Why Draw a Diagram Before Comparing Outcomes?
When a randomized trial is compared with an external control cohort, the treatment groups may come from different data sources and may have entered those sources for different reasons. Baseline prognosis can affect trial participation, treatment, and outcome. Trial participation can determine which treatment is available and how outcomes are recorded. If an analysis includes only patients whose outcomes were captured, that restriction can also change the relationships among variables. A causal diagram makes these assumptions and potential pathways explicit before a statistical adjustment is chosen.
A diagram is not a picture of the data and does not establish that its arrows are true. It is a compact statement of a proposed causal structure. Its value is to make the implications of that structure inspectable: which variables are causes, which variables are selection mechanisms, and which routes can connect treatment to outcome without representing the treatment effect of interest.
Specify the Nodes and Their Timing
For an external-control comparison, begin with distinct nodes for the variables that are often inadvertently combined. Let \(S\) indicate trial participation (\(S=1\)) versus membership in the external data source (\(S=0\)); let \(A\) indicate treatment received; let \(L\) denote baseline prognosis measured before treatment; let \(Y\) be the true clinical outcome over a defined follow-up period; and let \(R\) indicate whether that outcome is captured in the available data. These definitions matter: \(S\) is not the same as \(A\), and the true outcome \(Y\) is not the same as whether it was observed, \(R\).
A useful starting diagram may contain \(L\to S\), \(S\to A\), \(L\to Y\), \(A\to Y\), and \(S\to R\). These arrows are hypotheses, not universal facts. For example, \(S\to A\) is plausible when trial participants can receive the investigational treatment while external patients receive routine care. The arrow \(S\to R\) is plausible when trial follow-up is more systematic than routine-care recording. Add a cause only when there is a substantive reason to believe it affects the downstream variable; do not add arrows merely to make the diagram look comprehensive.
Other baseline causes may be needed. A variable \(U\), such as health-seeking behavior or access to specialist care, might affect both participation and outcome. A variable \(C\), such as calendar period, might affect which source a patient enters, treatment availability, and outcome. If a cause is important but not measured, show it as an unmeasured node rather than silently assuming it does not exist. The diagram should also respect time: a baseline covariate cannot be caused by a treatment received later, although post-treatment variables can be affected by treatment.
Constructing and Reading the Core Diagram
State the target population, treatment strategies, outcome, and follow-up period. Make clear whether treatment is assigned by randomization, routine care, or data-source membership.
Draw measured and plausible unmeasured pre-treatment causes of participation, treatment, and outcome. Keep their temporal status explicit.
Draw arrows into \(S\) for causes of source membership and arrows into \(A\) for causes of treatment. Add \(S\to A\) when source membership constrains treatment availability or assignment.
Draw causes of \(Y\), then separately draw causes of \(R\). If outcome occurrence prompts recording, \(Y\to R\) may be plausible; if analysis requires a recorded outcome, mark that the analysis conditions on \(R=1\).
Start at \(A\) and end at \(Y\). Identify paths that begin with an arrow into \(A\), pathways opened by conditioning on a collider, and routes created by outcome selection. Record which causal assumptions the diagram makes.
In the core structure, \(A\leftarrow L\to Y\) is a backdoor path: prognosis affects both treatment and outcome, so treatment and outcome can be associated even apart from a treatment effect. Similarly, \(A\leftarrow S\to Y\) is a noncausal path if participation affects both treatment and outcome, for example through differences in co-interventions or care intensity. The path \(A\leftarrow S\leftarrow L\to Y\) is another route linking treatment and outcome through baseline prognosis.
Worked Example: Prognosis Confounds a Treatment Comparison
Hypothetical data. Consider 400 patients: 200 with high baseline risk and 200 with low baseline risk. Within each risk group, compare patients receiving treatment (\(A=1\)) with those receiving control (\(A=0\)). Among high-risk patients, the event rates are 80% with treatment and 90% with control. Among low-risk patients, they are 10% with treatment and 20% with control. The treated group contains 90 high-risk and 10 low-risk patients; the control group contains 10 high-risk and 90 low-risk patients.
The diagram includes \(L\to A\), \(L\to Y\), and \(A\to Y\). The observed treated event rate is \(0.90(0.80)+0.10(0.10)=0.73\), or 73%. The observed control event rate is \(0.10(0.90)+0.90(0.20)=0.27\), or 27%. The crude risk difference, treated minus control, is \(0.73-0.27=0.46\), or 46 percentage points higher with treatment.
That crude contrast reverses the within-risk comparisons: in each risk group, the treated event rate is 10 percentage points lower. To illustrate the role of the diagram, standardize both groups to a hypothetical population with equal proportions of high- and low-risk patients. The standardized treated rate is \(0.50(0.80)+0.50(0.10)=0.45\); the standardized control rate is \(0.50(0.90)+0.50(0.20)=0.55\). Their difference is \(0.45-0.55=-0.10\), or 10 percentage points lower with treatment in this standardized comparison. This arithmetic does not prove that adjustment is valid; it shows why the open path through \(L\) matters and why the target population must be specified.
Participation, Selection, and Outcome Capture
Selection pathways require particular care because the analysis often uses only people who entered a source or had an outcome recorded. A collider is a node at which two arrowheads meet, such as \(L\to S\leftarrow U\). Without conditioning on \(S\), this path is blocked at \(S\). Restricting to one participation category, or otherwise conditioning on \(S\), can open the path and induce an association between \(L\) and \(U\). If \(U\) also causes \(Y\), the opened route can connect participation, treatment, and outcome.
Outcome capture can create a related problem. Suppose trial participation increases monitoring, and a clinical event makes recording more likely. The diagram then contains \(S\to R\leftarrow Y\). The node \(R\) is a collider on this path. Restricting analysis to \(R=1\) can associate \(S\) with \(Y\), even if the two were otherwise independent. Since \(S\) may also determine treatment availability, this selection can distort a treatment–outcome comparison. This is a statement about the selection mechanism, not a claim that recording causes the clinical event.
Worked Example: Conditioning on Captured Outcomes
Hypothetical illustration. Suppose \(S\) and a binary true outcome \(Y\) are independent, with each having probability 0.50. Outcome capture is complete if \(S=1\) or \(Y=1\), and absent only when \(S=0\) and \(Y=0\). Thus \(R=1\) for three of the four equally likely combinations.
Among patients with \(S=1\) and \(R=1\), both outcome values remain possible, so \(P(Y=1\mid S=1,R=1)=0.50\). Among patients with \(S=0\) and \(R=1\), capture requires \(Y=1\), so \(P(Y=1\mid S=0,R=1)=1\). The selected data therefore show a 50-percentage-point outcome difference between the participation groups, although \(S\) and \(Y\) were independent before selection. In the diagram, conditioning on \(R=1\) opens \(S\to R\leftarrow Y\). Real capture mechanisms are rarely this simple, but the example demonstrates why comparing only observed outcomes can be misleading.
A diagram can also show that the desired comparison is unsupported by the data. If \(S\) determines \(A\) exactly—for example, all trial participants receive treatment and all external patients receive control—there may be no patients receiving both treatments within a source category. This is a lack of treatment overlap, not a problem that can be repaired simply by drawing additional arrows or fitting a more elaborate model. The diagram signals that causal interpretation will depend on assumptions about comparability across sources.
A Short Identification Justification
The diagram’s role is to make assumptions visible, not to guarantee identification. For example, consider a discrete baseline prognostic variable \(L\) and a target population in which both treatment strategies are possible at every relevant level of \(L\). Let \(Y(a)\) denote the outcome that would occur under treatment \(a\). If consistency holds, conditional exchangeability \(Y(a)\perp A\mid L\) holds, and treatment positivity \(P(A=a\mid L=l)>0\) holds for every \(l\) with positive probability in the target population, then the observed data identify the mean potential outcome by standardization:
To see the justification, partition the target population by \(L\): \(P(Y(a)=1)=\sum_l P(Y(a)=1\mid L=l)P(L=l)\). Conditional exchangeability permits the conditional potential-outcome probability to be replaced by \(P(Y(a)=1\mid A=a,L=l)\). Consistency then equates this probability to \(P(Y=1\mid A=a,L=l)\) among those receiving \(a\). Positivity ensures that this observed conditional probability is defined at every target-population level receiving positive weight. If \(L\) is continuous, the sum is replaced by an integral over its target-population distribution. In an external-control setting, the exchangeability assumption may be especially demanding because unmeasured differences between trial and external patients can violate it. The formula should therefore be read as conditional on its assumptions, not as a consequence of drawing a DAG alone.
Worked Example: Checking a Proposed Diagram
Hypothetical setting. A team proposes arrows \(S\to A\), \(L\to S\), \(L\to Y\), \(A\to Y\), and \(S\to R\). It also plans to analyze only patients with \(R=1\). The team’s clinical review indicates that events themselves prompt additional documentation, so it adds \(Y\to R\).
The diagram now contains two noncausal routes worth inspecting: \(A\leftarrow S\leftarrow L\to Y\), which links treatment and outcome through prognosis, and \(A\leftarrow S\to R\leftarrow Y\), which is opened by restricting to \(R=1\). If differences in care associated with trial participation also affect the clinical outcome, an additional arrow \(S\to Y\) is warranted, creating the path \(A\leftarrow S\to Y\).
The proposed diagram is therefore not equivalent to a claim that the treatment effect is identified. It shows where assumptions are needed: whether relevant prognosis and source differences are represented, whether selection on capture can be avoided or addressed, and whether comparable treatment groups exist. A reviewer should ask the data team how each arrow is supported and whether any important common cause has been omitted.
Diagnostics, Alternatives, and Limitations
A DAG cannot be validated by a statistical test. It is a causal model informed by clinical knowledge, study design, data provenance, and the timing of measurements. Still, several checks can reveal an inadequate or internally inconsistent diagram:
- Check temporal order. An arrow from a future treatment to a genuinely pre-treatment prognostic variable is generally inconsistent with the stated timing.
- Check node definitions. If “trial participation” means enrollment but is used to represent treatment received, separate those concepts. If “outcome” actually means a coded record, distinguish the true clinical event from capture.
- Ask what caused each arrow’s source and target. A missing common cause of treatment and outcome can leave an open backdoor path; an unsubstantiated arrow can imply a pathway that is not clinically plausible.
- Inspect selection explicitly. Mark whether the analysis conditions on participation, linked records, or outcome capture. A selection node may be a collider, so restriction can open a path that was previously blocked.
- Check source-specific treatment availability. If treatment is nearly determined by trial versus external source, the comparison relies heavily on cross-source assumptions and may have little empirical overlap.
A verbal causal narrative is useful for discussing clinical mechanisms, but it can hide whether a variable is a common cause, mediator, or collider. A DAG makes those roles and pathways easier to audit. A regression model, propensity score, or weighting procedure can estimate an adjusted contrast under specified assumptions, but none determines which arrows are causally correct. Choosing variables for an adjustment set and implementing adjustment are separate tasks; a diagram is a tool for stating the causal structure, not a substitute for those decisions.
The main limitation is that distinct causal systems can be compatible with the same observed data. Unmeasured confounding, selection, measurement error, and changes in care over calendar time may remain uncertain even after careful diagramming. In addition, a DAG usually simplifies a complex clinical process: one node may conceal several mechanisms, and an arrow may vary across patients or time. Record the substantive rationale for consequential arrows, consider plausible alternative diagrams, and avoid presenting one diagram as if it were discovered from the data.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A baseline severity measure \(L\) affects treatment received \(A\) and outcome \(Y\). Write the corresponding path and explain why it matters.
- In a diagram \(S\to R\leftarrow Y\), what can happen when an analysis includes only patients with \(R=1\)?
- Why should trial participation \(S\) and treatment \(A\) be represented as separate nodes in an external-control comparison?
- State the additional condition, beyond consistency and conditional exchangeability, needed to use observed treatment-specific conditional outcome probabilities for every discrete prognosis level in a target population.
Solutions. (1) The path is \(A\leftarrow L\to Y\). It is a backdoor path because \(L\) is a common cause of treatment and outcome, so the observed association may include prognostic differences as well as a treatment effect. (2) \(R\) is a collider; conditioning on \(R=1\) can open the path and induce an association between \(S\) and \(Y\). If participation also affects treatment, this selection can distort the treatment–outcome comparison. (3) Participation describes entry into a trial or external source, whereas treatment describes what the patient receives. They need not be identical, and they can have different causes and effects on outcome capture. (4) Treatment positivity is required: \(P(A=a\mid L=l)>0\) for every level \(l\) with positive probability in the target population.