Efficiency Balanced Information Criterion for Item Selection

# 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 example
a <- function(x,y) x^y
 
# Let's see
integ(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 returns
returns <- 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 items
I1 <- 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. 
 


# 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.