The Decaying Missing-at-Random Framework: Model Doubly Robust Causal Inference with Partially Labeled Data

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.