Transcript PH 240A: Chapter 8 Mark van der Laan University of California Berkeley

PH 240A: Chapter 8
Mark van der Laan
University of California Berkeley
(Slides by Nick Jewell)
1
Counterfactual Pancreatic Cancer Responses
to Coffee Drinking Conditions
Group
Coffee
(E )
No Coffee
(E )
#
1
D
D
Np1
2
D
D
Np2
3
D
D
Np3
4
D
D
Np4
RRcausal 
p1  p2
p1  p3
ORcausal
p1  p2
1  ( p1  p2 )

p1  p3
1  ( p1  p3 )
ERcausal  ( p1  p2 )  ( p1  p3 )
2
Counterfactual Pancreatic Cancer Responses
to Coffee Drinking Conditions
Group
Coffee
(E )
No Coffee
(E )
#
1
D
D
Np1
2
D
D
Np2
3
D
D
Np3
4
D
D
Np4
Suppose Group 1 and 3 individuals are all coffee drinkers
and Group 2 and 4 individuals abstain, then . . .
3
Population Data
Pancreatic Cancer
Coffee
Drinking
(cups/day)
 1 (E)
0 (not E)
D
Not D
Np1
Np3
0
N ( p2  p4 )
Np1
N ( p2  p3  p4 )
N ( p1  p3 )
N ( p2  p4 )
OR  RR  
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Counterfactual Pancreatic Cancer Responses
to Coffee Drinking Conditions
Group
Coffee
(E )
No Coffee
(E )
#
1
D
D
Np1
2
D
D
Np2
3
D
D
Np3
4
D
D
Np4
Suppose Group 1 and 2 individuals are all abstainers
and Group 3 and 4 individuals drink coffee, then . . .
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Population Data
Pancreatic Cancer
Coffee
Drinking
(cups/day)
 1 (E)
0 (not E)
D
Not D
0
N ( p3  p4 )
Np1
Np1
Np2
N ( p3  p4 )
N ( p1  p2 )
N ( p2  p3  p4 )
OR  RR  0 !
6
All is Lost?
 Let’s quit—the first four weeks of
the class have been a total waste of
time . . .?
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Counterfactual Pancreatic Cancer Responses
to Coffee Drinking Conditions
Group
Coffee
(E )
No Coffee
(E )
#
1
D
D
Np1
2
D
D
Np2
3
D
D
Np3
4
D
D
Np4
Suppose individuals choose whether to drink coffee or not at random,
(say, toss a coin) then . . .
8
Population Data Under Random
Counterfactual Observation
Pancreatic Cancer
D
Coffee
Drinking
(cups/day)
1
Not D
(E)
N ( p1  p2 ) / 2
N ( p3  p4 ) / 2
N /2
0 (not E)
N ( p1  p3 ) / 2
N ( p2  p4 ) / 2
N /2
N (2 p1  p2  p3 ) / 2
N ( p2  p3  2 p4 ) / 2
OR  ORcausal
RR  RRcausal
ER  ERcausal
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Confounding Variables
 Randomization assumption
 Probability of observing a specific exposure
condition (eg coffee drinking or not) must not
depend on counterfactual outcome pattern (i.e.
vary across groups)
 Failure of randomization assumption
 Group 1 individuals are more likely to be males
than say Group 4 individuals
 If males are also more likely to drink coffee,
then we are more likely to observe the coffee
drinking counterfactual in Group 1 than Group 410
Confounding Variables
 For example, imagine a world where all
males, and only males, drank coffee
Pancreatic Cancer
D
Coffee
Drinking
(cups/day)
Not D
1
males
0
females
Even if coffee had no effect we would observe
an association (due to sex)
sex is a confounder11
Confounding Variables
C
E
?
D
 Conditions for confounding
 C must cause D
 C must cause E
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Stratification to Control Confounding
 Divide the population into strata
defined by different levels of C
 Within a fixed stratum there can be no
confounding due to C
 New issue: causal effects of E may vary
across levels of C
• Interaction or effect modification
 When no interaction, need methods to
combine common causal effects across C
strata (Chapter 9)
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Causal Graph Approach to Confounding
 Possible causal effects of childhood vaccination on
autism
 access to general medical care may affect autism
incidence and/or diagnosis
 Access to medical care increases vaccination
 Family SES influences access to medical care and also
ability to pay for vaccination
 Family medical history may affect risk for autism and
may also influence access to medical care
 Which of medical care access, SES, and family
history are confounders? Do we need to stratify
on all three?
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Directed Acyclic Graphs
 Nodes, directed graphs (edges have
direction)
C
B
A
D
 Directed paths: B-A-D
B-D-A, C-B-D
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Directed Acyclic Graphs
 Acyclic
 No loops: A cannot cause itself
 Mother’s Smoking Status
Child’s Respiratory Condition
Mother’s Smoking Status (t=0)
Mother’s Smoking Status (t=1)
Child’s Respiratory Condition (t=0)
Child’s Respiratory Condition (t=0)

node A at the end of a directed path starting at B is a descendant of B
(B is an ancestor of A)
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Directed Acyclic Graphs
 A node A can be a collider on a specific pathway if the path
entering and leaving A both have arrows pointing into A. A
path is blocked if it contains a collider.
B
F
A
C
D
 D is a collider on the pathway C-D-A-F-B; this path is
blocked
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Using Causal Graphs to Detect Confounding
 Delete all arrows from E that point to
any other node
 Is there now any unblocked backdoor
pathway from E to D?
 Yes—confounding exists
 No—no confounding
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Vaccination & Autism Example
SES
Medical Care
Access
Vaccination
Family
History
Autism
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Using Causal Graphs to Detect
Confounding
C
F
C
F
E
D
E
D
C
F
C
F
E
D
E
D
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Checking for Residual Confounding
 After stratification on one or more
factors, has confounding been
removed?
 Cannot simply remove stratification
factors and relevant arrows and check
residual DAG
 Have to worry about colliders
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Controlling for Colliders
 Stratification on a collider can induce an
association that did not exist previously
Rain
Sprinkler
Wet Pavement
Diet sugar (B)
Fluoridation (A)
Tooth Decay (D)
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Hypothetical Data on Water Flouridation,
High-Sugar Diet, and Tooth Decay
Tooth Decay (D)
Flouridation
A
A
High-Sugar Diet
High-Sugar Diet
D
D
B
160
40
B
120
80
B
80
120
B
40
160
OR
ER
2.67
0.2
2.67
0.2
6.00
0.4
6.00
0.4
1.00
0.00
Tooth Decay (D)
High-Sugar Diet
B
B
Fluoridation
Fluoridation
D
D
A
160
40
A
80
120
A
120
80
A
40
160
Pooled table
Fluoridation
High-Sugar Diet
A
A
B
200
200
B
200
200
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Hypothetical Data on Water Flouridation,
High-Sugar Diet, and Tooth Decay
Tooth
Decay (D)
Fluoridation
A
A
OR
ER
B
160
80
0.67
-0.083
B
120
40
No Tooth
Decay (D )
Fluoridation
A
A
OR
ER
B
40
120
0.67
-0.083
B
800
160
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Checking for Residual Confounding
 Delete all arrows from E that point to any
other node
 Add in new undirected edges for any pair
of nodes that have a common descendant in
the set of stratification factors S
 Is there still any unblocked backdoor path
from E to D that doesn’t pass through S ?
If so there is still residual confounding,
not accounted for by S .
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Vaccination & Autism Example
SES
Medical Care
Access
Vaccination
Family
History
Autism
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Vaccination & Autism Example:
Stratification on Medical Care Access
Family
History
SES
Vaccination
Autism
Still confounding: need to stratify additionally
on SES or Family History, or both
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Caution: Stratification Can
Introduce Confounding!
F
C
E
D
No Confounding
Stratification on C introduces confounding!
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Collapsibility
 No Confounding and No Interaction
C
D
Pooled
C
D
D
D
D
D
E
120
280 400
E
14
86
100
E
134
366
500
E
60
340 400
E
7
93
100
E
67
433
500
RRC  2.00
RRC  2.00
RR  2.00
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Collapsibility
 Confounding and No Interaction
C
D
E
E
216
12
Pooled
C
D
D
E
14
86
100
E
E
7
93
100
E
D
504 720
68
80
RRC  2.00
D
RRC  2.00
D
230 590
19
161
820
1800
RR  2.66
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Collapsibility and Confounding
 With the assumption of the causal graph below,
and in the absence of interaction, the conditions
for collapsibility wrt RR (and ER) are the same as
for no confounding (i.i either C & E are
independent, or C & E are independent, given E, or
both)
But note that collapsibility
cannot distinguish directions
of the arrows
C
E
?
C
D
E
?
D
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Collapsibility
 No Confounding and Interaction
C
D
Pooled
C
D
D
D
D
D
E
60
340 400
E
14
86
100
E
74
426
500
E
120
280 400
E
7
93
100
E
127 373
500
RRC  0.50
RRC  2.00
RR  0.58
has causal interpretation
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Collapsibility
 Confounding and Interaction
C
D
Pooled
C
D
D
D
D
D
E
108
612 720
E
14
86
100
E
122
69
8
820
E
24
56
E
7
93
100
E
31
14
9
180
80
RRC  0.50
RRC  2.00
RR  0.86
has no causal interpretation
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Collapsibility and OR
 Collapsibility and “No Confounding”
not quite the same thing for OR
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