What is a directed acyclic graph in epidemiology?

A directed acyclic graph is a diagram of causal assumptions in which each variable is a node, each arrow is a direct cause pointing one way, and no path returns to its starting point. In epidemiology, a directed acyclic graph is used to decide which variables to adjust for, by making the assumptions behind that choice visible and criticisable.

The formal apparatus arrived with Greenland, Pearl and Robins, “Causal diagrams for epidemiologic research” (Epidemiology, 1999, 10:37–48), which translated Pearl’s graphical machinery into the language of confounding and adjustment that epidemiologists already used. This page covers what the three words mean, what a directed acyclic graph does and does not encode, the three elementary structures, the back-door criterion, where the arrows come from, how a mishandled graph is spotted in a manuscript, what reviewers say about it, and the limitations worth stating in your own discussion.

What the three words mean

A directed acyclic graph earns each of its three words, and each word is a substantive restriction rather than notation.

Directed. Every edge is an arrow with a head and a tail, and A → Y asserts that A is a direct cause of Y relative to the other variables in the graph. A line without an arrowhead is not permitted; an undirected edge would say the two variables are associated without saying why, which is precisely the information the diagram exists to record.

Acyclic. No sequence of arrows leads from a variable back to itself. Feedback is therefore not drawn as a loop. Time-varying treatment and time-varying confounding — where treatment at baseline affects a covariate that then affects treatment at follow-up — are represented by unrolling time into separate nodes, A₀ → L₁ → A₁ → Y, which remains acyclic because the baseline and follow-up measurements are different variables. Anyone who has drawn a two-headed arrow between two nodes has drawn something that is not a directed acyclic graph; the correct representation of a mutual dependence is an unmeasured common cause, A ← U → Y.

Graph. The object is a graph in the mathematical sense, so paths, ancestors, descendants and separation are defined and can be computed. The consequence matters: adjustment sets are derived by algorithm from the drawing rather than argued for in prose.

What a directed acyclic graph encodes, and what it does not

A directed acyclic graph is nonparametric. An arrow carries no functional form, no sign and no magnitude: Smoking → Mortality says only that some individuals’ mortality would differ under a change in their smoking, not whether the effect is linear, positive, or large.

The strong claims in a causal diagram are the arrows that are absent. An arrow drawn between two nodes commits the author to almost nothing, because it merely allows an effect. A missing arrow asserts no direct effect at all — an exact, falsifiable null. This inverts most authors’ intuition, and it is the reason a densely connected graph is a weak one: it asserts almost nothing and identifies almost nothing.

A valid causal directed acyclic graph must also include every common cause of any two variables already drawn, whether or not that common cause was measured. Unmeasured common causes are drawn as U nodes and left in the figure. Omitting them does not make the assumption of no unmeasured confounding go away; it only makes the assumption invisible.

Three things a directed acyclic graph does not encode: effect size, effect modification (a graph cannot distinguish additive from multiplicative interaction), and measurement error unless the true variable and its measured proxy are drawn as separate nodes with an arrow between them.

The three elementary structures

Any path in a directed acyclic graph is built from three configurations, and the rules for conditioning differ across all three. Naming which one you are looking at is most of the practical work.

Chain, A → M → Y. M transmits the effect of A on Y. Conditioning on M blocks the path and removes the indirect effect from the estimate, which is why adjusting for a mediator shrinks a total effect towards the null.

Fork, A ← C → Y. C is a common cause and the path is open without conditioning, producing a non-causal association between A and Y. Conditioning on C blocks it. This is confounding in graphical form, and it is the only one of the three where adjustment helps.

Collider, A → S ← Y. Two arrowheads meet at S. The path is blocked by default, and conditioning on S opens it, inducing an association between A and Y where none existed. Conditioning includes stratification, restriction, sample selection, and complete-case analysis after differential dropout — see collider bias.

The collider rule has a second clause that is routinely forgotten: conditioning on a descendant of a collider also partially opens the path. A downstream proxy of a selection variable behaves like the selection variable itself, which is why “we adjusted for a correlate of admission rather than admission” is not a defence.

The back-door criterion and minimal sufficient adjustment sets

The back-door criterion, stated by Pearl (Biometrika, 1995, 82:669–688) and carried into epidemiological practice by Greenland, Pearl and Robins in 1999, gives the condition under which a set of covariates Z identifies the causal effect of A on Y: Z must block every path that enters A through an arrowhead — every “back-door” path — and Z must contain no descendant of A.

Two consequences follow that most covariate lists violate. First, “block every back-door path” is a statement about paths, not about variables, so a variable may be needed only because of one path and harmful because of another; the graph resolves that conflict, and intuition generally does not. Second, “no descendant of A” excludes not only mediators but anything the exposure causes, including variables measured after exposure that look like baseline characteristics because they were recorded at a single follow-up visit.

A graph typically admits several minimal sufficient adjustment sets, not one. The dagitty package and its browser tool at dagitty.net (Textor, van der Zander, Gilthorpe, Liśkiewicz and Ellison, International Journal of Epidemiology, 2016, 45:1887–1894, DOI 10.1093/ije/dyw341) enumerate them. Choosing among them is an empirical decision — which variables you measured best, with least missingness — and not a graphical one, and stating which set you chose and why is a two-sentence addition that most manuscripts omit.

If the graph admits no sufficient adjustment set, no regression rescues the analysis. That is a design finding, and it points towards instrumental variables, front-door adjustment, negative controls, or a different data source.

Where the arrows come from

The arrows in a directed acyclic graph come from substantive knowledge, not from the dataset, and this is the step at which most graphs are weakest. Two published methods exist for doing it defensibly.

ESC-DAGs (Ferguson, McCann, Katikireddi, Thomson, Green, Smith and Lewsey, International Journal of Epidemiology, 2020, 49:322–329) sets out a four-stage procedure — mapping each study’s conclusions onto an implied graph, translating those relationships into causal statements against sequential criteria (temporality, face validity, recourse to theory) and a counterfactual thought experiment, synthesising the study-level graphs into one integrated graph, and recombining conceptually similar nodes. The output is a graph with a traceable provenance for each arrow rather than a whiteboard sketch.

The disjunctive cause criterion (VanderWeele and Shpitser, Biometrics, 2011, 67:1406–1413) is the fallback when a full graph cannot be specified: adjust for every pre-exposure covariate that is a cause of the exposure, of the outcome, or of both. VanderWeele and Shpitser prove this set is sufficient whenever some subset of the observed covariates is sufficient, which makes it a defensible default rather than a shortcut. It is not a licence to include post-exposure variables, and it does not protect against M-bias.

What does not generate arrows: bivariate significance testing, change-in-estimate thresholds, stepwise selection, and variable importance from a predictive model. All four select on association with the outcome, and colliders and mediators are associated with the outcome by construction — see my effect disappeared after adjusting.

Testable implications, and why a saturated graph is worthless

A directed acyclic graph makes empirical predictions, which is the property that separates it from a picture of an opinion. Every pair of nodes not connected by an arrow implies a conditional independence that can be tested in the data, and dagitty lists these as “implied conditional independencies”.

The arithmetic is unforgiving. A graph in which every pair of nodes is connected — a saturated graph — implies zero conditional independencies and therefore has zero testable implications: no pattern in the data can contradict it. Identification is not what is lost — with every variable observed, adjusting for all variables preceding the exposure still satisfies the back-door criterion. What is lost is the check. A graph that constrains nothing cannot be shown to be wrong, and it transfers the whole burden of the adjustment set onto measurement rather than argument. Density is not caution; it is the abandonment of the method while keeping the figure.

This is not a theoretical worry. Tennant and colleagues reviewed 234 applied health research articles published between 1999 and 2017 that reported using directed acyclic graphs (International Journal of Epidemiology, 2021, 50:620–632), and found a median saturation of 46% of the arcs possible between the nodes on show (IQR 31–67%), with just 3% of graphs fully saturated. The review reads that number the other way round from the argument above: because omitting an arc asserts precisely no effect, Tennant and colleagues treat the missing arcs, not the drawn ones, as the assumptions requiring justification. Their eight recommendations are about reporting rather than density: state the focal relationship and estimand in the aims, make the DAG available, include unobserved variables, arrange arcs to flow in one direction, justify any arc you omit, state the DAG-implied adjustment set, report the estimate it produces, and justify any alternative set separately.

Testing the implied independencies will not prove your graph correct: many distinct graphs imply the same set of independencies, and every member of that Markov equivalence class fits the data identically. A failed test, however, falsifies the graph as drawn, and reporting one is a stronger methods paragraph than any amount of assertion.

How a mishandled directed acyclic graph is detected in a manuscript

Detecting a mishandled directed acyclic graph is a matter of cross-checking three lists that a reader can extract from the paper without any special access: the nodes in the figure, the covariates in the fitted model, and the variables in the baseline characteristics table.

The specific checks, in the order they tend to pay off:

The adjustment set does not follow from the graph. The commonest defect. The figure implies a minimal sufficient set of four variables; the model contains eleven, including the four. Nothing in the paper reconciles the two, and the extra seven are usually there because they were available.

The estimand is unstated. Total effect and direct effect require different adjustment sets, so a graph cannot be evaluated at all without knowing which is being estimated. The Tennant review found this to be one of the commonest reporting omissions: most articles using a causal diagram did not state the target estimand, and many did not report the adjustment set their own diagram implied.

Post-exposure variables appear as nodes on the back-door side. Check the measurement timing of every covariate against the exposure definition; a variable recorded “at study visit 2” is not a baseline confounder.

No U nodes. A graph with no unmeasured common causes asserts complete measurement of the causal structure — an extraordinary claim for observational data, and one most published graphs make silently by simply not drawing any unobserved variables.

Super-nodes. A box labelled “socioeconomic factors” or “clinical characteristics” hides the arrows that decide the analysis. A super-node cannot be adjusted for: the analyst adjusts for its constituent variables, and which of those are colliders or mediators is exactly what the box conceals.

Every coefficient in the results table is interpreted causally. The adjustment set identifies the exposure effect, not the covariate effects — the Table 2 fallacy, named by Westreich and Greenland (American Journal of Epidemiology, 2013, 177:292–298).

What a peer reviewer says when it is mishandled

Peer reviewers who work with causal diagrams have a small, recognisable repertoire of objections. The formulations below are illustrative — written to show the shape of each objection, not quoted from any particular report.

“The DAG in Figure 1 is presented, but the covariates in the adjusted model do not correspond to any minimal sufficient adjustment set implied by it.”

“The target estimand is not stated. It is unclear whether the authors intend a total or a controlled direct effect, and the two require different adjustment sets.”

“Several adjustment variables were measured after the exposure and may be affected by it.”

“The graph contains no unmeasured variables, which amounts to an assumption of no unobserved confounding. Please state and defend this assumption.”

“The DAG is close to saturated and therefore has no testable implications; it does not constrain the analysis.”

“Please supply the graph in reproducible form, for example the dagitty model code, so that the implied adjustment sets can be checked.”

“The associations for covariates in Table 2 are discussed causally, although the adjustment set was constructed for the exposure of interest only.”

The first of those is the one that decides papers, because it is verifiable from the manuscript alone and needs no additional data. A reviewer can redraw the graph, compute the adjustment sets, and observe that the model matches none of them — see reviewer says the finding is correlative, not causal for the argument that usually follows.

Contested points and honest limitations

Directed acyclic graphs have limitations that a strong discussion section states outright rather than leaving for a reviewer.

M-bias makes “prefer baseline variables” a heuristic, not a theorem. In the structure U₁ → Z ← U₂ with U₁ → A and U₂ → Y, the variable Z precedes the exposure, is associated with both exposure and outcome, and is nonetheless a collider whose adjustment introduces bias. Greenland set out the arithmetic in “Quantifying biases in causal models: classical confounding vs collider-stratification bias” (Epidemiology, 2003, 14:300–306). How much this matters in practice is genuinely contested: several authors argue that realistic M-bias is small relative to the confounding removed by adjusting for pre-exposure covariates, which is part of why the disjunctive cause criterion remains a reasonable default.

A graph cannot be validated by the data. Markov equivalence means that competing graphs are observationally indistinguishable. A directed acyclic graph earns credibility from the substantive literature behind its arrows, and from being published in a form others can dispute.

Timing errors are not visible in a static graph. Immortal time bias arises from how person-time is allocated relative to the exposure definition. A graph drawn with correct arrows can accompany an analysis that misallocates follow-up, so a correct diagram is not a certificate of a correct analysis.

Selection is representable, but only if you draw it. Hernán, Hernández-Díaz and Robins showed that selection bias has a common graphical structure — a selection node conditioned on by the design (Epidemiology, 2004, 15:615–625). Most published graphs simply omit that node, and secondary analyses of routinely collected data are the most exposed; see registry data limitations.

How to report a directed acyclic graph

Report a directed acyclic graph so that a reader can disagree with a specific arrow, which is the entire value of drawing one.

State the target estimand in one sentence before the figure: whose effect, on what outcome, in which population, total or direct. Publish the graph as a figure and supply the model code — dagitty syntax is a few lines of plain text and belongs in the supplement, where it can be re-run. List the minimal sufficient adjustment sets the graph implies, say which you used, and say why that one (measurement quality, missingness, sample size). Report the implied conditional independencies you tested and what happened.

Keep unmeasured common causes in the figure. A U node with an honest sentence beside it — which variable, in which direction it would bias the estimate — is worth more than a graph that looks complete. Where the analysis remains vulnerable, quantify: an E-value or a quantitative bias analysis states how strong an unmeasured confounder would need to be, and reviewers treat that as engagement rather than acknowledgement. Digitale, Martin and Glymour give a compact worked template in “Tutorial on directed acyclic graphs” (Journal of Clinical Epidemiology, 2022, 142:264–267).

Related

Confounding · Collider bias · Mediator versus confounder · Table 2 fallacy · Research methods concepts

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Frequently asked questions

What does DAG stand for in epidemiology?

DAG stands for directed acyclic graph: a diagram in which variables are nodes, arrows represent direct causal effects in one direction, and no sequence of arrows returns to its starting node. Epidemiologists use one to derive which covariates must be adjusted for in order to estimate a causal effect.

What is a causal diagram used for in observational research?

A causal diagram is used to derive an adjustment set. Drawing the assumed causes and effects lets the back-door criterion identify which covariates block every non-causal path between exposure and outcome, and which covariates would introduce bias if included. The diagram also makes those assumptions available for other researchers to dispute.

How do you explain a directed acyclic graph simply?

A directed acyclic graph is a map of what you believe causes what. Each variable is a dot, each arrow means “this directly affects that”, and arrows never loop back. The map is then used to work out which variables belong in the statistical model and which must be left out.

Why are DAGs used to choose confounders?

Directed acyclic graphs are used to choose confounders because confounding is a causal property, not a statistical one. Colliders and mediators are also associated with exposure and outcome, so no correlation-based procedure can distinguish them. The back-door criterion separates the three by reading the direction of the arrows.

What makes a graph “acyclic”, and why does it matter?

A graph is acyclic when no path of arrows leads from a variable back to itself. It matters because acyclicity is what makes causal order well defined. Feedback over time is represented by separate nodes for separate time points, such as treatment at baseline and treatment at follow-up, rather than by a loop.

Is a DAG the same as a path diagram or structural equation model?

A directed acyclic graph is closely related but weaker by design. A path diagram in structural equation modelling attaches coefficients and functional forms to its arrows; a directed acyclic graph is nonparametric and attaches nothing, so it commits only to which direct effects may exist and which are assumed absent.

Do journals require a DAG in observational papers?

Journals do not uniformly require one. The STROBE checklist for observational studies contains no item mandating a causal diagram, and reporting practice varies widely — reviews of applied health research have found that many papers citing a diagram never publish it, and that the target estimand often goes unstated. Reviewers of observational manuscripts nonetheless ask for one routinely, so supplying a graph and its model code pre-empts a predictable round of revision.