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Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference
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Proposes the sequential adjustment criterion (SAC), a sound and complete graphical criterion for sequential covariate adjustment in causal graphs, with a constructive method and an algorithm for minimal adjustment sets.
Problem
Existing graphical criteria for sequential covariate adjustment (sequential back-door, multi-outcome sequential back-door) are sound but incomplete, so some identifiable sequential adjustment cases are missed.
Approach
Defines a proper sequential back-door graph per time step and a sequential adjustment criterion that is shown to be sound and complete for sequential covariate adjustment; it also generalizes the adjustment criterion and mSBD, and provides a constructive set and a minimal-set algorithm.
Key results
- Proves the sequential adjustment criterion (SAC) is sound and complete for sequential covariate adjustment (Theorem 1).
- Shows mSBD implies SAC (Corollary 1) and adjustment criterion implies SAC (Theorem 2), so SAC strictly generalizes prior criteria.
- Constructive set Zan computable in O(m(|V|+|E|)) and minimal set algorithm minSCA also in O(m(|V|+|E|)) (Theorems 3 and 4).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace mSBD with the sequential adjustment criterion (SAC) | Sequential covariate adjustment · Sequential causal graphs | completeness | Helped |
| use the minimal sequential covariate adjustment algorithm (minSCA) | SAC adjustment-set construction · Sequential causal graphs | adjustment-set size | Helped |
Surprising findings
- The multi-outcome sequential back-door criterion (mSBD) is incomplete: an example graph (Fig. 2a) admits a valid sequential adjustment even though mSBD is not satisfied.
Limitations
- Results are theoretical and assume a known causal graph; no empirical estimation evaluation is reported.
- Completeness is stated relative to graphs and distributions compatible with G, with the ordering of treatments and outcomes fixed by a topological order.
Experimental setup
Baselines: sequential back-door criterion multi-outcome sequential back-door criterion adjustment criterion sequential ignorability SWIG twin network
Builds on: adjustment criterion multi-outcome sequential back-door criterion proper back-door graph sequential back-door criterion do-calculus
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