Scale, Causal Pies
and
Interaction
1h
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/
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Agenda
• Concepts
– Scale
– Causal Pies
– Interaction and Effect Modification
• Methods
– Regression and Scale
– Regression and Interaction
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SCALE
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3
30
The importance of scale
20
Females
Multiplicative scale
Absolute increase
Relative increase
Females: 30-20=10
Males: 20-10=10
Females: 30/20=1.5
Males: 20/10=2.0
Conclusion:
Same increase for
males and females
Conclusion:
More increase for
males
y, RD
RR (OR, HRR)
0
10
Males
Additive scale
T1
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T2
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Obesity and death
RD: The effect of obesity on death increases with age!
RR: The effect of obesity on death decreases with age!
RR=1.5
D
ep
re
ss
ed
RR=2.0
Happy
clam
Thin
asasa anail
20
30
40
50
60
70
Age
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Lessons learned
• Scale is important
– Use both additive and multiplicative
– When reporting RR or RD or similar, always
report reference risk
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CAUSAL PIES
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Causal pies
• Sufficient cause:
Three causes for a disease
– 1 to 3 (AIDS)
• Component cause:
Sufficient
Cause 1
Sufficient
Cause 2
Sufficient
Cause 3
A
A
A
– A to F (A=HIV, B=sex, E=injection)
• Necessary cause:
– A (HIV)
C
B
D
B
F
E
• Interaction
– A and B (smoke+radoncancer)
• Induction time:
– time to accumulate A to C (accumulate mutations cancer)
• Attributable fraction (AF)
– Sum>100% (remove E:33%, remove B:66%, Remove A:100%)
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Pies and Risk of lung cancer
N=1000
Cases Risk
U
Smoke
Radon
10
70
20
1%
7%
Smoke
+
-
+
13 %
3%
Radon
8%
1%
2%
Observable?
Risk Difference (RD)
Radon Smoke
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3%
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INTERACTION,
EFFECT MODIFICATION
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Definitions of interaction
• Risk factors A and B
A
B
B A
• No additive interaction:
B1.
B A =0
2. RDAB=RDA+RDB
3. RDA is independent of B (and vice versa)
• The 3 definitions are identical
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Comparing definitions of no
additive
interaction
risks
U
U
Radon
+
Smoke
Smoke
S
+
-
RDsmoke
-
S+R+U
R+U
S
S+U
U
S
RDradon
R
R
S+R
RDsmoke independent of radon
Radon
Radon Smoke
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RDradon independent
of smoke
RDsmoke,radon =
RDsmoke+RDradon
R
What happens if radon-smoke interaction in not 0?
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So far so good!
12
Interaction and scale
1%
U
Smoke
7%
Radon
2%
Radon Smoke
0%
Smoke
RDsmoke
RRsmoke
+
-
+
10 %
3%
7%
3.33
Radon
8%
1%
RDradon
RRradon
2%
2%
1.25
3.00
RRradon dependent
of smoke
10.00
RRsmoke,radon 
RRsmoke*RRradon
9%
7%
8.00
RRsmoke dependent of radon
Lesson learned:
No additive interaction  multiplicative interaction
Interaction is scale dependent
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Interaction versus Effect Modification
Interaction
• Risk factors (Actions)
Effect Modification
• Variables (No actions)
– Smoking
– Asbestos
– Sex
– Age
• Two risk factors acting
together
• The effect of a risk factor
modified by a variable
smoking and asbestos may act
together to produce lung cancer
The effect of smoking on heart disease
is different for men and women
The two definitions are mathematically equivalent,
only the type of variable differs
Both concepts are scale dependent!
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REGRESSION AND
INTERACTION
AND SCALE
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Regression and scale
• Linear models (linear-regression, -risk, -survival): additive
– No interaction if:
RDAB=RDA+RDB
or
RDA is independent of B
• “Other” models (logistic, Poisson, log-risk, Cox): multiplicative
• No interaction if:
RRAB=RRA*RRB
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or
RRA is independent of B
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Estimating interaction in regression
Linear model
U
A
U
B
U
A
B
Observable?
y  b0  b1 A  b2 B  b3 AB
Effects
y A  b1  b3 B
is independent of B if b3=0
Test
b3  0
Interaction if b30
ConfidenceInterval (easy or technical)
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U
1%
Smoke
7%
Regression example
Smoke
2%
Radon
+
-
RDsmoke
Radon Smoke
Radon
+
13 %
8%
3%
1%
10 %
7%
3%
Linear risk model (all variables=0/1)
𝐿𝑢𝑛𝑔𝐶𝑎𝑛𝑐𝑒𝑟 = 𝑏0 + 𝑏1 𝑠𝑚𝑜𝑘𝑒 + 𝑏2 𝑟𝑎𝑑𝑜𝑛 + 𝑏3 𝑠𝑚𝑜𝑘𝑒 ∙ 𝑟𝑎𝑑𝑜𝑛
𝐿𝑢𝑛𝑔𝐶𝑎𝑛𝑐𝑒𝑟 = 0.01 + 0.07𝑠𝑚𝑜𝑘𝑒 + 0.02𝑟𝑎𝑑𝑜𝑛 + 0.03𝑠𝑚𝑜𝑘𝑒 ∙ 𝑟𝑎𝑑𝑜𝑛
RDsmoke  0.07  0.03radon 
RDradon  ?
0.07 if radon=0
0.10 if radon=1
Stata: margins, dydx(smoke) at(radon=(0 1))
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Stratify or use interaction term
Two models
radon=0 radon=1
smoke 7 %
10 %
Co 1
Co 2
Co 3
Co 4
const
Model with interaction
radon=0 radon=1
smoke 7 %
10 %
Co 1
Co 2
Co 3
Co 4
const
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• Alt 1 : Two models (stratify on radon)
– Easy
– No test for interaction
– Inefficient (12 estimates)
• Alt 2: Model with interaction
– Technical (ci)
– Test for interaction
– Efficient (7 estimates)
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Summing up 1
• Scale (additive or multiplicative) is important
• Causal Pies (SCC)
– Multifactorial, Additive
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Summing up 2
• Interaction/ effect modification
– Same concept (action*action / action*immutable)
– Scale dependent
• Regression
– Linear models are additive
– “All” other models are multiplicative
– In both: estimate interaction as product term
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