# R code written by Melanie M. Wall - last updated 11/19/2010

# This program calculates a 95% confidence interval for
# TIA (Total information area index) for the 
# 2 parameter logistic IRT model.  The TIA is calculated
# as the integral of the total test information function across 
# the entire (-infinity, +infinity) latent trait.  
# It can equivalently be calculated
# as the sum of the p discrimination parameters (where p is the number of items in the set).
# The standard error (and confidence interval) for the TIA in this function
# is thus obtained based on the 
# the asymptotic distributions of the sum of discrimination parameters

# This program utilizes the 'ltm' latent trait modeling package, so it must first be loaded
# onto your system from the CRAN site for the program to work

library(ltm)

################Beginning of the function called  asym.confint.irt.info ##########

asym.confint.irt.info<-function(dataset){

###########################################################################
#### The following uses the ltm function to obtain the ML estimators of the 
#### discrimination and severity parameters for the observed data
###########################################################################

parms.of.observed = ltm(dataset ~ z1,IRT.param=TRUE)
coef.observed<-coef(parms.of.observed)
numvars<-dim(dataset)[2]

##########################################################################
### The following makes the plot of the total information curves
### The test information curve is the sum of item information curves and
### provides a visual way to examine where the item set, i.e. "test",
### yields the most precise estimate of the underlying trait.  

### Often the location of the peak is 
### referred to as the place along the trait where the item set is
### "most discriminating", e.g. (Reise and Waller 2002).
###  Reise, S. andWaller, N. (2002) Item response theory for dichotomous assessment data. In Drasgow,
### F. and Schmitt, N., editors, Measuring and Analyzing Behavior in Organizations. San Francisco:
### Jossey-Bass.  
#############################################################################

plotall<-plot(parms.of.observed , type = "IIC", items = 0,xlab = "Latent trait")


###############################################################################
###
### The following calculates the Total Information Area Index
###
### To get the total information across the whole trait, the information function in the ltm package uses the 
### integrate() function built into R and integrates from -10 to 10
### this Total Information Area Index can also be calculated as the sum of the 
### discrimination parameters
###############################################################################

tia.original<-information(parms.of.observed, range = c(-10,10))$InfoTotal

#### Here the asymptotic standard error of the total information is obtained
#### using the fact that it is simply the sum of the discrimination 
#### parameters and so we have a closed form for its variance

covdiscrm<-vcov(parms.of.observed)[(numvars+1):(2*numvars),(numvars+1):(2*numvars)] 
varofsum<-t(c(rep(1,numvars)))%*%covdiscrm%*%c(rep(1,numvars))

#### Here using the asymptotic distribution of the sum of the discrimination estimators
#### being normally distributed, we calculate a symmetric 95% confidence interval

confint<-c(tia.original-1.96*sqrt(varofsum), tia.original+1.96*sqrt(varofsum))

allout<-list(coef.observed=coef.observed,tia.original=tia.original, confint = confint)
return(allout)
}

########## End of the Function ############



################################################

#######Example - simulated data

###############################################
set.seed(123499)
n<-700
prob<-matrix(rep(0,n*12),ncol = 12)
symptoms<-matrix(rep(99,n*12),ncol = 12)
f<-rnorm(n,0,1)
disc<-c(1,1,1,1,1,1,1,1,1,1,1,1)
severity<-c(-1.5,-1.5,-1,-1,0,0,0,0,1,1,1.5,1.5)
for (i in c(1:n)){
prob[i,]<- 1/(1+exp(-disc*(f[i]-severity)))
symptoms[i,]<-rbinom(12,1,prob[i,])
}

#######This is the way to call the function 

asym.confint.irt.info(symptoms)

#Here are the results for the simulated data
# TIA = 11.62 95% CI = (10.65,12.76) 
#$coef.observed
#             Dffclt    Dscrmn
#Item 1  -1.55058371 1.2111926
#Item 2  -1.53095794 1.0543602
#Item 3  -1.24211903 0.8660448
#Item 4  -1.27343552 0.7020015
#Item 5   0.05502064 0.9345833
#Item 6  -0.19743686 1.1093184
#Item 7   0.08471676 0.8292031
#Item 8  -0.01683570 1.0667659
#Item 9   1.04264012 0.8543570
#Item 10  0.97723563 0.9642812
#Item 11  1.62733107 0.8793472
#Item 12  1.44888562 1.1536152
#
#$tia.original
#[1] 11.62063
#
#$confint
#[1] 10.56235 12.67890