Simulating Item Response Models

Defines an S3 method for simulation of item response models.

IRT.simulate(object, ...)

# S3 method for tam.mml
IRT.simulate(object, iIndex=NULL, theta=NULL, nobs=NULL, ...)

# S3 method for tam.mml.2pl
IRT.simulate(object, iIndex=NULL, theta=NULL, nobs=NULL, ...)

# S3 method for tam.mml.mfr
IRT.simulate(object, iIndex=NULL, theta=NULL, nobs=NULL, ...)

# S3 method for tam.mml.3pl
IRT.simulate(object, iIndex=NULL, theta=NULL, nobs=NULL, ...)

Arguments

object	An object of class tam.mml, tam.mml.2pl, tam.mml.mfr or tam.mml.3pl.
iIndex	Optional vector of item indices
theta	Optional matrix of theta values
nobs	Optional numeric containing the number of observations to be simulated.
...	Further objects to be passed

Value

Data frame with simulated item responses

Examples

#############################################################################
# EXAMPLE 1: Simulating Rasch model
#############################################################################

data(data.sim.rasch)

#** (1) estimate model
mod1 <- TAM::tam.mml(resp=data.sim.rasch )

#** (2) simulate data
sim.dat <- TAM::IRT.simulate(mod1)

if (FALSE) {
#** (3) use a subset of items with the argument iIndex
set.seed(976)
iIndex <- sort(sample(ncol(data.sim.rasch), 15))  # randomly select 15 items
sim.dat <- TAM::IRT.simulate(mod1, iIndex=iIndex)
mod.sim.dat <- TAM::tam.mml(sim.dat)

#** (4) specify number of persons in addition
sim.dat <- TAM::IRT.simulate(mod1, nobs=1500, iIndex=iIndex)

# Rasch - constraint="items" ----
mod1 <- TAM::tam.mml(resp=data.sim.rasch,  constraint="items",
                control=list( xsi.start0=1, fac.oldxsi=.5) )

# provide abilities
theta0 <- matrix( rnorm(1500, mean=0.5, sd=sqrt(mod1$variance)), ncol=1 )
# simulate data
data <- TAM::IRT.simulate(mod1, theta=theta0 )
# estimate model based on simulated data
xsi.fixed <- cbind(1:nrow(mod1$item), mod1$item$xsi.item)
mod2 <- TAM::tam.mml(data, xsi.fixed=xsi.fixed )
summary(mod2)

#############################################################################
# EXAMPLE 2: Simulating 2PL model
#############################################################################

data(data.sim.rasch)
# estimate 2PL
mod2 <- TAM::tam.mml.2pl(resp=data.sim.rasch, irtmodel="2PL")
# simulate 2PL
sim.dat <- TAM::IRT.simulate(mod2)
mod.sim.dat <- TAM::tam.mml.2pl(resp=sim.dat, irtmodel="2PL")

#############################################################################
# EXAMPLE 3: Simulate multiple group model
#############################################################################

# Multi-Group ----
set.seed(6778)
N <- 3000
theta <- c( stats::rnorm(N/2,mean=0,sd=1.5), stats::rnorm(N/2,mean=.5,sd=1)  )
I <- 20
p1 <- stats::plogis( outer( theta, seq( -2, 2, len=I ), "-" ) )
resp <- 1 * ( p1 > matrix( stats::runif( N*I ), nrow=N, ncol=I ) )
colnames(resp) <- paste("I", 1:I, sep="")
group <- rep(1:2, each=N/2 )
mod3 <- TAM::tam.mml(resp, group=group)

# simulate data
sim.dat.g1 <- TAM::IRT.simulate(mod3,
                   theta=matrix( stats::rnorm(N/2, mean=0, sd=1.5), ncol=1) )
sim.dat.g2 <- TAM::IRT.simulate(mod3,
                   theta=matrix( stats::rnorm(N/2, mean=.5, sd=1), ncol=1) )
sim.dat <- rbind( sim.dat.g1, sim.dat.g2)
# estimate model
mod3s <- TAM::tam.mml( sim.dat, group=group)

#############################################################################
# EXAMPLE 4: Multidimensional model and latent regression
#############################################################################

set.seed(6778)
N <- 2000
Y <- cbind( stats::rnorm(N), stats::rnorm(N))
theta <- mvtnorm::rmvnorm(N, mean=c(0,0), sigma=matrix(c(1,.5,.5,1), 2, 2))
theta[,1] <- theta[,1] + .4 * Y[,1] + .2 * Y[,2]  # latent regression model
theta[,2] <- theta[,2] + .8 * Y[,1] + .5 * Y[,2]  # latent regression model
I <- 20
p1 <- stats::plogis(outer(theta[, 1], seq(-2, 2, len=I), "-"))
resp1 <- 1 * (p1 > matrix(stats::runif(N * I), nrow=N, ncol=I))
p1 <- stats::plogis(outer(theta[, 2], seq(-2, 2, len=I ), "-" ))
resp2 <- 1 * (p1 > matrix(stats::runif(N * I), nrow=N, ncol=I))
resp <- cbind(resp1, resp2)
colnames(resp) <- paste("I", 1 : (2 * I), sep="")

# (2) define loading Matrix
Q <- array(0, dim=c(2 * I, 2))
Q[cbind(1:(2*I), c(rep(1, I), rep(2, I)))] <- 1
Q

# (3) estimate models
mod4 <- TAM::tam.mml(resp=resp, Q=Q, Y=Y, control=list(  maxiter=15))

# simulate new item responses
theta <- mvtnorm::rmvnorm(N, mean=c(0,0), sigma=matrix(c(1,.5,.5,1), 2, 2))
theta[,1] <- theta[,1] + .4 * Y[,1] + .2 * Y[,2]  # latent regression model
theta[,2] <- theta[,2] + .8 * Y[,1] + .5 * Y[,2]  # latent regression model

sim.dat <- TAM::IRT.simulate(mod4, theta=theta)
mod.sim.dat <- TAM::tam.mml(resp=sim.dat, Q=Q, Y=Y, control=list( maxiter=15))
}