To calculate the power of a one-way ANOVA, we use the noncentral F distribution F(dfB, dfE, λ) where the noncentrality parameter is
The noncentrality parameter is also equal to f2n where f is the effect size measure described in Effect Size for ANOVA.
Example 1: Find the power for the test in Example 2 of One-way ANOVA Basic Concepts.
We start by showing the results of the one-way ANOVA using Real Statistics’s data analysis tool in Figure 1.
Figure 1 – One-way ANOVA
Using the results in Figure 1, we now calculate the power in Figure 2.
Figure 2 – Power of a one-way ANOVA
We see that the power is 65.3%. We can achieve the same result using the first of the following two supplemental functions.
Real Statistics Functions: The following functions are provided in the Real Statistics Pack:
ANOVA1_POWER(f, n, k, type, α, m, prec) = the power of a one-way ANOVA where type = 1 (default), f = Cohen’s effect size and n = the sample size. If type = 2 then f = the RMSSE effect size instead and if type = 0 then f = the noncentrality parameter.
ANOVA1_SIZE(f, k, 1−β, type, α, m, prec) = the minimum sample size required to obtain power of at least 1−β (default .80) in a of a one-way ANOVA where type = 1 (default), f = Cohen’s effect size. If type = 2 then f = the RMSSE effect size 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).
For Example 1, ANOVA1_POWER(Q11,Q9,Q10) = .652582, as expected. The same result can be achieved using the formulas
Example 2: How big a sample is required to achieve power of 80% for a one-way ANOVA with 4 groups and a Cohen’s effect size of .3?
We can use Excel’s Goal Seek capability as shown in Figure 3.
Figure 3 – Sample size required for a one-way ANOVA using Goal Seek
Upon clicking on the OK button in the Goal Seek dialog box, the results in Figure 4 are obtained.
Figure 4 – Sample size required for a one-way ANOVA
Rounding up we see that a sample size of 125 (cell V7) is required. Actually this sample size will be just short of 80% power (note that cell V16 is less than 80%). The actual sample size required to achieve 80% is 126.
We achieve the same result by using the supplemental formula ANOVA1_SIZE(V9,V8).