One-sided sample at the 1% level

Question

A two-sided test at \alpha = 0.01 with 80\% power needs 500 users per arm. You switch to a one-sided test at the same \alpha = 0.01, keeping power and effect fixed. Roughly how many users per arm does the one-sided test need?

Accepted Answer

The two constants. At the 1% level, the two-sided critical value is z{\alpha/2} = 2.576 and the one-sided is z\alpha = 2.326. - Two-sided: (2.576 + 0.84)^2 = (3.416)^2 = 11.67. - One-sided: (2.326 + 0.84)^2 = (3.166)^2 = 10.02. The ratio. \frac{n{	ext{one-sided}}}{n{	ext{two-sided}}} = \frac{10.02}{11.67} \approx 0.859. Apply it. n{	ext{one-sided}} \approx 0.859 	imes 500 \approx 430 	ext{ per arm}. The saving is only about 14% here, smaller than at \alpha = 0.05, because at a stricter level the gap between the one-sided and two-sided critical values is a smaller fraction of the (larger)…

One-sided sample at the 1% level

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