Testing heterogeneous treatment effect with quantile regression under covariate-adaptive randomization

In economic studies and clinical trials, it is prevalent to observe heterogeneous treatment effects that vary depending on the relative locations of units in the distribution of responses. In this study, we propose using quantile regression to estimate and conduct inference for conditional quantile treatment effects (cQTEs) in covariate-adaptive randomized experiments. First, we present sufficient conditions for consistently estimating the cQTEs, concerning the bias due to omitting important covariates in the inference stage. Second, we derive the weak convergence of the quantile regression process and develop a covariate-adaptive randomized bootstrap (CAR-BS) for standard error estimation. Our theoretical results indicate that the Wald test adjusted by CAR-BS is valid in terms of the Type I error, for a large class of covariate-adaptive randomization procedures at different quantiles, regardless of the choice of covariates used in inference. We perform extensive numerical and empirical studies to demonstrate advantages of the new method in various settings.