What’s the effect size?

The first thing to not is that there is no such thing as “power” - in the sense that a sample or a test would have “its power”. Rather power is relative to the sample size and the effect size, that are the most important factors at least.

So what’s the effect size for the one-way ANOVA? Cohen in his 1992 paper suggested the f value for this situation:

\[f = \frac{\sigma_m}{\sigma}\]

That is, \(f\) is defined as the ratio of the sd of the population means divided by the within-population means (equal sample sizes assumed).

A small effect size value is estimated at 0.1 by Cohen. But it’s more convenient to use this helper function:

library(pwr)
cohen.ES("f", size = "small")

## 
##      Conventional effect size from Cohen (1982) 
## 
##            test = f2
##            size = small
##     effect.size = 0.02

cohen.ES("f", size = "medium")

## 
##      Conventional effect size from Cohen (1982) 
## 
##            test = f2
##            size = medium
##     effect.size = 0.15

cohen.ES("f", size = "large")

## 
##      Conventional effect size from Cohen (1982) 
## 
##            test = f2
##            size = large
##     effect.size = 0.35

Hang on, that’s actually \(f2\) given back, not \(f\). So the values of \(f\) are (Cohen, 1992):

• small: 0.10
• medium: 0.25
• large: 0.40

Compute the power estimate

Computing the power is quite straight forward. Assume there are \(k=3\) groups, and we would like to achieve a power of .80:

p <- pwr.anova.test(k = 3,
                    f = 0.25,
                    power = .8)
p

## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 3
##               n = 52.3966
##               f = 0.25
##       sig.level = 0.05
##           power = 0.8
## 
## NOTE: n is number in each group

So we would need about 52 persons - in each group.

Plot it

There’s a convenient plotting function in the pwr package available:

plot(p)

This graph shows the relation between sample size and power assuming that there are 3 groups, alpha is .05, and effect size is \(f = 0.25\).