  Propensity score methods are widely used for estimating treatment effects
from observational studies. A popular approach is to estimate propensity scores
by maximum likelihood based on logistic regression, and then apply inverse
probability weighted estimators or extensions to estimate treatment effects.
However, a challenging issue is that such inverse probability weighting methods
including doubly robust methods can perform poorly even when the logistic model
appears adequate as examined by conventional techniques. In addition, there is
increasing difficulty to appropriately estimate propensity scores when dealing
with a large number of covariates. To address these issues, we study calibrated
estimation as an alternative to maximum likelihood estimation for fitting
logistic propensity score models. We show that, with possible model
misspecification, minimizing the expected calibration loss underlying the
calibrated estimators involves reducing both the expected likelihood loss and a
measure of relative errors which controls the mean squared errors of inverse
probability weighted estimators. Furthermore, we propose a regularized
calibrated estimator by minimizing the calibration loss with a Lasso penalty.
We develop a novel Fisher scoring descent algorithm for computing the proposed
estimator, and provide a high-dimensional analysis of the resulting inverse
probability weighted estimators of population means, leveraging the control of
relative errors for calibrated estimation. We present a simulation study and an
empirical application to demonstrate the advantages of the proposed methods
compared with maximum likelihood and regularization.
