sim.rasch.dep {sirt}

Simulation of the Rasch Model with Locally Dependent Responses

Description

This function simulates dichotomous item responses where for some itemclusters residual correlations can be defined.

Usage

sim.rasch.dep(theta, b, itemcluster, rho)

Arguments

theta

Vector of person abilities of length N

b

Vector of item difficulties of length I

itemcluster

Vector of integers (including 0) of length I. Different integers correspond to different itemclusters.

rho

Vector of residual correlations. The length of vector must be equal to the number of itemclusters.

Value

An N \times I data frame of dichotomous item responses.

Note

The specification of the simulation models follows a marginal interpretation of the latent trait. Local dependencies are only interpreted as nuisance and not of substantive interest. If local dependencies should be substantively interpreted, a testlet model seems preferable (see mcmc.3pno.testlet).

See Also

To simulate the generalized logistic item response model see sim.raschtype. Ramsay's quotient model can be simulated using sim.qm.ramsay.

Marginal item reponse models for locally dependent item responses can be estimated with rasch.copula2, rasch.pairwise or rasch.pairwise.itemcluster.

Examples

#############################################################################
# EXAMPLE 1: 11 Items: 2 itemclusters with 2 resp. 3 dependent items
#             and 6 independent items
#############################################################################

set.seed(7654)
I <- 11                             # number of items
n <- 1500                           # number of persons
b <- seq(-2,2, len=I)               # item difficulties
theta <- stats::rnorm( n, sd=1 )        # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ c(3,5)] <- 1
itemcluster[c(2,4,9)] <- 2
# residual correlations
rho <- c( .7, .5 )

# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("I", seq(1,ncol(dat)), sep="")

# estimate Rasch copula model
mod1 <- sirt::rasch.copula2( dat, itemcluster=itemcluster )
summary(mod1)

# compare result with Rasch model estimation in rasch.copula
# delta must be set to zero
mod2 <- sirt::rasch.copula2( dat, itemcluster=itemcluster, delta=c(0,0),
            est.delta=c(0,0)  )
summary(mod2)

# estimate Rasch model with rasch.mml2 function
mod3 <- sirt::rasch.mml2( dat )
summary(mod3)

## Not run: 
#############################################################################
# EXAMPLE 2: 12 Items: Cluster 1 -> Items 1,...,4;
#       Cluster 2 -> Items 6,...,9; Cluster 3 -> Items 10,11,12
#############################################################################

set.seed(7896)
I <- 12                             # number of items
n <- 450                            # number of persons
b <- seq(-2,2, len=I)               # item difficulties
b <- sample(b)                      # sample item difficulties
theta <- stats::rnorm( n, sd=1 )        # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ 1:4 ] <- 1
itemcluster[ 6:9 ] <- 2
itemcluster[ 10:12 ] <- 3
# residual correlations
rho <- c( .55, .25, .45 )

# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("I", seq(1,ncol(dat)), sep="")

# estimate Rasch copula model
mod1 <- sirt::rasch.copula2( dat, itemcluster=itemcluster, numdiff.parm=.001 )
summary(mod1)

# Rasch model estimation
mod2 <- sirt::rasch.copula2( dat, itemcluster=itemcluster,
            delta=rep(0,3), est.delta=rep(0,3) )
summary(mod2)

# estimation with pairwise Rasch model
mod3 <- sirt::rasch.pairwise( dat )
summary(mod3)

## End(Not run)