Doubly Robust Estimation with Stabilized Weights for Binary Proximal Outcomes in Micro-Randomized Trials

TL;DR

This work tackles estimating excursion effects in micro-randomized trials with binary proximal outcomes under small samples and extreme randomization. It introduces DR-EMEE, a doubly robust estimator that combines stabilized and truncated per-decision IPW with outcome regression, extendable to machine learning nuisance estimators via cross-fitting. The authors prove double robustness, asymptotic normality, and semiparametric efficiency, along with finite-sample variance corrections and a projection-based variant DR-EMEE2. Through extensive simulations and real-data analyses (HeartSteps, PAMAP2, mHealth), DR-EMEE demonstrates reduced RMSE, improved coverage, and substantial efficiency gains over IPW and EMEE, validating its practical robustness for both randomized and observational MRT settings.

Abstract

Micro-randomized trials (MRTs) are increasingly used to evaluate mobile health interventions with binary proximal outcomes. Standard inverse probability weighting (IPW) estimators are unbiased but unstable in small samples or under extreme randomization. Estimated mean excursion effect (EMEE) improves efficiency but lacks double robustness. We propose a doubly robust EMEE (DR-EMEE) with stabilized and truncated weights, combining per-decision IPW and outcome regression. We prove double robustness, asymptotic efficiency, and provide finite-sample variance corrections, with extensions to machine learning nuisance estimators. In simulations, DR-EMEE reduces root mean squared error, improves coverage, and achieves up to twofold efficiency gains over IPW and five to ten percent over EMEE. Applications to HeartSteps, PAMAP2, and mHealth datasets confirm stable and efficient inference across both randomized and observational settings.

Figures (7)

• Figure 1: Conceptual overview of the proposed DR-EMEE framework. Classical IPW is unbiased but unstable due to extreme weights. EMEE improves efficiency by incorporating outcome regression but lacks robustness. By combining stabilization/truncation of weights with augmentation, DR-EMEE achieves double robustness, finite-sample stability, and asymptotic efficiency.
• Figure 2: Weight stabilization and truncation across randomization probabilities. Each row corresponds to a different randomization probability ($p_t=0.1,0.3,0.5$). Columns show the distribution of raw weights, stabilized weights, truncated weights, and the bias--variance--MSE tradeoff curves. Raw IPW weights are highly unstable when treatment is rare, while stabilization and truncation progressively reduce variance and eliminate heavy tails. The tradeoff curves highlight how moderate truncation minimizes MSE, demonstrating the finite-sample stability advantage of DR-EMEE across diverse scenarios.
• Figure 3: Summary of simulation results across sample sizes and randomization probabilities. (a) Relative efficiency, (b) RMSE, and (c) coverage probability across all scenarios. (d--f) Coverage probability stratified by randomization probability ($p_t=0.1,0.5,0.9$). (g--i) Incremental relative efficiency of DR-EMEE over EMEE. DR-EMEE consistently achieved higher efficiency and lower RMSE while maintaining valid coverage across all scenarios.
• Figure 4: Forest plot of PAMAP2 dataset results (S1--S4). DR-EMEE consistently exhibits tighter confidence intervals and reduced variance relative to IPW and EMEE, highlighting its robustness in real-world activity monitoring data.
• Figure 5: Forest plot of mHealth dataset results (S1--S4). DR-EMEE produces stable estimates with substantially narrower intervals than IPW and improved robustness relative to EMEE, even in a large-scale observational setting.

Theorems & Definitions (28)

• Lemma 1: pd-IPW identification
• Proposition 2: EMEE vs. pd-EMEE comparison
• Corollary 3: Stabilized numerator efficiency
• Lemma 4: Truncation bias bound
• Proposition 5: Asymptotic negligibility of truncation
• Corollary 6: Extreme probability robustness
• Theorem 7: Double Robustness of DR-EMEE
• Theorem 8: Consistency and Asymptotic Normality
• Lemma 9: Stochastic equicontinuity
• Theorem 10: Semiparametric Efficiency Bound