Hypothesis Testing in Sequentially Sampled Data: ART to Maximize Power Beyond iid Sampling

Dae Woong Ham, Jiaze Qiu

Abstract: Testing whether a variable of interest affects the outcome is one of the most fundamental problem in statistics and is often the main scientific question of interest. To tackle this problem, the conditional randomiza- tion test (CRT) is widely used to test the independence of variable(s) of interest (X) with an outcome (Y) holding other variable(s) (Z) fixed. The CRT uses “Model-X” inference framework that relies solely on the iid sampling of (X,Z) to produce exact finite-sample p-values that are constructed using any test statistic. We propose a new method, the adaptive randomization test (ART), that tackles the same indepen- dence problem while allowing the data to be adaptively sampled. Like the CRT, the ART relies solely on knowing the (adaptive) sampling dis- tribution of (X,Z). Although the ART allows practitioners to flexibly design and analyze adaptive experiments, the method itself does not guarantee a powerful adaptive sampling procedure. For this reason, we show substantial power gains obtained from adaptively sampling com- pared to the typical iid sampling procedure in a multi-arm bandit setting and an application in conjoint analysis. We believe that the proposed adaptive procedure is successful because it takes arms that may ini- tially look like “fake” signals due to random chance and stabilizes them closer to “null” signals and samples more/less from signal/null arms.