Transcript Principles of Data Analysis

3 Causal Models Part III:
DAGs and new approaches to bias
Matthew Fox
Advanced Epidemiology
Discussion:
How do we decide what are potential
candidates for an adjusted model?
Multivariate modeling example
Effect of E on D,
adjusted for:
crude
RR
95% CI
1.0
(0.8-1.3)
A
2.1
(1.3-3.7)
B
1.0
(0.6-1.6)
C
1.0
(0.6-1.6)
A and B
1.0
(0.6-1.8)
A and C
1.0
(0.6-1.7)
B and C
1.0
(0.6-1.8)
A, B and C
1.0
(0.5-2.1)
Discussion:
If a variable is not a confounder is it
reasonable to include it in a model to
see its effect on the outcome as well?
This Morning
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Counterfactual model
–
–

No confounding as partial exchangeability
–
–

–
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Separate from confounders
p1 + p3 = q1 + q3
Non-identifiably leads to collapsibility
–

4 types: doomed, causal, preventive, immune
Emphasize role of reference group
If crude = adjusted, collapse
Mantel-Haenszel if no interaction, SMR if yes
Odds ratio is not strictly collapsible
Statistical criteria can fail
Approaches to Confounding
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Multivariable analysis
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–
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Directed acyclic graphs
–
–

Limits of statistical criteria
Which variables are candidates for a model?
Visual approaches to detecting and
adjusting for confounding
Direct and Indirect effects
The structural nature of bias
Multivariate modeling –
Conventional Epi Approach

To identify set of variables to include:
–
Add largest confounder influence > criterion (10%)


–
–
Add next largest, if ∆ still > 10%, add next largest
Stop when the change is below 10%

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RRc <0.9 or >1.1
Stepwise, but based on change in estimate
Political variables?
Never a question of statistical significance
–
–
–
We are interested in the effect of an exposure
A variable may predict outcome, but not confound
In this case we typically just lose power
Multivariate modeling example
Effect of circumcision,
adjusted for:
crude
HR
95% CI
RRc
0.34
(0.54 - 4.66)
religion
0.53
(0.90 - 3.18)
1.57
# of partners
0.34
(0.55 - 4.76)
1.01
age
0.32
(0.52 - 5.12)
0.93
religion and age
0.60
(1.02 - 3.05)
1.13
religion and # of partners
0.50
(0.86 - 3.71)
0.94
age and # of partners
0.32
(0.52 - 5.26)
1.0
religion, age, and # of partners
0.57
(0.98 - 3.00)
0.95
Multivariate modeling example
Effect of circumcision, adjusted
for:
HR
95% CI
RRc
1/RRc
crude
0.34
(0.54 - 4.66)
religion
0.53
(0.90 - 3.18) 0.64 1.57
# of partners
0.34
(0.55 - 4.76) 0.99 1.01
age
0.32
(0.52 - 5.12) 1.08 0.93
religion and age
0.60
(1.02 - 3.05) 0.88 1.13
religion and # of partners
0.50
(0.86 - 3.71) 1.06 0.94
age and # of partners
religion, age, and # of partners
0.32
0.57
(0.52 - 5.26) 1.00 1.00
(0.98 - 3.00) 1.05 0.95
Problems with statistical approach

Ignores everything we know about the
relationship between variables in the model
–
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Can go wrong when causal structure is
complicated
–
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Removes control and thought
And it often is complicated
What is a better approach for identifying an
appropriate set of confounders?
–
Causal diagrams
The Causal Web Model (individual)
Tobacco
chewing
Nicotine
Utero-Placental
insufficiency
Placenta
Previa
Cigarette
Smoking
Carbon
monoxide
Abruptio
Placenta
Neonatal
Sepsis
Premature
Rupture of
Membrane
Household Smoke
(wood/charcoal)
Fetal Hypoxia
Preterm delivery
+ Small for
gestational age
Birth Asphyxia
Stillbirth
Early
neonatal
death
Common Causes
C
E
D
What is it?
If no effect of E on D, will E and D
be associated if we do nothing?
Does E cause D?
Indirect Effect
C
E
D
If we do nothing, will E and D be
associated?
Does E cause D?
Common effects
E
D
C
If E doesn’t cause D, if we do
nothing, will E and D be associated?
Will E and D be associated with C?
Would stepwise procedures include it?
Terminology - DAGs

Arc or Edge
–
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Arrows encode causal relations
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–
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No arrow = independence
Arrows indicate the flow of information
Parent-Child
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Line connecting two variables
Arrow from one node to another
Ancestor
B
C
E
D
Terminology - DAGs

Directed:
–
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Acyclic:
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All parent to child
No directed path forms a loop
Future cannot predict the past
Causal:
–
All arrows represent effects
B
C
E
D
Adopt convention time flows from
left to right. Heads of arrows
should always be to right of tails
Terminology - DAGs

A path
–
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Always leaving tail,
entering a head
Backdoor path
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Blocked path
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Directed path
–
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Unbroken route
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A non-causal path from E
to D that does not contain
any variable affected by E
Collider
–
–
Can’t go in 1 arrow head
& out 2nd head
Specific to a path
–
A path is blocked if there is
a collider
Or we control for a variable
on the path
Causal DAGs
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An arrow implies an effect
–
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A DAG is causal if the common causes of
any two variables are shown in the graph
–
–
–

A→Y means Pr[Ya=1=1]≠ Pr[Ya=0=1]
In other words, does not need to include every
variable
Start with E and D and look for all common causes
Then add common causes of those variables
Common causes imply association not
causation
We use DAGs to see if two
variables are d-separated


D = “directional”
D-separation means we can determine
causality
–

A and B will be d-separated if there is no
unblocked backdoor path from A to B
–

If our DAG is “faithful”
Unconditional independence
Two variables will also be d-separated if all
paths are blocked through control
–
Conditional independence
DAGs show association and
causation

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The DAG, if causal, says:
– Pr[Ya=1=1] = Pr[Ya=0=1] and
– Pr[Y=1|A=1] ≠ Pr[Y=1|A=0]
In other words:
–
–
A does not cause Y, so the true
effect of A on Y is null
In our crude data, A will be
associated with Y
No Causation
Association
C
A
Y
DAGs show association and
causation
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The DAG, if causal, says:
– Pr[Yc=1=1] ≠ Pr[Yc=0=1] and
– Pr[Y=1|C=1] ≠ Pr[Y=1|C=0]
In other words:
–
–
C does cause Y, so the true
effect of C on Y is not null
In our crude data, the
association between C and Y
will be the causal effect
Causation
Association
C
A
Y
DAGs show association and
causation


The DAG, if causal, says:
– Pr[Ac=1=1] ≠ Pr[Ac=0=1] and
– Pr[A=1|C=1] ≠ Pr[A=1|C=0]
In other words:
–
–
C does cause A, so the true
effect of C on A is not null
In our crude data, the
association between C and A will
be the causal effect
Causation
Association
C
A
Y
General Rule of DAGs:
We can trace a backdoor path
from E to D going in any
direction we like, except we
can’t go in the head
of one arrow and out the head
of another or through a variable
we control for statistically.
Rules of DAGs
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A path is blocked if:
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–
It contains a non-collider that’s been conditioned on
OR: It contains a collider not conditioned on and no
child of that collider has been conditioned on
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Conditioning on a child partly conditions on the parent
Two variables are d-separated if all
backdoor paths between them are blocked
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No confounding
To diagnose confounding, first
remove all arrows emanating from E
Are there any unblocked
backdoor paths from E to D?
Estimates of the effect of E on D
will be confounded if there is an
unblocked backdoor path from E to D
Note that each path we can trace really
shows common causes – look again.
Confounding IS common causes.
Beware conditioning on a collider
Conditioning on a collider
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
Conditioning on a collider opens the flow
of information
Only two reasons the ground can be wet
–
–
It can rain
Sprinkler is on 1 week schedule
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–
Unrelated to weather
The two are completely unrelated
Sprinkler
Rain
Ground is wet
Conditioning on a collider


We notice the ground is wet
This is equivalent to only looking in the
strata wet = 1
–
–
If we know it rained, is it more or less likely that the
sprinkler was on?
If we know the sprinkler was on, is it more or less
likely that it rained?
Sprinkler
Age
Rain
Ground is wet
As an example
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If 50% chance of rain and 50% chance of sprinkler
on, and both are independent:
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–

If I wake up and the ground is dry:
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If rain, chance of sprinkler is 50%
If no rain, chance of sprinkler is 50%, RR = 1
Perfect correlation (both did not occur)
If I wake up and ground is wet then:
–
–
–
–
If rain, chance of sprinkler on is 50%
If no rain, chance of sprinkler on is 100%, RR = 2
If sprinkler was on, chance of rain is 50%
If sprinkler was off, chance of rain is 100%, RR = 2
Put another way

If we had two independent variables that
perfectly predicted a third
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Then
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–
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A and B are independent binary variables and
C=A+B
If we look among C = 2, A and B must be 1
If we look among C = 0, A and B must be 0
If we look among C = 1, if A = 1 then B = 0 and if A
= 0, C = 1
So within C, A & B are perfectly correlated
Ways two variables can be associated
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Causation
–
Direct, indirect and reverse
Common causes
 Conditioning on a common effect
 Random variation

–
Not a part of DAGs which represent structural
relations
Find a set sufficient for
statistical control
A and C are sufficient
Could we have chosen other sets?
Circumcision and HIV DAG
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG:
Remove arrows from exposure
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG:
Any unblocked backdoor paths?
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG
New paths?
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG
New paths
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG
Identify {S} (sufficient set?)
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG
New paths?
Age
Religion
# sexual
partners
Circumcision
HIV
Circumcision and HIV DAG
All unblocked paths pass {S}?
Age
Religion
# sexual
partners
Circumcision
HIV
Which is the confounder, age or sexual
partners?
Age
Religion
# sexual
partners
Circumcision
HIV
This is why we separate
confounders from confounding
Could also have just chosen religion
Age
Religion
# sexual
partners
Circumcision
HIV
Multivariate modeling example II
Effect of birth weight on child
mortality, adjusted for:
Crude
HR
RRc
3.34
Breastfeeding
2.43
1.37
Maternal smoking
2.54
1.31
Breastfeeding and smoking
1.55
1.64
Direct and indirect effects
Breastfeeding
Maternal
smoking
Low birth weight
Child
Death
Should we adjust for A?
B
A
C
Exposure
Disease
Using a traditional definition of
confounding, is A a confounder?
B
A
C
Exposure
Disease
Is A associated with E?
Yes, because A and E share a
common cause (B)
B
A
C
Exposure
Disease
Is A associated with D?
Yes, because A and D share a
common cause (C)
B
A
C
Exposure
Disease
Is A on causal pathway from E→D?
No, so by traditional definitions,
this is a confounder
B
A
C
Exposure
Disease
Any unblocked backdoor paths
from E→D?
No, so despite the change in
estimates, the crude is correct
B
A
C
Exposure
Disease
Corresponds to this example from
earlier this morning
Type 1
Type 4
total
risk
RR
Type 1
Type 4
total
risk
RR
MHRR
E+
100
100
200
0.5
E250
250
500
0.5
1
C+
E+
60
40
100
0.6
E70
30
100
0.7
Type I
Type 4
total
0.86
CE+
40
60
100
0.40
E180
220
400
0.45
0.88
0.87
Multivariate modeling example III
Effect of E on D,
adjusted for:
crude
HR
95% CI
1.0
(0.8-1.3)
A
2.1
(1.3-3.7)
B
1.0
(0.6-1.6)
C
1.0
(0.6-1.6)
A and B
1.0
(0.6-1.8)
A and C
1.0
(0.6-1.7)
B and C
1.0
(0.6-1.8)
A, B and C
1.0
(0.5-2.1)
Multivariate modeling example III
Effect of Wearing Makeup on
Insomnia, adjusted for:
crude
HR
95% CI
1.0
(0.8-1.3)
Drivers license
2.1
(1.3-3.7)
Age
1.0
(0.6-1.6)
Sex
1.0
(0.6-1.6)
Age and Drivers license
1.0
(0.6-1.8)
Age and Sex
1.0
(0.6-1.7)
Drivers license and Sex
1.0
(0.6-1.8)
Age, Drivers license and Sex
1.0
(0.5-2.1)
M Bias
Sex
Age
Drive
Make up
Insomnia
M Bias
Sex
Drive
Age
Make up
Insomnia
Is it appropriate to adjust for A if we
don’t have data on B?
B
A
Exposure
Disease
A is a surrogate confounder
Now imagine B is the true vitamin
intake and A is a measure of
vitamin intake from nutritional
assessment?
B
A
Exposure
Disease
Control of a misclassified
confounder leads to only partial
control (residual confounding)
We are studying the causal effect
of E on D, so we adjust for B in a
regression model. Can we include
A just to see what effect is on D?
A
B
Exposure
Disease
We are interested in valid and precise
estimate of the effect an exposure on an
outcome. Everything else is noise.
Pros and Cons of DAGs

Advantages:
–
Flexible

–
Illustrates dependencies between variables


Represents states of individuals in populations
Implications for validity of measures of effect
Disadvantages:
–
No inherent technique to quantify associations
based on info in the graphs

No way to estimate effects based only on information in the
graphs