A Causal Diagram for Biomarker, Dropout, and Event

Why Draw the Data-Generating Story?

A repeated biomarker is not observed in isolation. It reflects a person’s underlying health at the time of measurement, while that health may also help determine who remains in follow-up and who experiences a terminal event. A directed acyclic graph (DAG) makes these proposed relationships explicit. It helps separate a biomarker’s role as a measurement of health from a possible role as a cause of later health or event risk.

The diagram is a statement of assumptions, not a finding produced automatically by the data. In particular, an arrow does not show that an effect has been estimated, and the absence of an arrow is also an assumption. The diagram is useful because it forces the analyst to specify which variables precede others and which pathways could connect them.

Define Nodes and Order Them in Time

Let \(i\) index individuals and let \(k\) index scheduled biomarker visits at times \(t_k\). Use the following nodes:

• \(H_{ik}\): latent health at visit time \(t_k\), where larger values are defined here to mean worse health.
• \(Y_{ik}\): the biomarker value at \(t_k\), which may be observed only for some individuals.
• \(D_{i,k+1}\): an indicator that the individual drops out between \(t_k\) and \(t_{k+1}\), before a terminal event is known to have occurred.
• \(E_{i,k+1}\): an indicator that the terminal event occurs in the interval after \(t_k\) and by \(t_{k+1}\).
• \(R_{i,k+1}\): an indicator that a biomarker is observed at \(t_{k+1}\).

The timing matters. \(H_{ik}\) precedes the next-interval indicators \(D_{i,k+1}\) and \(E_{i,k+1}\). If dropout or the terminal event has occurred, the next scheduled biomarker is not observed, so the observation indicator \(R_{i,k+1}\) depends on those prior follow-up outcomes. A simple version of the graph has these arrows:

The arrow \(H_{ik} \to Y_{ik}\) represents the biomarker as a measurement influenced by current latent health. The arrow \(H_{ik} \to H_{i,k+1}\) represents persistence in health over time. The arrows from current health to subsequent dropout and event represent the possibility that worsening health makes either outcome more likely. The final arrows represent observation availability, not biological effects: dropout and a terminal event prevent a later scheduled measurement from being recorded.

This is a deliberately compact graph. It omits baseline factors and other causes that may matter in a particular application. It also does not assert that dropout causes the terminal event. A shared cause, such as latent health, can make dropout and event occurrence dependent even when neither causes the other. The graph must be expanded if the scientific setting calls for additional nodes or arrows.

Constructing a Coherent Diagram

A reliable construction begins with time rather than with a search for every conceivable association. First list the variables and their time points. Then add arrows only when a direct dependence is part of the proposed scientific story. Finally, add observation indicators so that the graph distinguishes a value that exists from a value that is recorded.

The biomarker’s position requires particular care. If \(Y_{ik}\) is only a measurement of latent health, the association between \(Y_{ik}\) and a later event can arise because both depend on health. A direct arrow from \(Y_{ik}\) to \(E_{i,k+1}\) would instead assert an additional pathway: changing the biomarker itself, while holding its causes fixed, could affect the event. That claim needs scientific justification. If the biomarker could affect subsequent health, a possible pathway is \(Y_{ik} \to H_{i,k+1} \to E_{i,k+1}\). Whether that arrow belongs in the graph is a substantive assumption, not a decision made from the biomarker’s predictive strength alone.

What the Graph Implies About Dependence

A DAG can show why two outcomes may be associated without a direct arrow between them. In the compact graph, \(H_{ik}\) is a common cause of \(D_{i,k+1}\) and \(E_{i,k+1}\). If the graph includes no other shared causes and no direct dependence between those outcomes, it represents them as conditionally independent given \(H_{ik}\). In symbols, the proposed model has \(P(D_{i,k+1}, E_{i,k+1}\mid H_{ik}) = P(D_{i,k+1}\mid H_{ik})P(E_{i,k+1}\mid H_{ik})\). This conditional independence is a model assumption, not a general fact about dropout and events.

The following calculation shows how marginal dependence can still arise. For simplicity, let current health have two categories: \(H=1\) means worse health and \(H=0\) means better health. Suppose the graph’s conditional independence assumption holds for dropout and a later event given \(H\).

The probability calculation is an instance of the graph’s factorization. With \(H\) as the only shared parent and conditional independence given \(H\), sum the product of the conditional probabilities over the possible health states:

This follows by conditioning on \(H\), applying the assumed conditional independence within each health state, and adding the two mutually exclusive cases \(H=0\) and \(H=1\). If additional common causes or direct links are scientifically plausible, this factorization must be changed to match the expanded graph.

Diagnostics, Alternatives, and Limits

A graph is a disciplined description of assumptions, but it cannot verify unobserved causes by itself. Use the following checks before relying on one:

• Check temporal order. A later event cannot cause an earlier biomarker. If a proposed cycle appears, separate the variables by time, such as \(H_{ik}\) and \(H_{i,k+1}\), and reconsider the ordering.
• Distinguish event occurrence from event observation. A terminal event can stop later measurements; dropout can stop event ascertainment. The recorded event indicator may therefore need a separate observation or ascertainment node.
• Ask what each arrow means. An arrow from latent health to a measured biomarker is different from an arrow asserting that changing the biomarker would change future health.
• Look for implausible omissions. If treatment, baseline severity, or another factor plausibly affects several nodes, omitting it may make the displayed conditional independencies misleading.
• Do not equate fit with truth. Observed data can be inconsistent with some implications of a proposed graph, but agreement with observed patterns cannot establish that all unmeasured causes or arrows have been captured.

A DAG and a joint statistical model serve different purposes. The DAG clarifies the assumed structure and time ordering; it does not specify probability distributions, estimate a longitudinal trajectory, or provide an event-risk estimate. A joint model gives a statistical framework for the longitudinal and event processes, with parameters and distributional assumptions. Conversely, analyzing each process separately can be simpler, but may fail to represent dependencies suggested by a shared-health graph. The graph helps explain why dependence may matter; choosing and fitting a statistical model requires additional assumptions.

A common mistake is to read every association as a direct causal arrow. For example, “the biomarker predicts the event, therefore the biomarker causes the event” confuses predictive information with a causal pathway. The correction is to ask whether the biomarker is a measurement of latent health, a cause of subsequent health, or both, and to draw only the arrows justified by the scientific question. A second mistake is to collapse dropout and the terminal event into one generic “missingness” node. They have different meanings and may require different observation mechanisms.

Finally, a graph is not a substitute for careful data definitions. The timing of visits, the definition of dropout, and whether an event occurring after dropout is ascertained all affect the appropriate nodes and arrows. Write these definitions down before interpreting the diagram.