PAMLj

In PAMLj we can use both Partial Eta-Squared or Eta-Squared, because in the case of the \(R^2\) the two effect size are identical. For simplicity, we use Partial Eta-Squared.

Had we used Eta-Squared, we simply put the \(R^2\) to test both in the effect size and the \(R^2\) field, obtaining the same results.

G*Power

R

In R, we can use the pwr.f2.test command, just plugging in the effect size \(f^2\) and the effect DF \(u=5\). However, \(N\) should be computed as the residuals degrees of freedom \(v\) plus the \(u+1\), thus we have \(N=v+u+1\)

sqrt(.10)

## [1] 0.3162278

(res<-pwr::pwr.f2.test(f2=.1111,u=5, power=.6735,sig.level=.05))

## 
##      Multiple regression power calculation 
## 
##               u = 5
##               v = 88.99396
##              f2 = 0.1111
##       sig.level = 0.05
##           power = 0.6735

N<-round((res$v+res$u+1))
N

## [1] 95

SPSS

In SPSS, we can use Power Analysis -> Regression -> Univariate Linear to compare the results. The caveat is that SPSS requires the multiple partial correlation as input, which is simply the squared root of the partial eta-squared. So, in this case, we have \(pr=\sqrt{.10}=0.3162\).

Results

PAMLj

Results are perfectly in line with other software.

G*Power

R

In R, we can use the pwr.f2.test command, just plugging in the effect size \(f^2\) and the effect DF \(u=4\). The final \(N\) will results as \(N=v+u+1\)

f2=.05/(1-.30)
u<-4
m<-9
power<- .241296
(res<-pwr::pwr.f2.test(f2=f2,u=u, power=power,sig.level=.01))

## 
##      Multiple regression power calculation 
## 
##               u = 4
##               v = 84.60237
##              f2 = 0.07142857
##       sig.level = 0.01
##           power = 0.241296

N<-round((res$v+u+1))
N

## [1] 90

SPSS

In SPSS, we can use Power Analysis -> Regression -> Univariate Linear to compare the results. SPSS requires the two \(R^2\) as input. We also need to input the number of preditor of the full model, which are the degrees of freedom of the full model, and the number of predictors in the nested model, which are the degrees of freedom of the effect.

Results are in line with PAMLj.

PAMLj

We plug in the partial Eta-squared and the required parameters, and we obtain N=108, as in GPower.

G*Power

First notice that for this problem, GPower offers F test: Fixed effects ANOVA routine, which requires an \(f\) as the effect size . \(f\) is simply the square root of \(f^2\) so it can be computed from the partial Eta-squared as \(f=\sqrt{p\eta^2/(1-p\eta^2)}\) .

Plugging in the parameters we obtain:

R

As we have seen in Rosetta: GLM posthoc power, pwr.f2.test command will underestimate the power due to the fact that the non centrality parameter is smaller than the one used in G*Power. Nonetheless, we can get an approximated estimation of the sample size by computing the N employing the whole model DF \(m=35\), yielding \(N=v+m+1\)

f<-0.2450722
u<-4
m<-35
f2<-f^2
power=.475627
(res<-pwr::pwr.f2.test(f2=f2,u=4, power=power,sig.level=.05))

## 
##      Multiple regression power calculation 
## 
##               u = 4
##               v = 101.0071
##              f2 = 0.06006038
##       sig.level = 0.05
##           power = 0.475627

N<-round(res$v+m+1)
N

## [1] 137

As expected, the required sample size obtained in pwr.f2.test is larger than the one obtained in GPower.

SPSS

In SPSS, we can use Power Analysis -> Regression -> Univariate Linear to compare the results. In fact, the F-test in GLM is the same for regression and the ANOVA, and the number of predictors in SPSS is nothing else that the degrees of freedom of the effect and the model. The caveat is that SPSS requires the multiple partial correlation as input, which is simply the squared root of the partial eta-squared. So, in this case, we have \(pr=\sqrt{.0.0566}=0.238\).

SPSS gives the same results of PAMLj.