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

image7323

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.

One-way ANOVA Excel

Figure 1 – One-way ANOVA

Using the results in Figure 1, we now calculate the power in Figure 2.

Power one-way ANOVA

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

=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.

goal-seek-anova-size

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.

sample-size-anova

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. 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

  2. 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

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