From a Study Timeline to a Longitudinal DAG
A longitudinal DAG is useful when a study has several treatment decisions, changing health status, and incomplete follow-up. The difficulty is not simply drawing more arrows than in a baseline study. Each arrow must represent a plausible causal direction at the correct time. A graph that puts a consequence of treatment before the treatment, or treats dropout as if it were an ordinary baseline covariate, can lead to a misleading account of the data.
Consider a study in which medication is assigned at repeated visits, disease severity is measured between visits, and some participants stop attending. The task is to represent the assumed causal structure and then ask which paths connect a treatment at one visit to a later outcome. A DAG records assumptions; it does not establish those assumptions from observed associations.
For the examples below, \(A_t\) denotes medication at visit \(t\), \(L_t\) denotes disease severity measured before the next treatment decision, \(D_t\) denotes dropout during the interval after visit \(t\), and \(Y\) denotes the outcome of interest. \(U\) denotes a baseline or unmeasured cause when needed. A node \(D_t=1\) means dropout occurred in that interval. For a path to be considered open, the conditioning set must be stated: the same graph can contain both open and blocked paths under different choices.
The first construction rule is temporal: draw the study timeline and place each variable at the time it is measured or assigned. Then add arrows only from plausible causes to their effects. Within a visit, state the assumed order if treatment, severity measurement, and dropout assessment occur close together. An arrow is a causal assumption, not merely a report that two variables are associated.
Worked Problems
Worked Example: Medication, Severity, and a Later Outcome
Suppose baseline severity \(L_1\) is measured before the first medication decision \(A_1\). Severity \(L_2\) is measured later, before medication \(A_2\), and outcome \(Y\) is assessed after \(A_2\). Baseline health \(U\) may affect both treatment decisions and the outcome. Construct a graph and find open paths from \(A_2\) to \(Y\) when nothing is conditioned on.
A defensible time order is \(U,L_1,A_1,L_2,A_2,Y\). One possible edge list is \(U\to L_1,\ U\to A_1,\ U\to L_2,\ U\to A_2,\ U\to Y,\ L_1\to A_1,\ L_1\to L_2,\ L_1\to Y,\ A_1\to L_2,\ A_1\to A_2,\ A_1\to Y,\ L_2\to A_2,\ L_2\to Y,\ A_2\to Y\). Each arrow points forward in the stated order, so the graph is acyclic.
To find paths that could confound the relationship between \(A_2\) and \(Y\), start at \(A_2\) and follow an arrow pointing into it. For example, \(A_2\leftarrow L_2\to Y\) is open: \(L_2\) is a non-collider and is not conditioned on. This path represents prior severity influencing both the medication decision and the outcome. The path \(A_2\leftarrow A_1\to Y\) is also open, as is \(A_2\leftarrow U\to Y\). The path \(A_2\leftarrow L_2\leftarrow L_1\to Y\) is open as well.
These paths are not all the same kind of relationship. \(A_2\leftarrow A_1\to Y\) reflects an earlier treatment that predicts both later treatment and outcome; \(A_2\leftarrow L_2\to Y\) reflects a time-varying common cause. Meanwhile, \(A_1\to L_2\to Y\) is a directed path from earlier treatment to outcome, not a backdoor path from \(A_2\). The graph therefore distinguishes causal transmission from noncausal association without deciding, by itself, which statistical adjustment is appropriate.
Worked Example: Dropout as a Collider
Suppose earlier medication \(A_1\) affects later severity \(L_2\). Both severity and an unmeasured health factor \(U\) affect dropout \(D_2\); \(U\) also affects outcome \(Y\). Assume \(A_1\to L_2\), \(L_2\to D_2\), \(U\to D_2\), and \(U\to Y\). Identify the path from \(A_1\) to \(Y\) and determine what happens when the analysis includes only participants who did not drop out.
The relevant path is
At \(D_2\), both path arrows point inward, so \(D_2\) is a collider. If no variable on this path is conditioned on, the path is blocked at \(D_2\). Restricting the analysis to participants with \(D_2=0\) conditions on that collider, making the path open. Within the selected sample, \(A_1\) can then be associated with \(U\), and hence with \(Y\), through this path even if there was no such open path in the full population.
The conclusion depends on the assumed arrows. If severity does not affect dropout, or if \(U\) does not affect either dropout or outcome, this particular path is absent. The graph makes those assumptions visible so they can be examined rather than hidden inside a generic statement that follow-up was incomplete.
Worked Example: Repeated Treatment With Dropout at Two Intervals
Consider the order \(L_1,A_1,L_2,A_2,D_2,L_3,A_3,Y\). Earlier medication can affect later severity and later medication; severity can affect the next treatment and dropout; dropout prevents later measurements from being observed. Let \(U\) affect \(L_1,A_1,L_2,D_2,Y\). Add the arrows \(A_1\to L_2,A_1\to A_2,A_1\to Y,L_2\to A_2,L_2\to D_2,L_2\to Y,A_2\to L_3,A_2\to A_3,A_2\to Y,D_2\to L_3,D_2\to A_3,D_2\to Y,L_3\to A_3,L_3\to Y,A_3\to Y\), together with the stated arrows from \(U\).
Every listed arrow points forward in the displayed order, so this specification contains no directed cycle. The edge \(D_2\to L_3\) represents the fact that dropout determines whether the later severity measurement is available, not necessarily that dropout biologically changes disease. This distinction matters: a node can represent a measurement or observation process, and arrows into it may describe what determines whether data are recorded.
For the relationship between \(A_2\) and \(Y\), one open path without conditioning is \(A_2\leftarrow L_2\to Y\), and another is \(A_2\leftarrow A_1\to Y\). The path \(A_2\to L_3\to Y\) is instead directed from treatment to outcome. If the analysis is restricted to people with \(D_2=0\), paths involving the dropout node must be checked again; conditioning can change whether paths are open. In particular, if an unmeasured \(U\) also causes dropout and outcome, \(A_2\leftarrow L_2\to D_2\leftarrow U\to Y\) is blocked before selection at the collider \(D_2\), but conditioning on \(D_2\) opens it.
Two Graphical Results to Use While Constructing
Proof. Assign each node its position in the specified ordering. Along every directed arrow, the position strictly increases. If a directed cycle existed, following its arrows would strictly increase position at every step and yet return to the starting node. That would require the starting position to be strictly less than itself, which is impossible. Therefore no directed cycle exists. \(\square\)
This result is a practical audit, not a guarantee that the graph is causally correct. If a proposed arrow points backward in the order, first check whether the variables were placed at the wrong times or whether the assumed causal story is inconsistent. A true longitudinal process can have same-visit variables, but their order must be specified before the forward-arrow check can be used.
Derivation. By definition, a path is open only if each collider on it is conditioned on or has a conditioned descendant. If neither condition holds for \(C\), the path fails the openness criterion at \(C\) and is blocked there. If \(C\), or a descendant of \(C\), is conditioned on, this collider condition for openness is met. The whole path is then open exactly when every other collider meets the same condition and no non-collider on the path is conditioned on. Thus conditioning on dropout or on a consequence of dropout can change a path from blocked to open; it does not automatically open every path in the graph. \(\square\)
These are graphical statements. To interpret d-separation as a conditional independence, one additionally assumes the distribution is Markov with respect to the DAG. Neither graphical result proves that the chosen arrows capture the true causal structure.
How to Audit an Open Path
For a proposed treatment-outcome relationship, trace paths from the treatment node to the outcome node, not only directed paths. Record arrow direction at each step, identify colliders, and then apply the stated conditioning set. A useful working table is:
| Path feature | Question to ask | Interpretation |
|---|---|---|
| Arrow leaves treatment and eventually reaches outcome | Does every arrow point along the path? | A directed causal path in the assumed graph |
| Path begins with an arrow into treatment | Is it open under the conditioning set? | A possible backdoor association path |
| Two arrows meet at a node | Is the collider or a descendant conditioned on? | The path is blocked or activated depending on selection |
| Dropout node is conditioned on | Does dropout have multiple causes, including causes of outcome? | Selection may create an association path |
A path is not automatically a confounding path merely because it connects treatment and outcome. It must be open under the relevant conditioning set, and its causal interpretation depends on the graph. In particular, a path through a post-treatment severity measure may be part of the treatment effect, may connect treatment to later treatment, or may act as a backdoor path for a later treatment. Its role depends on which treatment contrast is under study.
Diagnostics, Alternatives, and Limits
Several warning signs suggest that a proposed graph needs revision. A backward arrow often signals a timing error or an ambiguous within-visit order. A dropout node with no parents may be implausible if severity or treatment plausibly affects retention. A graph that omits common causes can make a backdoor path appear absent. Conversely, adding arrows merely because variables are associated can create causal claims that are not supported by the study design.
A timeline or causal narrative is a useful alternative representation while planning a study: it can expose unclear measurement timing before the graph is drawn. A DAG makes the direction of assumed relationships and the role of colliders easier to inspect. Neither is a substitute for a statistical model. Regression adjustment can be described with equations, but a model formula alone may hide whether a variable is pre-treatment, post-treatment, or a selection variable. The graph and the model answer different questions and should be checked for consistency.
A DAG is also limited by what it represents. A standard DAG does not encode effect sizes, functional forms, or uncertainty about arrows unless these are added through a separate modeling framework. Two different causal systems can be compatible with the same observed distribution. An open path flags a possible source of association under the stated graph; it does not quantify the bias or show that the path is present in reality.
Check Your Understanding
For each problem, first identify the relevant arrows and conditioning set. Then compare your reasoning with the solution.
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A baseline factor \(U\) causes both \(A_1\) and \(Y\), with arrows \(U\to A_1\) and \(U\to Y\). Is \(A_1\leftarrow U\to Y\) open when nothing is conditioned on?
Solution: Yes. \(U\) is a non-collider on the path and is not conditioned on, so the path is open. -
In \(A_1\to L_2\to D_2\leftarrow U\to Y\), what happens to the path if the analysis conditions on \(D_2=0\), with no other nodes conditioned on?
Solution: It becomes open. \(D_2\) is a collider, and conditioning on it satisfies the collider condition for an open path. The other internal nodes are non-colliders and are not conditioned on. -
A graph is ordered \(L_1,A_1,L_2,A_2,Y\), but includes \(A_2\to L_2\). What should be checked?
Solution: The arrow points backward in the proposed order. Check whether the measurement timing is wrong or the causal assumption is inconsistent. If both directions \(L_2\to A_2\) and \(A_2\to L_2\) are included, they form a directed cycle, which is not a DAG. -
In \(A_2\leftarrow L_2\to Y\), is the path open when conditioning on \(L_2\)?
Solution: No. \(L_2\) is a non-collider on this path. Conditioning on a non-collider blocks the path. -
Why is \(A_1\to L_2\to Y\) not a backdoor path from \(A_1\) to \(Y\)?
Solution: Both arrows point along the path from \(A_1\) toward \(Y\). It is a directed path and represents a possible causal route under the graph, rather than a path beginning with an arrow into \(A_1\).