One-sided sample from a known two-sided count

Question

A two-sided test at \alpha = 0.05 with 80\% power needs 1{,}000 users per arm. You switch to a one-sided test at the same \alpha, keeping the power and effect fixed. Roughly how many users per arm does the one-sided test need?

Accepted Answer

The ratio of constants. Required sample scales with (z{	ext{crit}} + z\beta)^2: - Two-sided: (1.96 + 0.84)^2 = 7.84. - One-sided: (1.645 + 0.84)^2 = 6.18. \frac{n{	ext{one-sided}}}{n{	ext{two-sided}}} = \frac{6.18}{7.84} \approx 0.788. Apply it. n{	ext{one-sided}} \approx 0.788 	imes 1{,}000 \approx 788 	ext{ per arm}. So you save a bit over 200 per arm by committing to a direction in advance. Takeaway. At 80% power, the one-sided test needs about 79% of the two-sided sample: 1,000 per arm becomes roughly 788.

One-sided sample from a known two-sided count

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