Scaling from 80 Percent to Some Other Power

Here’s how you take a sample-size estimate that provides 80 percent power from one of the preceding

rules and scale it up or down to provide some other power:

For 50 percent power: Use only half as many participants — multiply the estimate by 0.5.

For 90 percent power: Increase the sample size by 33 percent — multiply the estimate by 1.33.

For 95 percent power: Increase the sample size by 66 percent — multiply the estimate by 1.66.

For example, if you know from doing a prior sample size calculation that a study with 70 participants

provides 80 percent power to test its primary objective, then a study that has

, or 93

participants will have about 90 percent power to test the same objective. The reason to consider

power of levels other than 80 percent is because of limited sample. If you know that 70 participants

provides 80 percent power, but you will only have access to 40, you can estimate maximum power you

are able to achieve.

Scaling from 0.05 to Some Other Alpha Level

Here’s how you take a sample-size estimate that was based on testing at the α = 0.05 level, and scale it

up or down to correspond to testing at some other α level:

For α = 0.10: Decrease the sample size by 20 percent — multiply the estimate by 0.8.

For α = 0.025: Increase the sample size by 20 percent — multiply the estimate by 1.2.

For α = 0.01: Increase the sample size by 50 percent — multiply the estimate by 1.5.

For example, imagine that you’ve calculated you need a sample size of 100 participants using α = 0.05

as your criterion for significance. Then your boss says you have to apply a two-fold Bonferroni

correction (see Chapter 11) and use α = 0.025 as your criterion instead. You need to increase your

sample size to 100 x 1.2, or 120 participants, to have the same power at the new α level.

Adjusting for Unequal Group Sizes

When comparing means or proportions between two groups, you usually get the best power for a given

sample size — meaning it’s more efficient — if both groups are the same size. If you don’t mind

having unbalanced groups, you will need more participants overall in order to preserve statistical

power. Here’s how to adjust the size of the two groups to keep the same statistical power:

If you want one group twice as large as the other: Increase one group by 50 percent, and reduce

the other group by 25 percent. This increases the total sample size by about 13 percent.

If you want one group three times as large as the other: Reduce one group by a third, and

double the size of the other group. This increases the total sample size by about 33 percent.

If you want one group four times as large as the other: Reduce one group by 38 percent and

increase the other group by 250 percent. This increases the total sample size by about 56 percent.