# Efficiency Balanced Information Criterion for Item Selection

November 1, 2013
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`# Han (2012) in the paper "An Efficiency Balanced Information Criterion # for Item Selection in Computerized Adaptive Testing" proposes a method# of evaluating potential items based on expected item potential information # as a function of maximum potential item information. # This method favors items which have lower a values to be initially# selected when there is greater uncertainty in the test but favors selection# of items with higher a parameters as the test progresses. # This small bit of code demonstrates how such a proceedure rescales# item information. # First we will define a few functions that we will use to construct our scale. # Birbaum approximates the theta which maximizes the information function at# a specific a, b, and c parameter level:tmax <- function(a,b,c,D=1.7)  b+1/(D*a)+log((1+sqrt(1+8*c))/2) # For example:tmax(a=2,b=2,c=.2) # This is the item information function for a 3PL (3 parameter logistic)iinfo <- function(theta,a,b,c,D=1.7)  ((D*a)^2*(1-c))/((c+exp(D*a*(theta-b)))*                 (1+exp(-D*a*(theta-b)))^2) iinfo(theta=0,a=1,b=0,c=.1) # Now we define a function which approximates the integration of function # "fun" from start to end.integ <- function(start,end, step, fun, ...) {  x <- seq(start+step/2,end-step/2,step)  sum(get(fun)(x, ...)*step)}# As step size goes to zero the integ function approaches true integration.# Of course that would mean infinite calculations which would be impossible# for any computer.  Thus a larger step size is a worse approximation but# uses less machine time. # For examplea <- function(x,y) x^y # Let's seeinteg(0,2,.00001, "a", y=0)integ(0,2,.00001, "a", y=1)# Looking good. # This is the big function that we are interested in:IE <- function(thetahat,SEE,a,b,c,D=1.7,step=.001) {  # thetahat is the current estimate of ability  # SSE is the current standard error of the estimate  # step is the number of steps used to estimate the integral   # We calculate the item information at the current thetahat  ii <- iinfo(thetahat,a=a,b=b,c=c,D=D)  # Now we calculate the "max" theta value for the item.  thetamax <- tmax(a=a,b=b,c=c,D=D)  # Now the max information for that item.  maxI <- iinfo(thetamax,a=a,b=b,c=c,D=D)  # The efficient information as defined by Han at the  # current theta is:  ie <- ii/maxI   # einfo is the expected information for a particular  # item integrated across the range thetahat-SEE to  # thetahat+SEE.  einfo <- integ(thetahat-SEE*2,                thetahat+SEE*2,                step=step,                "iinfo",               a=a,b=b,c=c,D=D)   # Finally we can rescale the expected item information  # by the maxI to find the expected item efficiency.  eie <- einfo/maxI   # This provides a list of returned values.  list(eie=eie,        ii=ii,       ie=ie,       maxI=maxI,        thetamax=thetamax,        einfo=einfo)} test <- IE(0,1,a=1,b=0,c=.1,step=.001)test # Let's see this criterion in action:theta <- seq(-3,3,.1) # Make a list of returnsreturns <- names(test)  for(v in returns) assign(v,NULL) # Let's create one last function that returns a list of # mappings for each of the ability levels. mapping <- function(theta=seq(-3,3,.1), SEE=.5,a=1,b=0,c=.1,step=.001) {  I1 <- list()  for(i in 1:length(theta)) {    res <- IE(theta=theta[i],SEE=SEE,a=a,b=b,c=c,step=step)    for(v in returns) I1[[v]][i] <- res[[v]]  }  I1} # Now let's imagine five different itemsI1 <- mapping(a=.5 , b=-1.5, c=.3, SEE=.5)I2 <- mapping(a=1  , b=-1  , c=.3, SEE=.5)I3 <- mapping(a=1.7, b=0   , c=.3, SEE=.5)I4 <- mapping(a=1  , b=1   , c=.3, SEE=.5)I5 <- mapping(a=1.5, b=1.5 , c=.3, SEE=.5) plot(theta , I3\$ii, type="n",     main="Item Information at ThetaHat     SEE=.5",     xlab="ThetaHat", ylab="Information")lines(theta, I1\$ii, lwd=2, col="red")lines(theta, I2\$ii, lwd=2, col="blue")lines(theta, I3\$ii, lwd=2, col="green")lines(theta, I4\$ii, lwd=2, col="purple")lines(theta, I5\$ii, lwd=2, col="black")# We can see that some items have much more information# than other items such that they would almost never# be selected.  Item 4 for instance is almost never expected# to yeild higher information.  # If we are less sure of our theta estimate we may instead# calculate our expected information.plot(theta , I3\$einfo, type="n",     main="Expected Item Information at ThetaHat     SEE=.5",     xlab="ThetaHat", ylab="Information")lines(theta, I1\$einfo, lwd=2, col="red")lines(theta, I2\$einfo, lwd=2, col="blue")lines(theta, I3\$einfo, lwd=2, col="green")lines(theta, I4\$einfo, lwd=2, col="purple")lines(theta, I5\$einfo, lwd=2, col="black")# In general this basically makes the peaks less extreme but# does not generally favor our items with lower a values. `
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`# If we want to see how our expected efficiency item# information value will do we can see that as well.# However, before we do that imagine first each of these# information functions divided by it's peak value.plot(c(0,theta) , c(0,I1\$eie), type="n",     main="Expected Efficiency Item Information at ThetaHat     SEE=.5",     xlab="ThetaHat", ylab="Information")lines(theta, I1\$eie, lwd=2, col="red")lines(theta, I2\$eie, lwd=2, col="blue")lines(theta, I3\$eie, lwd=2, col="green")lines(theta, I4\$eie, lwd=2, col="purple")lines(theta, I5\$eie, lwd=2, col="black")# Now we can see that item 1 (red) and 4 (purple) are favored by # this algorithm, though by standard item maximization or by # expected item maximization they would almost never have been# chosen. `
`# The authors suggest a summing or the Efficiency Information# and that of expected information might yeild a good solution.plot(c(0,theta) , c(0,I3\$eie+I3\$einfo), type="n",     main="Expected Efficiency Item Information at ThetaHat     SEE=.5",     xlab="ThetaHat", ylab="Information")lines(theta, I1\$eie+I1\$einfo, lwd=2, col="red")lines(theta, I2\$eie+I2\$einfo, lwd=2, col="blue")lines(theta, I3\$eie+I3\$einfo, lwd=2, col="green")lines(theta, I4\$eie+I4\$einfo, lwd=2, col="purple")lines(theta, I5\$eie+I5\$einfo, lwd=2, col="black")  # The argument is that as SEE gets small the information begins# to look much more like that of Item Information which is# appropropriate for later in the test.I1 <- mapping(a=.5 , b=-1.5, c=.3, SEE=.15)I2 <- mapping(a=1  , b=-1  , c=.3, SEE=.15)I3 <- mapping(a=1.7, b=0   , c=.3, SEE=.15)I4 <- mapping(a=1  , b=1   , c=.3, SEE=.15)I5 <- mapping(a=1.5, b=1.5 , c=.3, SEE=.15) plot(c(0,theta) , c(0,I3\$eie), type="n",     main="Expected Efficiency Item Information at ThetaHat     SEE=.15",     xlab="ThetaHat", ylab="Information")lines(theta, I1\$eie, lwd=2, col="red")lines(theta, I2\$eie, lwd=2, col="blue")lines(theta, I3\$eie, lwd=2, col="green")lines(theta, I4\$eie, lwd=2, col="purple")lines(theta, I5\$eie, lwd=2, col="black")# Now we can see that item 1 (red) and 4 (purple) are favored by # this algorithm, though by standard item maximization or by # expected item maximization they would almost never have been# chosen.  # The authors suggest a summing or the Efficiency Information# and that of expected information might yeild a good solution.plot(c(0,theta) , c(0,I3\$eie+I3\$einfo), type="n",     main="Expected Efficiency Item Information at ThetaHat      SEE=.15",     xlab="ThetaHat", ylab="Information")lines(theta, I1\$eie+I1\$einfo, lwd=2, col="red")lines(theta, I2\$eie+I2\$einfo, lwd=2, col="blue")lines(theta, I3\$eie+I3\$einfo, lwd=2, col="green")lines(theta, I4\$eie+I4\$einfo, lwd=2, col="purple")lines(theta, I5\$eie+I5\$einfo, lwd=2, col="black")  # We can see that item 1 is still favored though we expected# it to give us very little information.  Overall, the# method seems interesting but not yet ideal.`

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