rasch.evm.pcm: Estimation of the Partial Credit Model using the Eigenvector Method

Description

This function performs the eigenvector approach to estimate item parameters which is based on a pairwise estimation approach (Gardner & Engelhard, 2002). No assumption about person parameters is required for item parameter estimation. Statistical inference is performed by Jackknifing. If a group identifier is provided, tests for differential item functioning are performed.

Usage

rasch.evm.pcm(dat, jackunits = 20, weights = NULL, pid = NULL ,
    group=NULL , powB = 2, adj_eps = 0.3, progress = TRUE )

## S3 method for class 'rasch.evm.pcm':
summary(object,...)

## S3 method for class 'rasch.evm.pcm':
coef(object,...)

## S3 method for class 'rasch.evm.pcm':
vcov(object,...)

Arguments

dat

Data frame with dichotomous or polytomous item responses

jackunits

A number of Jackknife units (if an integer is provided as the argument value) or a vector in which the Jackknife units are already defined.

weights

Optional vector of sample weights

pid

Optional vector of person identifiers

group

Optional vector of group identifiers. In this case, item parameters are group wise estimated and tests for differential item functioning are performed.

powB

Power created in $B$ matrix which is the basis of parameter estimation

adj_eps

Adjustment parameter for person parameter estimation (see mle.pcm.group)

progress

An optional logical indicating whether progress should be displayed

object

Object of class rasch.evm.pcm

...

Further arguments to be passed

Value

A list with following entries
itemData frame with item parameters. The item parameter estimate is denoted by est while a Jackknife bias-corrected estimate is est_jack. The Jackknife standard error is se.
bItem threshold parameters
personData frame with person parameters obtained (MLE)
BPaired comparison matrix
DTransformed paired comparison matrix
coefVector of estimated coefficients
vcovCovariance matrix of estimated item parameters
JJNumber of jackknife units
JJadjReduced number of jackknife units
powBUsed power of comparison matrix $B$
maxKMaximum number of categories per item
GNumber of groups
descSome descriptives
difstatsStatistics for differential item functioning if group is provided as an argument

References

Choppin, B. (1985). A fully conditional estimation procedure for Rasch Model parameters. Evaluation in Education, 9, 29-42. Garner, M., & Engelhard, G. J. (2002). An eigenvector method for estimating item parameters of the dichotomous and polytomous Rasch models. Journal of Applied Measurement, 3, 107-128. Wang, J., & Engelhard, G. (2014). A pairwise algorithm in Rfor rater-mediated assessments. Rasch Measurement Transactions, 28(1), 1457-1459.

Examples

Run this code

#############################################################################
# EXAMPLE 1: Dataset Liking for Science
#############################################################################

data(data.liking.science)
dat <- data.liking.science

# estimate partial credit model using 10 Jackknife units
mod1 <- rasch.evm.pcm( dat , jackunits=10 )
summary(mod1)

# compare results with TAM
library(TAM)
mod2 <- TAM::tam.mml( dat )
r1 <- mod2$xsi$xsi
r1 <- r1 - mean(r1)
# item parameters are similar
dfr <- data.frame( "b_TAM"=r1 , mod1$item[,c( "est","est_jack") ] )
round( dfr , 3 )
  ##      b_TAM    est est_jack
  ##  1  -2.496 -2.599   -2.511
  ##  2   0.687  0.824    1.030
  ##  3  -0.871 -0.975   -0.943
  ##  4  -0.360 -0.320   -0.131
  ##  5  -0.833 -0.970   -0.856
  ##  6   1.298  1.617    1.444
  ##  7   0.476  0.465    0.646
  ##  8   2.808  3.194    3.439
  ##  9   1.611  1.460    1.433
  ##  10  2.396  1.230    1.095
  ##  [...]

# partial credit model in eRm package
library(eRm)
mod3 <- eRm::PCM(X=dat)
summary(mod3)
eRm::plotINFO(mod3)      # plot item and test information
eRm::plotICC(mod3)       # plot ICCs
eRm::plotPImap(mod3)     # plot person-item maps

#############################################################################
# EXAMPLE 2: Garner and Engelhard (2002) toy example dichotomous data
#############################################################################

dat <- scan()
   1 0 1 1   1 1 0 0   1 0 0 0   0 1 1 1   1 1 1 0   
   1 1 0 1   1 1 1 1   1 0 1 0   1 1 1 1   1 1 0 0

dat <- matrix( dat , 10 , 4 , byrow=TRUE)
colnames(dat) <- paste0("I" , 1:4 )

# estimate Rasch model with no jackknifing
mod1 <- rasch.evm.pcm( dat , jackunits=0 )

# paired comparison matrix
mod1$B
  ##          I1_Cat1 I2_Cat1 I3_Cat1 I4_Cat1
  ##  I1_Cat1       0       3       4       5
  ##  I2_Cat1       1       0       3       3
  ##  I3_Cat1       1       2       0       2
  ##  I4_Cat1       1       1       1       0

#############################################################################
# EXAMPLE 3: Garner and Engelhard (2002) toy example polytomous data
#############################################################################

dat <- scan()
   2 2 1 1 1   2 1 2 0 0   1 0 0 0 0   0 1 1 2 0   1 2 2 1 1   
   2 2 0 2 1   2 2 1 1 0   1 0 1 0 0   2 1 2 2 2   2 1 0 0 1

dat <- matrix( dat , 10 , 5 , byrow=TRUE)
colnames(dat) <- paste0("I" , 1:5 )

# estimate partial credit model with no jackknifing
mod1 <- rasch.evm.pcm( dat , jackunits=0 , powB=3 )

# paired comparison matrix
mod1$B
  ##          I1_Cat1 I1_Cat2 I2_Cat1 I2_Cat2 I3_Cat1 I3_Cat2 I4_Cat1 I4_Cat2 I5_Cat1 I5_Cat2
  ##  I1_Cat1       0       0       2       0       1       1       2       1       2       1
  ##  I1_Cat2       0       0       0       3       2       2       2       2       2       3
  ##  I2_Cat1       1       0       0       0       1       1       2       0       2       1
  ##  I2_Cat2       0       1       0       0       1       2       0       3       1       3
  ##  I3_Cat1       1       1       1       1       0       0       1       2       3       1
  ##  I3_Cat2       0       1       0       2       0       0       1       1       1       1
  ##  I4_Cat1       0       1       0       0       0       2       0       0       1       2
  ##  I4_Cat2       1       0       0       2       1       1       0       0       1       1
  ##  I5_Cat1       0       1       0       1       2       1       1       2       0       0
  ##  I5_Cat2       0       0       0       1       0       0       0       0       0       0

#############################################################################
# EXAMPLE 4: Partial credit model for dataset data.mg from CDM package
#############################################################################

library(CDM)
data(data.mg,package="CDM")
dat <- data.mg[ , paste0("I",1:11) ]

#*** Model 1: estimate partial credit model
mod1 <- rasch.evm.pcm( dat )
# item parameters
round( mod1$b , 3 )
  ##        Cat1   Cat2  Cat3
  ##  I1  -1.537     NA    NA
  ##  I2  -2.360     NA    NA
  ##  I3  -0.574     NA    NA
  ##  I4  -0.971 -2.086    NA
  ##  I5  -0.104  0.201    NA
  ##  I6   0.470  0.806    NA
  ##  I7  -1.027  0.756 1.969
  ##  I8   0.897     NA    NA
  ##  I9   0.766     NA    NA
  ##  I10  0.069     NA    NA
  ##  I11 -1.122  1.159 2.689

#*** Model 2: estimate PCM with pairwise package
library(pairwise)
mod2 <- pairwise::pair(daten=dat)
summary(mod2)
plot(mod2)
# compute standard errors
semod2 <- pairwise::pairSE(daten=dat,  nsample = 20)
semod2

#############################################################################
# EXAMPLE 5: Differential item functioning for dataset data.mg
#############################################################################

library(CDM)
data(data.mg,package="CDM")
dat <- data.mg[ data.mg$group %in% c(2,3,11) , ]
# define items
items <- paste0("I",1:11)
# estimate model
mod1 <- rasch.evm.pcm( dat[,items] , weights= dat$weight , group= dat$group )
summary(mod1)

#############################################################################
# SIMULATED EXAMPLE 6: Differential item functioning for Rasch model
#############################################################################

# simulate some data
set.seed(9776)
N <- 1000	# number of persons
I <- 10		# number of items
# simulate data for first group
b <- seq(-1.5,1.5,len=I)
dat1 <- sim.raschtype( rnorm(N) , b )
# simulate data for second group
b1 <- b
b1[4] <- b1[4] + .5 # introduce DIF for fourth item
dat2 <- sim.raschtype( rnorm(N,mean=.3) , b1 )
dat <- rbind(dat1 , dat2 )
group <- rep( 1:2 , each=N )
# estimate model
mod1 <- rasch.evm.pcm( dat , group= group )
summary(mod1)