# R: Power analysis for a 2 by 2 within-within ANOVA interaction effect [pwr::pwr.f2.test()]

Background

I would like to perform a power analysis for the interaction effect in a 2 by 2 within-within ANOVA design. I would like to crosscheck that I am performing this correctly using the pwr::pwr.f2.test() function in R.

Required inputs

The pwr::pwr.f2.test() function requires the following parameters:

• u: degrees of freedom for numerator

• v: degrees of freedom for denominator

• f2: the effect size Cohen's $$f^2$$

• sig.level: $$\alpha$$ level

• power: the desired power ($$1 - \beta$$)

We will omit v so the function will estimate this value.

Inputs for a $$2 \times 2$$ within-within design

1. u

The numerator degrees of freedom for an interaction effect in this design is given as:

$$u = A \times B = (a - 1)(b - 1)$$

Where $$a$$ is the number of levels of Factor $$A$$, and $$b$$ is the number of levels of Factor $$B$$. Therefore, u is:

$$u = (2-1) \times (2-1) = 1$$

1. f2

Let's assume I have estimated that the interaction effect should be $$\eta^2_p$$ = 0.2. I can convert this to $$f^2$$ using the following formula:

$$f^2 = \frac{\eta^2_p}{1 - \eta^2_p}$$

Therefore, f2 is:

$$f^2 = \frac{0.2}{1-0.2} = 0.25$$

1. sig.level

I will set $$\alpha = .05$$

1. power

I will set $$power = 0.80$$

Perform the analysis

We use the above inputs in the following power analysis:

pwr::pwr.f2.test(u = 1, f2 = 0.25, sig.level = 0.05, power = 0.8)

Multiple regression power calculation

u = 1
v = 31.42944
f2 = 0.25
sig.level = 0.05
power = 0.8


The required denominator degrees of freedom to detect our effect with 80% power is 31.42. To convert v to the total $$N$$ required we will do some simple algebra.

The denominator degrees of freedom for a $$2 \times 2$$ interaction effect is given as:

$$v = A \times B \times S = (a - 1)(b - 1)(N - 1)$$

Where $$N$$ is the total sample size. In our study this is:

$$v = (2 - 1)(2 - 1)(N - 1) = N - 1$$

$$N = v + 1$$

This means we simply add 1 to v to estimate the required sample size, and round up:

ceiling(31.42944 + 1)
[1] 33


The required sample size is therefore $$N = 33$$.

Question

Have I performed these steps correctly?

I think that all of the ANOVA functions in pwr are 1-way. If you have the right data, you should be able to do your calculation using Superpower. It's available as an R package and a shiny app (which is a bit like G*Power).