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From scientific experiments to online A/B testing, the previously observeddata often affects how future experiments are performed, which in turn affectswhich data will be collected. Such adaptivity introduces complex correlationsbetween the data and the collection procedure. In this paper, we prove thatwhen the data collection procedure satisfies natural conditions, then samplemeans of the data have systematic \emph{negative} biases. As an example,consider an adaptive clinical trial where additional data points are morelikely to be tested for treatments that show initial promise. Our surprisingresult implies that the average observed treatment effects would underestimatethe true effects of each treatment. We quantitatively analyze the magnitude andbehavior of this negative bias in a variety of settings. We also propose anovel debiasing algorithm based on selective inference techniques. Inexperiments, our method can effectively reduce bias and estimation error.

Why Adaptively Collected Data Have Negative Bias and How to Correct for It