tam.jml: 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, verbose=TRUE, control=list(), version=2)
# S3 method for tam.jml
summary(object, file=NULL, …)
# S3 method for 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 $ξ$ of item intercepts. 1st column: Index of $ξ$ parameter, 2nd column: Fixed value of $ξ$ parameter

xsi.inits

An optional vector of initial $ξ$ 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 $θ$ . The first column includes the index whereas the second column includes the fixed value. This argument can only be applied for version=1.

A design array $A$ for item category intercepts. For item $i$ and category $k$ , the threshold is specified as $\sum_{j} a_{i k j} ξ_{j}$ .

A design array for scoring item category responses. Entries in $B$ represent item loadings on abilities $θ$ .

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.

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).

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

item

Data frame with item parameters

xsi

Vector of item parameters $ξ$

errorP

Standard error of item parameters $ξ$

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

Design matrix $A$ of item intercepts

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 $ξ$ item parameters

control

Control list

…

References

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

Examples

Run this code

# NOT RUN {
#############################################################################
# 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 )

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

#############################################################################
# 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 )
# }

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