# Power for One-way ANOVA

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, iter, 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, iter, 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 iter (default 1,000).

For Example 1, ANOVA1_POWER(Q11,Q9,Q10) = .652582, as expected. The same result can be achieved using the formulas

=ANOVA1_POWER(Q12,Q9,Q10,2)

=ANOVA1_POWER(Q13,Q9,Q10,0).

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).

### 4 Responses to Power for One-way ANOVA

1. CJ says:

Why the NF_DIST return #NAME?
I used your example numbers here in my office365
I installed everything as your instruction, solver, real statistics pack.
Thanks.
CJ.

• Charles says:

CJ,
It sounds like the Real Statistics software is not yet installed properly; otherwise you shouldn’t see #NAME?
When you enter =VER() into any cell what do you see?
When you press Alt-TI do you see RealStat and Solver on the list of Excel addins with check marks next to them?
Charles

2. Eric says:

What would be the distribution to prepare a lambda table for critical values for “acceptable” power levels, for example, eighty percent and up?

• Charles says:

Eric,
For a given alpha (say .05), sample size and power (say 80%), you could use the Real Statistics tools to estimate acceptable lambda values. There would be some trial and error in this calculation (or you could use Goal Seek).
Charles