Interaction

Assoc. Prof. Pratap Singhasivanon
Faculty of Tropical Medicine, Mahidol University
Interactions
The definitions
• a situation where the risk or rate of disease in the presence
of 2 or more risk factors differs from the rate expected to
result from their individual effects
• rate can be greater than expected
- positive interaction or synergism
• rate can be less than expected
- negative interaction or antagonism
• an interaction (or effect modification) is formed when a third
variable modifies the relationship between an exposure and
outcome
Interaction
When the incidence rate of disease in
the presence of two or more risk
factors differs from the incidence rate
expected to result from their individual
effects
Interaction
The effect can be greater than what
we would expect (positive interaction)
or less than we would expect
(negative interaction)
Interaction (Effect Modification)
• Represents the phenomenon where the risk
associated with the presence of two risk
factors exceeds the risk we expect from the
combination of the component risk
X
R1
Y
R2
X and Y
>
R1 and R2
Interaction (Miettinen
1974)
SAMPLE BASED
(Statistical Interaction)
POPULATION BASED
(Effect Modification)
(Biological Interaction)
Statistical Interaction
• Model Dependent
• Depends on deviation from statistical model
(not biologic)
Additive Model
Multiplicative Model
Y
Absent
Present
Absent
R00
R01
Present
R10
R11
X


R10  Pr D X  Y  
R 01  Pr D X  Y  
R11  Pr D X  Y 

R10  Pr D X  Y 

RR11 = R11 / R00
RR01 = R01 / R00
RR10 = R10 / R00
Probability of disease in the presence of factors X and Y
Probability of disease in the presence of factors X only
Probability of disease in the presence of factors Y only
Probability of disease in the absence of both X and Y
Background RISK
Additive Model
1. In term of excess over “ONE”
( RR11  1)  ( RR101)  ( RR011)
Stage of “No interaction” on additive scale
2. HOGANS
T  R11  R10  R 01  R 00  0
Multiplicative Model
RR11  RR10  RR01
Stage of “No interaction” on
Multiplicative model
Example :
asbestos
+
smoke
-
+
50
10
-
5
1
Additive Model :
(50-1)  (10-1) + (5-1)
Presence of “Interaction” on
Additive model
ID/1000 PY
RR11 = 50/1 (smoking + asbestos)
RR10 = 10/1 (smoking alone)
RR10 = 5/1 (asbestos alone)
Multiplicative Model :
(50) = (10) * (5)
Presence of “Interaction” on
Multiplicative model
Example :
D
D
XY
X Y
X Y
XY
40
20
20
10
60
80
80
90
100
100
100
100
= .4
RR11 = .4/.1
=4
R10 = R01 = 20/100 = .2
RR10 = .2/.1
=2
R00 = 10/100
RR01 = .2/.1
=2
R11 = 40/100
= .1
Multiplicative Model :
RR11 = RR10 * RR01
4
=
2 * 2
No interaction on
Multiplicative Model
Additive Model :
(RR11-1)  (RR10-1)+(RR01-1)
3

1 + 1
T = R11-R10-R01+R00 = 0
= .40 - .20 - .20 + .10 = .10
There is
evidence of
interaction on
Additive Model
NOTES :
1. Neither model is right or wrong. They are
simply devices for modeling data and may
be more or less suitable for a particular
application.
2. Most statistical techniques are based on
multiplicative model.
The presence or absence of interaction pertains
to whether or not a particular effect measure
(RR, OR) varies in value over categories or strata
based on level of some factor(s).
Equivalent to an assessment regarding
interaction based on multiplicative model
Which of the 2 models we should use :
1. For addressing public health concerns
regarding disease frequency reduction,
deviation from additivity appears to be
most relevant
2. Contribution to the understanding of
disease etiology  multiplicative model
Blood Pressure
(Y)
smokers
Additive Model
(No interaction)
Non-smokers
Only change in intercepts
no change in slope
irrespective of the value
of Xi which is being held
constant
Age (X)
Height
(Y)
Urban
Interactive Model
Rural
Age (X)
There is change in both
intercepts and slope as
the level of Xi which is
held constant and varied
χ TOTAL   Wg ln(OR)   67.404
2
2
χ 2ASSO.  ( Wg )θ̂ 2  (49.290)(1.0898) 2  58.54
θ̂ 
2
 Wg ln(OR)
 Wg
χ HOMO.  67.404  58.54  8.86
2
53.717

 1.0898
49.290
P < 0.005
Conclude that the non uniformity of the observed OR’s is
unlikely to have occurred by chance; thus there is some
evidence of interaction.
males
females
VAC VAC
VAC VAC
D
10
14
D
22
12
D
191
15
D
155
17
201
29
177
29
P̂
0.05
(P1)
.483
(P2 )
.124
(P1 )
.414
(P2 )

RD .483-0.05 = .433
.144-.124 = .290
Males
Females
0.433
0.290
(p1q1 n1  p1q 2 n 2 )
0.08847
0.08979
3. Wg  (1 Var(R̂D g )
113.03
111.37
224.4
4. Wg (R̂D g )
48.94
32.30
81.24
2
5. Wg (RDg )
21.19
9.37
30.56
1. R̂D
2. Var( R̂D g )
2
χ TOTAL
 30.56
χ 2ASS  224.4(.362 ) 2
Total
(θ̂)R̂D  81.24 224.4  .362
 29.41
χ 2HOMO  30.56  29.41  1.15
Relative risk of oral cancer according to presence
or absence or two exposures :
smoking and alcohol consumption
smoking
No
Yes
No
1.00
1.53
Yes
1.23
5.71
alcohol
Relative risk of liver cancer for persons exposed to
Aflatoxin and/or Chronic Hepatitis B infection :
An example of interaction
Aflatoxin
Negative
Negative
Positive
1.00
3.4
7.3
59.4
HBs Ag
Positive
Deaths from lung cancer (per 100,000) among
individuals with and without exposure to
cigarette smoking and asbestos
Cigarette smoking
No
Yes
Asbestos Exposure
No
Yes
11.3
58.4
122.6
601.6
Age-Adjusted Odds Ratios Estimated from Logistic Models
with and without an Interaction between SMOKING and
ORAL CONTRACEPTIVE USE
No interaction Model
Interaction Model
OC Use
OC use
Cig/day
No
Yes
No
Yes
None
1.0
3.3 (2.0, 5.5)
1.0
3.6 (1.2, 11.1)
1 - 24
3.1 (2.0, 4.6)
10.1 (5.2, 19.5)
3.3 (2.2, 5.1)
3.7 (1.04, 13.0)
 25
8.5 (5.6, 12.8)
27.8 (14.4, 53.5)
8.0 (5.2, 12.4)
40.4 (19.4, 84.1)
Conceptual Framework of the definition of interaction
based on comparing expected and observed joint effects
A. When there is no interaction, the joint effect of risk
factors A and Z equals the sum of their independent
effects :
Z
A
A+Z
Expected
Observed
Conceptual Framework of the definition of interaction
based on comparing expected and observed joint effects
B. When there is positive interaction (synergism). The
observed joint effect of risk factors A and Z is greater
than that expected on the basis of summing the
independent effects of A and Z :
Z
A
A+Z

Expected
Observed
Excess due to positive interaction
Conceptual Framework of the definition of interaction
based on comparing expected and observed joint effects
C. When there is negative interaction (antagonism), the
observed joint effect of risk factors A and Z is smaller
than that expected on the basis of summing the
independent effects of A and Z :
Z
A
A+Z
*
Expected
Observed
*
“Deficit” due to negative interaction
Schematic representation of the meaning of the formula,
Expected ORA+Z+=Observed ORA+Z-+Observed ORA-Z+-1.0.
(5) OR = 7.0
(4) OR = 4.0
(3) OR = 3.0
(2) OR = 2.0
(1) OR = 1.0
BL
Baseline
I
A
A
A
Z
Z
Z
BL
BL
BL
BL
Baseline
+ excess
due to A
Baseline
+ excess
due to Z
Expected Joint
OR based on
adding
absolute
independent
excesses due
to A and Z*
Observed joint
OR> Expected
OR. Excess due to
I (Interaction) is
not explainable on
the basis of
excess due to
A and Z
* Note that when the independent relative odds for A and Z are added, the baseline is added twice;
thus, it is necessary to subtract 1.0 from the joint expected OR: that is, Expected ORA+Z+=(Excess
due to A + baseline) + (Excess due to Z + baseline) – baseline = ORA+Z- + ORA-Z+ - 1.0.
Factor B
_
+
Factor A
_
+
3.0
9.0
15.0
Incidence Rates
_
Factor
B
+
_Factor A+
3.0
9.0
15.0
21.0
Attributable Rates
_
Factor
B
+
Incidence Rates
_
Factor
B
+
_Factor A+
3.0
9.0
15.0
21.0
_Factor A+
0
6
12
Attributable Rates
_
Factor
B
+
_Factor A+
0
6
12
18
Evans County Study
 Prospective cohort study of CVD and
Cerebro vascular disease
 Logistic regression analysis of 10-year
mortality among the members of this cohort
(1904)
 9 variables were considered for inclusion of
the model
Example of Logistic Regression Analysis of a prospective
cohort study adapted from Evans County Study 1960-72
Variable
coefficient
Standard error
of coefficient
P value
40-69 years
0.08652
0.01153
< 0.001
Gender
0 = male ; 1 = female
1.49976
0.96723
0.121
Age X Gender
Males : 0 ; Females : 40-69
-0.04296
0.01699
0.011
Race
0 = white ; 1 = black
1.59382
0.96355
0.098
SBP
88-310 mmHg
0.01943
0.00208
< 0.001
Diabetes
0 = no or suspect ; 1 = yes
1.12325
0.26134
< 0.001
Cigarette Smoking
0 = never smoke ;
1 = present or past smoker
0.31739
0.15682
0.043
Cholesterol
94-546 mg/100 ml
0.00311
0.00152
0.041
Quelet index X 100
2.107 – 8.761
-1.06415
0.43158
0.014
Variable
Variable range
Age
SBP = systolic blood pressure
Quelet index = weight in pounds divided by the square of height in inches
Use of Logistic Regression Coefficient for calculation of the probability of
dying over a ten-year period for a hypothetical individual
Coefficient
(log Odds for 1 unit)
Value for hypothetical
individual
Product
Age
0.08652
50
4.32600
Gender
1.49976
0
0.00000
Age X Gender
-0.04296
0
0.00000
Race
1.59382
1
1.59382
SBP
0.01943
180
3.49740
Diabetes
1.12325
1
1.12325
Cigarette Smoking
0.31739
1
0.31739
Cholesterol
0.00311
350
1.08850
Quelet index X 100
-1.06415
3.2653
-3.47477
Variable
Intercept -6.37626
Maximum Likelyhood estimates of logistic parameters
(seven risk factors of coronary heart disease)
Variable
Parameter
Estimate (
ˆi
)
Standard Error
ˆi
of
X0 interce0pt
0
-13.2573
X1 age (yr)
1
.1216
.0473
X2 cholesterol (mg/dl)
2
.0070
.0025
X3 systolic BP (mm Hg)
3
.0068
.0060
X4 relative Weight
4
.0257
.0091
X5 hemoglobin (g%)
5
-.0010
.0098
X6 cigarettes
6
.4223
.1031
X7 ECG Abnormality
7
.7206
.4009
X4 = relative weight (100 x actual weight / median for sex-height group)
X6 = cigarettes per day (coded 0=never, 1=less than one pack, 2= one pack, 3= more than one pack)
X7 = ECG (coded  0=normal, 1=abnormal)
Prevalence of Down syndrome at Maternal Age
9
8
7
6
5
4
3
2
1
0
<20
20-24
25-29
30-34
Maternal Age
35-39
40+
Prevalence of Down syndrome at birth by birth order
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
Birth Order
4
5+
Hypothetical Examples of Unadjusted and Adjusted Relative Risks
According to Type of confounding (Positive or Negative)
Example No.
Type of Confounding
Unadjusted Relative
Risk
Adjusted Relative
Risk
1
Positive
3.5
1.0
2
Positive
3.5
2.1
3
Positive
0.3
0.7
4
Negative
1.0
3.2
5
Negative
1.5
3.2
6
Negative
0.8
0.2
7
Qualitative
2.0
0.7
8
Qualitative
0.6
1.8

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