Aspirin
•Anger
•Coronary Heart Disease (CHD)
Interaction = “Effect
modification”: The “effect” of
the risk factor -- anger – on
the outcome – CHD -- differs
depending on the presence or
absence of a third factor
(effect modifier) --aspirin. The
third factor (aspirin) modifies
the effect of the risk factor
(anger) on the outcome
(CHD).
Anger
Anger
CHD
CHD
How is “effect”* measured in
epidemiologic studies?
• If effect is measured on an additive or absolute scale
(attributable risks)  additive interaction assessment
(Attributable Risk model).
• If effect is measured on a relative (ratio) scale (relative
risks, odds ratios, etc.)  multiplicative interaction
assessment (Relative Risk model).
*For practical purposes in this lecture, “effect” refers to
associations that may or may not be causal.
Two strategies to evaluate interaction based on different,
but equivalent definitions:
• Effect modification (homogeneity/heterogeneity of
effects)
• Comparison between joint expected and joint observed
effects
First strategy to assess interaction:
Effect Modification
ADDITIVE (attributable risk) interaction
Hypothetical example of absence of additive interaction
Z
A
No
No
10.0
Yes
20.0
No
30.0
Yes
40.0
Yes
Incidence rate (%) ARexp to A (%)
Potential effect modifier
Potential risk factor of primary interest
First strategy to assess interaction:
Effect Modification
ADDITIVE (attributable risk) interaction
Hypothetical example of absence of additive interaction
Z
A
No
No
10.0
Yes
20.0
No
30.0
Yes
40.0
Yes
Incidence rate (%) ARexp to A (%)
10.0
10.0
Conclude: Because AR’s associated with A are not
modified by exposure to Z, there is no additive interaction.
First strategy to assess interaction:
Effect Modification
ADDITIVE (attributable risk) interaction
Hypothetical example of presence of additive interaction
Z
A
No
No
5.0
Yes
10.0
No
10.0
Yes
30.0
Yes
Incidence rate (%) ARexp to A (%)
5.0
20.0
Conclude: Because AR’s associated with A are modified
by exposure to Z, additive interaction is present.
45
40
35
30
25
20
15
10
5
0
Example 2
35
ARA
Z+
ARA
Z-
Incidence rate (%)
Incidence rate (%)
Example 1
30
25
ARA
20
15
10
5
Z+
Z-
ARA
0
A-
A+
A-
A+
Conclude:
Conclude:
-The stratum-specific effects (AR)
are homogeneous
-The stratum-specific effects (AR)
are heterogeneous
- Z does not modify the effect of A
- Z modifies the effect of A
-There is no (additive) interaction
-There is (additive) interaction
Example of Effect Modification (Interaction) in
a Clinical Trial with a Continuous Outcome
Average No. of New Nevi
Freckles, % Sunscreen Control Difference
10
24
24
0
20
20
28
-8
30
20
30
-10
40
16
30
-14
From: Szklo, Arch Dermatol 2000;136:1546 (Based on Gallagher et al, 2000)
New Nevi, No.
35
30
25
20
15
10
5
Sunscreen
Control
0
5
10
20
30
40
Freckles, %
Example of Freckling as an Interacting Variable
(Effect Modifier)
From: Szklo, Arch Dermatol 2000;136:1546 (Based on Gallagher et al, 2000)
First strategy to assess interaction:
Effect Modification
MULTIPLICATIVE (ratio-based) interaction
Hypothetical example of absence of multiplicative interaction
Z
A
Incidence rate (%)
No
No
10.0
Yes
20.0
No
25.0
Yes
50.0
Yes
RRA
2.0
2.0
Conclude: Because RR’s associated with A are not modified
by exposure to Z, there is no multiplicative interaction.
First strategy to assess interaction:
Effect Modification
MULTIPLICATIVE (ratio-based) interaction
Hypothetical example of presence of multiplicative interaction
Z
A
Incidence rate (%)
No
No
10.0
Yes
20.0
No
25.0
Yes
125.0
Yes
RRA
2.0
5.0
Conclude: Because RR’s associated with A are modified
by exposure to Z, multiplicative interaction is present.
Example 2
140
140
120
120
Incidence rate (%)
Incidence rate (%)
Example 1
100
80
60
40
20
Z+
Z-
100
80
60
40
20
0
0
A-
A+
Z+
Z-
A-
A+
Is this the best way to display the data?
NO!
To assess multiplicative effects, use a log scale:
Example 1
Example 2
1000
Incidence rate (%)
Incidence rate (%)
1000
100
Z+
10
Z-
1
100
Z+
10
Z-
1
A-
A+
A-
A+
Conclude:
Conclude:
-The stratum-specific effects (RR)
are homogeneous
-The stratum-specific effects (RR)
are heterogeneous
- Z does not modify the effect of A
- Z modifies the effect of A
-There is no (multiplicative)
interaction
-There is (multiplicative) interaction
Two strategies to evaluate interaction based on
different, but equivalent definitions:
• Effect modification (homogeneity/heterogeneity of
effects) 
• Comparison between joint expected and joint
observed effects
Second strategy to assess interaction:
(based on the calculation of “joint effects”)
Individual effects
A
+
Z
Expected joint effect
Observed joint effect
A+Z
No interaction
Observed joint effect
A+Z
+I
Synergism (Positive
Interaction)
Observed joint effect
A+Z
-I
Antagonism (Negative
Interaction)
The two definitions and strategies are completely
equivalent. It is impossible to conclude that there
is (or there is not) interaction using one strategy,
and reach the opposite conclusion upon use of
the other strategy!
Thus, when there is effect modification, the joint
observed and the joint expected effects will be
different.
Second strategy to assess interaction:
comparison of joint expected and joint observed effects
Additive interaction
Stratified
Factor Z Factor A Incidence (%)
ARA
ARvs(--)
Reference Reference
No
No
10.0
10.0
Yes
20.0
Reference
Yes
No
30.0
10.0
Yes
40.0
Second strategy to assess interaction:
comparison of joint expected and joint observed effects
Additive interaction
Stratified
Independent
effects of:
Factor Z Factor A Incidence (%)
ARA
ARvs(--)
Reference Reference
No
No
10.0
10.0
Yes
20.0
10.0  A
Reference
Yes
No
30.0
20.0  Z
10.0
Yes
40.0
30.0  A + Z
Second strategy to assess interaction:
comparison of joint expected and joint observed effects
Additive interaction
Stratified
ARA
Factor Z Factor A Incidence (%)
Reference
No
No
10.0
10.0
Yes
20.0
Reference
Yes
No
30.0
10.0
Yes
40.0
Observed
ARvs(--)
Reference
10.0
20.0
30.0
Joint observed ARA+Z+ = 30%
Joint expected ARA+Z+ = ARA+Z- + ARA-Z+= 30%
Conclude:
Because the observed joint AR is the same as that expected by
adding the individual AR’s, there is no additive interaction
(that is, the same conclusion as when looking at the stratified AR’s)
Second strategy to assess interaction:
comparison of joint expected and joint observed effects
Additive interaction
Factor Z Factor A Incidence (%)
No
No
5.0
Yes
10.0
Yes
No
10.0
Yes
30.0
Stratified
ARA
Observed
ARvs(--)
Reference Reference Expected
5.0
5.0
Reference
5.0
20.0
25.0 10.0
Joint observed AR = 25%
Joint expected AR = ARA+Z- + ARA-Z+= 10%
Conclude:
Because the observed joint AR is different from that expected by
adding the individual AR’s, additive interaction is present
(that is, the same conclusion as when looking at the stratified AR’s)
Second strategy to assess interaction:
comparison of joint expected and joint observed effects
Multiplicative interaction
Stratified
RRA
Factor Z Factor A Incidence (%)
Reference
No
No
10.0
2.0
Yes
20.0
Reference
Yes
No
25.0
2.0
Yes
50.0
RRvs(--)
Reference
2.0
2.5
5.0
Joint observed RRA+Z+ = 5.0
Joint expected RRA+Z+ = RRA+Z-  RRA-Z+= 2.0  2.5 = 5.0
Conclude:
Because the observed joint RR is same as that expected by
adding the individual RR’s in a multiplicative scale (equivalent
to multiplying the individual RR’s), multiplicative interaction is
not present(that is, the same conclusion as when looking at the stratified RR’s)
Second strategy to assess interaction:
comparison of joint expected and joint observed effects
Multiplicative interaction
Stratified
RRA
Factor Z Factor A Incidence (%)
No
No
10.0
Reference
2.0
Yes
20.0
Yes
No
25.0
Reference
5.0
Yes
125.0
RRvs(--)
Reference
2.0
2.5
5.0
12.5
Joint observed RRA+Z+ = 12.5
Joint expected RRA+Z+ = RRA+Z-  RRA-Z+= 2.0  2.5 = 5.0
Conclude:
Since the observed joint RR is different from that expected by
multiplying the individual RR’s, there is multiplicative interaction
(that is, the same conclusion as when looking at the stratified RR’s)
How can we assess interaction in
case-control studies?
Case-control study
First strategy to assess interaction:
Effect Modification
Additive interaction cannot be assessed in case-control
studies by using the effect modification
(homogeneity/heterogeneity) strategy, as no incidence
rates are available to calculate attributable risks in the
exposed
Prospective Study
Z
A
No
No
5.0
Yes
10.0
No
10.0
Yes
30.0
Yes
Incidence rate (%) ARexp to A (%)
5.0
20.0
Case-control study
First strategy to assess interaction:
Effect Modification
Layout of table to assess
MULTIPLICATIVE interaction
Factor Factor
Stratified
What does it
Z
A
Cases Controls
ORA
mean?
No
No
Effect of A in the
absence of Z
Yes
Yes
No
Effect of A in the
presence of Z
Yes
Odds Ratios for the association among isolated clubfoot, maternal
smoking, and a family history of clubfoot, Atlanta, Georgia, 1968-80
Family history
of clubfoot
Yes
No
Maternal
smoking
Cases
Controls
Yes
14
7
No
11
20
Yes
118
859
No
203
2,143
Honein et al. Family history, maternal smoking, and clubfoot: an
indication of gene-environment interaction. Am J Epidemiol
2000;152:658-65.
Hypothesis: Family History is a potential effect
modifier of the association between Maternal
Smoking and clubfoot
Use the first strategy (homogeneity/heterogeneity) to
evaluate the presence of multiplicative interaction
Odds Ratios for the association among isolated clubfoot, maternal
smoking, and a family history of clubfoot, Atlanta, Georgia, 1968-80
Family history
of clubfoot
Yes
No
Maternal
smoking
Cases
Controls
Stratified
ORmaternal smk
Yes
14
7
3.64
No
11
20
Yes
118
859
No
203
2,143
1.45
Honein et al. Family history, maternal smoking, and clubfoot: an indication of geneenvironment interaction. Am J Epidemiol 2000;152:658-65.
Conclude: Since the stratified ORs are different (heterogeneous),
there is multiplicative interaction.
Now evaluate the same hypothesis (that there is an interaction between
family history of clubfoot and maternal smoking) using the second
strategy: comparison between joint observed and joint expected
“effects”.
Case-control study
Second strategy to assess interaction:
comparison of joint observed and expected effects
Layout of table to assess both ADDITIVE
and MULTIPLICATIVE interaction
Note common reference category
Factor Factor
Z
A
Cases Controls
No
No
Yes
Yes
No
Yes
Under ADDITIVE MODEL:
ORvs-1.0
OR-+
OR+OR++
What does it
mean?
Reference
Indep effect of A
Indep effect of Z
Joint effect
Exp’d OR++ = OR+- + OR-+ - 1.0
Derivation of formula for expected joint OR
expected
observed
Expected AR  Inc  Inc  ( Inc  Inc )  ( Inc  Inc )
Inc  Inc
Inc  Inc



Inc Inc
Inc Inc
RR++
1.0
RR+-
1.0

Inc Inc

Inc Inc
RR-+
RR  RR  RR  1.0
If disease is “rare” (e.g., <5%):
OR  OR  OR  1.0
1.0
Derivation of formula: Expected OR++ = OR+- + OR-+ - 1.0
Intuitive graphical derivation:*
OR
3.5
2.0
2.5
EXCZ
EXCA
EXCA
1.0
EXCZ
BL
BL
BL
BL
OR--
OR-+
OR+-
Exp’d
OR++
Baseline
Baseline+Excess due to A
[EXCA+BL] + [EXCZ+BL] - BL
=
Baseline+Excess due to Z
OR-+ + OR+- – 1.0
*For a more formal derivation, see Szklo & Nieto, pp. 229-230 (not required).
OR
Observed OR++
3.5
2.0
3.5
2.5
1.0
OR--
OR-+
OR+-
Exp’d
OR++
Conclude:
If the observed joint OR is the same as the expected under the
additive model, there is no additive interaction
Observed OR++
6.0
OR
3.5
2.0
2.5
Excess due to the
joint effects of A
and Z
1.0
OR--
OR-+
OR+-
Excess due to
interaction
(“interaction term”)
Exp’d
OR++
Conclude:
If the observed joint OR is different than the expected under the
additive model, there is additive interaction
Second Strategy: Comparison between joint expected and joint observed effects - allows assessment of both ADDITIVE and MULTIPLICATIVE interactions-Odds Ratios for the association among isolated clubfoot, maternal
smoking, and a family history of clubfoot, Atlanta, Georgia, 1968-80
Family history
of clubfoot
Yes
No
Cases
Controls
Stratified
ORs
ORs using No/No
as the reference
category
Yes
14
7
3.64
20.30
No
11
20
Yes
118
859
No
203
2,143
Maternal
smoking
5.81
1.45
Expected under the
ADDITIVE model
6.26
1.45 + 5.81 – 1.0=
1.45
1.0 (reference)
Honein et al. Family history, maternal smoking, and clubfoot: an indication of geneenvironment interaction. Am J Epidemiol 2000;152:658-65.
Conclude: Since the observed joint
OR(20.3) is different from the joint OR
expected under the additive model (6.26),
there is additive interaction
Effect of Maternal Smoking only, i.e., in the
absence of Family History
Case-control study
Second strategy to assess interaction:
comparison of joint observed and expected effects
Layout of table to assess both ADDITIVE
and MULTIPLICATIVE interaction
Factor Factor
Z
A
Cases Controls
No
No
Yes
Yes
No
Yes
Under ADDITIVE MODEL:
Under MULTIPLICATIVE MODEL:
ORvs-1.0
OR-+
OR+OR++
What does it
mean?
Reference
Indep effect of A
Indep effect of Z
Joint effect
Exp’d OR++ = OR+- + OR-+ - 1.0
Exp’d OR++ = OR+-  OR-+
Odds Ratios for the association among isolated clubfoot, maternal
smoking, and a family history of clubfoot, Atlanta, Georgia, 1968-80
Family history
of clubfoot
Yes
No
Cases
Controls
Stratified
ORs
ORs using No/No
as the reference
category
Yes
14
7
3.64
20.30
8.42
No
11
20
5.81
1.45  5.81 =
Yes
118
859
No
203
2,143
Maternal
smoking
1.45
Expected under the
MULT. model
1.45
1.0 (reference)
Honein et al. Family history, maternal smoking, and clubfoot: an indication of geneenvironment interaction. Am J Epidemiol 2000;152:658-65.
Conclude: Since the observed joint
OR(20.3) is different from the joint OR
expected under the multiplicative
model(8.42), there is multiplicative
interaction
Effect of Maternal Smoking only, i.e., in the
absence of Family History
Back to the terms...
• Synergism or Synergy: The observed joint “effect”
is greater than that expected from the individual
“effects”.
Which is equivalent to saying that the “effect” of A in
the presence of Z is stronger than the “effect” of A
when Z is absent.
• Antagonism: The observed joint “effect” is smaller
than that expected from the individual “effects”.
Which is equivalent to saying that the “effect” of A in
the presence of Z is weaker than the “effect” of A
when Z is absent
Note: the expressions “synergism/antagonism” and “effect modification”
should ideally be reserved for situations in which one is sure of a causal
connection. In the absence of evidence supporting causality, it is
preferable to use terms such as “heterogeneity” or “positive/negative
interaction”.
Terminology
• Positive interaction = Synergism = “More than
additive effect” (for the additive model) or “More
than multiplicative effect” (for the multiplicative
model)
• Negative interaction = Antagonism = “Less than
additive/multiplicative effect”
Some investigators reserve the term “synergy” to
define a biologically plausible interaction
Further issues for discussion
• Quantitative vs. qualitative interaction
Odds Ratios for the association among isolated clubfoot, maternal
smoking, and a family history of clubfoot, Atlanta, Georgia, 1968-80
Family history
of clubfoot
Yes
No
Maternal
smoking
Cases
Controls
Stratified
ORmaternal smk
Yes
14
7
3.64
No
11
20
Yes
118
859
No
203
2,143
1.45
Honein et al. Family history, maternal smoking, and clubfoot: an indication of geneenvironment interaction. Am J Epidemiol 2000;152:658-65.
Quantitative
interaction:
both ORs are
in the same
direction(>1.0),
but they are
heterogeneous
Am J Epidemiol 1995;142:1322-9
Reproductive Health Study, retrospective study of 1,430 non-contraceptive parous
women, Fishkill, NY, Burlington, VT, 1989-90.
Smoking
Caffeine
No. pregnancies
No
No
301 mg/d
No
301 mg/d
575
90
76
83
Yes
Delayed
conception
>12 months
47
17
15
11
Stratified
ORA
95% CI
2.62
1.36-4.98
0.62
0.27-1.45
Odds ratios are not only different; they have different directions (>1, and
<1). Smoking modifies the effect of caffeine on delayed conception in a
qualitative manner, i.e., there is qualitative interaction.
When there is qualitative interaction in
one scale (additive or multiplicative),
it must also be present in the other
Z+
ZA-
ARA
Z+ Positive (>0)
Z- Negative (<0)
RRA
>1
<1
A+
Qualitative Interaction:
Effect Modifier
Risk Factor
Incidence/1000
ARA
RRA
Z+
A+
10.0
+5/1000
2.0
A-
5.0
Reference
1.0
A+
3.0
-3/1000
0.5
A-
6.0
Reference
1.0
Z-
Interaction in both scales
When there is qualitative interaction in
one scale (additive or multiplicative), it
must also be present in the other
Z+
ZA-
ARA
Z+ Positive (>0)
Z- Negative (<0)
RRA
>1
<1
A+
Qualitative Interaction:
Effect Modifier
Risk Factor
Incidence/1000
ARA
RRA
Z+
A+
10.0
+5/1000
2.0
A-
5.0
Reference
1.0
A+
3.0
-3/1000
0.5
A-
6.0
Reference
1.0
Z-
When there is qualitative interaction in
one scale (additive or multiplicative), it
must also be present in the other
“cross-over”
Z+
ZA-
ARA
Z+ Positive (>0)
Z- Negative (<0)
RRA
>1
<1
A+
Another type of qualitative interaction: “effect”of A is flat in one stratum
of the effect modifier; in the other stratum, an association is observed
Z+
ZA-
A+
ARA
Z+ Positive (>0)
Null (=0)
Z-
RRA
>1
=1
Example of qualitative interaction
(CHD)
CHD-free cumulative probabilities
Circulation 2000;101:2034-9
Low
Anger score:
Low (10-14)
Moderate (15-21
High (22-40)
Days of follow-up
Days of follow-up
CHD event-free survival probabilities
among normotensive individuals by trait
anger scores
CHD event-free survival probabilities
among hypertensive individuals by trait
anger scores
Normotensive persons
Anger score
Age-adjusted HR of CHD:
Hypertensive persons
Anger score
Low
Moderate
High
Low
Moderate
High
1.00
1.36
2.97
1.00
0.88
1.05
Further issues for discussion
• Quantitative vs. qualitative interaction
• Reciprocity of interaction
If Z modifies the effect of A on disease Y, then Z will
necessarily modify the effect of Z on disease Y
Reciprocity of interaction
The decision as to which is the “principal” variable and which is the
effect modifier is arbitrary, because if A modifies the effect of Z, then
Z modifies the effect of A.
Factor Z Factor A Incidence (%)
No
No
10.0
Yes
20.0
Yes
No
25.0
Yes
125.0
Stratified
RRA
2.0
5.0
RRvs(--)
Reference
2.0
2.5
12.5
Z modifies the effect of A
Factor A Factor Z Incidence (%)
No
No
10.0
Yes
25.0
Yes
No
20.0
Yes
125.0
Stratified
RRZ
2.5
6.25
RRvs(--)
Reference
2.5
2.0
12.5
A modifies the effect of Z
INTERACTION IS NOT CONFOUNDING
Matched case-control study (matching by gender) of the
relationship of risk factor X (e.g., alcohol drinking ) and
disease Y (e.g., esophageal cancer)
Pair No.
Case
Control
1 (male)
+
-
2 (male)
+
-
3 (male)
-
+
4 (male)
+
-
5 (male)
+
+
6 (female)
-
-
7 (female)
+
-
8 (female)
-
+
9 (female)
+
+
10 (female)
-
-
Total (Pooled) Odds Ratio
OR by
gender
INTERACTION IS NOT CONFOUNDING
Matched case-control study (matching by gender) of the
relationship of risk factor X (e.g., alcohol drinking ) and
disease Y (e.g., esophageal cancer)
Pair No.
Case
Control
1 (male)
+
-
2 (male)
+
-
3 (male)
-
+
4 (male)
+
-
5 (male)
+
+
6 (female)
-
-
7 (female)
+
-
8 (female)
-
+
9 (female)
+
+
10 (female)
-
-
Total (Pooled) Odds Ratio
OR by
gender
4/2= 2.0
INTERACTION IS NOT CONFOUNDING
Matched case-control study (matching by gender) of the
relationship of risk factor X (e.g., alcohol drinking ) and
disease Y (e.g., esophageal cancer)
Pair No.
Case
Control
1 (male)
+
-
2 (male)
+
-
3 (male)
-
+
4 (male)
+
-
5 (male)
+
+
6 (female)
-
-
7 (female)
+
-
8 (female)
-
+
9 (female)
+
+
10 (female)
-
-
Total (Pooled) Odds Ratio
OR by sex
3/1 = 3.0
1/1= 1.0
4/2= 2.0
Further issues for discussion
• Quantitative vs. qualitative interaction
• Reciprocity of interaction
• Interpretation and uses of interaction
– Additive interaction as “public health
interaction” (term coined by Rothman)
Joint effects of current cigarette smoking and low consumption of vitamin C (≤ 100
mg/day) with regard to adenocarcinoma of the salivary gland, San Francisco-Monterey
Bay area, California, 1989-1993
Current Smoking
Status
Low Vitamin C intake
(mg/day)
Odds Ratio
No
No
1.0
Yes
No
6.8
No
Yes
1.8
Yes
Yes
10.6
(Horn-Ross et al. Diet and risk of salivary gland cancer. Am J Epidemiol
1997;146:171-6)
Additive Model:
Expected joint Odds Ratio = 6.8 + 1.8 – 1.0= 7.6
Multiplicative Model:
Expected joint Odds Ratio = 6.8  1.8 = 12.4
Conclude: For Public Health purposes, ignore negative multiplicative interaction, and focus on
smokers for prevention of low vitamin C intake
Additive interaction as “Public Health interaction”
Incidence of disease “Y” by smoking and family history of “Y”
Incidence
per 100
5.0
10.0
20.0
Family
history (EM)
No
30.0
EM- effect modifier
RF- risk factor of interest
Yes
Smoking
(RF)
No
Yes
No
Stratified
ARSmk%
Stratified
RRSmk
5.0
2.0
Yes
10.0
1.5
Positive additive interaction
(synergism), but negative
multiplicative interaction
(antagonism)
Thus, if there are enough subjects who are positive for both variables and if
resources are limited, smokers with a positive family history should be regarded as
the main “target” for prevention  examine the prevalence of (Fam HIst+ and
Smk+ ) and estimate the attributable risk in the population
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction
Am J Epidemiol 1995;142:1322-9
Reproductive Health Study, retrospective study of 1,430 non-contraceptive parous
women, Fishkill, NY, Burlington, VT, 1989-90.
Smoking
Caffeine
No. pregnancies
No
No
301 mg/d
No
301 mg/d
575
90
76
83
Yes
Delayed
conception
>12 months
47
17
15
11
Stratified
ORA
95% CI
2.62
1.36-4.98
0.62
0.27-1.45
“…An interaction between caffeine and smoking is also biologically plausible. Several
studies have shown that cigarette smoking significantly increases the rate of caffeine
metabolism […]. The accelerated caffeine clearance in smokers may explain why we
failed to observe an effect of high caffeine consumption on fecundability among women
who smoked cigarettes.”
This interaction can be properly named, “synergy”, as it has a
strong biological plausibility
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction
– Statistical interaction (not causal)
• Differential confounding
Circulation 2000;101:2034-9
Normotensives
Anger score
Age-adjusted RR of CHD:
Hypertensives
Anger score
Low
Moderate
High
Low
Moderate
High
1.00
1.36
2.97
1.00
0.88
1.05
Another explanation: hypertensive subjects with high anger level may have a
lower level of confounder X2 than those who are at lower levels (negative
confounding). However, in normotensive individuals, the distributions of X2
are the same for low, moderate and high anger scores
Example of confounding resulting in apparent interaction
• No association between the exposure (e.g., chewing gum) and the disease (e.g.,
liver cancer)
• Unaccounted-for confounder (e.g., a genetic polymorphism G)
• Incidence of the disease by G:
G+ = 0.04
G- = 0.02
Prevalence
of G
Incidence
Relative
Risk
Exposed
0.8
[(0.8  0.04 ) + (0.2  0.02)]  100=
3.6%
1.6
Unexposed
0.1
[(0.10  0.04) + (0.90  0.02)] 
100 = 2.2%
1.0
Exposed
0.20
[(0.20  0.04) + (0.80  0.02)] 
100= 2.4%
1.0
Unexposed
0.20
[(0.20  0.04) + (0.80  0.02)] 
100= 2.4%
1.0
Men
Women
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction 
– Statistical interaction (not causal)
• Differential confounding across strata of the effect
modifier 
• Misclassification resulting from different sensitivity
and specificity values of the variable under study
across strata of the effect modifier
Example of effect of misclassification of overweight by
smoking category, on the Odds Ratios
Smoking Status
Smokers
Non-smokers
BMI status
Cases
Controls
Odds
Ratio
Overweight
200
100
2.25
Not overweight
800
900
Overweight
200
100
Not overweight
800
900
2.25
Smokers:
Values of indices
of validity different
between smokers
and non-smokers
Cases
Controls
Sensitivity
0.80
0.80
Specificity
0.85
0.85
Cases
Controls
Sensitivity
0.95
0.95
Specificity
0.98
0.98
Non-smokers:
Smokers
Non-differential
misclassification
within each
stratum
Non-Smokers
Overweight
Cases
Controls
ORMISCL
Overweight
Cases
Controls
ORMISCL
Yes
280
215
1.4
Yes
206
113
2.0
No
720
785
No
794
887
Cases
Controls
Odds RatioTRUE
Overweight
200
100
2.25
Not overweight
800
900
Overweight
200
100
Not overweight
800
900
Smoking Status
Smokers
Non-smokers
BMI status
2.25
Further issues for discussion
• Quantitative vs. qualitative interaction
• Reciprocity of interaction
• Interpretation and uses of interaction
– Additive interaction as “public health interaction”
– Biological interaction
– Statistical interaction (not causal)
• Differential confounding across strata of the effect
modifier
• Differential misclassification across strata of the effect
modifier
• The dose (amount of exposure) may be higher in one
stratum than in the other
Oral cancer odds ratios* related to consumption of diluted and undiluted
forms of liquor by liquor drinkers, Puerto Rico, 1992-1995
Usually drank liquor with
nonalcoholic mixers (n= 163)
Usually drank liquor straight
(undiluted) (n= 206)
Odds Ratio (95% CI)
Odds Ratio (95% CI)
>0 - <8
1.0 (reference)
3.2 (1.4, 7.2)
8 - <22
1.0 (0.3, 3.0)
4.2 (1.7, 10.5)
22 - <43
3.6 (1.2, 10.8)
7.9 (3.0, 21.3)
43 - <64
6.2 (1.2, 31.1)
8.3 (2.3, 29.4)
64 - <137
1.1 (0.2, 5.4)
23.5 (6.8, 81.5)
Drinks/week
*Adjusted for age, tobacco use, consumption of raw fruits and vegetables, and
educational level
Exposure duration and interaction
Gender
Smoking
Relative Risk
Man
Yes
3.0
No
1.0
Yes
1.5
No
1.0
Woman
When studying effects of smoking in men and women, the category “smoker”
is related to more cigarettes/day in men than in women. Thus, the observed
odds ratios may be heterogeneous because of different levels of smoking
exposure between men and women, and not because men are more
susceptible to smoking-induced disease.
Are you surprised??
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction
– Statistical interaction (not causal)
• Differential confounding across strata of the effect modifier 
• Differential misclassification across strata of the effect modifier 
• The dose (amount of exposure) may be higher in one stratum than in
the other 
• Biologic interaction:
– Consistent with pathophysiologic mechanisms (biologic
plausibility)
– Confirmed by animal studies
– What is best model from the biologic viewpoint?
No one knows for sure… Think about the specific
condition under study – Examples: trauma, cancer
Problem: Epidemiology usually assesses proximal cause X1  X2  X3  Y
70
Myocardial
infarct
Cerebral
infarct
Gangrene of
extremities
Abdominal
aortic
aneurism
60
Age in years
50
clinical horizon
40
Calcification
Complicated lesion- hemorrhage, ulceration,
thrombosis
30
Fibrous plaque
20
Intimal-medial thickening
Endothelial dysfunction (no anatomical
expression)
Fatty streaks
10
Normal artery
Schematic view of the development of atherosclerosis.
Based on McGill et al, in Atherosclerosis and Its Origins,
M Sandler & G Bourne, eds, p. 42, © 1963, Academic
Press
Natural History of of Disease
X1
XZ
X2 X3 X3
Y1  Y2  Y3 --> ….YZ-1  YZ
Proximal relationship:
Usual realm of epidemiologic studies
LDL
Smoking
ARIC
Normal
Fatty
streaks
Endothelial
dysfunction
Intima-media
thicknening
Sharrett AR, et al. Atherosclerosis 2004;172:143-149
Fibrous plaque: Clinical
•Stable
events
•Unstable
(CHD,
LEAD,
etc)
hemorrhage,
thrombosis
Natural History of of Disease
EM or RF 2
EM or RF 1
X1
XZ
X2 X3 X3
Y1  Y2  Y3 --> ….YZ-1  YZ
Proximal relationship:
Usual realm of epidemiologic studies
Atherosclerosis
High blood pressure: endothelial injury/trauma: additive model?
First event
Multiplication (and
migration) of medial
smooth muscle cells 
multiplicative?
Second event
Blood flow
intima
media
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction
– Statistical interaction (not causal)
• Differential confounding across strata of the effect modifier 
• Differential misclassification across strata of the effect modifier 
• The dose (amount of exposure) may be higher in one stratum than in
the other 
• Biologic interaction
• Matching and interaction
Matching and interaction
• In a matched case-control study, the interaction
between the exposure of interest and the
matching variable…
– Can be assessed under the multiplicative model, using
the effect modification strategy (i.e., looking at the
heterogeneity of the OR’s stratified according to the
matching variable)
– Cannot be assessed under the additive model,
because the expected joint OR is undefined:
Exp’d OR++ = OR+- + OR-+ - 1.0
Set to be 1.0, by definition
INTERACTION IS NOT CONFOUNDING
Matched case-control study (matching by gender) of the
relationship of risk factor X (e.g., alcohol drinking ) and
disease Y (e.g., esophageal cancer)
Pair No.
Case
Control
1 (male)
+
-
2 (male)
+
-
3 (male)
-
+
4 (male)
+
-
5 (male)
+
+
6 (female)
-
-
7 (female)
+
-
8 (female)
-
+
9 (female)
+
+
10 (female)
-
-
Total (Pooled) Odds Ratio
OR by sex
3/1 = 3.0
1/1= 1.0
4/2= 2.0
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction
– Statistical interaction (not causal)
• Differential confounding across strata of the effect modifier 
• Differential misclassification across strata of the effect modifier 
• The dose (amount of exposure) may be higher in one stratum than in
the other 
• Biologic interaction
• Matching and interaction
• Interaction and selection bias
Selection Bias: Effects of interaction on the pooled relative risk
(“main effect”) of outcome Y (e.g., atherosclerotic events)
associated with X (e.g., smoking) by prevalence of Z (e.g.,
infection)
Example 1
• RRX/Z+=3.0; RRX/Z-=3.0
• Thus, no interaction between Z and X
• Prevalence of Z+= 20%
Z+
RRX= 3.0
ZRRX= 3.0
Pooled (total reference population) RRX= 3.0
Example 1
• RRX/Z+=3.0; RRX/Z-=3.0
• Thus, no interaction between Z and X
• Prevalence of Z+= 20%
Not included or censored
Included and not censored
Z+
RRX= 3.0
ZRRX= 3.0
Relative Risk in those not lost to follow-up= 3.0 
representative of RR of the total reference population
Example 2
• RRX/Z+=3.0; RRX/Z-=1.0
• Thus, interaction between Z and X
• Prevalence of Z+= 20%
ZRRX= 1.0
Z+
RRX= 3.0
Pooled (total population) RRX= 1.4
Example 2
• RRX/Z+=3.0; RRX/Z-=1.0
• Thus, interaction between Z and X
• Prevalence of Z+= 20%
Not included or censored
Included and not censored
Z+
RRX= 3.0
ZRRX= 1.0
Relative Risk in those not lost to follow-up= 1.0  NOT
representative of RR of the total reference population
Effects of interaction on the pooled relative risk (“main effect”) of
outcome Y associated with X, by prevalence of the effect modifier Z
(RRX/Z+=3.0; RRX/Z-=1.0)
Z+ = 100%
Pooled RRX = 3.0
Z+ =0%
Pooled RRX = 1.0
Z+ = 50%
Z+ = 50%
Pooled RRX = 2.0
Pooled RRX = 1.2
Further issues for discussion
• Quantitative vs. qualitative interaction 
• Reciprocity of interaction 
• Interpretation and uses of interaction
– Additive interaction as “public health interaction” 
– Biological interaction
– Statistical interaction (not causal)
• Differential confounding across strata of the effect modifier 
• Differential misclassification across strata of the effect modifier 
• The dose (amount of exposure) may be higher in one stratum than in
the other 
• Biologic interaction
• Matching and interaction
• Interaction and selection bias
• Interaction and adjustment
Breast Cancer Incidence Rates
Interaction between age and ethnic background
“cross-over”
WW
BW
40
Age (years)
Adjustment and Interaction
Age
<50
A (e.g.,
B (e.g.,
exposed) unexposed)
N
Rate
N
Rate ARexp RR
(%)
(%)
100
20
200
10 10% 2.00
50+
200
50
100
40
10%
1.25
• Note that ARs are the same, but
RR’s are different
Multiplicative interaction
When Relative Risks are heterogeneous, the adjusted RR varies
according to the composition of the effect modifier in the standard
population
Age
Age
A
N
B
<50
100
Rate
(%)
20
50+
200
50
N
200
100
Rate ARexp RR
(%)
10 10% 2.00
40
10%
1.25
Standard Populations
Population B
Arbitrary
<50
200
1800
Minimum
variance
66.7
50+
100
200
66.7
Adj.
Rate
ARexp
RR
A
B
A
B
30%
20%
23%
A
B
13% 35% 25%
10%
10%
10%
1.5
1.8
1.4
• For younger standard populations (e.g., arbitrary), the “adjusted” RR will approximate
the rate seen in those who are <50 years old
• For older standard populations (e.g., minimum variance), the adjusted RR will
approximate the AR seen in those who are 50+ years old
• Because there is no interaction in the additive (AR) scale, the composition of the
standard population is irrelevant, and the adjusted ARs are always the same
regardless of the standard
Adjustment and Interaction
Age
A
N
B
<50
100
Rate
(%)
6
50+
200
30
N
200
100
Rate ARexp RR
(%)
3 3%
2.0
16
15%
2.0
• Note that RRs are the same, but
ARexp’s are different
Additive interaction
When Attributable Risks in the exposed are heterogeneous, the
adjusted AR varies according to the composition of the effect
modifier in the standard population
Age
Age
A
N
Population B
Arbitrary
<50
200
1800
Minimum
variance
66.7
50+
100
200
66.7
B
<50
100
Rate
(%)
6
50+
200
30
N
200
100
Rate ARexp RR
(%)
3 3%
2.0
16
15%
Standard Populations
Adj.
Rate
2.0
A
B
14%
7%
A
B
A
B
8.4% 4.2% 18%
9%
ARexp
7%
4.2%
9.0%
RR
2.0
2.0
2.0
• For younger standard populations (e.g., arbitrary), the “adjusted” AR will approximate
the rate seen in those who are <50 years old
• For older standard populations (e.g., minimum variance), the adjusted AR will
approximate the AR seen in those who are 50+ years old
• Because there is no interaction in the multiplicative scale, the composition of the
standard population is irrelevant, and the adjusted RRs are always the same,
regardless of the standard
Conclusion
• If heterogeneity is present… is there
interaction?
– What is the magnitude of the difference? (p-value?)
– Is it qualitative or just quantitative?
– Is it biologically plausible?
• If we conclude that there is interaction, what
should we do?
– Report the stratified measures of association …
The interaction may be the most important finding
of the study!