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Register a new userQuantile Treatment Effects and Bootstrap Inference under Covariate-Adaptive Randomization
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Yichong Zhang
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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In this paper, we study the estimation and inference of the quantile treatment effect under covariate-adaptive randomization. We propose two estimation methods: (1) the simple quantile regression and (2) the inverse propensity score weighted quantile regression. For the two estimators, we derive their asymptotic distributions uniformly over a compact set of quantile indexes, and show that, when the treatment assignment rule does not achieve strong balance, the inverse propensity score weighted estimator has a smaller asymptotic variance than the simple quantile regression estimator. For the inference of method (1), we show that the Wald test using a weighted bootstrap standard error under-rejects. But for method (2), its asymptotic size equals the nominal level. We also show that, for both methods, the asymptotic size of the Wald test using a covariate-adaptive bootstrap standard error equals the nominal level. We illustrate the finite sample performance of the new estimation and inference methods using both simulated and real datasets.
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Covariate-adaptive randomization schemes such as the minimization and stratified permuted blocks are often applied in clinical trials to balance treatment assignments across prognostic factors. The existing theoretical developments on inference after covariate-adaptive randomization are mostly limited to situations where a correct model between the response and covariates can be specified or the randomization method has well-understood properties. Based on stratification with covariate levels utilized in randomization and a further adjusting for covariates not used in randomization, in this article we propose several estimators for model free inference on average treatment effect defined as the difference between response means under two treatments. We establish asymptotic normality of the proposed estimators under all popular covariate-adaptive randomization schemes including the minimization whose theoretical property is unclear, and we show that the asymptotic distributions are invariant with respect to covariate-adaptive randomization methods. Consistent variance estimators are constructed for asymptotic inference. Asymptotic relative efficiencies and finite sample properties of estimators are also studied. We recommend using one of our proposed estimators for valid and model free inference after covariate-adaptive randomization.
Datasets from field experiments with covariate-adaptive randomizations (CARs) usually contain extra baseline covariates in addition to the strata indicators. We propose to incorporate these extra covariates via auxiliary regressions in the estimation and inference of unconditional QTEs under CARs. We establish the consistency, limiting distribution, and validity of the multiplier bootstrap of the regression-adjusted QTE estimator. The auxiliary regression may be estimated parametrically, nonparametrically, or via regularization when the data are high-dimensional. Even when the auxiliary regression is misspecified, the proposed bootstrap inferential procedure still achieves the nominal rejection probability in the limit under the null. When the auxiliary regression is correctly specified, the regression-adjusted estimator achieves the minimum asymptotic variance. We also derive the optimal pseudo true values for the potentially misspecified parametric model that minimize the asymptotic variance of the corresponding QTE estimator. We demonstrate the finite sample performance of the new estimation and inferential methods using simulations and provide an empirical application to a well-known dataset in education.
Analyses of environmental phenomena often are concerned with understanding unlikely events such as floods, heatwaves, droughts or high concentrations of pollutants. Yet the majority of the causal inference literature has focused on modelling means, rather than (possibly high) quantiles. We define a general estimator of the population quantile treatment (or exposure) effects (QTE) -- the weighted QTE (WQTE) -- of which the population QTE is a special case, along with a general class of balancing weights incorporating the propensity score. Asymptotic properties of the proposed WQTE estimators are derived. We further propose and compare propensity score regression and two weighted methods based on these balancing weights to understand the causal effect of an exposure on quantiles, allowing for the exposure to be binary, discrete or continuous. Finite sample behavior of the three estimators is studied in simulation. The proposed methods are applied to data taken from the Bavarian Danube catchment area to estimate the 95% QTE of phosphorus on copper concentration in the river.
Understanding treatment effect heterogeneity in observational studies is of great practical importance to many scientific fields because the same treatment may affect different individuals differently. Quantile regression provides a natural framework for modeling such heterogeneity. In this paper, we propose a new method for inference on heterogeneous quantile treatment effects that incorporates high-dimensional covariates. Our estimator combines a debiased $ell_1$-penalized regression adjustment with a quantile-specific covariate balancing scheme. We present a comprehensive study of the theoretical properties of this estimator, including weak convergence of the heterogeneous quantile treatment effect process to the sum of two independent, centered Gaussian processes. We illustrate the finite-sample performance of our approach through Monte Carlo experiments and an empirical example, dealing with the differential effect of mothers education on infant birth weights.
This paper examines methods of inference concerning quantile treatment effects (QTEs) in randomized experiments with matched-pairs designs (MPDs). Standard multiplier bootstrap inference fails to capture the negative dependence of observations within each pair and is therefore conservative. Analytical inference involves estimating multiple functional quantities that require several tuning parameters. Instead, this paper proposes two bootstrap methods that can consistently approximate the limit distribution of the original QTE estimator and lessen the burden of tuning parameter choice. Most especially, the inverse propensity score weighted multiplier bootstrap can be implemented without knowledge of pair identities.
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