Power

The package ‘Superpower’ is used due to its support for mixed designs (ours is a 2x2 within x between). We use medium effect size as justified in: Schmiedt-Fehr et al., 2016: DOI: 10.1016/j.clinph.2016.07.008, Reteig et al., 2019: 10.1016/j.cortex.2019.02.016.

library(Superpower)

## Warning: package 'Superpower' was built under R version 4.0.5

set.seed(2022)
mu <- c(-0.25, 0.25, 0.25, -0.25) #Cohen's f expected to be 0.25
#Justification: Schmiedt-Fehr et al., 2016: DOI: 10.1016/j.clinph.2016.07.008, Reteig et al., 2019: 10.1016/j.cortex.2019.02.016
#f value: Cohen, 1988
n <- 17 #Number of participants suggested by plotting
sd <- 1 #Typical parameters
r <- 0.5 #Typical parameters
string = "2w*2b" # 2 within x 2 between subjects
alpha_lev <- 0.05 #Desired alpha level
labelnames = c("age", "old", "young", "alpha", "low", "high") #Condition names

design_result <- ANOVA_design( #Design is compiled
  design = string,
  n = n,
  mu = mu,
  sd = sd,
  r = r,
  labelnames = labelnames
)

A simulation shows the cutoff point.

plot_power(design_result, min_n = 10, max_n = 250, desired_power = 80) # A simulation is run, the generated plot shows the cut-off point

## Achieved Power and Sample Size for ANOVA-level effects
##    variable                     label   n achieved_power desired_power
## 1     alpha Desired Power Not Reached 250            5.0            80
## 2       age Desired Power Not Reached 250            5.0            80
## 3 alpha:age    Desired Power Achieved  17           80.7            80

Using n of 17, the simulation results in power above 80.

simulation_result <- ANOVA_power(design_result,  #Simulation is confirmed 100x with n set to 17
                                 alpha_level = alpha_lev, 
                                 nsims = 100,
                                 verbose = FALSE,
                                 seed = 2022)

final_power <- simulation_result[["main_results"]][["power"]][3]

simulation_result

## Power and Effect sizes for ANOVA tests
##                 power effect_size
## anova_alpha         6     0.03330
## anova_age           3     0.02505
## anova_alpha:age    81     0.22122
## 
## Power and Effect sizes for pairwise comparisons (t-tests)
##                                            power effect_size
## p_age_old_alpha_low_age_old_alpha_high        29    0.502695
## p_age_old_alpha_low_age_young_alpha_low       50    0.518962
## p_age_old_alpha_low_age_young_alpha_high       7   -0.009527
## p_age_old_alpha_high_age_young_alpha_low       5    0.014159
## p_age_old_alpha_high_age_young_alpha_high     57   -0.531962
## p_age_young_alpha_low_age_young_alpha_high    35   -0.530269

The power can be extracted from the model.

final_power #This is the power value found with n at 17

## [1] 81

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