Extra power from going one-sided at fixed sample

Question

A two-sided test at \alpha = 0.05 has exactly 80\% power to detect a given effect at your current sample size. Without adding any data, you switch to a one-sided test at the same \alpha = 0.05, in the direction of the true effect. What is the power now?

Accepted Answer

Recover the effect in standard-error units. For a two-sided test with 80% power, the effect d (in standard-error units) satisfies d - z{\alpha/2} = z\beta, so d = 1.96 + 0.84 = 2.80. Recompute power for the one-sided test. The one-sided cutoff is lower, z\alpha = 1.645, so 	ext{power} = \Phi(d - z\alpha) = \Phi(2.80 - 1.645) = \Phi(1.155) \approx 0.88. Same data, same effect, but power rose from 0.80 to about 0.88 purely because the one-sided test clears a lower bar. The price is being blind to an effect in the opposite direction. Takeaway. At fixed sample and effect, switching to a one-side…

Extra power from going one-sided at fixed sample

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