Modified Bin and Union Method for Item Pool Design

# Reckase (2003) proposes a method for designing an item pool for a computer
# adaptive test that has been known as the bin and union method.  This method
# involves drawing a subject from a distribution of abilities. Then selecting
# the item that maximizes that subject's information from the possible set of 
# all items given a standard CAT proceedure. This is repeated until the test
# reaches the predifined stopping point.
 
# Then then next subject is drawn and a new set of items is drawn.  Items are
# divided into bins such that there is a kind of rounding.  Items which are
# sufficiently close to other items it terms of parameter fit are considered
# the same item and the two sets are unionized together into a larger pool.
 
# As more subjects are added more items are collected though at a decreasing
# rate as fewer new items become neccessary.
 
# In the original paper he uses a fixed length test though in a forthcoming 
# paper he and his student Wei He is also using a variable length test.
 
# I have modified his proceedure slightly in this simulation.  Rather than
# selecting optimal items for each subject based from the continuous pool
# of possible items I have the test look within the already constructed pool
# to see if any items are within bin length of the subject's estimated ability.
 
# If there is no item then I add an item that perfectly matches the subject's
# estimated ability. The reason I prefer this method is that I think it better
# represents the process that a CAT test typically must go through with items
# close to but rarely exactly at the level of the subjects. Thus the information
# for each subject will be slightly less as a result of this modified method
# relative to the original.
 
# As with the new paper this simulation uses a variable length test. My stopping
# rule is simple.  Once the test achieves a sufficiently high level of 
# information, then it stops.
 
# I have constructed this simulation as one with three nested loops.
# Over subjects within the item pool construction.
 
# It simulates the item pool construction a number of times to get the
# average number of items after each subject as well as a histogram
# of average number of items required at each difficulty level.
 
# I have also included a control for item exposure.  This control 
# dicatates that as the acceptable exposure rate is reduced, more items
# will be required since some are too frequently exposed.
 
# Overall this method is seems pretty great to me.  It allows for
# item selection criteria, stopping rules, and exposure controls
# to be easily modified to accomidate most any CAT design.
 
require("catR")
 
# Variable Length Test
 
# The number of times to repeat the simulation
nsim <- 10
 
# The number of subjects to simulate
npop <- 1000
 
# The maximum number of items
max.items <- 5000
 
# Maximum exposure rate of individual item
max.exposure <- .2
 
# Stop the test when information reaches this level
min.information <- 10
 
# How far away will the program reach for a new item (b-b_ideal)
bin.width <- .25
 
expect.a <- 1
 
p <- function(theta, b) exp(theta-b)/(1+exp(theta-b))
info <- function(theta, b, a=expect.a) p(theta,b)*(1-p(theta,b))*a^2
 
info(0,0)
 
# The choose.item funciton takes an input thetahat and searches
# available items to see if any already exist that can be used
# otherwise it finds a new item.
choose.item <- function(thetahat, item.b, items.unavailable, bin.width) {
  # Construct a vector of indexes of available items
  avail.n <- (1:length(item.b))
 
  # Remove any already make unusuable
  if (length(items.unavailable)>0) 
    avail.n <- (1:length(item.b))[-items.unavailable]
 
  # If there are no items available then generate the next item
  # equal to thetaest.
  if (length(avail.n)==0) 
    return(c(next.b=thetahat, next.n=length(item.b)+1))
 
  # Figure out how far each item is from thetahat
  avail.dist <- abs(item.b[avail.n]-thetahat)
 
  # Reorder the n's and dist in terms of proximity
  avail.n <- avail.n[order(avail.dist)]
  avail.dist <- sort(avail.dist)
 
  # If the closest item is within the bin width return it
  if (avail.dist[1]<bin.width) 
    return(c(next.b=item.b[avail.n[1]], next.n=avail.n[1]))
  # Otherwise generate a new item
  if (avail.dist[1]>=bin.width) 
    return(c(next.b=thetahat, next.n=length(item.b)+1))
}
 
# Define the simulation level vectors which will become matrices
Tnitems <- Ttest.length <- Titems.taken.N <- Titem.b<- NULL
 
# Loop through the number of simulations
for (j in 1:nsim) {
 
 
  # Seems to be working well
  choose.item(3, c(0,4,2,2,3.3), NULL, .5)
 
  # This is the initial item pool
  item.b <- 0
 
  # This is the initial number of items taken
  items.taken.N <- rep(0,max.items)
 
  # A vector to record the individual test lengths 
  test.length <- NULL
 
  # Number of total items after each individual
  nitems <- NULL
 
  # Draw theta from a population distribution
  theta.pop <- rnorm(npop)
 
  # Start the individual test
  for (i in 1:npop) {
    # The this person has a theta of:
    theta0 <- theta.pop[i]
 
    # Our initial guess at theta = 0
    thetahat <- 0
 
    print(paste("Subject:", i,"- Item Pool:", length(item.b)))
    response <- items.taken <- NULL
 
    # Remove any items that would have been overexposed
    items.unavailable <- (1:length(item.b))[!(items.taken.N < max.exposure*npop)]
 
    # The initial imformation on each subject is zero
    infosum <- 0
 
    # Loop through each subject
    while(infosum < min.information) {
 
      chooser <- choose.item(thetahat, item.b, items.unavailable, bin.width)
 
      nextitem <- chooser[2]
      nextb <- chooser[1]
        names(nextitem) <- names(nextb) <- NULL
 
      items.unavailable <- c(items.unavailable,nextitem)
      item.b[nextitem] <- nextb
 
      response <- c(response, runif(1)<p(theta0, nextb))
 
      items.taken <- c(items.taken, nextitem)
 
      it <- cbind(1, item.b[items.taken], 0,1)
 
      thetahat <- thetaEst(it, response)
 
      infosum <- infosum+info(theta0, nextb)
    }
 
    # Save individual values
    nitems <- c(nitems, length(item.b))
 
    test.length <- c(test.length, length(response))
 
    items.taken.N[items.taken] <- items.taken.N[items.taken]+1
  }
 
  # Save into matrices the results of each simulation
  Titem.b <- c(Titem.b, sort(item.b))
  Tnitems <- cbind(Tnitems, nitems)
  Ttest.length <- cbind(Ttest.length, test.length)
  Titems.taken.N <- cbind(Titems.taken.N, items.taken.N)
 
}
 
plot(apply(Tnitems, 1, max), type="n",
     xlab = "N subjects", ylab = "N items",
     main = paste(nsim, "Different Simulations"))
for (i in 1:nsim) lines(Tnitems[,i], col=grey(.3+.6*i/nsim))
 

# We can see that the number of items is a function of the number of
# subjects taking the exam.  This relationship becomes relaxed
# when the number of subjects becomes large and the exposure controls
# are removed.

hist(Titem.b, breaks=30)

 
hist(Ttest.length, breaks=20)