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, b.fixed=NULL, weights=NULL,
    conv=1e-05, maxiter=3000, progress=TRUE, b.init=NULL, zerosum=FALSE)

Value

Object of class rasch.pairwise with elements

b: Vector of item difficulties
item: Data frame of item parameters ($N$, $p$ and item difficulty)

Arguments

dat: An $N \times I$ data frame. 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.
b.fixed: Matrix for fixing item parameters. The first columns contains the item (number or name), the second column the parameter to be fixed.
weights: Optional Vector of sampling weights
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.
zerosum: Optional logical indicating whether item difficulties should be centered in each iteration. The default is that no centering is conducted.

Details

This is an adaptation of the algorithm 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. In detail, the estimation rests on observed frequency tables of items $i$ and $j$ and therefore on conditional probabilities $$ \frac{P(X_i=x, X_j=y)}{P(X_i + X_j=1 )} \quad \mbox{with} \quad x,y=0,1 \quad \mbox{and} \quad x+y=1 $$ If for some item pair $(i,j)$ a higher positive (or negative) correlation is expected (i.e. deviation from local dependence), then this pair is removed from estimation. Clearly, there is a loss in precision but item parameters can be less biased.

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.

Examples

Run this code

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

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 <- 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="")

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

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

if (FALSE) {
# Rasch MML estimation
mod4 <- sirt::rasch.mml2( dat )
summary(mod4)

# Rasch Copula estimation
mod5 <- sirt::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, stats::sd ), 3 )

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

  ##   > 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

# set item parameters of first item to 0 and of second item to -0.7
b.fixed <- cbind( c(1,2), c(0,-.7) )
mod5 <- sirt::rasch.pairwise.itemcluster( dat=dat, b.fixed=b.fixed,
             itemcluster=itemcluster )
# difference between estimations 'mod' and 'mod5'
dfr <- cbind( mod5$item$b, mod$item$b )
plot( mod5$item$b, mod$item$b, pch=16)
apply( dfr, 1, diff)
}

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