  Causal inference on the average treatment effect (ATE) using non-probability
samples, such as electronic health records (EHR), faces challenges from sample
selection bias and high-dimensional covariates. This requires considering a
selection model alongside treatment and outcome models that are typical
ingredients in causal inference. This paper considers integrating large
non-probability samples with external probability samples from a design survey,
addressing moderately high-dimensional confounders and variables that influence
selection. In contrast to the two-step approach that separates variable
selection and debiased estimation, we propose a one-step plug-in doubly robust
(DR) estimator of the ATE. We construct a novel penalized estimating equation
by minimizing the squared asymptotic bias of the DR estimator. Our approach
facilitates ATE inference in high-dimensional settings by ignoring the
variability in estimating nuisance parameters, which is not guaranteed in
conventional likelihood approaches with non-differentiable L1-type penalties.
We provide a consistent variance estimator for the DR estimator. Simulation
studies demonstrate the double robustness of our estimator under
misspecification of either the outcome model or the selection and treatment
models, as well as the validity of statistical inference under penalized
estimation. We apply our method to integrate EHR data from the Michigan
Genomics Initiative with an external probability sample.
