tam.jml: Joint Maximum Likelihood Estimation

In TAM: Test Analysis Modules

Joint Maximum Likelihood Estimation

Description

This function estimate unidimensional item response models with joint maximum likelihood (JML, see e.g. Linacre, 1994).

Usage

tam.jml(resp, group=NULL, adj=.3, disattenuate=FALSE, bias=TRUE,
    xsi.fixed=NULL, xsi.inits=NULL, theta.fixed=NULL, A=NULL, B=NULL, Q=NULL,
    ndim=1, pweights=NULL, constraint="cases", verbose=TRUE, control=list(), version=3)

## S3 method for class 'tam.jml'
summary(object, file=NULL, ...)

## S3 method for class 'tam.jml'
logLik(object, ...)

Arguments

resp	A matrix of item responses. Missing responses must be declared as NA.
group	An optional vector of group identifier
disattenuate	An optional logical indicating whether the person parameters should be disattenuated due to unreliability? The disattenuation is conducted by applying the Kelley formula.
adj	Adjustment constant which is subtracted or added to extreme scores (score of zero or maximum score). The default is a value of 0.3
bias	A logical which indicates if JML bias should be reduced by multiplying item parameters by the adjustment factor of (I-1)/I
xsi.fixed	An optional matrix with two columns for fixing some of the basis parameters \xi of item intercepts. 1st column: Index of \xi parameter, 2nd column: Fixed value of \xi parameter
xsi.inits	An optional vector of initial \xi parameters. Note that all parameters must be specified and the vector is not of the same format as xsi.fixed.
theta.fixed	Matrix for fixed person parameters \theta. The first column includes the index whereas the second column includes the fixed value. This argument can only be applied for version=1.
A	A design array A for item category intercepts. For item i and category k, the threshold is specified as \sum _j a_{ikj} \xi_j.
B	A design array for scoring item category responses. Entries in B represent item loadings on abilities \theta.
Q	A Q-matrix which defines loadings of items on dimensions.
ndim	Number of dimensions in the model. The default is 1.
pweights	An optional vector of person weights.
constraint	Type of constraint for means. Either "cases" (set mean of person parameters to zero) or "items" (set mean of item parameters to zero).
verbose	Logical indicating whether output should be printed during iterations. This argument replaces control$progress.
control	A list of control arguments. See tam.mml for more details.
version	Version function which should be used. version=2 is the former tam.jml2 function in TAM (<2.0). The default version=3 allows efficient handling in case of missing data.
object	Object of class tam.jml (only for summary.tam function)
file	A file name in which the summary output will be written (only for summary.tam.jml function)
...	Further arguments to be passed

Value

A list with following entries

item1	Data frame with item parameters
xsi	Vector of item parameters \xi
errorP	Standard error of item parameters \xi
theta	MLE in final step
errorWLE	Standard error of WLE
WLE	WLE in last iteration
WLEreliability	WLE reliability
PersonScores	Scores for each person (sufficient statistic)
ItemScore	Sufficient statistic for each item parameter
PersonMax	Maximum person score
ItemMax	Maximum item score
deviance	Deviance
deviance.history	Deviance history in iterations
resp	Original data frame
resp.ind	Response indicator matrix
group	Vector of group identifiers (if provided as an argument)
pweights	Vector of person weights
A	Design matrix A of item intercepts
B	Loading (or scoring) matrix B
nitems	Number of items
maxK	Maximum number of categories
nstud	Number of persons in resp
resp.ind.list	Like resp.ind, only in the format of a list
xsi.fixed	Fixed \xi item parameters
control	Control list
item	Extended data frame of item parameters
theta_summary	Summary of person parameters
...

Note

This joint maximum likelihood estimation procedure should be compatible with Winsteps and Facets software, see also http://www.rasch.org/software.htm.

References

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

See Also

For estimating the same class of models with marginal maximum likelihood estimation see tam.mml.

Examples

#############################################################################
# EXAMPLE 1: Dichotomous data
#############################################################################
data(data.sim.rasch)
resp <- data.sim.rasch[1:700, seq( 1, 40, len=10)  ]  # subsample
# estimate the Rasch model with JML (function 'tam.jml')
mod1a <- TAM::tam.jml(resp=resp)
summary(mod1a)
itemfit <- TAM::tam.fit(mod1a)$fit.item

# compare results with Rasch model estimated by MML
mod1b <- TAM::tam.mml(resp=resp )

# constrain item difficulties to zero
mod1c <- TAM::tam.jml(resp=resp, constraint="items")

# plot estimated parameters
plot( mod1a$xsi, mod1b$xsi$xsi, pch=16,
    xlab=expression( paste( xi[i], " (JML)" )),
    ylab=expression( paste( xi[i], " (MML)" )),
    main="Item Parameter Estimate Comparison")
lines( c(-5,5), c(-5,5), col="gray" )

# Now, the adjustment pf .05 instead of the default .3 is used.
mod1d <- TAM::tam.jml(resp=resp, adj=.05)
# compare item parameters
round( rbind( mod1a$xsi, mod1d$xsi ), 3 )
  ##          [,1]   [,2]   [,3]   [,4]   [,5]  [,6]  [,7]  [,8]  [,9] [,10]
  ##   [1,] -2.076 -1.743 -1.217 -0.733 -0.338 0.147 0.593 1.158 1.570 2.091
  ##   [2,] -2.105 -1.766 -1.233 -0.746 -0.349 0.139 0.587 1.156 1.574 2.108

# person parameters for persons with a score 0, 5 and 10
pers1 <- data.frame( "score_adj0.3"=mod1a$PersonScore, "theta_adj0.3"=mod1a$theta,
           "score_adj0.05"=mod1d$PersonScore, "theta_adj0.05"=mod1d$theta  )
round( pers1[ c(698, 683, 608), ],3  )
  ##       score_adj0.3 theta_adj0.3 score_adj0.05 theta_adj0.05
  ##   698          0.3       -4.404          0.05        -6.283
  ##   683          5.0       -0.070          5.00        -0.081
  ##   608          9.7        4.315          9.95         6.179

## Not run: 
#*** item fit and person fit statistics
fmod1a <- TAM::tam.jml.fit(mod1a)
head(fmod1a$fit.item)
head(fmod1a$fit.person)

#*** Models in which some item parameters are fixed
xsi.fixed <- cbind( c(1,3,9,10), c(-2, -1.2, 1.6, 2 ) )
mod1e <- TAM::tam.jml( resp=resp, xsi.fixed=xsi.fixed )
summary(mod1e)

#*** Model in which also some person parameters theta are fixed
# fix theta parameters of persons 2, 3, 4 and 33 to values -2.9, ...
theta.fixed <- cbind( c(2,3,4,33), c( -2.9, 4, -2.9, -2.9 ) )
mod1g <- TAM::tam.jml( resp=resp, xsi.fixed=xsi.fixed, theta.fixed=theta.fixed )
# look at estimated results
ind.person <- c( 1:5, 30:33 )
cbind( mod1g$WLE, mod1g$errorWLE )[ind.person,]

#############################################################################
# EXAMPLE 2: Partial credit model
#############################################################################

data(data.gpcm, package="TAM")
dat <- data.gpcm

# JML estimation
mod2 <- TAM::tam.jml(resp=dat)
mod2$xsi     # extract item parameters
summary(mod2)
TAM::tam.fit(mod2)    # item and person infit/outfit statistic

#* estimate rating scale model
A <- TAM::designMatrices(resp=dat, modeltype="RSM")$A
#* estimate model with design matrix A
mod3 <- TAM::tam.jml(dat, A=A)
summary(mod3)

#############################################################################
# EXAMPLE 3: Facet model estimation using joint maximum likelihood
#            data.ex10; see also Example 10 in ?tam.mml
#############################################################################

data(data.ex10)
dat <- data.ex10
  ## > head(dat)
  ##  pid rater I0001 I0002 I0003 I0004 I0005
  ##    1     1     0     1     1     0     0
  ##    1     2     1     1     1     1     0
  ##    1     3     1     1     1     0     1
  ##    2     2     1     1     1     0     1
  ##    2     3     1     1     0     1     1

facets <- dat[, "rater", drop=FALSE ] # define facet (rater)
pid <- dat$pid      # define person identifier (a person occurs multiple times)
resp <- dat[, -c(1:2) ]        # item response data
formulaA <- ~ item * rater      # formula

# use MML function only to restructure data and input obtained design matrices
# and processed response data to tam.jml (-> therefore use only 2 iterations)
mod3a <- TAM::tam.mml.mfr( resp=resp, facets=facets, formulaA=formulaA,
             pid=dat$pid,  control=list(maxiter=2) )

# use modified response data mod3a$resp and design matrix mod3a$A
resp1 <- mod3a$resp
# JML
mod3b <- TAM::tam.jml( resp=resp1, A=mod3a$A, control=list(maxiter=200) )

#############################################################################
# EXAMPLE 4: Multi faceted model with some anchored item and person parameters
#############################################################################

data(data.exJ03)
resp <- data.exJ03$resp
X <- data.exJ03$X

#*** (0) preprocess data with TAM::tam.mml.mfr
mod0 <- TAM::tam.mml.mfr( resp=resp, facets=X, pid=X$rater,
                formulaA=~ leader + item + step,
                control=list(maxiter=2) )
summary(mod0)

#*** (1) estimation with tam.jml (no parameter fixings)

# extract processed data and design matrix from tam.mml.mfr
resp1 <- mod0$resp
A1 <- mod0$A
# estimate model with tam.jml
mod1 <- TAM::tam.jml( resp=resp1, A=A1, control=list( Msteps=4, maxiter=100 ) )
summary(mod1)

#*** (2) fix some parameters (persons and items)

# look at indices in mod1$xsi
mod1$xsi
# fix step parameters
xsi.index1 <- cbind( 21:25, c( -2.44, 0.01, -0.15, 0.01,  1.55 ) )
# fix some item parameters of items 1,2,3,6 and 13
xsi.index2 <- cbind( c(1,2,3,6,13), c(-2,-1,-1,-1.32, -1 ) )
xsi.index <- rbind( xsi.index1, xsi.index2 )
# fix some theta parameters of persons 1, 15 and 20
theta.fixed <- cbind(  c(1,15,20), c(0.4, 1, 0 ) )
# estimate model, theta.fixed only works for version=1
mod2 <- TAM::tam.jml( resp=resp1, A=A1, xsi.fixed=xsi.fixed, theta.fixed=theta.fixed,
            control=list( Msteps=4, maxiter=100) )
summary(mod2)
cbind( mod2$WLE, mod2$errorWLE )

## End(Not run)

TAM documentation built on May 29, 2024, 2:20 a.m.