  Data fusion is an important way to realize powerful and generalizable
analyses across multiple sources. However, different capability of data
collection across the sources has become a prominent issue in practice. This
could result in the blockwise missingness (BM) of covariates troublesome for
integration. Meanwhile, the high cost of obtaining gold-standard labels can
cause the missingness of response on a large proportion of samples, known as
the semi-supervised (SS) problem. In this paper, we consider a challenging
scenario confronting both the BM and SS issues, and propose a novel
Data-adaptive projecting Estimation approach for data FUsion in the
SEmi-supervised setting (DEFUSE). Starting with a complete-data-only estimator,
it involves two successive projection steps to reduce its variance without
incurring bias. Compared to existing approaches, DEFUSE achieves a two-fold
improvement. First, it leverages the BM labeled sample more efficiently through
a novel data-adaptive projection approach robust to model misspecification on
the missing covariates, leading to better variance reduction. Second, our
method further incorporates the large unlabeled sample to enhance the
estimation efficiency through imputation and projection. Compared to the
previous SS setting with complete covariates, our work reveals a more essential
role of the unlabeled sample in the BM setting. These advantages are justified
in asymptotic and simulation studies. We also apply DEFUSE for the risk
modeling and inference of heart diseases with the MIMIC-III electronic medical
record (EMR) data.
