rasch.jml: Joint Maximum Likelihood (JML) Estimation of the Rasch Model

In sirt: Supplementary Item Response Theory Models

Joint Maximum Likelihood (JML) Estimation of the Rasch Model

Description

This function estimates the Rasch model using joint maximum likelihood estimation (Lincare, 1994). The PROX algorithm (Lincare, 1994) is used for the generation of starting values of item parameters.

Usage

rasch.jml(dat, method="MLE", b.init=NULL, constraints=NULL, weights=NULL,
    center="persons", glob.conv=10^(-6), conv1=1e-05, conv2=0.001, progress=TRUE,
    bsteps=4, thetasteps=2, wle.adj=0, jmliter=100, prox=TRUE,
    proxiter=30, proxconv=0.01, dp=NULL, theta.init=NULL, calc.fit=TRUE,
    prior_sd=NULL)

## S3 method for class 'rasch.jml'
summary(object, digits=3, ...)

Arguments

dat	An N \times I data frame of dichotomous item responses where N indicates the number of persons and I the number of items
method	Method for estimating person parameters during JML iterations. MLE is maximum likelihood estimation (where person with perfect scores are deleted from analysis). WLE uses weighted likelihood estimation (Warm, 1989) for person parameter estimation. Default is MLE.
b.init	Initial values of item difficulties
constraints	Optional matrix or data.frame with two columns. First column is an integer of item indexes or item names (colnames(dat)) which shall be fixed during estimation. The second column is the corresponding item difficulty.
weights	Person sample weights. Default is NULL, i.e. all persons in the sample are equally weighted.
center	Character indicator whether persons ("persons"), items ("items") should be centered or ("none") should be conducted.
glob.conv	Global convergence criterion with respect to the log-likelihood function
conv1	Convergence criterion for estimation of item parameters
conv2	Convergence criterion for estimation of person parameters
progress	Display progress? Default is TRUE
bsteps	Number of steps for b parameter estimation
thetasteps	Number of steps for theta parameter estimation
wle.adj	Score adjustment for WLE estimation
jmliter	Number of maximal iterations during JML estimation
prox	Should the PROX algorithm (see rasch.prox) be used as initial estimations? Default is TRUE.
proxiter	Number of maximal PROX iterations
proxconv	Convergence criterion for PROX iterations
dp	Object created from data preparation function (.data.prep) which could be created in earlier JML runs. Default is NULL.
theta.init	Initial person parameter estimate
calc.fit	Should itemfit being calculated?
prior_sd	Optional value for standard deviation of prior distribution for ability values if penalized JML should be utilized
object	Object of class rasch.jml
digits	Number of digits used for rounding
...	Further arguments to be passed

Details

The estimation is known to have a bias in item parameters for a fixed (finite) number of items. In literature (Lincare, 1994), a simple bias correction formula is proposed and included in the value item$itemdiff.correction in this function. If I denotes the number of items, then the correction factor is \frac{I-1}{I}.

Value

A list with following entries

item	Estimated item parameters
person	Estimated person parameters
method	Person parameter estimation method
dat	Original data frame
deviance	Deviance
data.proc	Processed data frames excluding persons with extreme scores
dp	Value of data preparation (it is used in the function rasch.jml.jackknife1)

References

Linacre, J. M. (1994). Many-Facet Rasch Measurement. Chicago: MESA Press.

Warm, T. A. (1989). Weighted likelihood estimation of ability in the item response theory. Psychometrika, 54, 427-450.

See Also

Get a summary with summary.rasch.jml.

See rasch.prox for the PROX algorithm as initial iterations.

For a bias correction of the JML method try rasch.jml.jackknife1.

JML estimation can also be conducted with the TAM (TAM::tam.jml) and immer (immer::immer_jml) packages.

See also marginal maximum likelihood estimation with rasch.mml2 or the R package ltm.

Examples

#############################################################################
# EXAMPLE 1: Simulated data from the Rasch model
#############################################################################

set.seed(789)
N <- 500    # number of persons
I <- 11     # number of items
b <- seq( -2, 2, length=I )
dat <- sirt::sim.raschtype( stats::rnorm( N, mean=.5 ), b )
colnames(dat) <- paste( "I", 1:I, sep="")

# JML estimation of the Rasch model (centering persons)
mod1 <- sirt::rasch.jml( dat )
summary(mod1)

# JML estimation of the Rasch model (centering items)
mod1b <- sirt::rasch.jml( dat, center="items" )
summary(mod1b)

# MML estimation with rasch.mml2 function
mod2 <- sirt::rasch.mml2( dat )
summary(mod2)

# Pairwise method of Fischer
mod3 <- sirt::rasch.pairwise( dat )
summary(mod3)

# JML estimation in TAM
## Not run: 
library(TAM)
mod4 <- TAM::tam.jml( resp=dat )

#******
# item parameter constraints in JML estimation
# fix item difficulties: b[4]=-.76 and b[6]=.10
constraints <- matrix( cbind( 4, -.76,
                              6, .10 ),
                  ncol=2, byrow=TRUE )
mod6 <- sirt::rasch.jml( dat, constraints=constraints )
summary(mod6)
  # For constrained item parameters, it this not obvious
  # how to calculate a 'right correction' of item parameter bias

## End(Not run)

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