Why Represent Population Differences Graphically?
A causal effect estimated in one population need not apply unchanged in another. The populations may differ in the prevalence of baseline characteristics, in how treatment is assigned, or in the mechanisms generating the outcome. The key question is not simply whether the populations look different. It is whether those differences leave a path from population membership to the outcome’s causal response after accounting for suitable covariates.
A selection diagram makes this question explicit. It extends a causal directed acyclic graph (DAG) with selection nodes indicating which variable-generating mechanisms may differ between the source and target populations. Graph separation can then identify covariate sets that block the relevant selection-related paths. This is a criterion for conditional transportability, not by itself a numerical transport estimate.
Selection Diagrams and S-Admissibility
For a simple diagram, \(S\) may be used for a single selection node. For instance, \(S \to C\) says that the distribution-generating mechanism for baseline covariate \(C\) may vary by population. An arrow \(S \to Y\) says that the outcome mechanism may vary directly by population. These arrows encode possible differences; they do not assert that a difference must be present in every data set.
To assess transport of effects of \(A\) on \(Y\), form the intervention graph \(G_{\bar A}\) by deleting all arrows pointing into \(A\). This represents intervening to set treatment rather than allowing its usual causes to determine it. A covariate set is ordinarily chosen from measured, pre-treatment variables, so it can be conditioned on without changing the meaning of the total effect of treatment.
The graphical criterion targets equality of conditional causal outcome distributions: under the assumptions represented by a correct selection diagram, \(Y(a)\) is independent of the selection nodes given \(Z\). In a two-population description, this means that, for values of \(z\) represented in both populations, the distribution of \(Y(a)\) given \(Z=z\) is the same in the source and target. The covariate distribution itself may still differ. S-admissibility therefore identifies which conditional causal responses can be shared; it does not say the populations have identical marginal effects.
Worked Examples of the Graphical Criterion
Worked Example: A Population Difference in Baseline Composition
Suppose the diagram has \(S \to C\), \(C \to Y\), and \(A \to Y\), where \(C\) is baseline disease severity. There are no other arrows into \(Y\), and no selection arrow points directly to \(Y\). The populations may have different severity distributions, while the treatment and outcome mechanisms conditional on severity are invariant.
In \(G_{\bar A}\), arrows into \(A\) are removed. In this example there are none to remove. The only path from the selection node to the outcome is \(S \to C \to Y\). Conditioning on \(C\), a non-collider on that path, blocks it. Thus \(Z=\{C\}\) is S-admissible. The empty set is not S-admissible because the path remains open without conditioning.
For illustration, suppose the source and target have the same conditional potential-outcome risks, with \(P(Y(0)=1\mid C=0)=0.10\), \(P(Y(1)=1\mid C=0)=0.20\), \(P(Y(0)=1\mid C=1)=0.30\), and \(P(Y(1)=1\mid C=1)=0.50\). The conditional risk differences are \(0.20-0.10=0.10\) for \(C=0\), and \(0.50-0.30=0.20\) for \(C=1\). These are risks and risk differences in the hypothetical outcome setting; the graph supports sharing each conditional response across populations, not claiming that the populations’ marginal risk differences are equal. Different proportions with severe disease can produce different marginal effects.
Worked Example: A Direct Selection Path That Cannot Be Blocked
Now suppose the graph contains \(S \to Y\) and \(A \to Y\), with no measured pre-treatment covariate on the direct selection-to-outcome path. In \(G_{\bar A}\), the arrow \(A \to Y\) remains, because intervention removes arrows into treatment, not arrows out of it. The path \(S \to Y\) is open and has no intermediate node to condition on. No pre-treatment covariate set can make \(Y\) d-separated from \(S\).
Consider hypothetical conditional risks that do not depend on any baseline covariates: in the source, \(P(Y(0)=1)=0.20\) and \(P(Y(1)=1)=0.40\); in the target, the corresponding risks are \(0.30\) and \(0.60\). The source risk difference is \(0.40-0.20=0.20\), while the target risk difference is \(0.60-0.30=0.30\). The differing contrasts are consistent with a population-sensitive outcome mechanism. Because the direct selection path remains open, these source responses cannot be justified as the target responses by conditioning on a covariate set in this diagram. Additional substantive assumptions or information would be needed.
Worked Example: A Collider Is Not an Admissible Substitute
Suppose the graph contains \(S \to D \to Y\), \(S \to C \leftarrow U \to Y\), and \(A \to Y\). Here \(D\) is a baseline covariate on a selection path, while \(C\) is a collider: arrows from \(S\) and an unmeasured cause \(U\) both point into it. Assume there are no other paths from \(S\) to \(Y\).
In \(G_{\bar A}\), \(A \to Y\) remains, and the selection-related paths are unchanged. Without conditioning, the path \(S \to C \leftarrow U \to Y\) is blocked at collider \(C\). The other path, \(S \to D \to Y\), is open. Conditioning on \(D\) blocks that path while leaving \(C\) unconditioned, so \(Z=\{D\}\) is S-admissible.
The tempting choice \(Z=\{C\}\) fails. It does not block \(S \to D \to Y\), because \(D\) remains an unconditioned non-collider. Moreover, conditioning on collider \(C\) opens \(S \to C \leftarrow U \to Y\). Thus \(C\) is not S-admissible. The calculation is a path check: with \(\{D\}\), the first path is blocked at \(D\) and the second at unconditioned \(C\); with \(\{C\}\), the first path remains open and the second becomes open.
Why the Criterion Works
The justification combines the intervention interpretation with the global Markov property of a DAG. The global Markov property states that d-separation in a graph implies the corresponding conditional independence in a distribution that is Markov with respect to that graph. For the causal interpretation, the diagram must also correctly represent the relevant causal mechanisms, and the selection nodes must adequately encode their possible population differences.
Delete arrows into \(A\), producing \(G_{\bar A}\). The resulting graph describes the causal system when treatment is externally set to \(a\).
If \(Z\) d-separates \(Y\) from all selection nodes in \(G_{\bar A}\), the Markov property gives \(Y \perp\!\!\!\perp \mathbf S \mid Z\) in the intervention graph.
Under the diagram’s invariance assumptions, the conditional distribution of the outcome under \(A=a\), given \(Z\), does not depend on the population. Consequently, the source conditional causal response can represent the target conditional response at the same \(Z\) values.
In potential-outcome notation, the implication is \(Y(a) \perp\!\!\!\perp \mathbf S \mid Z\), when the selection nodes encode population membership and the graph’s causal assumptions hold. This is conditional invariance, not a claim that selection is randomized, that all measured covariates have identical distributions, or that the marginal causal effect is automatically identical.
How to Use a Selection Diagram Carefully
A disciplined analysis begins by stating which mechanisms may differ, rather than drawing selection arrows merely because two populations have different observed averages. For each candidate covariate set, inspect every path from each selection node to the outcome in the intervention graph. A set is useful only if it blocks all such paths, and the variables used must be measured and meaningful in both populations.
The distinction between an arrow into treatment and an arrow into the outcome matters. A selection node may point to \(A\), indicating different treatment-assignment mechanisms. Because \(G_{\bar A}\) deletes incoming arrows to \(A\), that difference alone does not imply a difference in the outcome response under an intervention. By contrast, a selection path that remains connected to \(Y\) after intervention can threaten transportability.
- Do not infer invariance from similarity alone. Similar observed outcome rates do not establish that the outcome mechanisms are the same under intervention.
- Do not condition indiscriminately. Conditioning on a collider can open a selection-related path, as in the third example.
- Do not use a post-treatment variable as an ordinary baseline adjustment set. A mediator or treatment descendant can change the causal quantity being represented and does not qualify as the pre-treatment covariate set defined here.
- Do not confuse conditional transportability with a completed transport analysis. S-admissibility says which conditional causal responses are invariant. A target-population marginal effect also depends on how covariates are distributed in the target and on whether the needed values are supported.
Check Your Understanding
For each question, identify the relevant paths or assumptions before giving a conclusion.
- A selection node points to baseline \(L\), and \(L\) points to \(Y\). What set blocks the selection path, and what does that imply about conditional causal responses?
- Why is an arrow \(S \to A\) treated differently from an arrow \(S \to Y\) when constructing \(G_{\bar A}\)?
- In a path \(S \to C \leftarrow U \to Y\), what happens to the path when \(C\) is conditioned on?
- If \(Z\) is S-admissible, does that establish that the source and target marginal causal effects are equal? Explain.
- What additional concern arises if a proposed adjustment variable is measured after treatment?