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

In sirt: Supplementary Item Response Theory Models

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)

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.

Value

Object of class rasch.pairwise with elements

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

Note

No standard errors are provided by this function. Use resampling methods for conducting statistical inference.

Formulas for asymptotic standard errors of this pairwise estimation method are described in Zwinderman (1995).

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

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)

## Not run: 
# 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)

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.