In this paper, we propose a robust method to estimate the average treatment effects in observational studies when the number of potential confounders is possibly much greater than the sample size. We first use a class of penalized M-estimators for the propensity score and outcome models. We then calibrate the initial estimate of the propensity score by balancing a carefully selected subset of covariates that are predictive of the outcome. Finally, the estimated propensity score is used to construct the inverse probability weighting estimator. We prove that the proposed estimator, which has the sample boundedness property, is root-n consistent, asymptotically normal, and semiparametrically efficient when the propensity score model is correctly specified and the outcome model is linear in covariates. More importantly, we show that our estimator remains root-n consistent and asymptotically normal so long as either the propensity score model or the outcome model is correctly specified. We provide valid confidence intervals in both cases and further extend these results to the case where the outcome model is a generalized linear model. In simulation studies, we find that the proposed methodology often estimates the average treatment effect more accurately than the existing methods. We also present an empirical application, in which we estimate the average causal effect of college attendance on adulthood political participation. Open-source software is available for implementing the proposed methodology.
Most recently, @cite_33 @cite_14 proposed a penalized calibrated propensity score method and studied its robustness to model misspecification. Our work is closely related to @cite_33 , which can be seen as equivalent to directly plugging the initial estimator @math into the Horvitz-Thompson estimator with @math . However, this method does not balance the covariates as we did in Step 3. Corollary 3 of @cite_33 implies that the estimator has the rate of the convergence @math , which is slower than that of our estimator. In our proof, one can treat @math as the bias" of the Horvitz-Thompson estimator, which is eliminated by the covariate balancing step, whereas this term remains in @cite_33 . In the followup paper, @cite_14 removed this bias by constructing an AIPW estimator so that the resulting estimator is robust to model misspecification. However, our result is more general than that of @cite_14 . First, our Theorem , and Propositions and show that there exists a large class of estimators that is asymptotically normal under possible model misspecification. Second, our theory holds for generalized linear models as shown in Theorem , whereas @cite_14 's method is invalid if the propensity score model is misspecified.
@cite_14 @cite_33
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Consider the problem of estimating average treatment effects when a large number of covariates are used to adjust for possible confounding through outcome regression and propensity score models. The conventional approach of model building and fitting iteratively can be difficult to implement, depending on ad hoc choices of what variables are included. In addition, uncertainty from the iterative process of model selection is complicated and often ignored in subsequent inference about treatment effects. We develop new methods and theory to obtain not only doubly robust point estimators for average treatment effects, which remain consistent if either the propensity score model or the outcome regression model is correctly specified, but also model-assisted confidence intervals, which are valid when the propensity score model is correctly specified but the outcome regression model may be misspecified. With a linear outcome model, the confidence intervals are doubly robust, that is, being also valid when the outcome model is correctly specified but the propensity score model may be misspecified. Our methods involve regularized calibrated estimators with Lasso penalties, but carefully chosen loss functions, for fitting propensity score and outcome regression models. We provide high-dimensional analysis to establish the desired properties of our methods under comparable conditions to previous results, which give valid confidence intervals when both the propensity score and outcome regression are correctly specified. We present a simulation study and an empirical application which confirm the advantages of the proposed methods compared with related methods based on regularized maximum likelihood estimation. 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.