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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 (Garner & 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, digits=3, file=NULL, ...)

## 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

digits

Number of digits after decimals for rounding in summary.

file

Optional file name if summary should be sunk into a file.

...

Further arguments to be passed

Value

A list with following entries

item

Data 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.

b

Item threshold parameters

person

Data frame with person parameters obtained (MLE)

B

Paired comparison matrix

D

Transformed paired comparison matrix

coef

Vector of estimated coefficients

vcov

Covariance matrix of estimated item parameters

JJ

Number of jackknife units

JJadj

Reduced number of jackknife units

powB

Used power of comparison matrix B

maxK

Maximum number of categories per item

G

Number of groups

desc

Some descriptives

difstats

Statistics 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 R for rater-mediated assessments. Rasch Measurement Transactions, 28(1), 1457-1459.

See Also

See the pairwise package for the alternative row averaging approach of Choppin (1985) and Wang and Engelhard (2014) for an alternative R implementation.

Examples

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

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

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

## Not run: 
# 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
miceadds::library_install("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 <- sirt::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 <- sirt::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 <- sirt::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
miceadds::library_install("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$group %in% c(2,3,11), ]
# define items
items <- paste0("I",1:11)
# estimate model
mod1 <- sirt::rasch.evm.pcm( dat[,items], weights=dat$weight, group=dat$group )
summary(mod1)

#############################################################################
# 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 <- sirt::sim.raschtype( stats::rnorm(N), b )
# simulate data for second group
b1 <- b
b1[4] <- b1[4] + .5 # introduce DIF for fourth item
dat2 <- sirt::sim.raschtype( stats::rnorm(N,mean=.3), b1 )
dat <- rbind(dat1, dat2 )
group <- rep( 1:2, each=N )
# estimate model
mod1 <- sirt::rasch.evm.pcm( dat, group=group )
summary(mod1)

## End(Not run)

sirt

Supplementary Item Response Theory Models

v3.10-118
GPL (>= 2)
Authors
Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>)
Initial release
2021-09-22 17:45:34

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