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Register a new userRerandomization and Regression Adjustment
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Xinran Li
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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Randomization is a basis for the statistical inference of treatment effects without strong assumptions on the outcome-generating process. Appropriately using covariates further yields more precise estimators in randomized experiments. R. A. Fisher suggested blocking on discrete covariates in the design stage or conducting analysis of covariance (ANCOVA) in the analysis stage. We can embed blocking into a wider class of experimental design called rerandomization, and extend the classical ANCOVA to more general regression adjustment. Rerandomization trumps complete randomization in the design stage, and regression adjustment trumps the simple difference-in-means estimator in the analysis stage. It is then intuitive to use both rerandomization and regression adjustment. Under the randomization-inference framework, we establish a unified theory allowing the designer and analyzer to have access to different sets of covariates. We find that asymptotically (a) for any given estimator with or without regression adjustment, rerandomization never hurts either the sampling precision or the estimated precision, and (b) for any given design with or without rerandomization, our regression-adjusted estimator never hurts the estimated precision. Therefore, combining rerandomization and regression adjustment yields better coverage properties and thus improves statistical inference. To theoretically quantify these statements, we discuss optimal regression-adjusted estimators in terms of the sampling precision and the estimated precision, and then measure the additional gains of the designer and the analyzer. We finally suggest using rerandomization in the design and regression adjustment in the analysis followed by the Huber--White robust standard error.
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Completely randomized experiments have been the gold standard for drawing causal inference because they can balance all potential confounding on average. However, they can often suffer from unbalanced covariates for realized treatment assignments. Rerandomization, a design that rerandomizes the treatment assignment until a prespecified covariate balance criterion is met, has recently got attention due to its easy implementation, improved covariate balance and more efficient inference. Researchers have then suggested to use the assignments that minimize the covariate imbalance, namely the optimally balanced design. This has caused again the long-time controversy between two philosophies for designing experiments: randomization versus optimal and thus almost deterministic designs. Existing literature argued that rerandomization with overly balanced observed covariates can lead to highly imbalanced unobserved covariates, making it vulnerable to model misspecification. On the contrary, rerandomization with properly balanced covariates can provide robust inference for treatment effects while sacrificing some efficiency compared to the ideally optimal design. In this paper, we show it is possible that, by making the covariate imbalance diminishing at a proper rate as the sample size increases, rerandomization can achieve its ideally optimal precision that one can expect with perfectly balanced covariates while still maintaining its robustness. In particular, we provide the sufficient and necessary condition on the number of covariates for achieving the desired optimality. Our results rely on a more dedicated asymptotic analysis for rerandomization. The derived theory for rerandomization provides a deeper understanding of its large-sample property and can better guide its practical implementation. Furthermore, it also helps reconcile the controversy between randomized and optimal designs.
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We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional vector predictors. We derive finite sample theoretical guarantees and show that the excess prediction risk of our estimator is minimax optimal. Furthermore, our analysis reveals an interesting phase transition phenomenon that the optimal excess risk is determined jointly by the smoothness and the sparsity of the functional regression coefficients. A novel efficient optimization algorithm based on iterative coordinate descent is devised to handle the smoothness and group penalties simultaneously. Simulation studies and real data applications illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.
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