sirt (version 0.31-20)

rasch.pairwise.itemcluster: Pairwise Estimation of the Rasch Model for Locally Dependent Items

Description

This function uses pairwise conditional likelihood estimation for estimating item parameters in the Rasch model.

Usage

rasch.pairwise.itemcluster(dat, itemcluster = NULL, conv = 1e-05, 
    maxiter = 3000, progress = TRUE, b.init = NULL)

Arguments

dat

An $N$ times $I$ dataframe. Missing responses are allowed and must be recoded as NA.

itemcluster

Optional integer vector of itemcluster (see Examples). Different integers correspond to different item clusters. No item cluster is set as default.

conv

Convergence criterion in maximal absolute parameter change

maxiter

Maximal number of iterations

progress

A logical which displays progress. Default is TRUE.

b.init

Vector of initial item difficulty estimates. Default is NULL.

Value

• Object of class rasch.pairwise with elements
• b
  Vector of item difficulties
• item
  Data frame of item paramters ($N$, $p$ and item difficulty)

Details

This is an adaptation of the code of van der Linden and Eggen (1986). Only item pairs of different item clusters are taken into account for item difficulty estimation. Therefore, the problem of locally dependent items within each itemcluster is (almost) eliminated (see Examples below) because contributions of local dependencies do not appear in the pairwise likelihood terms.

References

van der Linden, W. J. & Eggen, T. J. H. M. (1986). An empirical Bayes approach to item banking. Research Report 86-6, University of Twente. Zwinderman, A. H. (1995). Pairwise parameter estimation in Rasch models. Applied Psychological Measurement, 19, 369-375.

See Also

rasch.pairwise, summary.rasch.pairwise, Pairwise marginal likelihood estimation (also labeled as pseudolikelihood estimation) can be conducted with rasch.pml3. Other estimation methods are implemented in rasch.copula2 or rasch.mml2. For simulation of locally dependent data see sim.rasch.dep.

Examples

############################################################
####################### EXAMPLE 1 ##########################
#   Example with locally dependent items
#   12 Items: Cluster 1 -> Items 1,...,4
#             Cluster 2 -> Items 6,...,9
#             Cluster 3 -> Items 10,11,12
#   Data is simulated using the sim.rasch.dep function
############################################################

set.seed(7896)
I <- 12                             # number of items
n <- 5000                           # number of persons
b <- seq(-2,2, len=I)               # item difficulties
bsamp <- b <- sample(b)             # sample item difficulties
theta <- 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 <- sim.rasch.dep( theta , b , itemcluster , rho )
colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")

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

# use item cluster in rasch pairwise estimation
mod <- rasch.pairwise.itemcluster( dat = dat , itemcluster = itemcluster )
summary(mod)

# Rasch MML estimation
mod4 <- rasch.mml2( dat )
summary(mod4)

# Rasch Copula estimation
mod5 <- rasch.copula2( dat , itemcluster = itemcluster )
summary(mod5)

# compare different item parameter estimates
M1 <- cbind( "true.b"=bsamp ,  "b.rasch" = mod4$item$b ,
        "b.rasch.copula" =  mod5$item$thresh , 
        "b.rasch.pairwise" = mod3$b  ,
        "b.rasch.pairwise.cluster" = mod$b 
      )
# center item difficulties
M1 <- scale( M1  , scale=FALSE )
round( M1 , 3 )
round( apply( M1 , 2 , sd ) , 3 )

#  Below the output of the example is presented.
#  It is surprising that the rasch.pairwise.itemcluster is pretty close
#  to the estimate in the Rasch copula model.

##   > M1 <- scale( M1  , scale=F )
##   > round( M1 , 3 )
##       true.b b.rasch b.rasch.copula b.rasch.pairwise b.rasch.pairwise.cluster
##   I1   0.545   0.561          0.526            0.628                    0.524
##   I2  -0.182  -0.168         -0.174           -0.121                   -0.156
##   I3  -0.909  -0.957         -0.867           -0.971                   -0.899
##   I4  -1.636  -1.726         -1.625           -1.765                   -1.611
##   I5   1.636   1.751          1.648            1.694                    1.649
##   I6   0.909   0.892          0.836            0.898                    0.827
##   I7  -2.000  -2.134         -2.020           -2.051                   -2.000
##   I8  -1.273  -1.355         -1.252           -1.303                   -1.271
##   I9  -0.545  -0.637         -0.589           -0.581                   -0.598
##   I10  1.273   1.378          1.252            1.308                    1.276
##   I11  0.182   0.241          0.226            0.109                    0.232
##   I12  2.000   2.155          2.039            2.154                    2.026
##   > round( apply( M1 , 2 , sd ) , 3 )
##                     true.b                  b.rasch           b.rasch.copula       
##                      1.311                    1.398                    1.310   
##      b.rasch.pairwise    b.rasch.pairwise.cluster 
##                 1.373                       1.310