5.3.2 sufficient cause em

1. Outline
1. What does causal inference entail?
2. Using directed acyclic graphs
a. DAG basics
b. Identifying confounding
c. Understanding selection bias
3. Causal perspective on effect modification
a. Brief recap of effect modification (EM)
b. Linking EM in our studies to reality
c. Types of interaction
d. Causal interaction / EM
1. Sufficient cause model (“causalpies”)
2. Potential outcomes model (“causal types”)
e. Choosing which measure of interaction to estimate and report
4. Integrating causal concepts into your research

2. Potential Outcomes Model & EM
(i.e. “counterfactual framework”)
• Reference to what would have happened if, contrary to fact,
the exposure had been something other than what it actually
was
• Ya: individual outcome when exposure =a
• E[Ya]: expectation of the outcome in a population in which all
people experienced exposure a
• E[Y1-‐Y0]:population causal effect
– Expected difference in the outcome in the population if all
experienced exposure 1 vs. all experienced exposure 0
with everything else the same

3. Potential Outcomes Model & EM
• Recall the counterfactual “types” of individuals regarding the
effect of exposure on outcome
• Counterfactual outcomes: 1 = disease, 0 = no disease
Outcome by exposure Proportion of types
Type Exposed Unexposed Description in population
1 1 1 Doomed p1
2 1 0 Causal p2
3 0 1 Preventive p3
4 0 0 Immune p4

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Recap: Causal perspective on
measures of association
• AR isolates incidence in which the exposure was a
cause
• RR does not and the magnitude depends on the
incidence due to causes without the exposure
• In conclusion, magnitude or strength of relative
measures depend on causes of disease other than
the exposure of interest, while absolute measures
do not

5. 24
Potential Outcomes Model & EM
• Counterfactual “types” can beexpanded
to involve two exposures (X and Z)
• Each row represents the counterfactual
outcomes that would be observed for a
“type” of person given each potential
combination of exposure to two factors,
X and Z
• We imagine that our study population
can be categorized into these types but
we cannot know the proportion of each
type because counterfactuals are
unobserved
• Counterfactual outcomes – 1=disease,
0=no disease

6. 25
Potential Outcomes Model & EM
• Types with no causal interaction
• At least one factor never has an
effect, so there can be no
interaction
• Examples:
– People of type 1: get the disease
regardless of values of X and Z
(doomed)
– People of type 6: disease is caused by
exposure to X=1, Z has no effect
– What are type 4 people?

7. 26
Potential Outcomes Model & EM
• Types with causal interaction
• Don’t know what the effect of X
will be without knowing the value
of Z (and vice versa)
• Examples:
– People of type 8: get the disease only
when X and Z are present
– People of type 5: X=0 with Z=1
blocks disease

8. • Take home point: When EM is present on the additive
scale, causal effect modification is implied (if you can make
a set of other assumptions)
• To show this, we will:
1. Get the AR for Z (AR01) assuming no causal types
2. Get the AR for X (AR10) assuming no causal types
3. Get the AR for X and Z (AR11) assuming no causal types
4. Show that AR11 = AR01+ AR10 when there are no causal types in the
population (i.e. the AR is homogeneous)
5. By this logic, we can infer that if we observe AR11 ≠ AR01+ AR10,
there are at least some causal types in our population (i.e. the AR is
not homogeneous)
What we are about to do…

9. 28
Potential Outcomes Model & EM
• So far we have discussed individual
counterfactual risks
• Now, let’s consider the average risk of Y in
a study population
– Causal risks of outcome for combinations of
exposures X and Z
– Notation: RXZ
• Sum types that have value 1 down each
column:
– R11 = p1+p2+p3+p4+p5+p6+p7+p8
– R01 = p1+p2+p3+p4+p9+p10+p11+p12
– R10 = p1+p2+p5+p6+p9+p10+p13+p14
– R00 = p1+p3+p5+p7+p9+p11+p13+p15

10. 29
Potential Outcomes Model & EM
• What would we expect if there were no
interaction?
– R11 = p1+p4+p6
– R01 = p1+p4+p11
– R10 = p1+p6+p13
– R00 = p1+p11+p13
• AR01 = R01 – R00
• AR01 = (p1+p4+p11) – (p1+p11+p13)
• AR01 = p4-‐p13
• How would you describe this in words?

11. 30
Potential Outcomes Model & EM
• What would we expect if there were no
interaction?
– R11 = p1+p4+p6
– R01 = p1+p4+p11
– R10 = p1+p6+p13
– R00 = p1+p11+p13
• AR10 = R10 – R00
• AR10 = (p1+p6+p13) – (p1+p11+p13)
• AR10 = p6-‐p11
• How would you describe this in words?

12. 31
Potential Outcomes Model & EM
• What would we expect if there were no
interaction?
– R11 = p1+p4+p6
– R01 = p1+p4+p11
– R10 = p1+p6+p13
– R00 = p1+p11+p13
• AR11 = R11 – R00
• AR11 = (p1+p4+p6) – (p1+p11+p13)
• AR11 = (p4+p6) – (p11+p13)
• How would you describe this in words, and
with respect to AR01 and AR10?

13. 32
Potential Outcomes Model & EM
• We just showed that if there are no interacting causal
‘types’
AR11 = AR01+ AR10
• What do we conclude if we observe AR11 ≠ AR01+ AR10?
– There are at least some interacting types
– (If we can assume no confounding in estimation of the AR)
• What do we conclude if we observe AR11 = AR01+ AR10?
– Either there are no interacting types, or they are canceling each
other out (example – a type 3 and a type 14 would cancel)

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Potential Outcomes Model & EM
• When AR is not homogeneous indicates causal interaction
• When AR is homogeneous, RR is not homogeneous
– When R11 – R01 = R10 – R00 we know that R11/R01 > or < R10/R00
• Cannot make statements about interaction as a causal
concept based on multiplicative interaction
– Effect modification only (measure of association modification)
• What about when you observe no interaction using AR as
the measure of association?
– It only indicates that there was no net effect of interacting causal
types

15. 34
Potential Outcomes Model & EM
• If you see possible causal interaction using observational
data:
• 1) There are at least some people in your population of an
interactive “type” with respect to your X, Z and Y
– Depicted in a causal pie containing both X and Z, or
– Depicted in the potential outcomes framework
• 2) Population experienced exposures such that you could
detect this interaction

16. • Take home point: when EM is present on the additive
scale or absent on the relative scale, causal effect
modification is implied (if you can make a set of other
assumptions).
• To show this, we just:
1. Got the AR for Z (AR01) assuming no causal types
2. Got the AR for X (AR10) assuming no causal types
3. Got the AR for X and Z (AR11) assuming no causal types
4. Showed that AR11 = AR01+ AR10 when there are no causal types in
the population (i.e. the AR is homogeneous)
5. By this logic, we inferred that if we observe AR11 ≠ AR01+ AR10,
there are at least some causal types in our population (i.e. the AR is
not homogeneous)
What we just did…