One-sided savings at 90% power

Question

A study is powered at 90\% with a two-sided test at \alpha = 0.05. You switch to a one-sided test at the same \alpha, keeping the power and effect fixed. Roughly what fraction of the two-sided sample does the one-sided test need?

Accepted Answer

The two constants. For 90% power, z\beta = 1.28. - Two-sided: (1.96 + 1.28)^2 = (3.24)^2 = 10.51. - One-sided: (1.645 + 1.28)^2 = (2.925)^2 = 8.56. The ratio. \frac{n{	ext{one-sided}}}{n{	ext{two-sided}}} = \frac{8.56}{10.51} \approx 0.82. So the one-sided test needs about 82% of the two-sided sample. The saving is a little smaller than at 80% power, because the z\beta term is now larger and the gap between 1.96 and 1.645 is a smaller share of the total. Takeaway. The one-sided saving shrinks as you demand more power: at 90% power the one-sided test needs about 82% of the two-sided sample, …

One-sided savings at 90% power

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