Regression-adjusted average treatment effect estimates in stratified and sequentially randomized experiments
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Stratified and sequentially randomized experiments are widely used in fields such as agriculture, industry, medical science, economics, and finance. Baseline covariates are often collected for each unit in the experiments. Many researchers use linear regression to analyse the experimental results for improving efficiency of treatment effect estimation by adjusting the minor imbalances of covariates in the treatment and control group. Our work studies the asymptotic properties of regression adjustment in stratified and sequentially randomized experiments, under the randomization-based inference framework. We allow both the number of strata and their sizes to be arbitrary, provided the total number of experimental units tends to infinity and each stratum has at least two treated and two control units. Under slightly stronger but interpretable conditions, we re-establish the finite population central limit theory for a stratified random sample. Based on this theorem, we prove in our main results that, with certain other mild conditions, both the stratified difference-in-means and the regression-adjusted average treatment effect estimator are consistent and asymptotically normal. The asymptotic variance of the latter is no greater and is typically lesser than that of the former when the proportion of treated units is asymptotically the same across strata or the number of stratum is bounded. The magnitude of improvement depends on the extent to which the within-strata variation of the potential outcomes can be explained by the covariates. We also provide conservative variance estimators to construct large-sample confidence intervals for the average treatment effect, which are consistent if and only if the stratum-specific treatment effect is constant. Our theoretical results are confirmed by a simulation study and we conclude with an empirical illustration.
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