With high power the winner's curse nearly vanishes

Question

A well-powered study reports only results with $p < 0.05$ (two-sided), and its true effect is a large three standard errors in size.

Show that the winner's curse is mild here, and explain why high power protects against inflated estimates.

Accepted Answer

The setup. In standard-error units the estimate is N(3, 1). Significance requires \hat	heta 1.96. Power. \mathbb{P}(\hat	heta 1.96) = \mathbb{P}\big(Z 1.96 - 3\big) = \mathbb{P}(Z -1.04) = \Phi(1.04) \approx 0.851. So about 85% of runs are significant: this is a well-powered study, and the wrong-tail probability is negligible. Inflation. The expected reported estimate among significant runs is \mathbb{E}[\hat	heta \mid \hat	heta 1.96] = 3 + \frac{\varphi(1.04)}{\Phi(1.04)} = 3 + \frac{0.232}{0.851} \approx 3.27. The reported edge averages about 3.27 against a true edge of 3.0, an inflatio…

With high power the winner's curse nearly vanishes

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