Joint Maximum Likelihood (JML) Estimation of the Rasch Model

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 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.

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
if (FALSE) {
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
}