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Doubly Robust Proximal Causal Learning for Continuous Treatments
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Proposes a kernel-based proximal doubly robust (PKDR) estimator for continuous treatment effects under proximal causal learning, with a single min-max optimization for nuisance functions and an O(n^{-4/5}) MSE rate.
Problem
Doubly robust estimation of average causal effects of continuous treatments under unmeasured confounding in proximal causal learning, where the indicator-based PIPW/DR estimators are infeasible for continuous treatments and nuisance bridge functions require per-treatment optimization.
Approach
Replace the indicator (Dirac delta) in the proximal PIPW and DR estimators with a bandwidth kernel, derive the DR form as a smooth influence-function approximation, and estimate the treatment bridge function q for all treatments with one min-max optimization using an estimated policy density.
Key results
- On the doubly robust misspecification study, PKDR achieves cMSE 0.11 (n=500) and 0.08 (n=1000) in Scenario 1, and is best or near-best across Scenarios 2-3, e.g. 0.16 vs. 3.32 for POR-based baselines in Scenario 2 (n=500).
- On the high-dimensional synthetic study (n=1000), PKDR attains cMSE 0.08 versus 0.12 for MINMAX and 0.74 for PMMR.
- The MSE convergence rate is O(n^{-4/5}) with optimal bandwidth h=O(n^{-1/5}).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PKDR instead of PKIPW | Continuous-treatment proximal causal estimation · Doubly Robust Scenario 1, n=1000 | cMSE: 0.11_{\pm 0.08} -> 0.08_{\pm 0.04} | Helped |
| Use PKDR instead of PKIPW | Continuous-treatment proximal causal estimation · Doubly Robust Scenario 2, n=1000 | cMSE: 0.20_{\pm 0.14} -> 0.24_{\pm 0.19} | Hurt |
| Use PKDR instead of PKIPW | Continuous-treatment proximal causal estimation · Doubly Robust Scenario 2, n=500 | cMSE: 0.16_{\pm 0.11} -> 0.16_{\pm 0.09} | No clear effect |
| Use PKDR instead of outcome-bridge-only POR | Continuous-treatment proximal causal estimation · Doubly Robust Scenario 3 | cMSE: n=500: 0.20_{\pm 0.12} -> 0.20_{\pm 0.15}; n=1000: 0.22_{\pm 0.14} -> 0.19_{\pm 0.10} | Mixed |
| Use PKDR instead of outcome-bridge-only methods | Continuous-treatment proximal causal estimation · Doubly Robust Scenario 4 | cMSE: n=500: PKDR 3.38_{\pm 5.03}, DFPV 1.00_{\pm 0.22}; n=1000: PKDR 3.56_{\pm 3.67}, DFPV 1.02_{\pm 0.23} | Hurt |
| Use PKDR instead of the strongest baseline | Continuous-treatment proximal causal estimation · High Dimension, n=1000 | cMSE: MINMAX 0.12_{\pm 0.07} -> PKDR 0.08_{\pm 0.04} | Helped |
| Use PKDR instead of the strongest baseline | Continuous-treatment proximal causal estimation · High Dimension, n=2000 | cMSE: NMMR 0.05_{\pm 0.04} -> PKDR 0.09_{\pm 0.04} | Hurt |
| Use PKDR instead of outcome-bridge-only POR | Continuous-treatment proximal causal estimation · Abortion & Crime | cMSE: POR 0.01_{\pm 0.00} -> PKDR 0.02_{\pm 0.00} | Hurt |
| Increase the kernel bandwidth scale c | PKIPW and PKDR · Synthetic data sensitivity analysis | cMSE | Mixed |
| Use an estimated policy function and a single min-max optimization to estimate q_0 for all treatments | PKDR nuisance-function estimation · Continuous treatments | optimization runs | Helped |
Surprising findings
- In the scenario where both bridge functions are misspecified, all methods suffer large errors, and PKDR's variance is very large (3.38 ± 5.03 at n=500).
- The DFPV method performs best in misspecified Scenario 4 despite being a baseline rather than the proposed method.
Limitations
- Estimator requires estimating the policy (propensity) density, which can cause large variance when misspecified.
- MSE rate O(n^{-4/5}) is slower than the parametric rate O(n^{-1}).
- Both-misspecified scenario yields large errors for all methods.
Experimental setup
Datasets: synthetic low-dimensional proximal data (Scenarios 1-4) synthetic high-dimensional proximal data (dim X=100) legalized abortion and crime dataset
Benchmarks: synthetic low-dimensional proximal data synthetic high-dimensional proximal data legalized abortion and crime dataset
Metrics: causal mean squared error (cMSE)
Baselines: PMMR KPV DFPV MINMAX NMMR POR
Builds on: proximal causal learning proximal doubly robust estimator PMMR KPV minimax bridge estimation
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