# Modified Bin and Union Method for Item Pool Design

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# 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)

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