The function:

This is a very simple data simulator for a 2PL Model. This is just to get you started, from here is easy to add function parameters for indicating item locations and slopes or person distribution characteristics.

1. The function accepts only two parameters:
   • The number of items
   • The number of persons
2. The function creates a list containing four objects:
   • A vector of item locations
   • A vector of item slopes
   • A vector of person locations
   • A matrix of simulated responses

The code:

twopl.sim         <- function( nitem = 20, npers = 100 ) {

i.loc         <- rnorm( nitem )
p.loc         <- rnorm( npers )
i.slp         <- rlnorm( nitem, sdlog = .4 )

temp          <- matrix( rep( p.loc, length( i.loc ) ), ncol = length( i.loc ) )

logits        <- t( apply( temp  , 1, '-', i.loc) )
logits        <- t( apply( logits, 1, '*', i.slp) )

probabilities <- 1 / ( 1 + exp( -logits ) )

resp.prob     <- matrix( probabilities, ncol = nitem)

obs.resp      <- matrix( sapply( c(resp.prob), rbinom, n = 1, size = 1), ncol = length(i.loc) )

output        <- list()
output$i.loc  <- i.loc
output$i.slp  <- i.slp
output$p.loc  <- p.loc
output$resp   <- obs.resp

output
}

Example:

This is a simple example that uses the IRToys (irtoys) package to estimate the model parameters after simulating the data. Do try this at home too!

#install.packages('irtoys')

# Note: 'irtoys' might not load right away in OSX. You can try the following sequence in that case:
# install.packages('mvtnorm')
# install.packages('msm')
# install.packages('ltm',type='source')
# install.packages('irtoys',type='source')

library(irtoys)

###### Running Simulation ######

data2pl <- twopl.sim(nitem=20,npers=10000)

sim.loc.param <- data2pl$i.loc
sim.slp.param <- data2pl$i.slp

###### Estimation Using irtoys ######

analysis.2pl  <- est(data2pl$resp, model = '2PL', engine = 'ltm')

est.slp.param <- analysis.2pl[,1]
est.loc.param <- analysis.2pl[,2]

layout(matrix(c(1,2),nrow=1))

plot(est.slp.param,sim.slp.param)
plot(est.loc.param,sim.loc.param)