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The "significance filter" refers to focusing exclusively on statisticallysignificant results. Since frequentist properties such as unbiasedness andcoverage are valid only before the data have been observed, there are noguarantees if we condition on significance. In fact, the significance filterleads to overestimation of the magnitude of the parameter, which has beencalled the "winner's curse". It can also lead to undercoverage of theconfidence interval. Moreover, these problems become more severe if the poweris low. While these issues clearly deserve our attention, they have beenstudied only informally and mathematical results are lacking. Here we studythem from the frequentist and the Bayesian perspective. We prove that therelative bias of the magnitude is a decreasing function of the power and thatthe usual confidence interval undercovers when the power is less than 50%. Weconclude that failure to apply the appropriate amount of shrinkage can lead tomisleading inferences.

The Significance Filter, the Winner's Curse and the Need to Shrink