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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. 

Abstract Semi-supervised (SS) inference has received much attention in recent years. Apart from a moderate-sized labeled data, $$\mathcal L$$, the SS setting is characterized by an additional, much larger sized, unlabeled data, $$\mathcal U$$. The setting of $$|\mathcal U\ |\gg |\mathcal L\ |$$, makes SS inference unique and different from the standard missing data problems, owing to natural violation of the so-called ‘positivity’ or ‘overlap’ assumption. However, most of the SS literature implicitly assumes $$\mathcal L$$ and $$\mathcal U$$ to be equally distributed, i.e., no selection bias in the labeling. Inferential challenges in missing at random type labeling allowing for selection bias, are inevitably exacerbated by the decaying nature of the propensity score (PS). We address this gap for a prototype problem, the estimation of the response’s mean. We propose a double robust SS mean estimator and give a complete characterization of its asymptotic properties. The proposed estimator is consistent as long as either the outcome or the PS model is correctly specified. When both models are correctly specified, we provide inference results with a non-standard consistency rate that depends on the smaller size $$|\mathcal L\ |$$. The results are also extended to causal inference with imbalanced treatment groups. Further, we provide several novel choices of models and estimators of the decaying PS, including a novel offset logistic model and a stratified labeling model. We present their properties under both high- and low-dimensional settings. These may be of independent interest. Lastly, we present extensive simulations and also a real data application. 

Summary A fundamental challenge in semi-supervised learning lies in the observed data’s disproportional size when compared with the size of the data collected with missing outcomes. An implicit understanding is that the dataset with missing outcomes, being significantly larger, ought to improve estimation and inference. However, it is unclear to what extent this is correct. We illustrate one clear benefit: root-$$n$$ inference of the outcome’s mean is possible while only requiring a consistent estimation of the outcome, possibly at a rate slower than root $$n$$. This is achieved by a novel $$k$$-fold, cross-fitted, double robust estimator. We discuss both linear and nonlinear outcomes. Such an estimator is particularly suited for models that naturally do not admit root-$$n$$ consistency, such as high-dimensional, nonparametric or semiparametric models. We apply our methods to estimating heterogeneous treatment effects. 

Extracellular vesicles (EVs) have gained considerable attention as vital circulating biomarkers since their structure and composition resemble the originating cells. The investigation of EVs’ biochemical and biophysical properties is of great importance to map them to their parental cells and to better understand their functionalities. In this study, a novel frequency-dependent impedance measurement system has been developed to characterize EVs based on their unique dielectric properties. The system is composed of an insulator-based dielectrophoretic (iDEP) device to entrap and immobilize a cluster of vesicles followed by utilizing electrical impedance spectroscopy (EIS) to measure their impedance at a wide frequency spectrum, aiming to analyze both their membrane and cytosolic charge-dependent contents. The EIS was initially utilized to detect nano-size vesicles with different biochemical compositions, including liposomes synthesized with different lipid compositions, as well as EVs and lipoproteins with similar biophysical properties but dissimilar biochemical properties. Moreover, EVs derived from the same parental cells but treated with different culture conditions were characterized to investigate the correlation of impedance changes with biochemical properties and functionality in terms of pro-inflammatory responses. The system also showed the ability to discriminate between EVs derived from different cellular origins as well as among size-sorted EVs harbored from the same cellular origin. This proof-of-concept approach is the first step towards utilizing EIS as a label-free, non-invasive, and rapid sensor for detection and characterization of pathogenic EVs and other nanovesicles in the future.