1. Causal Diagrams for Epidemiological Research
Eyal Shahar, MD, MPH
Professor
Division of Epidemiology & Biostatistics
Mel and Enid Zuckerman College of Public Health
The University of Arizona

2. What is it and why does it matter?
A tool (method) that:
clarifies our wordy or vague causal thoughts about the research topic
helps us to decide which covariates should enter the statistical model—and which should not
unifies our understanding of confounding bias, selection bias, and information bias

3. What is the key question in a non-randomized study?
When estimating the effect of E (“exposure”) on D (“disease”), what should we adjust for? or
Confounder selection strategy

4. Adjusting for Confounders Common Practice
The “change-in-estimate” method
List “potential confounders”
Adjust for (condition on) potential confounders
Compare adjusted estimate to crude estimate
(or “fully adjusted” to “partially adjusted”)
Decide whether “potential confounders” were “real confounders”
Decide how much confounding existed
Premise: The data informs us about confounding.
Are we asking too much from the data?

5. Adjusting for Confounders Common Practice
What is “a potential confounder”?
Typically, “a cause of the disease that is associated with
the exposure”
Confounder E D
What is the effect of a confounder?
Contributes to the crude (observed, marginal) association
between E and D

6. Adjusting for Confounders Common Practice
Extension to multiple confounders C1 C2 C3 E D E D E D C4 C5 C6 E D E D E D

7. Adjusting for Confounders Common Practice Problems
A sequence of isolated, independent, causal diagrams
but C1, C2, C3, C4, C5,.. might be connected causally
Unidirectional arrow = a causal direction
but what is the meaning of the bidirectional arrow?
Even with a single confounder, the “change-in-estimate” method could fail

8. Adjusting for Confounders Problems
An example where the “change-in-estimate” method fails U1 U2 C E D
The crude estimate may be closer to the truth than the C-adjusted estimate
To be explained

9. Alternative A Causal Diagram
A method for selecting covariates
Extension of the confounder triangle
Premises displayed in the diagram
New terms:
Path
Collider on a path
Confounding path

10. Selected references
Pearl J. Causality: models, reasoning, and inference Cambridge University Press
Greenland S et al. Causal diagrams for epidemiologic research. Epidemiology 1999;10:37-48
Robins JM. Data, design, and background knowledge in etiologic inference. Epidemiology 2001;11:
Hernan MA et al. A structural approach to selection bias. Epidemiology 2004;15:
Shahar E. Causal diagrams for encoding and evaluation of information bias. J Eval Clin Pract (forthcoming)

11. A Causal Diagram Notation and Terms
An arrow=causal direction between two variables E D
An arrow could abbreviate both direct and indirect effects U1 E D E D
could summarize U2 U3

12. A Causal Diagram Notation and Terms
A path between E and D: any sequence of causal arrows that connects E to D E D E U1 U2 D E U1 U2 D E U1 U2 D

13. A Causal Diagram Notation and Terms
Circularity (self-causation) does not exist: Directed Acyclic Graph E U1 D U2
A collider on the path between E and D E U1 U2 D
E and U2 collide at U1

14. A Causal Diagram Notation and Terms
A confounding path for the effect of E on D: Any path between E and D that meets the following criteria:
The arrow next to E points to E
There are no colliders on the path C U1 V1 U2 V2 U3 E D
In short: a path showing a common cause of E and D

15. C U1 V1 U2 C V2 U3 U1 V1 E D U2 V2 C U3 U1 V1 E D U2 V2 U3 E D
The paths below are NOT confounding paths for the effect of E on D C U1 V1 U2 C V2 U3 U1 V1 E D U2 V2 C U3 U1 E V1 D U2 V2 U3 E D

16. What can affect the association between E and D
What can affect the association between E and D? (Why do we observe an association between two variables?)
Causal path: E causes D
Causal path: D causes E
Confounding paths
Adjustment for colliders on a path from E to D E D D E C E D
Later…

17. Why does a confounding path affect the crude (marginal) association between E and D?
Intuitively:
Association= being able to “guess” the value of one variable (D) from the value of another (E)
ED allows us to guess D from E (and E from D)
A confounding path allows for sequential guesses along the path C U1 V1 U2 V2 U3 E D

18. How can we block a confounding path between E and D?
Condition on a variable on the path (on any variable)
Methods for conditioning
Restriction
Stratification
Regression C U1 V1 U2 V2 U3 E D

19. A point to remember C U1 V1 U2 V2 U3 E D
We don’t need to adjust for confounders (the top of the triangle.) Adjustment for any U or V below will do.
U and V are surrogates for the confounder C C U1 V1 U2 V2 U3 E D

20. Example
If the diagram below corresponds to reality, then we have several options for conditioning
For example:
On C and U2
Only on U2
Only on U3 C U1 V1 U2 V2 U3 E D

21. What can affect the association between E and D?
Causal path: E causes D
Causal path: D causes E
Confounding paths
Adjustment for colliders on a path from E to D E D D E C E D
NOW!

22. Conditioning on a Collider A Trap
A collider may be viewed as the opposite of a confounder
Collider and confounder are symmetrical entities, like matter and anti-matter
Confounder
Collider C U1 V1 U2 V2 U3 E D

23. Conditioning on a Collider A Trap
A path from E to D that contains a collider is NOT a confounding path. There is no transfer of “guesses” across a collider.
A path from E to D that contains a collider does NOT generate an association between E and D
Conditioning on the collider, however, will turn that path into a confounding path.
Why?

24. Conditioning on a Collider A TrapV1 U1 U2 V2 U3 E D
The horizontal line indicates an association (the possibility of “guesses”) that was induced by conditioning on a collider

25. Properties of a Collider Intuitive Explanation
A dataset contains three variables for N cars:
Brake condition (good/bad)
Street condition in the owner’s town (good/bad)
Involved in an accident in the owner’s town? (yes/no)
Brake condition
(good, bad)
Accident
(yes, no)
Street condition
(good, bad)
Accident is a collider.
Brake condition and street condition are not associated in the dataset. We cannot use the data to guess one from the other.

26. Properties of a Collider Intuitive Explanation
Why can’t we make a guess from the data?
Let’s try. Suppose we are told:
Car A has good brakes and car B has bad brakes.
This information tells us nothing about the street condition in each owner’s town.
Brake condition
Street condition
Car A
Good ?
Car B
Bad
Intuition: a common effect (collider) does not induce an association between its causes (colliding variables)

27. Properties of a Collider Intuitive Explanation
If, however, we condition (stratify) on the collider “accident”, we can make some guesses about the street condition from the brake condition.
Stratum #1
Accident = yes
Brake condition
Accident
Street condition
Car A
Good
Yes
Bad
(a guess)
Car B ?

28. Properties of a Collider Intuitive Explanation
Similarly, in the other stratum
Stratum #2
Accident = no
Brake condition
Accident
Street condition
Car A
Good No
(a guess)
Car B
Bad ?

29. Properties of a Collider
In summary:
Conditioning on a collider creates an association between the colliding variables and, therefore, may open a confounding path
Before conditioning on C
After conditioning on C U1 U2 U1 U2 C C E D E D

30. Derivations
The “change-in-estimate” method could fail if we condition on colliders, and thereby open confounding paths
To (rationally) select covariates for adjustment, we must commit to a causal diagram (premises)
(But we often say that we don’t know and can’t commit, and hope that the change-in-estimate method will work.)
Causal inference, like all scientific inference, is conditional on premises (which may be false)—not on ignorance

31. Derivations U1 U2 U1 U2 C C E D E D
Do not condition on colliders, if possible
If you condition on a collider,
Connect the colliding variables by a line
Check if you opened a new confounding path
Condition on another variable to block that new path
Conditioning on C alone
Conditioning on C and (U1 or U2) U1 U2 U1 U2 C C E D E D

32. Practical advice Study one exposure at a time
A model that may be good for exposure A might not be good for exposure B (even if B is in the model)
Never adjust for an effect of the exposure
Never adjust for an effect of the disease
Never select covariates by stepwise regression
Never look at p-values to decide on confounding
(actually, never look at p-values…)

33. Extension to other problems of causal inquiry
Causation always remains uncertain, even if we deal with a single confounder
Unbeknown to us the
reality happens to be
We draw U1 U2 C C E D E D
And naively condition on C
And our adjustment
may fail

34. Extension to other problems of causal inquiry
Estimating the “direct” effect by conditioning on an intermediary variable, I I E D
We should remember that variable I may be a collider D U E I D U I E

35. Extension to other problems of causal inquiry
Causal diagrams explain the mechanism of selection bias
Example:
What happens if we estimate the effect of marital status on dementia in a sample of nursing home residents?
Assume: no effect
both variables affect “place of residence” (home, or nursing home)

36. Extension to other problems of causal inquiry
Marital status
Dementia
Place of residence
(home, nursing home)
By studying a sample of nursing home residents, we are conditioning on a collider (on a “sampling collider”) and might create an association between marital status and dementia in that stratum

37. Extension to other problems of causal inquiry
Marital status
Dementia
Place of residence
(home, nursing home)
“Stratification”
Home
Nursing home
Marital status
Dementia

38. Extensions: control selection bias (Source: Hernan et al, Epidemiology 2004)
Source cohort: no effect
Estrogen MI
DS because disease status affects
selection. Diseased members of the
cohort are over-sampled (cases)
relative to non-diseased (controls)
S (0,1)
Selection into a case-control sample
S=1 E D
S (0,1)
Estrogen MI
Suppose: F is hip fracture F
Suppose: EF
Suppose: Controls preferentially
selected from women with hip fracture

39. Extensions: control selection bias (Source: Hernan et al, Epidemiology 2004)
Estrogen MI E D F
S (0,1)
S=0
(remainder of the source cohort)
S=1
(our case-control sample)
HRT MI E D
Association of E and D was created

40. Extensions: information bias (LAST EXAMPLE)D
Estrogen use
Endometrial cancer D*
Diagnosed
endometrial cancer ? Z
Frequency of exams
Vaginal bleeding

41. Summary Points
The “change-in-estimate” method could fail if we condition on colliders, and thereby open confounding paths
The theory of causal diagrams extends the idea of a confounder to the multi-confounder case
Unification of confounding bias, selection bias, and information bias under a single theoretical framework

42. “Back-door algorithm”
Sufficient set for adjustment
Minimally sufficient set
Differential losses to follow-up
Time-dependent confounders
Interpretation of hazard ratios
Conditioning on a common effect always induced an association between its causes, but this association could be restricted to some levels of the common effect

43. ? Age (young, old) Smoking drive (low, high) Sex Physical activity
Asthma
(yes, no)
Smoking
status ?
FEV1

44. ? Age (young, old) Smoking drive (low, high) Sex Physical activity
Asthma
(yes, no)
Smoking
status ?
FEV1

45. ? Age (young, old) Smoking drive (low, high) Sex Physical activity
Asthma
(yes, no)
Smoking
status ?
FEV1

46. ? ? Pneumonia Ulcer Hospitalization Status hospitalized
not hospitalized
Coughing ?
Abdominal
Pain
Stratification
other patients
hospitalized patients
Pneumonia
Ulcer
Coughing ?
Abdominal
Pain

47. Example: Do men have higher systolic blood pressure than women
Example: Do men have higher systolic blood pressure than women? (In other words: estimate the gender effect on systolic blood pressure)
The following table summarizes the answer to this question from two regression models
Association with gender
Mean SBP in men
Mean SBP in women
Mean difference
(coefficient)
Inference
Marginal (crude)
123.8 mmHg
122.1 mmHg
+1.7 mmHg
Men’s SBP is higher
Conditional on (“adjusted for”) BMI and WHR
121.2 mmHg
124.3 mmHg
-3.1 mmHg
Women’s SBP is higher
So, which is the true estimate and which is biased?

48. WHR
Gender
SBP
BMI Z1 Z2 .

49. U
WHR
Gender
SBP
BMI Z1 Z2 .

50. U
WHR
Gender
SBP
BMI Z1 Z2 .