To compute statistical power for multiple regression we use Cohen’s effect size f2 which is defined by
f2 = .02 represents a small effect, f2 = .15 represents a medium effect and f2 = .35 represents a large effect.
To calculate the power of a multiple regression, we use the noncentral F distribution F(dfReg, dfRes, λ) where dfReg = k, dfRes = n − k − 1 and the noncentral parameter λ (see Noncentral F Distribution) is
Example 1: What is the power of a multiple regression on a sample of size 100 with 10 independent variables when α = .05?
We show the calculation in Figure 1.
Figure 1 – Statistical Power
Real Statistics Functions: The following functions are provided in the Real Statistics Pack:
REG_POWER(effect, n, k, type, α, m, prec) = the power for multiple regression where type = 1 (default), effect = Cohen’s effect size f2 and n = the sample size. If type = 2 then effect = the R2 effect size instead and if type = 0 then effect = the noncentrality parameter λ.
REG_SIZE(effect, k, 1−β, type, α, m, prec) = the minimum sample size required to obtain power of at least 1−β (default .80) for multiple regression where type = 1 (default) and effect = Cohen’s effect size f2. If type = 2 then effect = R2 instead.
Here α = significance level (default = .05). The calculation of the infinite sum for the noncentral F distribution stops when the level of precision exceeds prec (default 0.000000001) or the number of terms in the infinite sum exceeds m (default 1,000).
We can therefore calculate the power for Example 1 using the formula
Example 2: What is the size of the sample required to achieve 90% power for a multiple regression on 8 independent variables where R2 = .2, α = .05?
We see from Figure 2 that the sample size required is 85 and the actual power achieved is 90.26%.
Figure 2 – Sample size required
Real Statistics Data Analysis Tool: Statistical power and sample size can also be calculated using the Power and Sample Size data analysis tool.
For Example 1, we press Ctrl-m and double click on the Power and Sample Size data analysis tool. Next we select the Multiple Regression on the dialog box that appears as Figure 3.
Figure 3 – Statistical Power and Sample Size dialog box
Finally we fill in the dialog box that appears as shown in the upper part of Figure 4. When we press the OK button the results shown in the lower part of Figure 4 appear.
Figure 4 – Multiple Regression Power dialog box