Item Equating with same Group – SAT, ACT example

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# Item equating is the practice of making the results from two 
# different assessments equivalent.  This can be done by either
# 1. having the same group take both assessments
# 2. having equivalent groups take the different assessments
# 3. having non-equivalent groups which use common items take the different
#    assessments.
# In this post I will cover topic 1.

# For this code I will use the catR package to generate my assessments 
# and responses.
# Let's attempt the first proceedure:
# First let's generate our item parameters for assessment 1.
# Let's create an item bank with 100 items for the assessments
nitems <- 100
bank1 <- createItemBank(model="2PL", items=nitems)$itemPar
bank2 <- createItemBank(model="2PL", items=nitems)$itemPar
# Now let's generate a 1000 person population sample to take our assessment
npeep <- 1000
theta <- rnorm(npeep)
# Calculate the score on both assessments
resp1 <- resp2 <- matrix(0, nrow=npeep, ncol=nitems)
for (i in 1:npeep) {
  resp1[i,Pi(theta[i],bank1)$Pi32] <- 32
ACT[ACT<4]  <- 4
ACT <- round(ACT) # ACT rounds to whole numbers
# The standard deviation is 100 and average score around 500 for the ACT
SAT <- (score2-mean(score2))/sd(score2)*100 + 500
SAT[SAT>800] <- 800
SAT[SAT<200]  <- 200
SAT <- round(SAT/10)*10 # SAT rounds to nearest 10
# Now let's see if we can transform our SAT scores to be on our ACT scale
(A <- sd(ACT)/sd(SAT))
(B <- mean(ACT)-A*mean(SAT))
SATscaled <- A*SAT + B
# The results from taking parrellel tests should fall on a linear form
plot(ACT,SATscaled, main="SAT results placed on ACT scale")
# This is the easiest method of equating two tests.  
# However, it is not usually the most practical since it is costly to get the same
# group of individuals to take two different tests.  In addition, there
# may be issues with fatigue which could be alleviated somewhat if for half of the
# group the first assessment was given first and for a different half 
# the second assessment assessment was given first.
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