Why Draw a Censoring Diagram?
A censoring assumption is a claim about how the process that ends observation relates to the event process. A directed acyclic graph (DAG) makes that claim easier to inspect: it records proposed causes and the directions of their effects, so that common causes of event and censoring can be made visible. The diagram is not a test of the assumption. Its value is to clarify which variables would need to be measured, which adjustments are plausible, and where uncertainty remains.
Let \(T\) denote the event time that would be observed with complete follow-up, and let \(C\) denote the censoring time. The observed time is \(Y=\min(T,C)\); an event is observed by the end of follow-up when \(T\leq C\). Independent-censoring assumptions concern the relationship between the event process and censoring process, possibly conditional on measured information. The earlier material on marginal and conditional independent censoring gives the formal versions; here the aim is to represent their possible causes.
For a simple baseline diagram, use \(E\) for a feature of the event process and \(K\) for a feature of the censoring process. These may be binary indicators by a fixed time, or more generally summaries of the processes; \(T\) and \(C\) are their corresponding event and censoring times. A diagram such as \(X\to E\) and \(X\to K\) represents a shared cause \(X\). The arrow \(X\to E\) does not mean that \(X\) is the event time; it means that \(X\) is proposed to influence the event process.
Reading Paths and Conditioning
A fork \(E\leftarrow X\to K\) is the basic common-cause pattern. If \(X\) is measured and the causal story is adequate, conditioning on \(X\) can block this path. In a DAG, a path is blocked by conditioning on a non-collider along the path; a collider is a node where two arrowheads meet, such as \(E\to S\leftarrow K\). Conditioning on a collider, or on a descendant of a collider, can instead open a path that was previously blocked.
This graphical language needs assumptions of its own. Under the Markov property, the joint distribution factorizes according to the graph, and d-separation implies conditional independence. The converse generally requires an additional faithfulness assumption: a missing association can occur by cancellation even when a path is present. Thus a diagram is a working causal model, not a machine that reads independence directly from the data.
Worked Example: A Measured Common Cause
Suppose a hypothetical follow-up study records a baseline frailty category \(X\). By a fixed horizon, let \(E=1\) indicate that the event has occurred and \(K=1\) indicate that censoring has occurred first. For illustration, assume \(P(X=0)=P(X=1)=0.5\), and, within each category, \(E\) and \(K\) are independent. In category 0, each has probability 0.2; in category 1, each has probability 0.8. The proposed graph is \(E\leftarrow X\to K\), with no remaining common cause after \(X\) is fixed.
Within category 0, \(P(E=1,K=1\mid X=0)=0.2(0.2)=0.04\). Within category 1, the joint probability is \(0.8(0.8)=0.64\). Averaging over the two categories gives
Both marginal probabilities are \(0.5(0.2)+0.5(0.8)=0.5\), so their product is \(0.25\), not \(0.34\). The indicators are dependent marginally even though they are independent within each \(X\) category. In this example, conditioning on the measured common cause removes the association represented by the fork; it does not make the two unconditional distributions independent.
The general reason is the covariance decomposition. For binary \(E\) and \(K\), conditional independence makes the expected conditional covariance zero, but variation in their conditional means can still produce marginal covariance:
When the first term is zero, the second need not be. If the categories with higher event probability also have higher censoring probability, that second term is positive, as in the example. This is why marginal and conditional independence are not generally nested assumptions. Conditional independence may hold while marginal independence fails. Conversely, marginal independence may hold while conditional independence fails, for example through cancellation across strata. Neither statement, on its own, logically implies the other.
Measured Causes, Unmeasured Causes, and Time
Worked Example: A Common Cause That Was Not Measured
Consider a hypothetical study in which, even after fixing the recorded baseline information \(X=x\), an unrecorded disease-severity feature \(U\) remains relevant. Within that \(X\) stratum, suppose \(U\) is equally likely to be 0 or 1. Let \(E\) indicate an event by a horizon and \(K\) indicate censoring before it. Conditional on \(U\), assume \(E\) and \(K\) are independent and each has probability 0.1 if \(U=0\), or 0.7 if \(U=1\). The graph contains \(U\to E\) and \(U\to K\); the common cause is not blocked by conditioning on \(X\).
Within the stratum \(X=x\), \(P(E=1\mid X=x)=P(K=1\mid X=x)=0.5(0.1)+0.5(0.7)=0.4\). Their joint probability is
The product of the conditional marginal probabilities is \(0.4(0.4)=0.16\), so \(E\) and \(K\) remain dependent given \(X=x\). The unmeasured \(U\) creates a common-cause path within the measured stratum. If \(U\) cannot be measured, adjustment for \(X\) alone does not establish conditional independent censoring. The diagram flags the vulnerability; it does not tell us the size of the resulting bias.
In event-history studies, causes can change over time. A patient's evolving health \(H_t\), for instance, may affect both the subsequent event intensity and the chance of leaving follow-up. A time-indexed sketch might include \(H_t\to\) later event process and \(H_t\to\) later censoring process. If variables at successive times affect one another, draw the graph in temporal order: \(H_t\to H_{t+1}\), not a cycle between the two. Whether a changing variable is measured at the right times, and how it should enter an analysis, are separate questions; the diagram helps identify the proposed pathways but does not prescribe an estimator.
A further caution is that conditioning can create rather than remove dependence. Suppose \(E\) and \(K\) are initially independent causes of a selection indicator \(S\), represented as \(E\to S\leftarrow K\). Restricting analysis to \(S=1\) conditions on their common effect and opens a path between them. In censoring work, selection might arise from inclusion in a restricted analytic subset or complete-case criteria. The exact selection mechanism must be drawn rather than assumed away.
Worked Example: Conditioning on a Common Effect
For a simple hypothetical illustration, let \(E\) and \(K\) be independent Bernoulli variables, each with probability 0.5. Let \(S=1\) when at least one of \(E\) or \(K\) equals 1. Before selection, \(P(E=1,K=1)=0.25=P(E=1)P(K=1)\). Among observations with \(S=1\), the possible pairs are \((1,0),(0,1),(1,1)\), each having conditional probability \(1/3\), because each had probability \(0.25\) before selection.
Thus, among selected observations, \(P(E=1\mid S=1)=2/3\), \(P(K=1\mid S=1)=2/3\), and \(P(E=1,K=1\mid S=1)=1/3\). Since \(1/3\ne (2/3)(2/3)=4/9\), the variables are associated after conditioning on \(S\), despite being independent before selection. In practice, do not condition on an inclusion or follow-up variable merely because it is available: first determine whether it is a common effect of event- and censoring-related causes.
Building and Using a Diagram
A useful diagram is specific enough to expose assumptions, but not so elaborate that its arrows become uninterpretable. A practical construction is:
State the event of interest, the time origin, and what ends observation. Keep the latent event time \(T\) distinct from the observed time \(Y=\min(T,C)\).
Include baseline characteristics, evolving health, access to care, study procedures, and administrative factors when they could affect either process. Label which causes are measured.
Draw arrows in temporal order. Look for common causes with paths to both processes, and note whether conditioning on a proposed variable blocks a path or opens a collider path.
Specify whether the intended claim is marginal independence or independence conditional on measured history. State the conditioning information explicitly; do not let a diagram substitute for that statement.
For example, a graph containing only \(X\to E\) and \(X\to K\) supports considering independence conditional on \(X\), if the graph is a sufficiently complete account and the model assumptions hold. Adding an unmeasured \(U\to E\) and \(U\to K\) shows why that conditional claim may still fail. An arrow directly between the processes, or another open path between them, also signals that the proposed conditioning set may not suffice.
Diagnostics, Alternatives, and Limits
DAGs are most useful alongside other evidence. Compare censoring frequencies across measured baseline groups and over follow-up; inspect whether recorded reasons for censoring vary with prognostic factors; and check whether important predictors of the event process are available in the censoring information. These checks can reveal inconsistencies with a proposed diagram, but observed data cannot rule out unmeasured common causes. A non-significant association test is not proof of independence, especially with limited data or coarse measurement.
| Approach | What it contributes | What it cannot establish alone |
|---|---|---|
| DAG | Displays proposed causal pathways, measured causes, and possible collider conditioning | Whether arrows are correct or all important causes were measured |
| Empirical comparisons | Show how observed censoring varies across recorded groups or time | Independence from latent event times or absence of unmeasured causes |
| Model for censoring | Summarizes associations with included covariates and can support adjustment strategies | Correct specification, adequate measurement, or positivity just from fitting the model |
| Sensitivity analysis | Explores how conclusions change under departures from the assumed censoring mechanism | Which departure is true without further information |
A common mistake is to see \(X\to E\) and \(X\to K\), adjust for \(X\), and conclude that censoring is independent. The correction is to ask whether every relevant common-cause path has been blocked, whether the measured \(X\) captures the causes adequately, and whether adjustment itself conditions on a collider or an inappropriate post-baseline variable. Another mistake is to call marginal independence “stronger” than conditional independence, or vice versa, as a general logical claim. The two conditions are not generally ordered: each can hold while the other fails.
Finally, a DAG is qualitative. It does not quantify bias, prove that the censoring mechanism is ignorable, or resolve which causal structure is correct. Its conclusions depend on the variables included, the arrows proposed, and the time ordering. Use it to make assumptions explicit and to identify what evidence or sensitivity analysis is needed—not to certify an analysis.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a graph \(E\leftarrow X\to K\), what role does \(X\) play, and what does conditioning on it aim to do?
- Can conditional independence given \(X\) hold while marginal independence fails? Explain briefly.
- In \(E\to S\leftarrow K\), why can restricting analysis to \(S=1\) induce an association between \(E\) and \(K\)?
- What does a DAG fail to establish about independent censoring, even when it contains no open common-cause path after adjustment?
Solutions. (1) \(X\) is a common cause; conditioning on it aims to block the fork path between \(E\) and \(K\), subject to the graph and measurement assumptions. (2) Yes. If the conditional means of \(E\) and \(K\) vary together across \(X\), the marginal covariance can be nonzero even when the conditional covariance is zero. (3) \(S\) is a collider. Conditioning on a collider opens a path between its causes, which were otherwise independent along that path. (4) A DAG cannot prove that the graph is correct or complete, rule out unmeasured common causes, or establish independence from the observed data; those require substantive justification and appropriate diagnostics or sensitivity analysis.