  Consider estimation of average treatment effects with multi-valued treatments
using augmented inverse probability weighted (IPW) estimators, depending on
outcome regression and propensity score models in high-dimensional settings.
These regression models are often fitted by regularized likelihood-based
estimation, while ignoring how the fitted functions are used in the subsequent
inference about the treatment parameters. Such separate estimation can be
associated with known difficulties in existing methods. We develop regularized
calibrated estimation for fitting propensity score and outcome regression
models, where sparsity-including penalties are employed to facilitate variable
selection but the loss functions are carefully chosen such that valid
confidence intervals can be obtained under possible model misspecification.
Unlike in the case of binary treatments, the usual augmented IPW estimator is
generalized by allowing different copies of coefficient estimators in outcome
regression to ensure just-identification. For propensity score estimation, the
new loss function and estimating functions are directly tied to achieving
covariate balance between weighted treatment groups. We develop practical
numerical algorithms for computing the regularized calibrated estimators with
group Lasso by innovatively exploiting Fisher scoring, and provide rigorous
high-dimensional analysis for the resulting augmented IPW estimators under
suitable sparsity conditions, while tackling technical issues absent or
overlooked in previous analyses. We present simulation studies and an empirical
application to estimate the effects of maternal smoking on birth weights. The
proposed methods are implemented in the R package mRCAL.
