Mediated Paths Through Post-Treatment Covariates

Tracing Routes From Treatment to Outcome

In a longitudinal study, an initial treatment can affect a later health measure through changes in an intermediate covariate. It may also affect the outcome along routes that do not pass through that covariate. For example, a treatment might change a biomarker that influences the outcome, while also affecting the outcome through a different biological process. A causal diagram helps keep these routes distinct.

The distinction is about the structure of a causal route, not about statistical association after adjustment. In particular, seeing a path through an intermediate variable does not by itself quantify mediation, and seeing a path that bypasses the variable does not automatically identify a direct effect. The graph records possible causal connections under its assumptions; further assumptions and an estimand are needed to make numerical causal claims.

Consider a graph with arrows \(A\to L\), \(L\to Y\), and \(A\to Y\). There are two directed paths from \(A\) to \(Y\): \(A\to L\to Y\), which goes through \(L\), and \(A\to Y\), which bypasses \(L\). If the graph also contains \(L\to A_1\to Y\), then \(A\to L\to A_1\to Y\) is another path through \(L\). The classification is always relative to the specified intermediate covariate: a path may pass through one post-treatment variable while bypassing another.

What a Path Does and Does Not Establish

A directed path gives a possible route for an effect to propagate. It does not guarantee that the net effect along that route is nonzero. A structural coefficient may be zero, or effects along different routes may cancel. Conversely, if the graph is a correctly specified causal DAG and there is no directed path from \(A\) to \(Y\), changing \(A\) cannot change \(Y\) through the modeled causal mechanisms.

Proof. Order the nodes so that every parent precedes its child, which is possible because the graph is finite and acyclic. Under two intervention values for \(A\), the structural values of nodes that are not descendants of \(A\) are identical: proceed in this order, noting that their equations and all their parent values are unchanged. Since there is no directed path from \(A\) to \(Y\), \(Y\) is not a descendant of \(A\). Its parents therefore have identical values under both interventions, so its structural equation gives the same value of \(Y\). This argument is for each fixed setting of the disturbances; it also implies equality of outcome distributions if the disturbances have the same distribution under both interventions. \(\square\)

The result is conditional on the model: the graph must include the relevant causal relationships, and the intervention must not alter other structural equations. An omitted causal route or an intervention that changes additional mechanisms falls outside the claim. A path that is present establishes only a potential route, not a guaranteed effect. For example, its structural coefficients could be zero.

Path Contributions in Linear Models

For an additive linear structural model, there is a precise way to calculate the total effect from the directed paths. Let the graph be finite and acyclic, and suppose each non-treatment node \(V_j\) satisfies \(V_j=\sum_{i\in pa(j)} b_{ji}V_i+U_j\), where \(pa(j)\) is its set of parents and \(b_{ji}\) is the coefficient on the arrow \(i\to j\). Assume the disturbances do not change under intervention on \(A\), and that the equations contain no interactions or nonlinear terms.

Proof. For each node \(V_j\), let \(d_j\) be its change under \(do(A=a+1)\) minus its value under \(do(A=a)\), with disturbances held fixed. The intervention sets \(d_A=1\). For every other node, subtracting its two structural equations cancels \(U_j\) and gives \(d_j=\sum_{i\in pa(j)} b_{ji}d_i\). In topological order, expand this recursion: each contribution to \(d_j\) is a contribution to a parent's change multiplied by the coefficient on the final arrow into \(j\). Starting with \(d_A=1\), each resulting term at \(Y\) is therefore the product of the coefficients along one directed path from \(A\) to \(Y\). Every such path is generated once, because its last arrow has exactly one parent and the earlier part of the path is generated in that parent's expansion. Summing the terms gives the stated result; if there is no path, the recursion produces no terms from \(d_A\), so \(d_Y=0\). \(\square\)

The path-product result is specific to additive linear equations. With nonlinearities or interactions, a path's contribution can depend on treatment and covariate values; there need not be a single coefficient product that summarizes it. The graph still classifies routes, but quantifying a pathway requires a clearly defined causal contrast and a model or identification strategy appropriate to that contrast.

Interpreting the Distinction Carefully

In everyday causal language, a route through an intermediate covariate is often called mediated, while a route bypassing it is sometimes called direct. These labels describe routes in the graph. In particular, bypassing \(L\) does not require a single arrow \(A\to Y\): a path such as \(A\to A_1\to Y\) also bypasses \(L\). Calling that route “direct” without qualification can be misleading.

A total treatment effect compares outcomes under treatment interventions without separating routes. A path-specific effect instead asks for a contrast that attributes change to a selected set of routes, such as those that pass through \(L\). The latter needs a precise definition. The diagram by itself does not say how to combine or intervene on the pathways, or whether the requested contrast is identifiable from observed data.

In particular, conditioning on \(L\) in a regression is not automatically a way to isolate the paths bypassing \(L\). Adjustment can block some routes through \(L\), but the resulting conditional association is not, without further assumptions, a causal effect along the remaining routes. Conditioning can also create bias in some graphs, as collider analysis illustrates. The relevant alternative is not “look at the graph” versus “adjust for the mediator”; it is to define the causal estimand and justify the assumptions and method used to identify it.

Common Mistakes and Limitations

• Equating a path with a nonzero effect. A path can have zero coefficients or cancel against other routes. The graph indicates possible transmission, not effect size.
• Calling every bypassing path a single direct arrow. A path can bypass \(L\) while passing through other variables. State which variable is being bypassed.
• Treating adjustment for \(L\) as mediation analysis. A regression coefficient conditional on \(L\) is not automatically a causal effect through routes that bypass \(L\).
• Reading an incomplete graph as proof of no effect. The no-path conclusion holds only if the causal graph and intervention assumptions are adequate for the question.

Path tracing is useful for clarifying mechanisms, identifying which routes a proposed causal question concerns, and checking simplified linear models. Its limitations matter: uncertain arrows, omitted causes, nonlinear effects, and interactions can prevent a diagram or a path-product calculation from supporting a numerical pathway claim. The total effect and a path-specific effect are different targets; neither should be substituted for the other without justification.