The Decaying Missing-at-Random Framework: Model Doubly Robust Causal Inference with Partially Labeled Data
Yuqian Zhang, Abhishek Chakrabortty, Jelena Bradic
TL;DR
This work tackles causal effect estimation when outcomes are partially labeled in large observational studies by introducing a decaying missing-at-random framework that couples labeling bias with missing outcomes. It develops two robust estimators, bias-reduced SS (BRSS) and de-coupled BRSS (DC-BRSS), that achieve model-DR and sparsity-DR properties under high-dimensional confounding and decaying labeling, with adaptive rate double robustness governed by the effective sample size $Na_N$. BRSS leverages targeted nuisance losses and asymmetric cross-fitting to reduce bias under misspecification, while DC-BRSS decouples labeling from treatment, enabling fully nonparametric treatment-propensity estimation via IPW integration. Theoretical results establish CAN inference under weaker sparsity and overlap conditions and provide adaptive rates that depend on the decaying labeling mechanism; extensive simulations and a pseudo-random ACIC dataset demonstrate robustness to labeling bias and model misspecification. The framework thus offers practical tools for reliable causal inference in semi-supervised, high-dimensional settings, with implications for generalizability and data integration in observational studies.
Abstract
In modern large-scale observational studies, data collection constraints often result in partially labeled datasets, posing challenges for reliable causal inference, especially due to potential labeling bias and relatively small size of the labeled data. This paper introduces a decaying missing-at-random (decaying MAR) framework and associated approaches for doubly robust causal inference on treatment effects in such semi-supervised (SS) settings. This simultaneously addresses selection bias in the labeling mechanism and the extreme imbalance between labeled and unlabeled groups, bridging the gap between the standard SS and missing data literatures, while throughout allowing for confounded treatment assignment and high-dimensional confounders under appropriate sparsity conditions. To ensure robust causal conclusions, we propose a bias-reduced SS (BRSS) estimator for the average treatment effect, a type of 'model doubly robust' estimator appropriate for such settings, establishing asymptotic normality at the appropriate rate under decaying labeling propensity scores, provided that at least one nuisance model is correctly specified. Our approach also relaxes sparsity conditions beyond those required in existing methods, including standard supervised approaches. Recognizing the asymmetry between labeling and treatment mechanisms, we further introduce a de-coupled BRSS (DC-BRSS) estimator, which integrates inverse probability weighting (IPW) with bias-reducing techniques in nuisance estimation. This refinement further weakens model specification and sparsity requirements. Numerical experiments confirm the effectiveness and adaptability of our estimators in addressing labeling bias and model misspecification.