sirt (version 0.36-30)

linking.haberman: Linking in the 2PL/Generalized Partial Credit Model

Description

This function does the linking of serval studies which are calibrated using the 2PL or the generalized item response model according to Haberman (2009). This method is a generalization of log-mean-mean linking from tow to several studies.

Usage

linking.haberman(itempars, conv = 1e-05, maxiter = 1000, progress = TRUE)

Arguments

itempars

A data frame with four or five column. The first four columns contain in the order: study name, item name, $a$ parameter, $b$ parameter. The fifth column is an optional weight for every item and every study.

conv

Convergence criterion.

maxiter

Maximum number of iterations.

progress

An optional logical indicating whether computational progress should be displayed.

Value

• A list with following entries
• transf.itempars
  Data frame with linear transformation functions for item parameters
• transf.personpars
  Data frame with linear transformation functions for person parameters
• joint.itempars
  Estimated joint item parameters $a_i$ and $b_i$

Details

For $t=1,\ldots,T$ studies, item difficulties $b_{it}$ and item slopes $a_{it}$ are available. For dichotomous responses, these parameters are defined by the 2PL response equation $$log P(X_{pi}=1| \theta_p ) = a_i ( \theta_p - b_i )$$ while for polytomous responses the generalized partial credit model holds $$logit \frac{P(X_{pi}=k| \theta_p )}{P(X_{pi}=k-1| \theta_p )} = a_i ( \theta_p - b_i + d_{ik} )$$ The parameters ${ a_{it} , b_{it} }$ of all items and studies are linearly transformed using equations $a_{it} \approx A_t \cdot a_i$ and $b_{it} / A_t \approx B_t + b_i$. For identification reasons we define $A_1=1$ and $B_1$ = 0. The optimization function (which is a least squares criterion; see Haberman 2009) seeks the transformation parameters $A_t$ and $B_t$ with an alternating least squares method. Note that every item $i$ and every study $t$ can be weighted (specified in the fifth column of itempars).

References

Haberman, S. J. (2009). Linking parameter estimates derived from an item respone model through separate calibrations. ETS Research Report ETS RR-09-40. Princeton, ETS.

See Also

See the plink package for a wide diversity of linking methods. Mean-mean linking, Stocking-Lord and Haebara linking in the generalized logistic item response model can be conducted with equating.rasch.

Examples

#############################################################################
# EXAMPLE 1: Item parameters data.pars1.rasch and data.pars1.2pl
#############################################################################

# Model 1: Linking three studies calibrated by the Rasch model
data(data.pars1.rasch)
mod1 <- linking.haberman( itempars=data.pars1.rasch )

# Model 1b: Linking these studies but weigh these studies by
#     proportion weights 3 : 0.5 : 1  (see below).
#     All weights are the same for each item but they could also
#     be item specific.
itempars <- data.pars1.rasch
itempars$wgt <- 1
itempars[ itempars$study == "study1","wgt"] <- 3
itempars[ itempars$study == "study2","wgt"] <- .5
mod1b <- linking.haberman( itempars=itempars )

# Model 2: Linking three studies calibrated by the 2PL model
data(data.pars1.2pl)
mod2 <- linking.haberman( itempars=data.pars1.2pl )

#############################################################################
# EXAMPLE 2: Linking longitudinal data
#############################################################################
data(data.long)

#******
# Model 1: Scaling with the 1PL model

# scaling at T1
dat1 <- data.long[ , grep("T1" , colnames(data.long) ) ]
resT1 <- rasch.mml2( dat1 )
itempartable1 <- data.frame( "study"="T1" , resT1$item[ , c("item" , "a" , "b" ) ] )
# scaling at T2
dat2 <- data.long[ , grep("T2" , colnames(data.long) ) ]
resT2 <- rasch.mml2( dat2 )
summary(resT2)
itempartable2 <- data.frame( "study"="T2" , resT2$item[ , c("item" , "a" , "b" ) ] )
itempartable <- rbind( itempartable1 , itempartable2 )
itempartable[,2] <- substring( itempartable[,2] , 1, 2 )
# estimate linking parameters
mod1 <- linking.haberman( itempars= itempartable )

#******
# Model 2: Scaling with the 2PL model

# scaling at T1
dat1 <- data.long[ , grep("T1" , colnames(data.long) ) ]
resT1 <- rasch.mml2( dat1 , est.a=1:6)
itempartable1 <- data.frame( "study"="T1" , resT1$item[ , c("item" , "a" , "b" ) ] )

# scaling at T2
dat2 <- data.long[ , grep("T2" , colnames(data.long) ) ]
resT2 <- rasch.mml2( dat2 , est.a=1:6)
summary(resT2)
itempartable2 <- data.frame( "study"="T2" , resT2$item[ , c("item" , "a" , "b" ) ] )
itempartable <- rbind( itempartable1 , itempartable2 )
itempartable[,2] <- substring( itempartable[,2] , 1, 2 )
# estimate linking parameters
mod2 <- linking.haberman( itempars= itempartable )