Research map / Probabilistic and scientific modeling / Treatment effects
Continuous Treatment Effects with Surrogate Outcomes
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Proposes a doubly robust, sample-splitting local-linear estimator of continuous dose-response curves that uses labeled and unlabeled data with surrogate outcomes under MAR missingness, and establishes its asymptotic normality.
Problem
Estimating continuous treatment (dose-response) effects when primary outcomes are partially missing not at random given covariates, so labeled-only analyses are biased and inefficient.
Approach
Derives a doubly robust pseudo-outcome for the dose-response function using labeled and unlabeled data and surrogates, regresses it on treatment via local linear smoothing with cross-fitting, and shows oracle efficiency and asymptotic normality with variance reduction over labeled-only methods.
Key results
- In simulations with n=500 and 2000 (M=500 reps), the doubly robust estimator beats the plug-in estimator when nuisance estimation error is large (small alpha), per the RMSE-vs-alpha figure.
- Theory shows asymptotic normality of the local linear DR estimator with variance sigma^2(a) int K^2 / f(a), and under MCAR the unlabeled data plus surrogates reduce the second variance term by a factor of rho versus labeled-only methods.
- The plug-in estimator outperforms the doubly robust one when nuisance estimates are very accurate (large alpha), due to accumulated error in pseudo-outcome construction and bandwidth selection.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the doubly robust estimator instead of the plug-in estimator when nuisance estimation error is large | Continuous Treatment Effects with Surrogate Outcomes · Simulation, estimating θ(1) with large nuisance estimation error | RMSE | Helped |
| Use the doubly robust estimator instead of the plug-in estimator when nuisance estimates are highly accurate | Continuous Treatment Effects with Surrogate Outcomes · Simulation, estimating θ(1) with accurate nuisance estimates | RMSE | Hurt |
| Add unlabeled data and surrogate outcomes | Continuous treatment effect estimation · MCAR setting | Asymptotic variance | Helped |
Surprising findings
- When nuisance estimation is accurate, the plug-in estimator beats the doubly robust estimator, because DR accumulates error from pseudo-outcomes, bandwidth selection and local linear regression.
Limitations
- Cannot handle the case where unlabeled data has only covariates and treatment is missing (generalizability setting).
- Local linear estimator is centered on a smoothed bias term; uncertainty is reported for the biased target rather than theta(a).
- Simulations use manually set nuisance errors and a single low-dimensional DGP; real data example only in supplement.
Experimental setup
Datasets: simulated data
Metrics: RMSE
Baselines: plug-in estimator based on outcome regression
Builds on: doubly robust estimation for continuous treatments stability framework for linear smoothers semiparametric efficiency theory
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