Power Analysis

I use the pwr package based on Cohen 1988 paper on Power Analysis. Precicely, I use the function for general linear models pwr.f2.test() that takes 5 parameters:

• u - numerator degrees of freedom = p-1, where p = N of predictors, including the intercept
• v - denominator degrees of freedom = n-p, where n = N of observations, p = N of predictors, including the intercept
• effect size (F2)
• significance level
• power

Having any 4 out of 5 parameters filled in, the function returns the fifth one. So we need to have v being NULL and understand the values of the other parameters

1. U

For u we need to know the number of predictors. Our most extensive model with the interaction of density and homogeneity effects has the following predictors:

IVs:

• network density
• homogeneity of community members(let’s say age)
• density*homogeniety interaction

Controls:

• community size
• community date of launch
• n of products
• n of reviews

Based on this, u(DF numerator) = u = 8-1 = 7

2. Effect size

Cohen (1988) suggests F2 values of 0.02, 0.15, and 0.35 represent small, medium, and large effect sizes for general linear models. I will take those values for now, probably, we should rely on the effect sizes from the previous studies for convincing argumentation

3. Significance Level

I will try most commonly used 0.05, 0.01, and 0.001

4. Power

I take 0.8 since this value is the most often recomended one

Calculations:

#library
library(pwr)
#we need to find v to understand the sample size
#small effect size (f2 = 0.02) and p = 0.05
pwr.f2.test(u = 7, v = NULL, f2 = 0.02, sig.level = 0.05, power = 0.8)
#n of observations = u + p = 717 + 8 = 725

#small effect size (f2 = 0.02) and p = 0.01
pwr.f2.test(u = 7, v = NULL, f2 = 0.02, sig.level = 0.01, power = 0.8)
#n of observations = u + p = 991 + 8 = 999

#small effect size (f2 = 0.02) and p = 0.001
pwr.f2.test(u = 7, v = NULL, f2 = 0.02, sig.level = 0.001, power = 0.8)
#n of observations = u + p = 1344 + 8 = 1352

#medium effect size (f2 = 0.15) and p = 0.05
pwr.f2.test(u = 7, v = NULL, f2 = 0.15, sig.level = 0.05, power = 0.8)
#n of ovservations = u + p = 95 + 8 = 103

#medium effect size (f2 = 0.15) and p = 0.01
pwr.f2.test(u = 7, v = NULL, f2 = 0.15, sig.level = 0.01, power = 0.8)
#n of ovservations = u + p = 133 + 8 = 141

#medium effect size (f2 = 0.15) and p = 0.01
pwr.f2.test(u = 7, v = NULL, f2 = 0.15, sig.level = 0.01, power = 0.8)
#n of ovservations = u + p = 183 + 8 = 191

#large effect size (f2 = 0.35) and p = 0.05
pwr.f2.test(u = 7, v = NULL, f2 = 0.35, sig.level = 0.05, power = 0.8)
#n of ovservations = u + p = 41 + 8 = 49

#large effect size (f2 = 0.35) and p = 0.01
pwr.f2.test(u = 7, v = NULL, f2 = 0.35, sig.level = 0.01, power = 0.8)
#n of ovservations = u + p = 58 + 8 = 66

#large effect size (f2 = 0.35) and p = 0.001
pwr.f2.test(u = 7, v = NULL, f2 = 0.35, sig.level = 0.001, power = 0.8)
#n of ovservations = u + p = 81 + 8 = 89