Sunday, December 1, 2013

More Explorations with catR

# For the purposes of simulating computerized adaptive tests
# the R package catR is unparallelled.
# catR is an excellent tool for students who are curious about
# how a computerized adaptive test might work. It is also useful
# for testing companies that are interested in seeing how
# their choices of number of items, or model, stopping rule,
# or quite a few of the other options which are available
# when designing a specific computerized adaptive test.
# In this post I will explore some of the features of the 
# function randomCAT, an extremely powerful function 
# that simulates an entire response pattern for an individual.
# In a previous post I explore  some of the other function 
# in catR in order to step by step demonstrate how to use 
# the package to simulate a test.
# First let's generate an item bank. 
# Items specifies how many items to generate
# Model specifies which model to use in generating the items
# a,b,c Priors are specifying distributions to draw
# the parameters from for each item.
# The final set of arguments is for specifying
# what range of theta values the bank will initially
# draw item parameters for.  Theta values are the typical
# latent traits for which item response theory is concerned
# with estimating.
Bank <- createItemBank(items = 500, model = "3PL", 
                       thMin = -4, thMax = 4,
                       step = 0.05)
# We may want to examine the object we have created called "Bank"
# Within the Bank object of class "itBank" there is three named
# attributes.
# itemPar lists the item parameters for those items which have been
# generated.  We could see a histogram of difficulty parameters (b) by
# targeting within the Bank object:
hist(Bank$itemPar[,2], breaks=30, 
     main="Distribution of Item Difficulties",
     xlab="b parameter")
# We can also see how much information a particular item would add # accross a range of ability levels.  This information is already # available within the Bank object under the names infoTab and # theta.   # Plot the first item's information plot(rep(Bank$theta,1),Bank$infoTab[,1],      type="l", main="Item 1's information",      xlab="Ability (theta)", ylab="Information")   # Plot the first 3 items # By specifying type = "n" this plot is left empty nitems = 3 plot(rep(Bank$theta,nitems),Bank$infoTab[,1:nitems], type="n",      main=paste0("First ",nitems," items' information"),      xlab="Ability (theta)", ylab="Information") # Now we plot the for (i in 1:nitems) lines(Bank$theta,Bank$infoTab[,i],                           col=grey(.8*i/nitems))
# We can see how different items can have information that # spans different ability estimates as well as some items # which just have more information than other items.     # Plotting all 500 items (same code as previously but now # we specify the number of items as 500) nitems = 500 plot(rep(Bank$theta,nitems),Bank$infoTab[,1:nitems], type="n",     main=paste0("First ",nitems," items' information"),     xlab="Ability (theta)", ylab="Information") for (i in 1:nitems) lines(Bank$theta,Bank$infoTab[,i],                           col=grey(.8*i/nitems)) # This plot may look nonsensical at first.  Be it actually # provides some useful information.  From it you can see the # maximum amount of information available for any one # item at different levels of ability.  In the places where # there is only one very tall item standing out we may be # concerned about item exposure since subjects which seem to # be in the area of that item are disproportionately more likely # to get the same high info item than other other subjects # in which the next highest item is very close in information # to the max item.   # To see the max information for each ability we can add a line. lines(Bank$theta,apply(Bank$infoTab, 1, max), col="blue", lwd=2)   # We might also be interested in seeing how much information # on average a random item chosen from the bank would provide # or in other words what is the expected information from a # random item drawn from the bank at different ability levels. lines(Bank$theta,apply(Bank$infoTab, 1, mean), col="red", lwd=2)   # Or perhaps we might want to see what the maximum average information # for a 20 item test might be. So we calculate the average information # for the top 20 items at different ability levels. maxmean <- function(x, length=20) mean(sort(x, decreasing=T)[1:length]) maxmean(1:100) # Returns 90.5, seems to be working properly   lines(Bank$theta,apply(Bank$infoTab, 1, maxmean), col="orange", lwd=3)  
# Now this last line is very interesting because it reflects # per item the maximum amount of information this bank can provide # given a fixed length of 20. Multiply this curve by 20 and it will give # us the maximum information this bank can provide given a 20 item test # and a subject's ability.   # This can really be thought of as a theoretical maximum for which # any particular CAT test might attempt to meet but on average will # always fall short.   # We can add a lengend legend(-4.2, .55, c("max item info", "mean(info)",                     "mean(top items)"),        lty = 1, col = c("blue","red","orange"),  adj = c(0, 0.6))       library("reshape") library("ggplot2")   # Let's seperate info tab infoTab <- Bank$infoTab   # Let's add three columns to info tab for max, mean, and mean(top 20) infoTab <- cbind(infoTab,                  apply(Bank$infoTab, 1, max),                  apply(Bank$infoTab, 1, mean),                  apply(Bank$infoTab, 1, maxmean))     # Melt will turn the item information array into a long object items.long <- melt(infoTab)   # Let's assign values to the first column which are thetas items.long[,1] <- Bank$theta   # Now we are ready to name the different columns created by melt names(items.long) <- c("theta", "item", "info")   itemtype <- factor("Item", c("Item","Max", "Mean", "Mean(Max)")) items.long <- cbind(items.long, type=itemtype) items.long[items.long$item==501,4] <- "Max" items.long[items.long$item==502,4] <- "Mean" items.long[items.long$item==503,4] <- "Mean(Max)"   # Now we are ready to start plotting # Assign the data to a ggplot object a <- ggplot(items.long, aes(x=theta, y=info, group=item))   # Plot a particular instance of the object a + geom_line(colour = gray(.2)) +     geom_line(aes(colour = type), size=2 ,             subset = .(type %in% c("Max", "Mean", "Mean(Max)")))  
# Now let's look at how the randomCAT function works. # There are a number of arguments that the randomCAt function # can take.  They can be defined as lists which are fed # into the function.   # I will specify only that the stoping rule is 20 items. # By specifying true Theta that is telling random CAT what the # true ability level we are estimating. res <- randomCAT(trueTheta = 3, itemBank = Bank,                  test=list(method = "ML"),                  stop = list(rule = "length", thr = 20)) # I specify test (theta estimator) as using ML because the # default which is Bayesian model is strongly centrally # biased in this case.   # Let's examine what elements are contained with the object "res" attributes(res)   # We can see our example response pattern. thetaEst <- c(0, res$thetaProv)   plot(1:21, thetaEst, type="n",      xlab="Item Number",      ylab="Ability Estimate",      main="Sample Random Response Pattern") # Add true ability line   abline(h=3, col="red", lwd=2, lty=2) # Add a line connecting responses   lines(1:21, thetaEst, type="l", col=grey(.8)) # Add the response pattern to   text(1:21, thetaEst, c(res$pattern, "X")) # Add the legend   legend(15,1,"True Ability", col="red", lty=2, lwd=2)  
# Plot the sample item information from the set of items selected. plot(rep(Bank$theta,20),Bank$infoTab[,res$testItems], type="n",      main="High information items are often selected",      xlab="Ability (theta)", ylab="Information") for (i in 1:500) lines(Bank$theta,Bank$infoTab[,i], col=grey(.75)) # Now we plot the for (i in res$testItems) lines(Bank$theta,Bank$infoTab[,i],                                lwd=2, col=grey(.2))  
# Now let's see how randomCat performs with a random draw # of 150 people with different ability estimates.   npers <- 150   # Specify number of people to simulate   theta <- rnorm(npers) # Draw a theta ability level vector   thetaest <- numeric(npers) # Creates an empty vector of zeros to hold future estimates # of theta   # Create an empty item object items.used <- NULL   # Create an empty object to hold b values for items used b.values <- NULL     for (i in 1:npers) {   # Input the particular theta[i] ability for a particular run.   res <- randomCAT(trueTheta = theta[i],                    itemBank = Bank,                    test=list(method = "ML"),                    stop = list(rule = "length", thr = 20))   # Save theta final estimates   thetaest[i] <- res$thFinal   # Save a list of items selected in each row of items.used   items.used <- rbind(items.used, res$testItems)   # Save a list of b values of items selected in each row   b.values <- rbind(b.values, res$itemPar[,2])   }   # Let's see how our estimated theta's compare with our true plot(theta, thetaest,      main="Ability plotted against ability estimates",      ylab="theta estimate") 
# To get a sense of how much exposure our items get 
itemTab <- table(items.used)
# We can see we only have 92 items used for all 150 subjects
# taking the cat exam.
# On average each item used is exposed 32 times which means
# over a 20% exposure rate on average in addition to some items
# have much higher exposure rates.
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