Why Draw a Causal Diagram?
In a competing-risks study, treatment may affect both the event of interest and an event that prevents it from occurring. A diagram makes these relationships explicit. It helps distinguish a baseline variable that confounds treatment comparisons from an event-free risk set that selects people according to treatment and prognosis. These are different problems: baseline confounding can often be addressed by appropriate adjustment, while conditioning on a post-treatment selection variable can create an association that was absent before treatment.
A directed acyclic graph (DAG) is a set of variables represented by nodes and causal directions represented by arrows. “Acyclic” means that following arrows cannot lead back to the starting node. A DAG is a set of causal assumptions, not a picture of correlations discovered in a data set. Its usefulness depends on whether the chosen variables and arrows adequately represent the scientific setting.
For a fixed follow-up horizon \(\tau\), let the event category \(J_\tau\) indicate whether the first event by \(\tau\) is the event of interest, a competing event, or neither. This categorical representation respects the fact that a person cannot experience both types as their first event by that horizon. In a time-indexed DAG, the event category can be represented separately at successive times, with prior event status determining who remains at risk. A diagram that uses separate indicators for target and competing events should make their mutually exclusive support clear; treating them as unconstrained independent outcomes can misrepresent the process.
The arrows in a DAG encode assumptions about direct causal influence among the represented variables. They do not show the size of an effect, and the absence of an arrow is a substantive claim that there is no direct causal effect at that level of representation. A DAG also cannot, by itself, demonstrate that all relevant common causes have been measured.
Confounding and Treatment Effects on Both Event Types
A common baseline structure has a covariate \(C\) that affects treatment choice and prognosis. If \(C\) affects the target-event process, the competing-event process, or both, it can create a noncausal treatment–outcome association. A useful starting diagram is:
Here \(J_\tau\) is categorical, so the arrow \(A \to J_\tau\) allows treatment to change the probabilities of the target event, the competing event, or remaining event-free. The backdoor path \(A \leftarrow C \to J_\tau\) is open before adjustment. If \(C\) is measured before treatment, adjusting for it can block this path, subject to the usual requirements for causal identification, including adequate measurement, consistency, and positivity. The graph does not justify adjustment for every available variable: adjusting for a post-treatment variable can change the question or introduce bias.
The target-event probability by \(\tau\) is a probability of experiencing the target event first by that horizon. A competing event is not simply missing follow-up: its occurrence changes which first event is observed. Thus, a DAG should represent competing events as part of the outcome process, rather than silently recoding them as ordinary independent censoring.
Worked Example: Baseline Confounding of a Target-Event Risk
Hypothetical data. Suppose \(C\) identifies high-risk and low-risk patients. There are 100 high-risk and 100 low-risk patients. In the high-risk group, 80 receive treatment and 20 do not; in the low-risk group, 20 receive treatment and 80 do not. Within each risk group, the target-event probability is 30% for high-risk patients and 5% for low-risk patients, regardless of treatment. The competing-event probability is 40% and 10%, respectively, also regardless of treatment. The target and competing events are mutually exclusive.
The diagram contains the open backdoor path \(A \leftarrow C \to J_\tau\). Treatment is more common in the high-risk group, and high risk predicts both event types. To see the resulting target-event association, calculate the crude risks.
Among untreated patients, the corresponding calculation is
The crude difference is \(0.25-0.10=0.15\), or 15 percentage points. But within each \(C\) group, the treatment-specific risks are equal by construction. Standardizing both treatment groups to a 50% high-risk, 50% low-risk distribution gives
for either treatment level. In this hypothetical example, the crude difference is entirely attributable to the different distributions of \(C\), not to a treatment effect. The competing-event probabilities are part of the same risk structure: standardizing them similarly gives \(0.50(0.40)+0.50(0.10)=0.25\) under either treatment. The example illustrates why the diagram must include causes of both event types when they affect treatment assignment.
Selection by Event-Free Survival
A different pathway appears when analysis is restricted to people who remain event-free up to a time \(t\). Let \(S_t=1\) indicate being event-free at \(t\), and let \(U\) be an unmeasured prognostic factor. If treatment and \(U\) both affect event-free survival, the diagram includes \(A \to S_t \leftarrow U\). Before conditioning, \(S_t\) is a collider on this path. Restricting analysis to \(S_t=1\) opens the path between \(A\) and \(U\); if \(U\) also affects a later target event, the selected treatment groups can differ in prognosis even when treatment was initially randomized.
The logic can be justified by considering the distribution among selected individuals. Suppose \(A\) and \(U\) are initially independent binary variables, each with probability \(1/2\). Suppose the probabilities of remaining selected are 0.9 when \((A,U)=(0,0)\), 0.4 when \((1,0)\), 0.4 when \((0,1)\), and 0.1 when \((1,1)\). These probabilities do not factor into separate contributions from \(A\) and \(U\), so selection can change their joint distribution.
Because the four prior joint probabilities are each \(0.25\), the common factor cancels when the conditional probabilities are normalized. The four selection weights sum to \(0.9+0.4+0.4+0.1=1.8\). Thus, among selected individuals,
The same marginal probability applies to \(U=1\), so their selected covariance is
The covariance was zero before selection and is negative afterward. This is the numerical form of the collider-path argument. It does not mean that every restriction to an event-free risk set creates bias for every target; it means that selection must be represented and justified rather than assumed harmless.
Worked Example: A Spurious Difference Among Survivors
Continue the hypothetical selection setup. Suppose \(Y\) is a later target event, and among people selected at time \(t\), its probability is 40% when \(U=1\) and 10% when \(U=0\), with no direct effect of \(A\). Among selected treated people, the probability of \(U=1\) is
Among selected untreated people, it is \(0.4/(0.9+0.4)=4/13\approx0.3077\). The later target-event risks in the selected groups are therefore
The selected-group difference is about \(-0.0323\), or \(-3.23\) percentage points, despite the stipulated absence of a direct treatment effect on \(Y\). The difference arises because selection changed the distribution of \(U\). The calculation concerns people selected at \(t\); it does not establish a causal effect for the original population.
Drawing a Useful Diagram
A disciplined drawing process helps prevent both omitted confounding and mistaken adjustment. Begin with the scientific question and the time ordering, then include treatment, baseline common causes, the event process, and any selection variables used in the analysis. Use separate nodes for distinct time points when necessary. For example, a baseline variable \(C\), treatment \(A\), event-free status \(S_t\), and later target event \(Y\) have a clear temporal order; a single undated node called “health” may not.
Specify the target event, the competing event, and the follow-up horizon or time-indexed event status. Decide whether the node represents a categorical first-event outcome or a specific event-free state.
Include measured baseline variables that affect treatment and prognosis, and represent treatment arrows to both target and competing event processes when scientifically plausible.
Identify open backdoor paths from treatment to outcome. Then check whether the proposed adjustment blocks those paths without conditioning on a collider or inappropriate post-treatment variable.
If the analysis conditions on surviving event-free, being observed, or entering a risk set, draw that condition as a node or selection rule and trace its incoming arrows.
The central path rule is simple but powerful: a noncollider blocks a path when conditioned on, whereas a collider blocks it when not conditioned on. Consequently, adjusting for a common cause can close a confounding path, but adjusting for a common effect can open a path. Descendants of a collider can also open the path when conditioned on. These rules concern the graph’s structure; they do not guarantee that the graph itself is correct.
Reading a Randomized-Treatment Diagram
Randomization removes baseline confounding of treatment assignment in expectation, but does not make treatment irrelevant to event-free selection. Treatment can affect both event types, and prognostic factors can affect both event types. Conditioning on surviving event-free may then associate randomized treatment with prognosis in the selected risk set. The following example separates an overall by-horizon comparison from a survivor-only comparison.
Worked Example: Treatment and Both First-Event Types
Hypothetical trial. Suppose 200 participants are randomized to each treatment group. By one year, 24 treated participants experience the target event first, 36 experience a competing event first, and 140 remain event-free. In the control group, the corresponding counts are 32, 16, and 152. These categories sum to 200 in each group.
A suitable one-year diagram has randomized \(A\) pointing to the categorical first-event outcome \(J_1\), with a prognostic variable \(U\) also pointing to \(J_1\). In an expanded time-indexed diagram, treatment may affect both the target and competing event processes, while \(U\) may affect both. Randomization supports independence of \(A\) from baseline \(U\) before selection; it does not imply independence after conditioning on a later event-free status.
The observed target-event probabilities are
Their difference is \(0.12-0.16=-0.04\), or \(-4\) percentage points in this hypothetical trial. The competing-event probabilities are \(36/200=0.18\) and \(16/200=0.08\). These are distinct categories in the outcome, not ordinary losses to follow-up. A comparison restricted to the event-free participants would instead compare selected groups, with denominators 140 and 152. The DAG warns that such a restriction can alter the distribution of \(U\), even though treatment was randomized. These counts alone do not show whether selection bias occurred; that depends on the causal structure and the analysis being interpreted.
Diagnostics, Alternatives, and Limits
A DAG is most useful as a diagnostic of reasoning, not as an empirical test. Ask whether any plausible common cause of treatment and either event type is missing; whether an arrow contradicts time order; whether a post-treatment variable has been mistaken for a baseline confounder; and whether the analysis conditions on event-free survival or another common effect. If different reasonable diagrams imply different adjustment sets, the causal conclusion depends on assumptions that should be made explicit.
Regression models and weighting procedures can implement adjustment under specified assumptions, but a fitted model does not reveal whether the adjustment set is causally appropriate. A DAG makes the assumed pathways visible; it does not supply effect estimates or validate an unmeasured-confounding assumption. Cause-specific hazard analyses focus on the event rate among people currently event-free, while a by-horizon event probability uses the observed competing-event process as part of the path to the outcome. The appropriate interpretation depends on the scientific question; neither analysis can be read off a graph without specifying its target.
A common mistake is to treat a competing event as independent censoring and then interpret the remaining target-event comparison as if it were automatically the total effect on target-event probability. The correction is to show the competing event in the outcome process and ask what conditioning or censoring operation the analysis performs. Another mistake is to adjust for event-free survival because it “balances” the risk sets. If survival is affected by both treatment and prognosis, conditioning on it can open a collider path instead.
Check Your Understanding
Answer each question using the path rules and event-process distinctions developed above. Solutions follow the questions.
- A baseline variable \(C\) causes both treatment \(A\) and the target event \(Y\). What path does this create, and what role can adjustment for \(C\) play?
- In the graph \(A \to S \leftarrow U \to Y\), what happens to the path between \(A\) and \(Y\) when analysis conditions on \(S=1\)?
- A trial reports 18 target events among 150 treated participants and 24 among 150 controls by a fixed horizon. Calculate the target-event risk difference, treated minus control. Why should competing events remain part of the outcome accounting?
- Why does randomization not automatically prevent selection bias in a comparison restricted to people who remain event-free?
Solutions. (1) The open backdoor path is \(A \leftarrow C \to Y\). Conditioning on the measured baseline common cause \(C\) blocks that path, provided the relevant identification assumptions hold. (2) \(S\) is a collider. The path is blocked when unconditioned, but conditioning on \(S=1\) opens it; \(A\) can then become associated with \(U\), which predicts \(Y\). (3) The risks are \(18/150=0.12\) and \(24/150=0.16\). The difference is \(0.12-0.16=-0.04\), or \(-4\) percentage points. Competing events determine the observed first-event category and are not automatically independent censoring. (4) Randomization balances baseline factors before selection, but if treatment and prognosis both affect event-free status, conditioning on that status can open a collider path and change the prognostic composition of the groups.