C.28.CM:4.4 - Inspect the whole relevant structure

Follow possible paths between the proposed cause and outcome. Look for common causes, intermediate mechanisms, measurement processes and selection of cases. A common cause can coexist with a causal path. The role of a node is relative to a path and question, not a permanent label attached to the variable.

For a directed acyclic graph (DAG), the following path test makes that inspection precise. A path joins distinct nodes along edges, ignoring arrow direction when finding the path. At an internal node, two arrowheads meeting there make it a collider on that path; other internal nodes are noncolliders.

Given a conditioning set W disjoint from the endpoints, a path is open when every noncollider on it is outside W and every collider is itself in W or has a descendant in W. Otherwise it is blocked. If every path between two variable sets is blocked, the sets are d-separated by W. For a distribution satisfying the graph’s Markov property, d-separation implies the corresponding conditional independence. An open path permits dependence; it does not guarantee it for every parameter choice. Inferring graph structure from observed independences needs further assumptions, often faithfulness (no extra independences beyond those entailed by the graph), as well as adequate data.

Apply this rule to all relevant paths, including those opened by the way the sample was selected. Conditioning on an incident being reported can matter even when “reported” is absent from the regression.

Consider the complete small graph:

Z → X → M → Y
Z → Y
X → S ← Y
S → D

There are three simple X-to-Y paths. X → M → Y carries the proposed causal mechanism; X ← Z → Y carries a common cause; X → S ← Y is blocked at S before conditioning. Conditioning on Z blocks the common-cause path and retains the causal path. Conditioning on M blocks that causal path. Conditioning on S, or its descendant D, opens the collider path. Inspecting only the fork would miss that selection effect.

For a total effect in an appropriate causal DAG, the back-door criterion provides one sufficient adjustment rule: choose measured covariates that are not descendants of X and block every path entering X through an arrowhead. This leaves the directed causal paths available. The resulting identification still relies on the model’s causal interpretation, suitable data and support for the required comparisons. Failure of this sufficient criterion is not proof that the effect is unidentified. Use C.28 for the identification question rather than inventing an adjustment from one recognizable three-node shape.

The ordinary d-separation rule above applies to DAGs. If feedback matters, distinguish times: an outcome at t may affect workload at t+1. A finite time-unfolded model still needs its initial conditions and omitted influences justified. An equilibrium with simultaneous causal feedback needs an appropriate cyclic model, its solution conditions and its separation rule. Do not erase feedback merely to obtain an acyclic picture.