Equating in the Generalized Logistic Rasch Model

This function does the linking in the generalized logistic item response model. Only item difficulties (\(b\) item parameters) are allowed. Mean-mean linking and the methods of Haebara and Stocking-Lord are implemented (Kolen & Brennan, 2004).

Usage

equating.rasch(x, y, theta=seq(-4, 4, len=100),
       alpha1=0, alpha2=0)

Arguments

x

Matrix with two columns: First column items, second column item difficulties

y

Matrix with two columns: First columns item, second column item difficulties

theta

Vector of theta values at which the linking functions should be evaluated. If a weighting according to a prespecified normal distribution \(N( \mu,\sigma^2)\) is aimed, then choose theta=stats::qnorm( seq(.001, .999, len=100), mean=mu, sd=sigma)

alpha1

Fixed \(\alpha_1\) parameter in the generalized item response model

alpha2

Fixed \(\alpha_2\) parameter in the generalized item response model

Value

B.est

Estimated linking constants according to the methods Mean.Mean (Mean-mean linking), Haebara (Haebara method) and Stocking.Lord (Stocking-Lord method).

descriptives

Descriptives of the linking. The linking error (linkerror) is calculated under the assumption of simple random sampling of items

anchor

Original and transformed item parameters of anchor items

transf.par

Original and transformed item parameters of all items

References

Kolen, M. J., & Brennan, R. L. (2004). Test Equating, Scaling, and Linking: Methods and Practices. New York: Springer.

See also

For estimating standard errors (due to inference with respect to the item domain) of this procedure see equating.rasch.jackknife.

For linking several studies see linking.haberman or invariance.alignment.

A robust alternative to mean-mean linking is implemented in linking.robust.

For linking under more general item response models see the plink package.

Examples

#############################################################################
# EXAMPLE 1: Linking item parameters of the PISA study
#############################################################################

data(data.pisaPars)
pars <- data.pisaPars

# linking the two studies with the Rasch model
mod <- sirt::equating.rasch(x=pars[,c("item","study1")], y=pars[,c("item","study2")])
  ##   Mean.Mean    Haebara Stocking.Lord
  ## 1   0.08828 0.08896269    0.09292838

if (FALSE) {
#*** linking using the plink package
# The plink package is not available on CRAN anymore.
# You can download the package with
# utils::install.packages("plink", repos="http://www2.uaem.mx/r-mirror")
library(plink)
I <- nrow(pars)
pm <- plink::as.poly.mod(I)
# linking parameters
plink.pars1 <- list( "study1"=data.frame( 1, pars$study1, 0 ),
                     "study2"=data.frame( 1, pars$study2, 0 ) )
      # the parameters are arranged in the columns:
      # Discrimination, Difficulty, Guessing Parameter
# common items
common.items <- cbind("study1"=1:I,"study2"=1:I)
# number of categories per item
cats.item <- list( "study1"=rep(2,I), "study2"=rep(2,I))
# convert into plink object
x <- plink::as.irt.pars( plink.pars1, common.items, cat=cats.item,
          poly.mod=list(pm,pm))
# linking using plink: first group is reference group
out <- plink::plink(x, rescale="MS", base.grp=1, D=1.7)
# summary for linking
summary(out)
  ##   -------  group2/group1*  -------
  ##   Linking Constants
  ##
  ##                        A         B
  ##   Mean/Mean     1.000000 -0.088280
  ##   Mean/Sigma    1.000000 -0.088280
  ##   Haebara       1.000000 -0.088515
  ##   Stocking-Lord 1.000000 -0.096610
# extract linked parameters
pars.out <- plink::link.pars(out)
}