Author(s): Bradic, Jelena; Wager, Stefan; Zhu, Yinchu | Abstract: Many popular methods for building confidence intervals on causal effects under high-dimensional confounding require strong "ultra-sparsity" assumptions that may be difficult to validate in practice. To alleviate this difficulty, we here study a new method for average treatment effect estimation that yields asymptotically exact confidence intervals assuming that either the conditional response surface or the conditional probability of treatment allows for an ultra-sparse representation (but not necessarily both). This guarantee allows us to provide valid inference for average treatment effect in high dimensions under considerably more generality than available baselines. In addition, we showcase that our results are semi-parametrically efficient.
We also note two recent papers that consider estimators that resemble ours. consider an estimator that, in the spirit of , first fit a penalized covariate-balancing propensity model, and then re-fit without penalty those coefficients that correspond to features that are relevant to outcome modeling. Meanwhile, augments a penalized covariate-balancing propensity model in an outcome regression; it turns out that his covariate-balancing mechanism designed to address the issue of misspecification is also helpful for relaxing sparsity requirements. Neither paper, however, achieves sparsity double robustness as discussed here; rather, they require both the outcome parameter vector @math and the propensity parameter vector @math to be ultra-sparse---or, if there is misspecification they require the population minimizers of both the outcome and propensity loss functions to be ultra-sparse. Under the framework of @cite_4 @cite_2 , are classified as examples of model double robustness, which means that one of the models (either conditional mean or propensity score) is misspecified. Rate double robustness requires that the product of the @math -norms of the estimation errors in two models is of the order @math .
@cite_4 @cite_2
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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. 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.