Adaptive and Efficient Learning with Blockwise Missing and Semi-Supervised Data

Data fusion enables powerful and generalizable analyses across multiple sources. However, different data collection capacities across different sources lead to blockwise missingness (BM), which poses challenges in practice. Meanwhile, the high cost of obtaining gold-standard labels leaves the majority of samples unlabeled, known as the semi-supervised (SS) problem. In this paper, we propose a novel Data-adaptive Estimation approach for data FUsion in the SEmi-supervised setting (DEFUSE) that handles both BM and SS issues in the presence of distributional shifts across data sources under a missing at random (MAR) mechanism}. DEFUSE starts with a complete-data-only estimator derived from the primary data source, and uses data-adaptive and distributional-shift-adjusted procedures to successively incorporate the data with BM covariates and the large unlabeled sample to effectively reduce the estimation variance without incurring bias. To further avoid bias due to fusion of misaligned data violating of the MAR assumption, a screening method is developed to identify and exclude data sources that are not aligned with the primary source. Compared to existing approaches, DEFUSE offers two main improvements. First, it offers a new data-adaptive control variate approach to handle BM, which achieves intrinsic efficiency and robustness against distributional shifts. Second, it reveals a more essential role for the unlabeled sample in the BM regression problem, leading to improved estimation. These advantages are theoretically guaranteed and empirically supported by simulation studies and two real-world biomedical applications.