Lagged Treatment Effects and Carryover Paths

Why Treatment History Can Matter

In a longitudinal study, a treatment may affect an outcome after a delay, or its influence may persist by changing later covariates and treatments. A diagram that connects each outcome only to the treatment assigned immediately before it can miss these possibilities. The key is to represent both when an effect can occur and which causal route carries it.

For example, an earlier treatment may change a biomarker measured later, and that biomarker may affect a subsequent outcome. The treatment then has a later effect through a mediated path. Alternatively, the earlier treatment may also have an arrow directly to that outcome. At the resolution of the graph, these represent distinct causal routes and should not be conflated.

Use \(A_t\) for treatment at time \(t\), \(L_t\) for a time-varying covariate measured at time \(t\), and \(Y_t\) for an outcome at time \(t\). The timeline must specify their order. For instance, if \(L_1\) is measured after \(A_0\) but before \(A_1\), it can be a mediator of the effect of \(A_0\) on \(A_1\) or on a later outcome.

Reading Direct and Mediated Carryover

Consider a time ordering in which \(A_0\) is assigned, then \(L_1\) is measured, then \(A_1\) is assigned, and finally \(Y_2\) is observed. The arrow \(A_0 \to Y_2\) represents direct carryover at this graph’s resolution. The path \(A_0 \to L_1 \to Y_2\) represents carryover mediated through \(L_1\). The path \(A_0 \to A_1 \to Y_2\) represents carryover through the later treatment, while \(A_0 \to L_1 \to A_1 \to Y_2\) passes through both the covariate and later treatment.

The presence of a directed path indicates a route through which an effect may be transmitted under a causal model compatible with the graph. It does not guarantee that the effect is nonzero: coefficients along a path could be zero, or contributions from different paths could cancel. Conversely, if there is no directed path from \(A_0\) to \(Y_2\), an intervention on \(A_0\) cannot affect \(Y_2\) in an acyclic structural causal model whose equations follow the graph.

Path Contributions in a Linear Structural Model

The distinction between an edge and a path can be made quantitative under a linear structural model. Assume a finite acyclic DAG and one structural equation for each node \(V\):

where \(\operatorname{pa}(V)\) is the set of parents of \(V\), the coefficients are fixed, and the exogenous terms \(\varepsilon_V\) are held fixed when comparing interventions. Assume the treatment node \(A_s\) is set by intervention, so its structural equation is replaced by that intervention. The model is additive and has no treatment-by-covariate interactions.

Proof. Order the nodes so every parent precedes its child; such an order exists because the graph is finite and acyclic. Compare two interventions that set \(A_s\) to values differing by \(\delta\), while keeping every exogenous term fixed. Write \(\Delta V\) for the value under the second intervention minus the value under the first. At \(A_s\), \(\Delta A_s=\delta\). For any other node \(V\), subtracting its two structural equations gives

Apply this recursion in topological order. For a node immediately downstream of \(A_s\), its change includes \(\beta_{A_sV}\delta\), the contribution from the one-edge path. At any later node, each parent’s contribution is multiplied by the coefficient of the edge from that parent to the node. Thus each contribution arriving at \(V\) is a contribution from a path ending at one of its parents, multiplied by the final edge coefficient. Every directed path to \(V\) has exactly one such final edge, so all paths are counted once. If no parent carries a nonzero change from \(A_s\), the recursion gives \(\Delta V=0\). This proves the formula for \(Y_t\). \(\square\)

Delayed Effects and Persistent Effects

A delayed effect occurs when an earlier treatment affects a variable only at a later time in the represented sequence. For example, if \(A_0\) has no arrow to \(Y_1\) but has an arrow to \(Y_2\), the graph allows a delayed influence on \(Y_2\). “Delayed” does not imply that the effect must be absent at every intermediate time in reality; it describes the causal structure represented by the chosen time points and variables.

A persistent effect can be represented by arrows from one treatment to multiple later nodes, such as \(A_0 \to L_1\) and \(A_0 \to Y_2\), or by a chain of later descendants such as \(A_0 \to L_1 \to Y_2\). The first representation includes a direct carryover arrow to \(Y_2\); the second represents persistence through the covariate. The paths may coexist, and the model should include those supported by the substantive causal assumptions.

The graph does not require that treatment at a later time erase earlier treatment effects. Nor does it imply that every earlier treatment has an effect at every future time. Whether a carryover arrow belongs in the graph is a substantive modeling decision informed by the timing, scientific knowledge, and variables measured.

Justification. Compare two interventions on \(A_s\), using the same exogenous variables in both. In topological order, any node other than \(A_s\) whose parents have unchanged values has the same structural equation inputs and therefore the same value. If there is no directed path from \(A_s\) to \(Y_t\), every parent of \(Y_t\), and recursively every ancestor of \(Y_t\), is unchanged by the intervention. Hence \(Y_t\) is unchanged for each fixed exogenous realization. The two interventions therefore induce the same distribution of \(Y_t\) when the exogenous variables have the same distribution. \(\square\)

Assumptions, Diagnostics, and Alternatives

A carryover DAG is meaningful only relative to its time ordering and causal assumptions. The treatment must precede the later variable it is said to affect; a later measurement cannot cause an earlier assignment. The graph must be acyclic, so feedback within a time point must be represented using a more detailed temporal ordering or a different model. As in other causal diagrams, an omitted common cause may make an apparent causal relation misleading; an arrow is not justified merely because two variables are associated.

The structural-equation path calculation above requires more than the graph itself: it assumes linear additive equations with fixed coefficients and no interactions. With nonlinear models or interactions, the treatment effect can depend on the covariate values and generally cannot be reduced to a sum of coefficient products. The qualitative distinction between direct edges and mediated paths remains, but numerical path contributions require a model appropriate to that setting.

Useful checks when drawing or interpreting carryover include:

• Write the event and measurement order first; check that each proposed arrow points forward in time.
• For each later variable, ask whether an earlier treatment could affect it directly, through measured intermediates, or through both routes.
• Check whether the graph’s intermediate nodes are sufficient for the intended level of description. An omitted mediator can make a route look direct.
• Do not infer a nonzero effect merely from a directed path, or infer no effect merely because a particular fitted coefficient is near zero.

An immediate-treatment-only graph is a simpler alternative: it includes effects of \(A_t\) on outcomes after time \(t\), but excludes arrows from earlier treatments to later variables unless those effects pass through included descendants. This can be useful if carryover is scientifically implausible, but it is a substantive restriction, not a neutral simplification. A richer longitudinal DAG can make lagged direct arrows and mediated paths explicit. Neither graph is automatically preferable; the choice depends on the causal assumptions and time resolution.

The distinction matters because later covariates may carry part of an earlier treatment’s influence. A graph with only a direct carryover arrow and a graph with only a mediated path can imply the same total effect while describing different mechanisms. In particular, conditioning on a post-treatment covariate changes which paths are being considered; causal adjustment decisions require the path structure, not merely a list of variables associated with the outcome.