1  Create Data for Activities

1.1 Install packages

A few R package are needed for these activities. The install.package() commands below only needs to be run once per machine.

install.packages("tidyverse", dep = TRUE)
install.packages("broom", dep = TRUE)
install.packages("MASS", dep = TRUE)

1.2 Activate packages

The library() commands below need to be run during each R session – they activate the packages:

library(tidyverse)
library(broom)
library(MASS)

1.3 Create Demonstration Data

Classical test theory is a population-level theory. Consequently, in these demonstrations we use a large number of test takers (i.e., 1,000,000 test takers). In the code below we create true score and random meansurement errors. The true scores have mean of 100 and a standard deviation of 10; which is a variance of 100. Errors have a mean of zero and a standard deviation of 5; which is a variance of 25. The standard deviation of 5 for the errors represents the Standard Error of Measurement - as will become evident later in these activities. An assumption of classical test theory is that the errors are uncorrelated with true scores. We create data below based on this assumption – a covariance of zero is consistent with a correlation of zero.

# Mean of 100 for true score and mean of 0 for errors
means = c(100,0)

# Specify variances and covariances. 
# A covariance of zero means the correlation between
# true scores and errors is zero. The 100 indicates a
# variance of 100 for true scores. The 25 indicates
# a variance of 25 for errors.
covariance_matrix = matrix(c(100,0, 0,25), nrow = 2)

# Create the score as per specifications
set.seed(1) # random number seed
scores <- mvrnorm(n = 1000000,
                  mu = means,
                  Sigma = covariance_matrix,
                  empirical = TRUE)

true  <- scores[,1]
error <- scores[,2]
observed <- true + error

As noted, an assumption of classical test theory is that true scores and errors are uncorrelated. We confirm this below:

cor(true, error)

[1] 4.75564e-15