  We investigate how to improve efficiency using regression adjustments with
covariates in covariate-adaptive randomizations (CARs) with imperfect subject
compliance. Our regression-adjusted estimators, which are based on the doubly
robust moment for local average treatment effects, are consistent and
asymptotically normal even with heterogeneous probability of assignment and
misspecified regression adjustments. We propose an optimal but potentially
misspecified linear adjustment and its further improvement via a nonlinear
adjustment, both of which lead to more efficient estimators than the one
without adjustments. We also provide conditions for nonparametric and
regularized adjustments to achieve the semiparametric efficiency bound under
CARs.
