Selection Bias Concepts
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/talks
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Questions
Given measured appropriate variables:
Can you adjust for confounding?
Yes
Can you adjust for selection bias?
Depends on the definition
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Contents
• Background
– Define bias
• Selection bias
– as effect modification
– as collider stratification bias
– DAG structure
(old concept)
(new concept)
• Examples
• Size and direction of bias
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Bias definition
• Bias
– Frequency:
– Effect:
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expected risk ≠ true risk
association ≠ causal effect
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Selection bias concepts
Concept
DAG structure
Effect responders
Effect
≠
modification
Effect non responders
Differential response bias
Collider
Differential loss to follow up
stratification
Healthy worker bias
bias
Berkson’s bias (case control)
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Selection bias as effect
modification
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Selection bias: Risk
• Selection of responders 
– The prevalence is different among
• the responders compared to the full population
• the responders compared to the non responders
Population
R0
Non responders
R1
Responders
Rp
Rp is the weighted mean of R0 and R1
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Effect modification
• Selection of responders 
– The effect of E on D is different among
• the responders compared to the full population
• the responders compared to the non responders
Population
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RR0
Non responders
RR1
Responders
RRp
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Problems
• Is not a bias, RR0 and RR1 are the true effects
• Is effect modification by selection variable S
• Leads to the conclusion that:
1.
2.
3.
Biolocical effects are protected from bias
The bias can not be adjusted for
RRp is the average of RR0 and RR1
Not true for
collider
stratification bias
“DAG” structure:
S
E
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D
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Selection bias as collider
stratification bias
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Example with paths
• Study
S
calcium supp.
E
D
milk
bone density
S
C
calcium supp.
family history
E
D
milk
bone density
Structure:
Collider stratification
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– Milk on bone density
– Exclude Calcium supplements
Path
1 ED
2 E[S]D
Type
Status
Causal
Open
Noncausal Open
2 E[S][C]D Noncausal Closed
Lessons learned:
Biological effect not protected
May adjust for selection bias
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Examples
S
C
respond
education
E
D
alcohol
CHD
S
C
loss to follow up
smoking
E
D
drug
disease
S
C
working
health
E
D
dust
lung disease
• Differential response
– Survey: Alcohol and CHD
• Differential loss to follow up
– Randomized trial: drug and disease
• Healthy worker effect
– Cross-section: Melt hall dust and
lung disease
Note: no confounding
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Selection bias structure
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Paths
1. Causal
2. Confounding
An open non-causal path without colliders
3. Selection bias
A non-causal path that is open due to
conditioning on a collider
C
A
E
A B D
Causal
BCVs?
C
B
E
D
Confounding
A
B
E
D
Selection bias
Collider stratification bias
• Selection bias = Collider stratification bias
• Selection bias, Path definition
– A non causal path that is open due to
conditioning on a collider
S
S
E
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S
D
E
C
D
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B
E
D
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Selection bias examples
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Folic acid and cardiac malformation
C
Selection: Study only live born
Live born
E
D
Folic acid
Card. Mal.
Bias?
Yes, E[C]D is open
S
Grief
Selection: Non grieving parents volonteer
C
Bias?
Live born
E
D
Folic acid
Card. Mal.
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Yes, E[C]D is (partially) open
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Education and unfaithfulness
• Study the effect
among couples
in a relationship
(not divorced)?
R
divorced
S
E
sensation
seeking
education
Path
1 ED
2 ERD
3 ERSD
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D
unfaithful
Type
Causal
Noncausal
Noncausal
Population
Open
Closed
Closed
Sample
Open
Open
Open
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Size and Direction of bias
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Example 1, full table
(Adjusted) RRs
Response
D
1
E
0
1
0
0.9
0.3
0.9
0.3
Response=
54 %
R
3.0
1.0
RR
E
2.0
E
1
0
Responders
D
1
0
sum
114
86
286
514
400
600
1000
RR=
OR=
RD=
2.0
2.4
0.14
D
0
2.0 2.0
True and biased RRs
Proportion responding in 1,1 group
Population
D
1
E
1
0
1
0
103
26
257
154
RR=
OR=
RD=
2.0
2.4
0.14
Non respond
D
E
1
0
1
0
11
60
29
360
RR=
OR=
RD=
2.0
2.4
0.14
Example 2
Response
D
E
1
0
1
0
0.9
0.9
0.3
0.3
Response=
42 %
Pattern:
Only D influence
response
R
1.0
3.0
RR
E
2.0
D
1
0
1.6 2.3
Result:
RR (and RD) biased, OR unbiased
ODS, Case-Control
Example 3
Response
D
E
1
0
1
0
0.9
0.45
0.45
0.225
Response=
38 %
Pattern:
Both E and D
influence
response
R
2.0
2.0
RR
E
1.0
D
1
0
1.0 0.3
Result:
Surprise: responders are unbiased
Theory: bias in at least one stratum
Example 4
Response
D
E
1
0
1
0
0.45
0.225
0.9
0.45
Response=
56 %
Pattern:
Both E and D
influence
response
R
2.0
0.5
RR
E
2.0
D
1
0
2.2 3.6
Result:
Surprise: both strata biased upwards
True RR is not a weighted average
Example 5
Response
D
E
1
0
1
0
0.99
0.495
0.66
0.33
Response=
51 %
R
2.0
RR
E
Response
D
E
1
0
1
0
0.5
0.25
0.3333
0.1667
Response=
26 %
Pattern:
Both E and D
influence
response
1.5
2.0
D
1
0
1.9 0.1
R
2.0
1.5
RR
E
2.0
D
1
0
1.9 1.8
Result:
Same DAG, different results
The DAG does not fully determine the
selection!
Summing up
• Selection bias as “effect modification”:
– Is not a bias, should not be called selection bias
– Has properties different from proper selection bias
• Selection bias as “collider stratification”:
– Structure defined in DAG,
– Distinct from confounding
– Consistent with
•
•
•
•
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Differential response bias
Differential loss to follow up
Healthy worker bias
Berkson’s bias (case control)
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Litterature
• Hernan and Robins, Causal Inference
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