## Monday, October 28, 2013

### 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)

Created by Pretty R at inside-R.org