1. Causal Directed Acyclic Graphs (DAG) (Causal Diagrams) 2013 Eyal Shahar, MD, MPH Professor

2. What is a causal diagram?
Components
Variables
Unidirectional arrows D A C E B

3. Rules: displaying variables
Called “nodes” or “vertices”
Should be clearly understood by others
Variables, not values of variables
“Smoking status” is okay; “smoking” is not
Displayed along the time axis (left to right)
but sometimes we ignore this rule

4. Rules: drawing arrows A B A B C A B An arrow No bidirectional arrows
From a postulated cause to its postulated effect
No bidirectional arrows
An arrow with a question mark
The research question at hand
An arrow without a question mark
Background theory or axiomatic A B C ? A B

5. Rules: drawing arrows A B C At=1 Bt=2 At=3 Bt=4 Directed Acyclic Graph
Circularity does not exist
A future effect cannot be a cause of its cause in the past
So-called “circularity”
Directed acyclic graph with time-indexed variables A B C
At=1
Bt=2
At=3
Bt=4

6. Ddx=diagnosed esophagus status I1 I2
Example: a causal diagram for gastroesophageal reflux and esophageal disease S1 S2 ? T R1 R2 D1 D2
R=reflux
S=symptoms
T=treatment
I=imaging
D=esophagus status
Ddx=diagnosed esophagus status I1 I2
D1dx
D2dx

7. How does a causal diagram help in research?
Decodes causal assertions
All of science is about causation!
Clarifies our wordy or vague causal thoughts about the research topic
Connects “association” with “causation”
Helps us decide which covariates should enter the statistical model—and which should not
Unifies our understanding of confounding bias, colliding bias, information bias (and three other, less well, known biases)
Can depict and explain all types of bias

8. PubMed search (through 2012)
“Causal diagrams”: 83 titles
“Directed acyclic graph”: 137 titles (some irrelevant)
Still not widely known
Rarely used

9. Some references
Pearl J. Causality: models, reasoning, and inference Cambridge University Press (2009, second edition)
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, Shahar DJ. Causal diagrams, information bias, and thought bias. Pragmatic and Observational Research 2010:1;33-47
Shahar E, Shahar DJ: Causal diagrams and three pairs of biases. In: Epidemiology –Current Perspectives on Research and Practice (Lunet N, Editor) :pp (reading material for this module)

10. A natural path between two variables
Formally: a sequence of arrows, regardless of their direction, that connects two variables (and does not pass more than once through each variable)
Informally: “can walk from A to Z, or from Z to A, on bridges” A Z A B C D Z A B C D Z A B C D E Z

11. Types of natural paths between two variables
Causal paths
Confounding paths
Colliding paths

12. A causal path between two variables (also called “directed path”)
A natural path between A and Z, in which all the arrows point in the same direction (hence, “directed path”)
“A is a cause of Z” or “Z is a cause of A A Z A Z A B C Z C D A B Z A B Z

13. “Direct” versus “indirect” causal pathB Z
“direct” causal path
“Direct” is often (maybe always) over-simplification
Is it really direct?
No intermediary exists?
Better terminology: “causal paths in which no intermediary variables are known or displayed”
Overall (total) effect: by all directed paths (combined)

14. A confounding path between two variables
A natural path between A and Z that contains a shared cause of A and Z on this path (a confounder) C C X A Z A Z
Alternative display A C Z A C X Z

15. A colliding path between two variables
A natural path between A and Z that contains at least two arrowheads that “collide” at some variable along this path (a collider on the path) L A Z K M A Z C
Alternative display A C Z A K M L Z

16. Side point: collider (and confounder) are path-specific terms
A variable called a collider (or a confounder) on one path need not be a collider (or a confounder) on another path B D C A Z
C is a collider on one path (ABCDZ) and a confounder on another path (ACZ) 16

17. Identify and name each natural path between A and ZQ S P R A Z K L M

18. A bridge to “association”
What is “association”?
Mathematical phenomenon
Ability to guess the value of one variable based on the value of another variable
Are there “spurious associations”?
Mathematical relation between variables is never “spurious”
Poor word choice
“The association of A with Z is spurious.” What does the writer have in mind, though?
What creates associations?
A causal structure

19. A bridge between natural paths and associations
Which natural paths between A and Z contribute to the marginal (crude) association between A and Z?
Causal paths
Confounding paths
Which natural paths between A and Z do not contribute to an association between A and Z?
Colliding paths
Open paths
Blocked paths

20. Identify open paths and blocked paths (between A and Z) in this diagram

21. When does an association between A and Z reflect the effect of A on Z?
When only causal paths contribute to the association between A and Z
When confounding paths do not exist, or are somehow blocked
Almost true: not a sufficient condition

22. How do we block a confounding path?
By conditioning on some variable along the path
What is “conditioning” on a variable?
Restricting the variable to one of its values
Various forms of “adjustment”
Standardization
Stratification and a weighted average (Mantel-Haenszel)
Adding an independent variable to a regression model

23. Conditioning on a variable…
Dissociates a variable from its causes and its effects A X B V Y C Z
Turns an open natural path into a blocked path A V Z A V Z

24. Deconfounding = blocking a confounding pathX A ? Z A ? Z C
But what if? X A ? Z

25. Induced paths
Conditioning on a collider creates (or contributes to) the association between the colliding variables L A Z K M A Z C
Why?
Later…

26. Induced paths An induced path may contain
Only dashed lines
Dashed lines and arrows
Colliders
An induced path may be blocked or open
An induced path is blocked
If there is at least one collider on the path
An induced path is open
If there are no colliders on the path

27. Blocked induced paths Blocked natural path Blocked induced path A C EZ A C E Z B D B D
Blocked natural path
Blocked induced path A C E Z A C E Z B D B D

28. Open induced paths Blocked natural path Open induced path C C A B Z A

29. Confounding bias and colliding bias
A confounding path contributes to the (marginal) association between A and Z
This unwanted contribution is called confounding bias
An open induced path contributes to the (conditional) association between A and Z
This unwanted contribution is called colliding bias

30. Can we block an open induced path? --Yes
Open induced paths
We can eliminate these paths
by conditioning on C C C A B Z A B Z A C E Z A C E Z B D B D

31. Key questions Why does a collider block a path?
Why don’t we observe an association between colliding variables?
Why does conditioning on a collider create an association between the colliding variables?
Blocked path
Open induced path A Z A Z C C

32. Intuitive explanation
A sample of N patients
Variables
M: meningitis status (yes, no)
S: stroke status (yes, no)
V: vital status (alive, dead)
Assume: causal reality is fully described in the diagram M S V

33. Is there a marginal (crude) association between meningitis status and stroke status?
No, we cannot guess stroke status from meningitis status (or vice versa)
Intuition: a common effect (vital status) cannot induce an association between its (past) causes
There is no transfer of guesses across a collider
A colliding path is a blocked path

34. Suppose we condition on V (vital status)…
Stratum 1 (V=alive) Stratum 2 (V=dead)
Alive patients Dead patients Pt
Stroke status
Vital status
Meningitis status 1 No
Alive ? Pt
Stroke status
Vital status
Meningitis status 2 No
Dead ?
My guess: “No”
My guess: “Yes”
We can make some guesses after conditioning
M (meningitis status) and S (stroke status) are associated within the strata of V (the collider)

35. Before and after conditioning…
Blocked path
Open induced path M S M S V V

36. Theorem and implications
Colliding variables will be associated within at least one stratum of their collider
Implications
a Mantel-Haenszel summary measure of association will differ from the crude, if we summarize across a collider
A regression coefficient will change if we “adjust” for a collider

37. Goal: estimate a measure of effect (causation) by a measure of association
Association is estimating causation (AZ) when:
The association between A and Z is due only to AZ
direct and indirect paths combined
Methods
Display variables and causal assumptions in a causal diagram
Block all confounding paths between A and Z
Do not create open induced paths between A and Z
or eliminate them, if created

38. Confounding bias (again)
The most widely known
Historical definitions and identification methods
“Lack of exchangeability”
“Mixed effects”
“Non-collapsibility”
“Change-in-estimate”
A fair amount of confusion
The basic causal structure C ? A Z

39. So what is a confounder? A B C D Z C B D A Z
A confounder is a common cause of the exposure (A) and the disease (Z) A B C D Z
Confounder
Note: we can block the path by conditioning on B or C or D. C B D A Z

40. Endless complexity Exposure: E0 (baseline exposure)
Disease: D2 (follow-up)
Question: Which is the confounder?
Q−3
Q−2
Q−1 Q0
E−3
E−2
E−1 E0 E1
D−2
D−1 D0 D1 D2

41. The basic causal structure
Colliding bias
Formerly known as “selection bias”
Confusing names and types
“No representativeness”
“Biased sample”
“Convenient sampling”
“Control-selection bias”
“Survival bias”
“Informative censoring”
The basic causal structure ? A Z C

42. But there are many more versionsX Y X C C ? ? A Z A Z X C ? C A Z A Z

43. Confounder versus colliderA Z
Collider

44. confounding bias and colliding bias: an antithetical pair
Confounder
Collider ? C A Z ? A Z C
Bias
No bias ? C A Z ? A Z C
No bias
Bias

45. Even more impressive in text…
Confounder
Collider
Main attribute
common cause
common effect
Association
contributes to the association between its effects
does not contribute to the association between its causes
Type of path
open path
blocked path
Effect of conditioning
Bias before conditioning?
Yes,
confounding bias No
Bias after conditioning?
colliding bias

46. What is selection bias? A type of colliding bias
Should be called “sampling colliding bias”

47. Types of colliding bias
Sampling colliding bias
Every study is restricted to selected people
Inevitable conditioning on “selection status” (S)
Sometimes, this unavoidable conditioning creates colliding bias
Analytical colliding bias
Restricted analysis: computing association for one stratum of a collider
Stratified analysis: computing association for each stratum of a collider
Adjustment by analysis
Computing a weighted average across the collider
Adding the collider to a regression model, as a covariate

48. Sampling colliding bias: a wrong sampling decision
What happens if we estimate the effect of marital status (A) on dementia status (Z) in a sample of nursing home residents?
Restricting recruitment to nursing home residents
Assumptions
No effect of A on Z
Both variables affect “place of residence” (P) (nursing home or elsewhere)

49. Causal diagrams (marital status) A (dementia status) Z (maritalP P P P S S
(Selection status)

50. Sampling colliding bias: a wrong sampling decision
What happens if we estimate the effect of coughing status (A) on abdominal pain status (Z) in a sample of hospitalized patients?
Restricting recruitment to hospitalized patients
Assumptions
Displayed in the diagram (next slide)
H is hospitalization status

51. Causal diagram S H H H H pneumonia status ulcer status ? A (coughingZ
(abdominal pain
status)

52. Basic causal diagrams for every case-control study
The key feature of a case-control study
Disease status affects selection into the case-control sample
Diseased people are much more likely to be selected than disease-free people ? ? A Z A Z S
(selection status) S
No bias, unless we mistakenly create
an open path between A and S!

53. Sampling colliding bias: a wrong sampling decision
Research question: What is the effect of smoking status (A) on cancer status (Z)?
Design: Hospital-based case-control study
Controls: patients with cardiovascular disease (CVD)

54. Causal diagram: smoking and cancer
status) A
(cancer
status) Z ?
Background knowledge
CVD status
Always exists in a
case-control study
Sampling decision for controls S
Note: CVD and Z collide at S

55. Colliding bias (AKA control selection bias)
(smoking
status) A
(cancer
status) Z ?
CVD status S

56. Willing to participate?
Sampling colliding bias: Willingness to participate in a case-control study
(smoking
status) A
(cancer
status) Z ?
Background knowledge
Willing to participate? S

57. Control (or case) selection bias
Two main mechanisms ? A Z A Z B B
Sampling/participation
of controls (or cases) S
Sampling/participation
of controls (or cases) S
Remember:
ZS always exists
We always condition on S

58. Types of colliding bias
Sampling colliding bias
Every study is restricted to selected people
Inevitable conditioning on “selection status” (S)
Sometimes, this unavoidable conditioning creates colliding bias
Analytical colliding bias
Restricted analysis: computing association for one stratum of a collider
Stratified analysis: computing association for each stratum of a collider
Adjustment by analysis
Computing a weighted average across the collider
Adding the collider to a regression model, as a covariate

59. Analytical colliding bias: restricted analysis
Research question: what is the effect of dietary fibers on colon polyp?
Design: a cross-sectional study
Analysis: restricted to people who have not developed yet colon cancer

60. Causal diagram (Dietary fibers) (Colon polyp status) A ? Z
Assumed knowledge
Assumed knowledge
Colon cancer status
Note: A and Z collide at colon cancer status

61. Analytical colliding bias
(Dietary fibers) A
(Colon polyp status) Z ?
Colon cancer status
Despite “intuition” we should not restrict the sample to
cancer-free people

62. Analytical colliding bias: adjustment
“We adjusted for everything, but the kitchen sink”
Traditional steps
Add a laundry list of covariates to the regression model
See what happens to the exposure coefficient
Use the “change-in-estimate” method
Change in the coefficient = Evidence for confounding
Report the “adjusted” coefficient as a better (less confounded) measure of effect
Prone to colliding bias

63. Analytical colliding bias
Research question: what is the overall effect of gender on blood pressure?
Design: a cross-sectional study
Analysis
Crude mean difference in systolic blood pressure
“Adjusted” mean difference (conditioned on waist circumference)

64. Mean difference (mmHg)
Results
Analysis
Mean SBP men (mmHg)
Mean SBP women (mmHg)
Mean difference (mmHg)
Crude
123.8
122.1
1.7
“Adjusted” for waist circumference
-3.1
Why do the estimates differ?
Which estimate should be reported?
Is the adjusted estimate less biased?

65. Is abdominal fat (measured by waist circumference) a confounder?
No! A
(Gender) Z
(Blood pressure)

66. Revised diagram No need to “adjust for” abdominal fatC A
(Gender) Z
(Blood pressure)
No need to “adjust for” abdominal fat
“Adjustment” could have:
blocked a causal path
created colliding bias

67. Could have blocked a causal path…
(abdominal fat) C A
(Gender) Z
(Blood pressure)

68. Could have created colliding bias…
(abdominal fat) C A
(Gender) Z
(Blood pressure)

69. Advice on multivariable regression
Do not adjust for an effect of the exposure
Do not adjust for an effect of the outcome
Select covariates according to theory (causal diagram), not mechanistically (change in estimate, stepwise regression)
“Every variable is adjusted for all others” is almost always false
Confounding is not a reciprocal property

70. Key points
The essence of epidemiology (and all of science) is causal theories
Your theories (about causation) are not “A is associated with Z”
“Possessing a cigarette lighter is associated with lung cancer” is true, but who cares? That’s not causal knowledge
Your theories about bias are not “intuition” about bias; they are causal theories, too.
Almost every theory in science is about causation, which means an arrow between variables

71. Key points
Magnitude of bias is more important than merely its presence
Small bias may be ignored
Magnitude of bias may be difficult to estimate
The bias-variance tradeoff