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Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect
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Derives graphical criteria for valid adjustment sets for the weighted controlled direct effect (WCDE), its nonparametric influence function, and shows the optimal (minimum asymptotic variance) adjustment set is the set of outcome parents excluding treatment, validated with AIPW simulations and bnlearn networks.
Problem
Identifying valid adjustment sets for estimating the weighted controlled direct effect (WCDE) from observational data and choosing the one that minimizes asymptotic variance among regular asymptotically linear estimators, which differs from ATE adjustment theory due to mediator-confounder interactio
Approach
Establishes necessary and sufficient graphical conditions for WCDE valid adjustment sets, derives the nonparametric influence function of WCDE, and shows via four variance-reducing operations that the optimal adjustment set is the non-treatment parents of the outcome (confounder, mediator, and informative non-descendant parents).
Key results
- On synthetic data (50 replications, n=250 to 2000), the O-set {G1,G2} achieves lowest variance and MSE versus other valid adjustment sets (e.g., 0.0029 vs 0.0102 variance at n=250).
- On ASIA, the criterion-identified set {bronc} attains lowest variance and MSE across all sample sizes from n=250 to 10000.
- On SIGNALING, the O-set {PKA, Erk} achieves smallest variance for n>=1000 but its MSE is not always the lowest, reflecting a bias-variance trade-off at small n.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the O-set {G1, G2} instead of adjustment set {G1, B2} | WCDE estimation on synthetic Fig [ref] · Synthetic Fig [ref], n=250 | variance: 0.0102 -> 0.0029 | Helped |
| Use the O-set {G1, G2} instead of adjustment set {G1, G2, B2} | WCDE estimation on synthetic Fig [ref] · Synthetic Fig [ref], n=250 | variance: 0.0050 -> 0.0029 | Helped |
| Remove redundant backdoor variable B2 from {G1, G2, B2} (ablation) | WCDE adjustment set on synthetic Fig [ref] · Synthetic Fig [ref], n=250 | variance: 0.0050 -> 0.0029 | Helped |
| Use the O-set {G1, G2} instead of adjustment set {B1, G2} | WCDE estimation on synthetic Fig [ref] · Synthetic Fig [ref], n=250 | variance: 0.0100 -> 0.0027 | Helped |
| Remove redundant mediator B1 from {G1, B1, G2} (ablation) | WCDE adjustment set on synthetic Fig [ref] · Synthetic Fig [ref], n=250 | variance: 0.0059 -> 0.0027 | Helped |
| Use O-set {bronc} instead of adjustment set {smoke} | WCDE estimation on ASIA · ASIA, n=250 | variance: 0.0210 -> 0.0143 | Helped |
| Use O-set {bronc} instead of adjustment set {bronc, smoke} | WCDE estimation on ASIA · ASIA, n=250 | variance: 0.0176 -> 0.0143 | Helped |
| Use O-set {Erk, PKA} instead of adjustment set {Erk, Mek, PKA} | WCDE estimation on SIGNALING · SIGNALING, n=250 | MSE: 0.00476 -> 0.00618 | Hurt |
| Use O-set {Erk, PKA} rather than the best-performing adjustment set at small sample sizes | WCDE estimation on SIGNALING · SIGNALING, n=250 and n=500 | MSE | Mixed |
| Use the O-set {Erk, PKA} for larger samples | WCDE estimation on SIGNALING · SIGNALING, n ≥ 1000 | variance | Helped |
| Use the O-set {meldug_3, middel_3, mikro_3} | WCDE estimation on MILDEW · MILDEW, n ≥ 500 | variance | Helped |
Surprising findings
- The optimal adjustment set for WCDE differs from ATE adjustment sets, including mediators that are parents of the outcome and excluding backdoor variables not in the outcome's parents.
- At small sample sizes (n<=500) on SIGNALING, the O-set is not the lowest-MSE set, showing finite-sample bias-variance trade-offs.
Limitations
- Assumes the causal DAG is known and correct; errors from causal discovery are not analyzed.
- Only the case where all observed mediators are fixed is treated; partial mediator fixing is left for future work.
- Necessity of the CDE identification condition is not established; the CDE criterion is sufficient but not shown necessary.
Experimental setup
Datasets: ASIA (bnlearn) SIGNALING (bnlearn) MILDEW (bnlearn) synthetic nonlinear SEM
Benchmarks: synthetic nonlinear SEM ASIA SIGNALING MILDEW
Metrics: variance MSE
Baselines: adjustment sets G1,B2 adjustment sets G1,G2,B2 adjustment sets B1,G2 adjustment sets G1,B1,G2
Builds on: controlled direct effect identification (VanderWeele) adjustment criterion (Shpitser/Perković) causal partition taxonomy (Maasch et al.) augmented inverse probability weighting
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