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An introduction to selection bias and over-adjustment.
Conditioning on an Effect
Selection Bias and
Patrick F. McArdle
What is Selection Bias?
A Structural Approach to Selection Bias
Path X-S-Y is blocked by the collider S.
Path X-S-Y is open upon conditioning or stratifying on S.
Example: S represents selection into the study, where by definition we only observe one strata; S = 1.
Selection Bias: A non-causal association due to observing only one strata of a common effect.
Conditioning on a Common Effect Induces an Association
Assume smoking and exposure to asbestos are unrelated. One does not cause the other and they share no common causes.
Assume for a case of lung cancer, we know the person is not a smoker. We would then have a stronger belief they were exposed to asbestos.
When we condition on a common effect (lung cancer), knowing something about one cause tells us something about the other.
Collider Stratification Bias
Conditioning on a common effect
Odds Ratio = (20*6)/(20*6) = 1
Card example from Merchant: Community Dent Oral Epidemiol 2002; 30: 399–404
Odds Ratio = 2.4
Odds Ratio = 2.5
Modeling Selection Bias
Add a variable S to your DAG to indicate selection into the study.
Include variables and relationships that may influence selection.
By definition, the study is only concerned with those who end up in the analysis, S=1.
Inappropriate Selection of Controls in a Case Control Study
Example: Case control study of the effect of Estrogen (E) on Myocardial Infarction (MI)
E HF S
Controls are selected from a population of Hip Fracture (HF) patients, without history of MI.
Estrogen is believed to have a protective effect on hip fractures.
Two diseases that are unassociated in the population are associated within hospital patients if both diseases lead to hospitalization.
Can their be selection bias in a prospective cohort study?
Example: We wish to study the effect of air brush painting on lung cancer.
Study Design: Enroll employees of a auto repair shop. Follow painters (E) and mechanics (E) prospectively to either death or lung cancer initiation.
Prospective Cohort Study DAG
Is there Selection Bias in this case?
The outcome occurs after selection into the study so one might assume there would be no selection bias.
Selection Bias can still occur if Y and S share a common cause.
Differential Loss to Follow-up
Example: Longitudinal study of antiretroviral therapy (AT) on AIDS.
Even in the absence of a true effect of AT on AIDS, those on the therapy are more likely to have side effects and drop out of the study.
Those will greater immunosuppression are both more likely to develop AIDS and more likely to drop out of the study due to disease symptoms impairing study participation.
Example: Study of the effect on the Heimlich maneuver on mortality.
Heimlich Deatht=1 Deatht=2
The Heimlich only has an immediate effect. It does not have an effect at later chocking incidents.
When using the hazard rate ratio, this is the risk of dying at a later time given you have survived to that time.
The Heimlich will be associated with long-term mortality if choking risks are not accounted for.
How do we reduce selection bias?
Proper study design
Can we statistically adjust away
the effect of selection bias?
Selection Bias is a Problem
Regardless of how well a study is designed, selection bias can still occur from loss to follow-up, self selection of participants and missing data.
In these cases, selection bias needs to be explicitly accounted for in the analysis.
Advanced Topic: Going beyond standard DAGs
One would assume based upon the DAG that selection bias is present.
Formally, a statistical association will appear between two causes in at least one strata of a common effect.
What if that strata is S=0? We are only concerned with S=1.
For more: VanderWeele, Robins. AJE (2007) 166(9):1096-1104
DAGs as Shorthand Notation
DAGs are very useful as shorthand notation to state a priori causal assumptions.
The accompanying Structural Equation Models provide the parametric form.
In particular, standard DAGs do not identify the presence of interactions.
DAGs + SEM
E = α0
U = δ0
S = β0 + β1*E + β2*U + β3*E*U
Y = γ0 + γ1*U
α0 = 0.50
δ0 = 0.50
γ0 = 0.01 γ1 = 0.05
β0 = 0.50 β1 = 0.10, β2 = 0.25, β3 =0.05
Absence of Selection Bias in S=1 due to Interaction
Observed Estimate in the Selected (S=1)
True Causal Effect
DAGs can be used to identify the presence of selection bias in the absence of interaction.
DAGs and their parametric form are always enough to identify bias.
Conditioning on a collider is a sufficient but not necessary condition for selection bias. Selection bias can occur (under non-null conditions) even when no open path is created.
Is selection bias either present or not?
No, selection bias could depend on the choice of effect measure.
Selection Bias Dependent Upon Choice of Effect Measure
Add a true effect to the previous example.
Risk Difference is not biased, but risk ratio and odds ratio are.
Alternative to conditional estimation:
Inverse Probability Treatment Weighting
Create a pseudo sample to more accurately reflect the underlying populations.
A method that could reduce effects of loss to follow-up.
Summary: Four Structural Explanations of a Statistical Association
Over Adjustment Bias
X Z Y
What is the effect of conditioning on Z*?
Will the estimate of the causal effect of X on Y be biased?
Controlling for an intermediate variable, or the descendant of an intermediate variable, on the causal pathway between exposure and outcome.
The net effect is to increase bias.
There are no free lunches. You can not just add covariates to the model and assume they will always reduce bias.
Among subjects selected into the study,
More on this next lecture out of the study due to disease symptoms impairing study participation.
Formally, a statistical association will a