rasch.jml: 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, 
    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)

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

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

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.

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

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?

object

Object of class rasch.jml

...

Further arguments to be passed

Value

A list with following entries
itemEstimated item parameters
personEstimated person parameters
methodPerson parameter estimation method
datOriginal data frame
devianceDeviance
data.procProcessed data frames excluding persons with extreme scores
dpValue of data preparation (it is used in the function rasch.jml.jackknife1)

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 $\frac{I-1}{I}$.

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

Run this code

#############################################################################
# SIMULATED 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 <- sim.raschtype( rnorm( N ) , b )
colnames(dat) <- paste( "I" , 1:I , sep="")

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

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

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

# JML estimation in TAM
library(TAM)
mod4 <- TAM::tam.jml2( 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 )
mod4 <- rasch.jml( dat , constraints = constraints )
summary(mod4)
# For constrained item parameters, it this not obvious
# how to calculate a 'right correction' of item parameter bias

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