Causal roles and adjustment sets

Whether a control variable belongs in a regression depends on its relationships with other variables. DAGassist labels each variable with a causal role, then builds the adjustment sets those roles imply. This article explains every role, which ones to adjust for, and how DAGassist builds the minimal and canonical adjustment sets. Our causal role classification framework follows Cinelli et al. (2022)’s taxonomy of good, bad, and neutral controls.

library(DAGassist)

A DAG with every role

To see every role at once, we write a small DAG in dagitty syntax. Each letter stands for one role: X is the exposure and Y the outcome.

roles_dag <- dagitty::dagitty("dag {
  X [exposure]
  Y [outcome]
  X -> Y
  Z -> X
  Z -> Y
  Z -> V -> Y
  Z -> W
  X -> M -> Y
  M -> N
  X -> C <- Y
  C -> Q
  Y -> D
  E -> X
  F -> Y
}")

Reading the roles table

Roles depend only on the DAG, so this step needs no data:

DAGassist(roles_dag, show = "roles", verbose = FALSE)
#> DAGassist Report: 
#> 
#> Roles:
#> variable  role        Exp.  Out.  conf  med  col  dOut  dMed  dCol  dConfOn  dConfOff  NCT  NCO
#> X         exposure    x                                                                        
#> Y         outcome           x                                                                  
#> Z         confounder              x                                                            
#> M         mediator                      x                                                      
#> C         collider                           x    x     x                                      
#> W         Dconf_off                                                          x                 
#> D         dOut                                    x     x                                      
#> E         nct                                                                          x       
#> F         nco                                                                               x  
#> N         Dmediator                                     x                                      
#> Q         dOut                                    x     x     x                                
#> V         nco                                                       x                       x  
#> 
#> Legend hidden because verbose = FALSE. Re-run with verbose = TRUE to see role definitions.

Each column is a role, and an x marks the roles a variable plays. The table below defines each role and the causal inference implications of adjusting for each when the target is the total effect of X on Y. Note that a variable can play more than one role, but the role column reports the most identification-relevant one. For example, if a variable were a descendant of a mediator and also a neutral control on the outcome, its primary role would be mediator because mediators shift estimands and neutral controls do not.

Column	Role	Example	What it is	Adjust for it?
conf	Confounder	Z	A common cause of X and Y	Yes. It opens a back-door path (X <- Z -> Y) that must be blocked.
dConfOn	Descendant of a confounder, on a back-door path	V	Caused by a confounder, and causes Y	Safe. It blocks the part of the back-door path that runs through it.
NCO	Neutral control on the outcome	F	Causes Y but not X	Safe, and usually improves precision.
NCT	Neutral control on the treatment	E	Causes X but not Y	Safe for bias, but usually widens standard errors.
dConfOff	Descendant of a confounder, off the back-door path	W	Caused by a confounder, but doesn’t affect Y	Irrelevant. It neither removes nor adds bias.
med	Mediator	M	On a causal path, X -> M -> Y	No. It removes part of the effect, turning the total effect into a direct effect.
dMed	Descendant of a mediator	N	Caused by a mediator	No. It partly controls for the mediator.
col	Collider	C	Caused by both X and Y	No. It opens a non-causal path between X and Y.
dCol	Descendant of a collider	Q	Caused by a collider	No. It partly opens the same path.
dOut	Descendant of the outcome	D	Caused by Y	No. It biases the estimate.

Minimal and canonical adjustment sets

An adjustment set is a set of controls that blocks every back-door path from X to Y without including anything X affects. DAGassist finds these sets with dagitty (Textor et al. 2016) and fits one model for each:

• A minimal set contains no redundant variables: remove any one, and some back-door path reopens. A DAG can have several minimal sets, and DAGassist fits a model for each (“Minimal 1”, “Minimal 2”, and so on).
• The canonical set contains every variable that causes X or Y, except those the exposure affects (Zander et al. 2014). It is unique, and it is a valid adjustment set whenever any valid set exists.

For roles_dag:

dagitty::adjustmentSets(roles_dag, type = "minimal")
#> { Z }
dagitty::adjustmentSets(roles_dag, type = "canonical")
#> { E, F, V, Z }

The minimal set is just the confounder. The canonical set adds V and the two neutral controls, and leaves out everything marked “do not adjust” as well as the irrelevant W.

Role	Example	Minimal	Canonical
Confounder	Z	✓	✓
Descendant of a confounder, on a back-door path	V	*	✓
Neutral control on the outcome	F		✓
Neutral control on the treatment	E		✓
Descendant of a confounder, off the back-door path	W
Mediator, descendant of a mediator	M, N	✗	✗
Collider, descendant of a collider	C, Q	✗	✗
Descendant of the outcome	D	✗	✗

✓ = included. Blank = left out, but harmless to add. ✗ = left out because adjusting for it biases the total effect.

* V could replace Z in a minimal set only if every back-door path through Z also ran through V. In roles_dag, Z also affects Y directly, so it can’t.

Why the two sets can give different answers

Both sets remove confounding, so both should recover the same effect. They differ in precision. The turnout example on the homepage shows this: its canonical set adds a neutral control on the outcome (elect_comp) and a neutral control on the treatment (industry) to the minimal set {age, state}.

DAGassist(turnout_dag,
          lm(turnout ~ income + state + age + polint + industry + elect_comp,
             data = turnout_data),
          show = "models", type = "text", verbose = FALSE)

Term	Original	Minimal 1	Canonical
income	0.281***	0.493***	0.492***
	(0.016)	(0.016)	(0.015)
state	0.331***	0.324***	0.332***
	(0.017)	(0.019)	(0.018)
age	0.275***	0.273***	0.267***
	(0.017)	(0.020)	(0.019)
polint	0.420***
	(0.014)
industry	-0.017		-0.010
	(0.015)		(0.016)
elect_comp	0.500***		0.506***
	(0.014)		(0.015)
Num.Obs.	5000	5000	5000
R2	0.596	0.423	0.525

• p-value legend: + < 0.1, * < 0.05, ** < 0.01, *** < 0.001.
• Controls (minimal): {age, state}.
• Controls (canonical): {age, elect_comp, industry, state}.

The minimal and canonical estimates agree (0.49), and both differ sharply from the original model, which controls for the mediator polint. Here the canonical standard error is slightly smaller (0.015 vs. 0.016). If the DAG is right, the minimal and canonical estimates should differ only by sampling noise, so a large gap between them suggests the DAG is wrong somewhere. The robustness article shows how to probe that.

References

Cinelli, Carlos, Andrew Forney, and Judea Pearl. 2022. “A Crash Course in Good and Bad Controls.” Sociological Methods & Research, ahead of print. https://doi.org/10.1177/00491241221099552.

Textor, Johannes, Benito van der Zander, Mark S. Gilthorpe, Maciej Liśkiewicz, and George T. H. Ellison. 2016. “Robust Causal Inference Using Directed Acyclic Graphs: The R Package ‘Dagitty’.” International Journal of Epidemiology 45 (6): 1887–94. https://doi.org/10.1093/ije/dyw341.

Zander, Benito van der, Maciej Liśkiewicz, and Johannes Textor. 2014. “Constructing Separators and Adjustment Sets in Ancestral Graphs.” Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence.