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

anovainteractionrrepeated measuresstatistical-power

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?

Best Answer

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).

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