Causal graphs and d-separation

A path ignores arrow direction when listing adjacent nodes, but the directions determine whether each internal node is a collider. A collider has both path arrows pointing into it. A path is blocked by conditioning on a noncollider along it, or by a collider for which neither the collider nor any descendant is conditioned upon. It is active only if every noncollider remains unconditioned and every collider has itself or a descendant conditioned upon. The same node can be a collider on one path and a noncollider on another.

Two node sets are d-separated given a third when every path between them is blocked by that third set. This rule handles chains, forks, and collider structures in one language. Conditioning can remove an association by blocking a fork, but it can also create one by opening a collider path. Thus adjusting for more measured variables is not automatically safer. The role of a variable must be evaluated relative to a specific path and a specific causal question, rather than assigned a permanent label from its correlation with the outcome.

A distribution that factorizes according to a DAG satisfies the graph’s d-separation conditional independences. This is the global Markov implication. The reverse direction does not follow automatically: numerical cancellations can create additional independences not encoded by the graph. Faithfulness rules out those extra cancellations and is a separate assumption used by some causal-discovery procedures. A failure to detect dependence in a finite sample is weaker still; it can reflect limited power, measurement error, or an unsuitable test.

For a structural causal model, intervening on A replaces its structural assignment equation while leaving the other mechanisms fixed. In the graph this removes arrows entering A, whereas ordinary conditioning retains the original selection mechanism. The distinction explains why a conditional outcome distribution can differ from an intervention distribution. If an important common cause is omitted, a simple fully observed DAG factorization may not justify the intended intervention interpretation. A credible graph is grounded in temporal and scientific knowledge as well as data.

Suppose A and U are independent binary variables, and a selection indicator C equals one whenever at least one is one. Before selection, observing A gives no information about U. Among selected units, however, an untreated unit must have U equal to one, whereas a treated unit may have either U value. Conditioning on the common effect C therefore creates a negative association between A and U. If the outcome is U itself, this appears as a treated-outcome contrast despite A having no effect on the outcome mechanism.

This example is about selection, not a claim that every collider adjustment is forbidden regardless of the rest of the graph. An adjustment set must be judged by whether all relevant paths remain appropriately blocked. Nor can a DAG alone assign a numerical magnitude to selection bias; the magnitude here comes from the specified binary mechanisms. Combining a small explicit probability model with the path rule helps distinguish the logical role of a graph from the numerical role of its conditional distributions.