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Introduction to Directed Acyclic Graphs
Causal Inference Shared Interest Group
January 19th, 2016
Overview/Goals
• Provide introduction on the use of DAGs for confounder
selection in nonexperimental studies
– What DAGs offer; limitations
– Overview of DAG terminology and rules
– Overview of d-separation criteria for assessing open and closed paths
– Share approaches for selecting minimum adjustment set
• Share a few examples of DAGs and when/why other
approaches to covariate selection may fail
• Have a broad discussion on the use of DAGs; hope for
this to be a conversation
Disclaimers/Focus
• Most examples can be found in:
– Modern Epidemiology 3rd edition (Chapter 12)
– Williamson et al. (2014) Introduction to casual diagrams for
confounder selection
– Hernan et al. (2002) Causal knowledge as a prerequisite for
confounding evaluation: an application to birth defects epidemiology
• Will not be covering mathematical proofs supporting use
of DAGs for causal inference or grounding in
counterfactual definition of causation
• Will focus on use of DAGs for confounder selection; many
other uses (e.g. selection bias, instrumental variable selection,
measurement error, mediation/moderation, etc. – see Ch. 12 Modern Epi)
What is a DAG?
What is a DAG?
• A casual diagram (e.g. graph) depicting the investigator's
assumptions about casual relations among the exposure,
outcome and covariates…that’s it!
• Arrows are used to denote directionality
• Contain only unidirectional arrows (single headed)
• Must be acyclic; contain no feedback loops
• Include all endogenous variables even if unknown (U)
Why DAGs?
• Because we are often interested in estimating casual
affects from observational data
– Not an easy task as sample associations are directly
observable, but causation is not
– Four possible casual structures can contribute to an
association between X and Y
• 1.) X may cause Y; what we are interested in
• 2.) Y may cause X; fairly easy to rule out
• 3.) X and Y may share a common cause we have failed to condition
on; this is classical confounding
• 4.) we have conditioned on a collider (variable affected by X and Y)
• *Associations may also be due to chance (i.e. sampling variability)
Why DAGs?
• DAGs provide a powerful and intuitive tool for deducing the
statistical associations implied in a hypothesized causal
network (without requiring any mathematics)
• Thus, they help us to select which variables we need to
condition on (e.g. control for) to obtain unbiased estimates in
nonexperimental studies
• If our DAG is correct, and all important factors can be
measured, we can identify the minimum adjustment set
necessary to achieve “conditional exchangeability”
– e.g. exposure is essentially randomized within levels of covariates
– And causal inference can be made from the observed data
Why DAGs
• Approaches to confounder selection based on statistical
associations may fail to ID important confounders and support
adjusting on non-confounders
– Automatic variable selection; implicit assumption is that not all
variables selected will be confounders, but all important confounders
will be selected
– Change-in-estimate criteria; implicit assumption is that any variable
associated with a change is worth adjusting for
– Traditional confounder definition – i.) associated w/ exposure in source
population, ii) associated with outcome among the unexposed, and iii.)
not on the casual pathway
– Intermediate, collider, and instrumental variables can all behave
statistically like confounders – need knowledge to guide selection!
Limitations of DAGs
• How can we know the true underlying causal
structure/network to draw the DAG?
– Can’t be known; If knew we won’t need the study!
– However, true casual structure exists even if we do not
know it…
• All casual inference based on statistical models are implicitly based
on some casual structure
• DAGs simply make these assumptions more explicit
• Can compare competing casual models to assess their
compatibility with the observed data
• From the observed statistical associations can also deduce causal
structures compatible with the observations/data
Limitations of DAGs
• Drawing DAGs can be challenging as they need to be
developed in the context of all available evidence
– Often requires multidisciplinary input; and well developed theory/area
• Latent variables can pose problems; causes and affects harder
to deduce; can’t measure directly also harder to condition on
• Effect modification “difficultish” to represent since each
variable represented by a single node
– Often represented by single node conceptually representing both
variables; may necessitate multiple DAGs; current area of research
• DAGs provide no information on the effect sizes or functional
forms
– Simply nonparametric graphical representations of casual networks
DAG Terminology
• Variables are nodes
• Arrows are edges and
imply direction
• Paths are unbroken
sequences of arrows
linking nodes regardless
of direction
– X > Y > Z
– U > Y > Z
– U > Y < X
X
U
Y Z
DAG Terminology
• X is said to directly affect Y if there is an arrow from X to Y
• X indirectly affects Z if there is a unidirectional path from X to
Z (X>Y>Z)
• Y is an “intermediate” variable between X and Z
• Y is also said to “intercept” the path from (X > Y > Z) or from
(U > Y > Z)
DAG Terminology
• Children of X are variables directly affected by X; {Y}
• Parents of Y are variables that directly affect Y; {X, U}
• Descendants of X are all variables either directly or indirectly
affected by X; {Y, Z}, but not U
• Ancestors of Z are all variables that either directly or indirectly
affect Z; {Y, X, U}
DAG Terminology
• Directed paths are the special case where all edges in path
flow head-to-tail (unidirectional); these are causal pathways
– Ex. (X > Y), (X > Y > Z), (U > Y), (U > Y > Z)
• All undirected paths are non-causal; (U > Y < X)
• A variable on the path where two arrows meet is called a
“collider”; child of both the variable before and after it
DAG Terminology
• Nodes where multiple arrows meet are called “colliders”;
node has multiple parents (U > Y < X)
• Traditional terminology is that Y is a common effect of U and
X; e.g. Y is child to both
• Association is not transmitted across common effects!
– Because two factors share a common effect doesn’t mean associated
– No need to condition on collider; path is closed
• Colliders cause special problems for causal inference
Collider bias
• Conditioning on a collider results in collider bias because it
opens a backdoor path across the colliding nodes
• Basically, what happens is that two variables that were
originally marginally independent become dependent
conditional on Z
• Thus, not only is adjustment not needed, but can be harmful
• Let’s look at a hypothetical example…
Example Hernan 2002
• X represents being on a diet
• Y represents recent diagnosis of non-diet-related cancer
• Z represent recent weight loss > 5kg (1=yes, 0=no)
• Assume X does not cause Y; Z common cause of X and Y
• X and Y are marginally independent in the source population
– Table 1 below
• RRXY = (100/(100+100)) / (200/(200+200)) = 1
• Knowing that someone was dieting does not change prob. of cancer
Y=1 Y=0
X=1 100 100
X=0 200 200
Example Hernan 2002
• Now we condition on recent weight loss (e.g. collider)
• Given that someone lost weight, it becomes more likely that
she had cancer if is she was not dieting
– Data here are made-up, but the above conditional association is
intuitive
• Thus, for those that lost weight, being on a diet and being
recently diagnosed with cancer are inversely related
• RRXY|C=1 = 0.79 RRXY|C=0 = 0.92
Z=1
D=1 D=0
X=1 55 25
X=0 70 10
Z=0
D=1 D=0
X=1 45 75
X=0 130 190
Examples of confounding and collider bias
- Top: No effect of A on Y; no effect of A
on Y observed; path blocked at collider
C4; no bias
- Middle: No effect of A on Y; potential
association between A and Y observed;
conditioning on collider C4; bias
- Bottom: No effect A on Y; A < C4 > Y
represents classical confounding, but
adjustment opens up A < C1 > C4 < C3
< C2 > Y path; A Y association
observed; bias (need also adjust C1)
d-separation criteria
• Stands for “direction” separation criteria a.k.a. “directed graph
separation rules”; this is one of the reasons we care about DAGs!
• Two variables are d-connected if there is an open path between
them; they are d-separated if all paths are closed
• If two variables are d-separated then by definition they will be
unassociated; no path from X to Y
• If two variables are d-connected there is an open path implying a
marginal association between them
– Open directed pathways between X and Y are what we want to estimate!
– We also want to control for all undirected open pathways between X and Y
d-separation criteria
• Often defined in terms of unconditional and conditional separation
• Unconditional d-separation
– Path is open if there are no colliders on the path
– If collider on path; closed unconditionally ; collider block path
– Thus, all directed paths are open (as can have no collider)
• Conditional d-separation
– Conditioning on a non-collider Z, blocks the path at Z
– Conditioning on a collider, or decedent of a collider, open the path
• Combining these criteria allows us to identify the set of covariates
that will block confounding paths!
Williamson d-separation summary
Let’s look at some examples
• Interested in association b/w
smoking and adult asthma
• Here truth assumed to be
known
• Smoking is not a cause of
asthma
• Do we need to adjust for
childhood asthma?
– Smk < cAsthma > aAsthma is
open
– Estimate will be biased
– Need to condition in cAsthma
– Classic confounding bias
Let’s look at some examples
• What if the true casual
structure were different?
• What do we condition on
here?
• Nothing – path blocked at
collider parent smk > cAsthma
< atopy; unbiased
• If adjust for cAsthma here
open up backdoor path; bias
• Would have to also condition
on parental smoking; unbiased
– Minimum set = {parent
smoking and cAsthma}
Let’s look at some examples
• What of the casual
structure were different?
• What to condition on?
• Path from personal smoking
< cAsthma < atopy >
aAsthma is now open; bias
• Colliders are path specific!
• Minimal sufficient
adjustment sets
– {atopy}
– {cAsthma, parent smoking}
Bit more realistic example; still simple
- Might need some rules to figure out what to condition on
here!
DAGs can get highly complex!
-Definitely going to need some rules here!
DAGs can get highly complex!
-And here!
Pearl’s Rules to ID Minimal Adjustment Set
Pearl’s Rules to ID Minimal Adjustment Set
Pearl’s Rules to ID Minimal Adjustment Set
• Rules provide a formulaic approach to identifying a
minimal adjustment set
• Still need to start by guessing at the set
• Still likely to be a bit confusing at first
• Lucky there are some terrific open-source software
packages that will do this for us!
– DAGitty: web platform – very easy to use
– dagR package in R
DAGitty Example
• I made this example in about 60 sec.
• Red shows biasing path
Example of when other approaches may fail
• Example taken from Hernan et al. (2002) Causal Knowledge as
a Prerequisite for Confounding Evaluation
• Examined relation between folic acid supplementation and
neural tube defects in the Slone Epidemiology Birth Defects
Study (1992-1997 to rule out impact of fortification)
• Cases were mothers who’s infants had neural tube defects;
controls delivered infants with non-folic acid related defects
• Exposure was folic acid supplementation (yes/no)
• C is a potential confounder, unrevealed for now, but known to
not be on the casual pathway from exposure to disease
• For simplicity assume all variables measured w/o error
Example of when other approaches may fail
• Automatic variable selection: add if p-value < 0.10; covariate meets
this criteria
• Change-in-estimate criteria: changes OR by ~15%; covariate meets
criteria
• Traditional confounding definition: meets all criteria
• Unadjusted OR = 0.65 (0.45; 0.94); inversely related
• Adjusted OR = 0.80 (0.53; 1.20) get some attenuation
• What is the covariate? Whether pregnancy ends in still
birth/therapeutic abortion
– Do we think it is a confounder? Common cause of supplementation use
and neutral tube defects? Not likely, but commonly adjusted for
– Theory suggest loss of birth common cause of low folic acid; collider
– Analogous to restricting subjects to just live births!
Good Resources
• Introductory papers:
– Williamson E, Aitken Z, Lawrie J, Dharmage S, Burgess J,
Forbes A. Introduction to causal diagrams for confounder
selection.Respirology. 2014;19(3):303–311.
– Hernán MA, Hernández-Díaz S, Werler MM, Mitchell AA.
Causal knowledge as a prerequisite for confounding
evaluation: an application to birth defects
epidemiology. Am J Epidemiol. 2002;155(2):176–84.
– Shrier I, Platt R. Reducing bias through directed acyclic
graphs. Bmc Med Res Methodol. 2008;8(1):70.
doi:10.1186/1471-2288-8-70.
• Chapter 12: Modern Epidemiology 3rd edition

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Introduction to Directed Acyclic Graphs.pptx

  • 1. Introduction to Directed Acyclic Graphs Causal Inference Shared Interest Group January 19th, 2016
  • 2. Overview/Goals • Provide introduction on the use of DAGs for confounder selection in nonexperimental studies – What DAGs offer; limitations – Overview of DAG terminology and rules – Overview of d-separation criteria for assessing open and closed paths – Share approaches for selecting minimum adjustment set • Share a few examples of DAGs and when/why other approaches to covariate selection may fail • Have a broad discussion on the use of DAGs; hope for this to be a conversation
  • 3. Disclaimers/Focus • Most examples can be found in: – Modern Epidemiology 3rd edition (Chapter 12) – Williamson et al. (2014) Introduction to casual diagrams for confounder selection – Hernan et al. (2002) Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology • Will not be covering mathematical proofs supporting use of DAGs for causal inference or grounding in counterfactual definition of causation • Will focus on use of DAGs for confounder selection; many other uses (e.g. selection bias, instrumental variable selection, measurement error, mediation/moderation, etc. – see Ch. 12 Modern Epi)
  • 4. What is a DAG?
  • 5. What is a DAG? • A casual diagram (e.g. graph) depicting the investigator's assumptions about casual relations among the exposure, outcome and covariates…that’s it! • Arrows are used to denote directionality • Contain only unidirectional arrows (single headed) • Must be acyclic; contain no feedback loops • Include all endogenous variables even if unknown (U)
  • 6. Why DAGs? • Because we are often interested in estimating casual affects from observational data – Not an easy task as sample associations are directly observable, but causation is not – Four possible casual structures can contribute to an association between X and Y • 1.) X may cause Y; what we are interested in • 2.) Y may cause X; fairly easy to rule out • 3.) X and Y may share a common cause we have failed to condition on; this is classical confounding • 4.) we have conditioned on a collider (variable affected by X and Y) • *Associations may also be due to chance (i.e. sampling variability)
  • 7. Why DAGs? • DAGs provide a powerful and intuitive tool for deducing the statistical associations implied in a hypothesized causal network (without requiring any mathematics) • Thus, they help us to select which variables we need to condition on (e.g. control for) to obtain unbiased estimates in nonexperimental studies • If our DAG is correct, and all important factors can be measured, we can identify the minimum adjustment set necessary to achieve “conditional exchangeability” – e.g. exposure is essentially randomized within levels of covariates – And causal inference can be made from the observed data
  • 8. Why DAGs • Approaches to confounder selection based on statistical associations may fail to ID important confounders and support adjusting on non-confounders – Automatic variable selection; implicit assumption is that not all variables selected will be confounders, but all important confounders will be selected – Change-in-estimate criteria; implicit assumption is that any variable associated with a change is worth adjusting for – Traditional confounder definition – i.) associated w/ exposure in source population, ii) associated with outcome among the unexposed, and iii.) not on the casual pathway – Intermediate, collider, and instrumental variables can all behave statistically like confounders – need knowledge to guide selection!
  • 9. Limitations of DAGs • How can we know the true underlying causal structure/network to draw the DAG? – Can’t be known; If knew we won’t need the study! – However, true casual structure exists even if we do not know it… • All casual inference based on statistical models are implicitly based on some casual structure • DAGs simply make these assumptions more explicit • Can compare competing casual models to assess their compatibility with the observed data • From the observed statistical associations can also deduce causal structures compatible with the observations/data
  • 10. Limitations of DAGs • Drawing DAGs can be challenging as they need to be developed in the context of all available evidence – Often requires multidisciplinary input; and well developed theory/area • Latent variables can pose problems; causes and affects harder to deduce; can’t measure directly also harder to condition on • Effect modification “difficultish” to represent since each variable represented by a single node – Often represented by single node conceptually representing both variables; may necessitate multiple DAGs; current area of research • DAGs provide no information on the effect sizes or functional forms – Simply nonparametric graphical representations of casual networks
  • 11. DAG Terminology • Variables are nodes • Arrows are edges and imply direction • Paths are unbroken sequences of arrows linking nodes regardless of direction – X > Y > Z – U > Y > Z – U > Y < X X U Y Z
  • 12. DAG Terminology • X is said to directly affect Y if there is an arrow from X to Y • X indirectly affects Z if there is a unidirectional path from X to Z (X>Y>Z) • Y is an “intermediate” variable between X and Z • Y is also said to “intercept” the path from (X > Y > Z) or from (U > Y > Z)
  • 13. DAG Terminology • Children of X are variables directly affected by X; {Y} • Parents of Y are variables that directly affect Y; {X, U} • Descendants of X are all variables either directly or indirectly affected by X; {Y, Z}, but not U • Ancestors of Z are all variables that either directly or indirectly affect Z; {Y, X, U}
  • 14. DAG Terminology • Directed paths are the special case where all edges in path flow head-to-tail (unidirectional); these are causal pathways – Ex. (X > Y), (X > Y > Z), (U > Y), (U > Y > Z) • All undirected paths are non-causal; (U > Y < X) • A variable on the path where two arrows meet is called a “collider”; child of both the variable before and after it
  • 15. DAG Terminology • Nodes where multiple arrows meet are called “colliders”; node has multiple parents (U > Y < X) • Traditional terminology is that Y is a common effect of U and X; e.g. Y is child to both • Association is not transmitted across common effects! – Because two factors share a common effect doesn’t mean associated – No need to condition on collider; path is closed • Colliders cause special problems for causal inference
  • 16. Collider bias • Conditioning on a collider results in collider bias because it opens a backdoor path across the colliding nodes • Basically, what happens is that two variables that were originally marginally independent become dependent conditional on Z • Thus, not only is adjustment not needed, but can be harmful • Let’s look at a hypothetical example…
  • 17. Example Hernan 2002 • X represents being on a diet • Y represents recent diagnosis of non-diet-related cancer • Z represent recent weight loss > 5kg (1=yes, 0=no) • Assume X does not cause Y; Z common cause of X and Y • X and Y are marginally independent in the source population – Table 1 below • RRXY = (100/(100+100)) / (200/(200+200)) = 1 • Knowing that someone was dieting does not change prob. of cancer Y=1 Y=0 X=1 100 100 X=0 200 200
  • 18. Example Hernan 2002 • Now we condition on recent weight loss (e.g. collider) • Given that someone lost weight, it becomes more likely that she had cancer if is she was not dieting – Data here are made-up, but the above conditional association is intuitive • Thus, for those that lost weight, being on a diet and being recently diagnosed with cancer are inversely related • RRXY|C=1 = 0.79 RRXY|C=0 = 0.92 Z=1 D=1 D=0 X=1 55 25 X=0 70 10 Z=0 D=1 D=0 X=1 45 75 X=0 130 190
  • 19. Examples of confounding and collider bias - Top: No effect of A on Y; no effect of A on Y observed; path blocked at collider C4; no bias - Middle: No effect of A on Y; potential association between A and Y observed; conditioning on collider C4; bias - Bottom: No effect A on Y; A < C4 > Y represents classical confounding, but adjustment opens up A < C1 > C4 < C3 < C2 > Y path; A Y association observed; bias (need also adjust C1)
  • 20. d-separation criteria • Stands for “direction” separation criteria a.k.a. “directed graph separation rules”; this is one of the reasons we care about DAGs! • Two variables are d-connected if there is an open path between them; they are d-separated if all paths are closed • If two variables are d-separated then by definition they will be unassociated; no path from X to Y • If two variables are d-connected there is an open path implying a marginal association between them – Open directed pathways between X and Y are what we want to estimate! – We also want to control for all undirected open pathways between X and Y
  • 21. d-separation criteria • Often defined in terms of unconditional and conditional separation • Unconditional d-separation – Path is open if there are no colliders on the path – If collider on path; closed unconditionally ; collider block path – Thus, all directed paths are open (as can have no collider) • Conditional d-separation – Conditioning on a non-collider Z, blocks the path at Z – Conditioning on a collider, or decedent of a collider, open the path • Combining these criteria allows us to identify the set of covariates that will block confounding paths!
  • 23. Let’s look at some examples • Interested in association b/w smoking and adult asthma • Here truth assumed to be known • Smoking is not a cause of asthma • Do we need to adjust for childhood asthma? – Smk < cAsthma > aAsthma is open – Estimate will be biased – Need to condition in cAsthma – Classic confounding bias
  • 24. Let’s look at some examples • What if the true casual structure were different? • What do we condition on here? • Nothing – path blocked at collider parent smk > cAsthma < atopy; unbiased • If adjust for cAsthma here open up backdoor path; bias • Would have to also condition on parental smoking; unbiased – Minimum set = {parent smoking and cAsthma}
  • 25. Let’s look at some examples • What of the casual structure were different? • What to condition on? • Path from personal smoking < cAsthma < atopy > aAsthma is now open; bias • Colliders are path specific! • Minimal sufficient adjustment sets – {atopy} – {cAsthma, parent smoking}
  • 26. Bit more realistic example; still simple - Might need some rules to figure out what to condition on here!
  • 27. DAGs can get highly complex! -Definitely going to need some rules here!
  • 28. DAGs can get highly complex! -And here!
  • 29. Pearl’s Rules to ID Minimal Adjustment Set
  • 30. Pearl’s Rules to ID Minimal Adjustment Set
  • 31. Pearl’s Rules to ID Minimal Adjustment Set • Rules provide a formulaic approach to identifying a minimal adjustment set • Still need to start by guessing at the set • Still likely to be a bit confusing at first • Lucky there are some terrific open-source software packages that will do this for us! – DAGitty: web platform – very easy to use – dagR package in R
  • 32. DAGitty Example • I made this example in about 60 sec. • Red shows biasing path
  • 33. Example of when other approaches may fail • Example taken from Hernan et al. (2002) Causal Knowledge as a Prerequisite for Confounding Evaluation • Examined relation between folic acid supplementation and neural tube defects in the Slone Epidemiology Birth Defects Study (1992-1997 to rule out impact of fortification) • Cases were mothers who’s infants had neural tube defects; controls delivered infants with non-folic acid related defects • Exposure was folic acid supplementation (yes/no) • C is a potential confounder, unrevealed for now, but known to not be on the casual pathway from exposure to disease • For simplicity assume all variables measured w/o error
  • 34. Example of when other approaches may fail • Automatic variable selection: add if p-value < 0.10; covariate meets this criteria • Change-in-estimate criteria: changes OR by ~15%; covariate meets criteria • Traditional confounding definition: meets all criteria • Unadjusted OR = 0.65 (0.45; 0.94); inversely related • Adjusted OR = 0.80 (0.53; 1.20) get some attenuation • What is the covariate? Whether pregnancy ends in still birth/therapeutic abortion – Do we think it is a confounder? Common cause of supplementation use and neutral tube defects? Not likely, but commonly adjusted for – Theory suggest loss of birth common cause of low folic acid; collider – Analogous to restricting subjects to just live births!
  • 35. Good Resources • Introductory papers: – Williamson E, Aitken Z, Lawrie J, Dharmage S, Burgess J, Forbes A. Introduction to causal diagrams for confounder selection.Respirology. 2014;19(3):303–311. – Hernán MA, Hernández-Díaz S, Werler MM, Mitchell AA. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. Am J Epidemiol. 2002;155(2):176–84. – Shrier I, Platt R. Reducing bias through directed acyclic graphs. Bmc Med Res Methodol. 2008;8(1):70. doi:10.1186/1471-2288-8-70. • Chapter 12: Modern Epidemiology 3rd edition